Nonequilibrium dynamics and relative phase evolution of two-component Bose-Einstein condensates

Transcription

1 Nonequilibrium dynamics and relative phase evolution of two-component Bose-Einstein condensates Russell Anderson Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy Centre for Atom Optics and Ultrafast Spectroscopy Swinburne University Australia May, 1

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3 Nonequilibrium dynamics and relative phase evolution of two-component Bose-Einstein condensates Russell Anderson Centre for Atom Optics and Ultrafast Spectroscopy Swinburne University of Technology, Melbourne, Australia A thesis submitted for the degree of Doctor of Philosophy Supervisory Committee: Professor Andrei Sidorov (chair), Doctor Brenton Hall, Professor Peter Hannaford Abstract In the years following the first Bose-Einstein condensation of dilute atomic gases in 1995, several experiments demonstrated many fundamental properties of two-component bec Subsequent to these pioneering experiments, almost a decade passed with little experimental activity, relative to investigations of single-component condensates In somewhat of a renaissance, the initial observations of two-component bec have been revisited in recent years, consolidating and clarifying results of the early works, and illuminating new and surprising behaviour of these rich quantum fluids This thesis presents new experimental results on the relative phase evolution of two- component Bose-Einstein condensates (becs) We approach our study of this system from the dual perspective of binary superfluidity and trapped atom interferometry We observe

4 iv nonequilibrium dynamics of the condensate that are deeply connected to the physics of interpenetrating superfluids These dynamics have consequences for interferometers whose two paths are different internal spin states of the condensate atoms Using trapped condensates for precision measurement draws upon the benefits of confined atom interferometry long interrogation times, precise position control to measure near-field potentials in addition to the unique coherence properties of becs Precision measurement with trapped particles, however, is obstructed by the inherent inhomogeneities associated with the confinement itself Spatial variation of the external potentials and atomic density can give rise to dephasing during the free evolution and imperfect splitting and recombination of the interferometric paths In this thesis, we consider the dephasing of the two-component mean field order parameter, which tends to diminish the interference signal attained with the condensate The promise of becs for atom interferometry lies within using nonclassical many body states to reduce the fundamental noise limit imposed upon measuring the projection of atomic pseudospin Before the many body quantum properties of two-component condensates can be used to their full potential, the ideal classical limit must be achieved and investigated As in optical interferometry, the classical ideality of an interference measurement is contingent upon control of the spatial modes the matter wave fields, or single particle wave functions Whilst much attention is currently focused on preparing nonclassical many body states for entanglement and squeezing of the condensate pseudospin, this thesis focuses on the ancillary yet important issue of spatial inhomogeneity of the condensate mean field Reversal of the spatially inhomogeneous mean field evolution is demonstrated, employing the techniques of nuclear magnetic resonance A new method is demonstrated to probe pseudospin-1 condensates involving microwave adiabatic passage The technique permits simultaneous, spatially resolved absorption imaging of the spatial mode of each component Controlled manipulation of the components into Zeeman states with different magnetic moments allows spatial separation using a Stern- Gerlach separation technique, providing a spatially resolved projection of the local condensate pseudospin from a single absorption image We propose an interferometric technique which employs this imaging method to directly image the relative phase The miscibility of the two components depends on the relative magnitude of the interactions between various collision partners The boundary of this miscibility is found to be less pronounced than expected from the commonly applied results of homogeneous superfluids The effects of kinetic energy are shown to be significant at the threshold of miscibility, leading to a breakdown in the Thomas-Fermi approximation Magnetic dipole transitions amongst alkali ground states are often used to prepare, manipulate, and interrogate pseudospin-1 condensates We present a thorough investigation of

5 v the two-photon magnetic dipole transitions between the F = 1, m F = 1 and F =, m F = +1 states, and the F = 1, m F = +1 and F =, m F = 1 states of 87 Rb Taking into account all 8 Zeeman levels of the electronic ground state reveals markedly different radiative shifts and effects beyond the rotating wave approximation, compared to the three-level formalism commonly used for these transitions This has implications for Ramsey interferometry using the two-photon magnetic dipole transitions The unchartered experimental territory offered by two-component condensates is vast, and future directions, supplementary to the work in this thesis, are elucidated

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7 Declaration This is to certify that this thesis contains no material which I have submitted for the award of any other degree or diploma, except where due reference is made in the text, to the best of my knowledge contains no material previously published or written by another person except where due reference is made in the text; and where the work is based on joint research or publications, I have disclosed the relative contributions of the respective workers or authors Russell Anderson Date

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9 Acknowledgements My graduate research and thesis preparation has been supported by many people to whom I owe thanks Andrei Sidorov has been my primary supervisor, and a great mentor during the past four and a half years Andrei provided an alternative and complementary perspective to my own, often the catalyst to our answering the big research questions of this thesis His timely and practical support has encouraged my development as a researcher I was also supervised by Brenton Hall, whose technique has influenced the way in which I design and conduct experiments During the many enjoyable hours we shared in and out of the laboratory, Brenton played a vital role in the progress and research outcomes of this project Thanks to the third member of my supervisory panel and the director of caous, Peter Hannaford Peter is always approachable, attentive, and I have benefited from his helpful guidance Thanks to everyone who I worked with at Swinburne, particularly Chris Ticknor, for encouraging me to simulate, and his great sense of humour I was fortunate to begin my graduate research working on an operational Bose-Einstein condensation apparatus: the atom chip 1 experiment at Swinburne built by Brenton, Andrei, and previous PhD students of the lab, Falk Scharnberg and Shannon Whitlock I offer special thanks to Shannon, who volunteered his time and effort bringing me up to speed with the apparatus before handing over the keys Towards the end of my PhD I had the pleasure of working with the current PhD students, Michael Egorov and Valentin Ivannikov, in whose trusty hands the keys now reside I had the great opportunity to visit the laboratory of David Hall at Amherst College during 8 David was a generous host and is a great scientist to work with His pioneering and continuing work on two-component Bose-Einstein condensates have shaped my understanding of the field and the results of this thesis Many thanks to all the friendly folk of Amherst who helped make my stay so enjoyable and rewarding; in particular, Adam Kaufman and Dylan Bianchi In addition to my supervisory panel, careful reading and useful comments were provided

10 x by Matthew Jeppesen and Adam Kaufman Thanks to Lincoln Turner, Nick Robins, Chris Vale, Peter Drummond, and Mandip Singh for many useful discussions Many thanks to my friends, especially those who showed me the way, by completing their theses during the preparation of my own Thanks to my parents, Judy and Max for their support Finally, and especially, I thank my beautiful and loving partner Micah

11 Contents List of Figures xv 1 Introduction 11 Multi-component superfluidity 1 Nonequilibrium bec dynamics 13 Wavefunction engineering 14 Interferometry with pseudospin Outline of this thesis Two-level systems 1 Two-level system formalism Density matrix 1 Optical resonance of two-level atoms 3 Radiative forces on atoms 31 Dipole force 3 Radiation pressure force 4 Adiabatic passage 5 Bloch vector formalism 6 Ramsey interferometry Magnetic-dipole coupling in alkali hyperfine ground levels 31 Hyperfine ground levels 3 Zeeman states of the hyperfine ground levels 33 Magnetic dipole transition matrix elements 331 Rabi frequency convention 34 Single photon radiofrequency transitions 341 Analytic Rabi solutions in a spin-f system 34 Radiofrequency dressed-state energies 35 Beyond the rotating wave approximation 36 Two-photon mw-rf transitions in 87 Rb 5 S1/ 361 Possible state combinations 36 Coupling strengths and ac Zeeman shifts 363 Fidelity of the two-photon transition : leakage to other states 364 Ramsey interferometry bec

12 xii Contents 4 Two-component bec : experiment 41 Swinburne atom chip 411 Atom chip apparatus 41 Optical pumping 413 Imaging 414 Computer control 415 Microwave and radiofrequency apparatus 4 Magnetic dipole transitions 41 Magnetic dipole Rabi oscillations in a spinor bec 4 Radiofrequency Landau-Zener transitions in a spinor bec 43 Microwave outcoupling spectroscopy of a bec 44 One- and two-photon Rabi spectroscopy 45 Two-photon Rabi oscillations 43 Dual state imaging 5 Two-component bec : theory 51 Coupled Gross-Pitaevskii equations 5 Stability, miscibility and excitations 53 Thomas-Fermi approximation 531 Miscible regime : a1 a11 a 53 Immiscible regime : a1 > a11 a Ground states of Rb pseudospin- 1 bec 54 Simulation of the coupled Gross-Pitaevskii equations 541 Gross-Pitaevskii ground state computation 55 Many-body theory of two-component bec 56 Bloch vector density 6 Relative phase evolution of a two-component bec 61 Historical development 6 Ramsey interferometry with a pseudospin- 1 bec 61 Net interference signal 6 Local spin projection 63 Phase and frequency domains 63 Collisional shift analysis of relative phase 64 Rephasing the nonlinear dynamics of a pseudospin- 1 bec 641 Spin Echo 65 Phase imaging : interferometric reconstruction 66 Future directions 661 Self rephasing 66 Wave function engineering 663 Spin squeezing 7 Conclusion

13 Contents xiii Appendices A Alternative adiabatic pulses 167 A1 Majorana spin flips 171 B Three level Raman transitions 175 B1 Multi-level, polychromatic rotating wave approximation 175 B Adiabatic elimination of the intermediate state 177 C Optical Pumping of the D line in Rubidium C1 Density matrix formalism 181 C Dual state imaging : relative detection efficiency 184 D Relative phase reconstruction using linear condensate densities 189 Author index to references 191 References 197

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15 List of Figures Ringing in the width of a single-component bec Component separation of a two-component bec Twisting and untwisting of the pseudospin of a bec Proposal for vortex creation by Williams and Holland Ramsey interferometry with pseudospin- 1 condensate Pseudospin- 1 bec interferometry: Comparison of different spatial modes Energy eigenvalues of the two-level system Decaying Rabi oscillations of an ensemble of two-level systems Population transfer during linear chirp adiabatic passage Analytic approximation to linear adiabatic passage fidelity Schematic of pulse sequence for Ramsey interferometry Relative coupling strengths of magnetic dipole transitions in 87 Rb 5 S1/ Resonant Rabi oscillations amongst Zeeman states of spin-1 and spin- Radiofrequency coupling of combined atom-field states rf radiative shifts of 5 S1/ levels beyond the rotating wave approximation Two-photon coupling of the 1, 1, +1 states Two-photon coupling of the 1, +1, 1 states Two-photon Rabi oscillations for various intermediate state detunings Leakage to states other than 1, 1 or, 1 during two-photon Rabi oscillations Current carrying structure of Swinburne atom chip i Schematic of the Swinburne atom chip i apparatus Efficacy of optical pumping of atoms into the 1, 1 state Perceived condensate width increase due to recoil from repump photons rf-rabi oscillations of a F = spinor bec in a magnetic trap Landau-Zener transitions in an F = spinor bec Dressed state energies of the F = Zeeman states Energy landscape for microwave spectroscopy of a magnetically trapped bec Microwave spectroscopy of a magnetically trapped 1, 1 bec Rabi spectroscopy for various pulse durations Navigating the Rabi landscape Rabi spectroscopy of the 1, 1, transition in an optically trapped bec Bias magnetic field calibration using microwave Rabi spectroscopy

16 xvi list of figures Two-photon Rabi oscillations of a magnetically trapped pseudospin- 1 bec Broadening of intermediate state transition: effect on two-photon coupling Broadening of intermediate state transition: effect on π/-pulse Simultaneous dual-state imaging schematic Relative detection efficiency during dual-state absorption imaging Microwave Landau-Zener transitions from 1, 1 to, Simultaneous dual-state imaging of 1, +1, 1 pseudospin- 1 bec Simultaneous dual-state imaging schematic Healing length and healing time of a two-component condensate Density configurations for two-component Thomas-Fermi ground states Two-component ground states: comparison of Thomas-Fermi approximation with Gross-Pitaevskii simulation Pulse sequence for Ramsey interferometry of a pseudospin- 1 bec 137 Simulated column densities and relative phase evolution 139 Column density of state as a function of Ramsey interferometry time 141 The temporal Ramsey interference signal of a pseudospin 87 Rb bec 14 Axial spin projection during Ramsey interferometry of a pseudospin- 1 bec 145 Ramsey interferometry in the phase domain 146 Axially dependent Ramsey fringe frequency 149 Temporally resolved spin echo of a pseudospin 87 Rb bec 153 Axial spin projection during spin-echo of a pseudospin- 1 bec 154 Net transverse spin during spin-echo and Ramsey interferometry experiments 155 Spin echo using Carr-Purcell pulse sequence of multiple π-pulses 157 Simulation of interferometric phase retrieval method 158 Self rephasing of a pseudospin- 1 condensate 16 Predicted nonlinearity χ during mean field dynamics of a condensate with 3 14 atoms A1 Population remaining in initial state after hyperbolic secant pulse 17 A Comparison of fastest adiabatic full passage using different coupling pulses 171 A3 Alternative adiabatic pulses 17 B1 Three level system used to demonstrate adiabatic elimination and rotating wave approximation for polychromatic fields C1 Optical pumping with σ + light from the initial state F =, m F = C Optical pumping with σ + light from the initial state F =, m F = +1 C3 Relative detection efficiency using absorption of σ + probe light

17 Chapter 1 Introduction Bose-Einstein condensation of dilute atomic gases in 1995 heralded a new era of atomic, molecular and optical physics [1,, 3] These macroscopic matter waves, in which every atom occupies the same single-particle quantum state, were originally composed of a single spin state of a particular atomic species Shortly after the experimental realisation of a BoseEinstein condensate (bec), several groups sought to liberate its spin degree of freedom, thus creating spinor and pseudospinor condensates The first of these multi-component condensates was composed of two hyperfine states of 87 Rb, created by the group led by Carl Wieman and Eric Cornell at jila in 1997 [4] These two-component becs offered a new platform for studying several seemingly disparate fields of physics, including superfluidity, precision measurement, and entanglement Following the creation of binary condensates in 87 Rb, the group of Wolfgang Ketterle at mit used an optical dipole trap to confine a bec of sodium atoms in all spin projections of the F = 1 hyperfine level [5] Multi-component becs can also be composed of distinct condensates of different atomic species This thesis focuses on two-component condensates comprised of different spin states of the same atomic species, although many results concerning their ground states and excitation spectra (Chapter 5) apply to mixtures of becs in two different species A two-component order parameter can be used to describe two-component bec Ψ1 (r) n1 (r) e iϕ1 (r) =, Ψ(r) Ψ (r) n (r) e iϕ (r) (11) where each component (i = 1, ) has the single-particle wave function (or mean field) Ψi (r), density n i (r), and phase ϕ i (r) Every atom in a single-component bec shares the same quantum mechanical phase, and pioneering studies investigated this macroscopic quantum coherence Early experiments

18 introduction demonstrated matter-wave interference by spatially splitting a single bec in half, allowing the two resulting condensates to interfere upon their expansion in free space In a two-component condensate, every atom need not share the same internal superposition, the extreme case being when every atom occupies only one of the two components1 However, when every atom shares the same internal superposition, the description encompassed by Eq (11) defines a spatially dependent pseudospinor, a macroscopic superposition whose relative phase is ϕ(r) ϕ (r) ϕ1 (r) (1) This quantity determines the outcome of interference experiments performed with pseudospin- 1 becs, which is the study of this thesis Marrying the physics of bec with spin- 1 particles yields a unique two-level system, for which a great deal of control can be exercised well surmised by Jack [6]: In contrast to condensed matter systems where there is often a proliferation of possible decoherence mechanisms that cannot be isolated from each other, atomic Bose-Einstein condensates are relatively clean systems with few possible decoherence mechanisms and are thus a promising test bed for understanding decoherence and quantum superpositions in general The fruits of knowledge from several decades of experience with spin- 1 systems, particularly nuclear magnetic resonance (nmr), can be applied to this novel system The constituent states of pseudospin condensates considered in this thesis are in different hyperfine levels of the same alkali atom Spin changing collisions are inelastic, and occur relatively seldom compared to the elastic collisions which mediate the condensate self-interaction Collisions which change the spin state of a collision partner result in the release of the hyperfine splitting energy (typically GHz in alkali hyperfine ground states) as kinetic energy, and subsequent loss of atoms from any trap in which the condensate is confined The only process which changes the spin state of atoms that remain in the condensate is electromagnetic coupling, commonly magnetic dipole coupling using microwave (mw) and radiofrequency (rf) radiation (Chapter 3) This distinguishes pseudospin- 1 bec from spinor condensates com- posed of all spin projections of a given hyperfine level, in which spin-changing collisions can coherently transfer population between the condensate components, leading to different symmetries, phase-space, and topological structures than pseudospin- 1 condensates [7, 8] 1 Condensates of two different atomic species are exclusively of this type, as it is impossible for an atom to be in a superposition of each species The many-body wave function of a two-component bec is discussed in detail in 55

19 11 11 multi-component superfluidity 3 Multi-component superfluidity Single-component Bose-condensed gases share many properties with the quantum phase of liquid 4 He (below 17 K) understood to be a superfluid since 1938 [9, 1, 11, 1, 13] they are both described by a complex order parameter [14] The realisation of the superfluid 3 He in the 197 by Osheroff, Richardson and Lee [15, 16] gave rise to a new phase of matter, as the Cooper pairs which comprise superfluid 3 He have non-zero spin3 (S = 1) and orbital angular momentum (L = 1), and are thus described by a multi-component order parameter More- over, there is a strong correlation between the spin and orbital angular momenta, leading to exotic phases of condensed matter, as commented on in Reference [18] The low temperature phases of 3 He have led to a unique multitude of fundamentally new concepts and have thereby enlarged our knowledge of the possible states of condensed matter Two-component bec is a propitious platform with which to study many aspects of bi- nary superfluid mixtures The superfluid properties of two-component becs are relevant to their ground states, as outlined in 5 53 Many experiments performed with two-compo- nent becs involve the dynamic behaviour that results when the condensate is prepared in a state other than its ground state, as introduced in the following Section Another key feature of superfluidity is the ability for vortices to be supported Vortex states in two-component condensates will be discussed in Section 13 1 Nonequilibrium bec dynamics Recent discoveries have illuminated the rich dynamics inherent to two-component conden- sates The role these dynamics play in using becs to perform precision measurements will be discussed in this thesis Experiments in this work confirm the nonlinear evolution of two-component becs decreases the fidelity with which they can be used to make precision measurements Various schemes are investigated in this work to control the dynamics, mitigating these interesting yet deleterious effects As a prelude to two-component bec dynamics and the history of their experimental enquiry, we consider an example of single-component bec dynamics The order param- eter Ψ(r) for a bec of a weakly interacting dilute gas is governed by the Gross-Pitaevskii equation (gpe) in the low temperature limit [19, ] The ground state wave function is the 3 Akin to electrons in a superconductor, the fermionic spin- 1 3 He atoms must form composite bosons, Cooper pairs, to exhibit superfluidity however, the Cooper pairs which comprise a conventional superconductor have zero spin [17]

20 4 introduction stationary state of the Gross-Pitaevskii equation which minimises the Gross-Pitaevskii energy functional [cf Eq (548), the two-component Gross-Pitaevskii energy functional] In the Thomas-Fermi approximation (Section 53), the ground state of a condensate in a spherically symmetric harmonic potential V = 1 mω r has density n(r) = Ψ(r) = 1 a R r max, ), ( ) (1 R 8πa 3 a a where ω is the trap frequency, and a = (13) ħ/mω is the harmonic oscillator unit length The spatial extent of the condensate is given by the Thomas-Fermi radius R = a( 15Na 1/5 ), a (14) which depends upon the number of atoms N, the curvature of the confining potential (via a), and the mean field interaction between the atoms in the condensate, determined by the s-wave scattering length a4 A positive s-wave scattering length results in a mutual repulsion of condensate atoms; a larger positive s-wave scattering length results in a condensate ground state which is larger in size for the same atom number and harmonic trap frequency Now consider the effect of instantaneously changing either the scattering length a, or trap frequency ω5 Immediately following such a change, every atom still shares the antecedent wave function, but this wave function is no longer the ground state of the new potential The condensate is out of equilibrium, and its spatial mode will evolve nontrivially Such a scenario was realised in the experiment of Matthews et al [1], who were able to rapidly vary the strength of the mean field interaction in a condensate of 87 Rb atoms The s-wave scattering length depends on the partners of a binary collision, which they altered by rapidly changing the internal spin state of a magnetically trapped condensate A bec of atoms in the F = 1, m F = 1 state was converted into a bec of atoms in the F =, m F = +1 state via resonant microwave coupling between the two Zeeman states (Chapter 3) Although the intrastate scattering length differs by only a few percent, the F =, m F = +1 conden- sate was prepared in an excited spatial mode, and a ringing of the condensate width was observed (Fig 11) for tens of milliseconds (several periods of the trapping potential) The discontinuous change in the mean field interaction was qualitatively described by the GrossPitaevskii equation, permitting an estimate of the ratios of the intra-state scattering lengths a11 /a Despite the seemingly single-component nature of this experiment, it marks the first in a series of trailblazing experiments investigating two-component bec which were 4 As seen by Eq (13), the peak density n(r = ) also depends on these three quantities 5 Rapid changes of N, resulting from a mechanism which outcouples atoms from the bec, for example, can deform the wave function, and are not considered here

21 1 nonequilibrium bec dynamics 5 axial width μm radial width μm time ms time ms Figure 11 Adapted from Reference [1]: Oscillation in the radial and axial widths of a condensate following an instantaneous change in the scattering length The size of the condensate is measured after a fixed evolution time in a trap whose radial trapping period is 94 ms and subsequent ballistic expansion The solid line is a fit of a model to the data, based on a simplified version of the Gross-Pitaevskii equation performed at jila in Boulder, Colorado, by the group led by Carl Wieman and Eric Cornell [1,, 3, 4, 5] In addition to the nonequilibrium dynamics of a single-component bec, Reference [1] was also the landmark work on pseudospin-1 condensates, demonstrating coherent radiative coupling between two internal states of a condensate, evident as Rabi oscillations in the populations of each component Nonequilibrium dynamics of a two-component bec can be understood by again considering the effect of a sudden change in the interactions between condensate atoms Pseduospin-1 becs are commonly prepared using a single-component condensate, initially in its ground state Electromagnetic radiation is applied which couples the internal spin states of each atom, simultaneously populating both components There are now three possible s- wave collisions between condensate atoms, inter- and intra-state collisions, corresponding to the repulsive mean field interaction between each component and itself These interactions depend on the local density, and three s-wave scattering lengths which generally differ As such, the two spatial modes created by the coupling pulse are unlikely to resemble the combined two-component ground state wave functions, 6 and dynamics of the excited state will ensue When both states of a pseudospin-1 bec were equally populated using a π -pulse in the famous experiment by Hall et al [], the two components of the condensate were seen to 6 The ground state wave functions of two-component condensates will be discussed at length in Chapter 5

22 6 a) introduction b) vertical position (µm) , 1 1, 1 5, Figure time (ms) 8 1 1, 1 Adapted from Reference []: Component separation of a two-component bec prepared out of equilibrium (a) Relative centre of mass motion between each com- ponent in a potential whose equilibrium position differs for each state (b) Absorption images of each component for a trap with coincident potential minima The component in the F =, m F = +1 state experiences a less repulsive mean field interaction, forming a crater surrounded by a shell of the F = 1, m F = 1 component spatially separate For certain evolution times, a ball and shell structure appeared in the density of the two components (shown in Fig 1), and oscillations in the centre of mass of each component were observed under certain trapping conditions By alternately observing each component of the condensate between experimental realisations for various evolution times, the relative motion of the components was found to be heavily damped Theoretical simulation of the jila system [6, 7] predicted excitations longer lived than those observed Recent experimental observations [8], including those of this thesis [9], illustrate that collective excitations can indeed persist for hundreds of milliseconds, in excellent agreement with the coupled Gross-Pitaevskii equations The rapid damping of external dynamics in the early jila experiments has been ascribed to a technical problem of some kind [8] These pioneering experiments showed that changing the internal state of a pseudospinor bec, initially prepared in the ground state of one of its components, takes the condensate out of equilibrium, and for the relative scattering lengths of the 1, 1, +1 combination, the components will spatially separate

23 wavefunction engineering 7 Pz (arb) 13 ms 5 14 ms 7 ms 1 65 ms ms 135 ms 16 ms 18 ms Pz time (ms) Figure 13 Adapted from Reference [5]: Twisting and untwisting of the pseudospin of a bec A continuous microwave drive is applied to a bec, rotating its pseudospin and causing Rabi oscillations of the population in each of the F = 1, m F = 1 and F =, m F = +1 states The coupling is inhomogeneous along the length of the condensate, causing a twist in the local pseudospin, damping of the Rabi oscillations The external dynamics act to untwist the pseudospin, and the Rabi oscillations recur at 18 ms Simulations of the coupled GrossPitaevskii equations are in good agreement with the observed long lived coherent evolution of the two-component bec 13 Wavefunction engineering Before discussing the first interferometric study of pseudospin- 1 bec [3], I will describe two works which exposed the rich interplay between external and internal dynamics of pseudospin- 1 bec Both involved spatially dependent coupling of the two spin states: manipulating the pseudospin differently at each point in the condensate Such control of the two-component order parameter can be thought of as an example of wave function engineering In Reference [5], Matthews et al realised a spatially dependent coupling using a fea- ture of the trap in which they confined the condensate The time-orbiting potential (top) trap involves a time-dependent magnetic field, which affects each of the F = 1, m F = 1 and F =, m F = +1 states differently The top trap consists of a linear quadrupole magnetic field, in addition to a uniform bias field rotating in the plane perpendicular to the quadrupole

24 8 introduction axis The atoms are confined in a harmonic potential by the time-averaged magnetic field they experience The F = 1, m F = 1 and F =, m F = +1 states have opposite total angular momentum, resulting in a different effective static magnetic field for each state This acts to change the properties of the two-photon transition that couples the states, and can also result in a vertical offset of the potential minimum confining each component (see Fig 1) A detailed discussion of these effects is presented in the thesis of Matthews [3, Chapter 3] This feature of the top trap was exploited in [5] to create a complex spin texture in a two-component bec The coupling conditions were controlled such that a gradient of the bare Rabi frequency and detuning ( 1) existed across the length of the condensate (Ω R /π varied by 6 Hz about its mean value of 5 Hz across the 54µm axial extent of the condensate) Continuous application of this coupling drive to a condensate initially in the ground state of F = 1, m F = 1 led to Rabi oscillations in the population of each component As the pseudospin rotated differently along the length of the condensate, it became increasingly twisted, and the Rabi oscillations decayed almost entirely after 5 ms Remarkably, a revival of the Rabi oscillations was observed at 18 ms, as the external dynamics serve to untwist the pseudospin Furthermore, both the internal and external dynamics of a pseudospin-1 bec were shown to be long lived, relatively undamped, and in excellent agreement with the coupled Gross-Pitaevskii equations (cgpes) The promise this work holds for using this system for interferometry is twofold First, the fact that Rabi oscillations are observed at a significant fraction of a second is evidence for long-lived phase coherence Second, the revival of Rabi oscillations indicates that the dephasing in this system is reversible We return to both these features in Chapter 6, where the twisting and untwisting of the pseudospin is reported on in the absence of continuous coupling The generation of vortices in becs of dilute gases has garnered much experimental and theoretical attention [31] Initial proposals to create vortex states in bec involved transferring orbital angular momentum to a condensate from either (i) laser light of a Laguerre-Gaussian beam involved in a transition between internal states of the condensate, or (ii) mechanical rotation of the trapping potential The first observation of vortex modes [4] was the realisation of an elegant proposal [3] which combined the above two techniques, in that it involves transitions between external modes of a two-component bec, in concert with dynamic rotation of the trapping potential A condensate was exposed to a weak off-resonant laser beam, offset from and rotating about the centre of the condensate Rotating the beam around the condensate provided a differential gradient to the two states, changing the detuning of the microwave transition throughout, and yielding a novel resonance condition The combined rotating laser and microwave drive coupled the ground state wave function of one of the states to a vortex mode of the other: an axially symmetric ring of the density, with a π phase winding around the centre of vanishing density, and a single quantum of angular momentum

25 13 wavefunction engineering 9 π density phase 1 Figure 14 Adapted from Reference [3]: Proposal for vortex creation by Williams and Holland The theoretical densities and phases of states 1 and [n i and ϕ i of the twocomponent order parameter in Eq (11)] are shown across two dimensions after a vortex mode has been prepared in state, which has one unit of angular momentum This proposal was successfully implemented using a novel excitation scheme by Matthews et al [4] The predicted transverse structure of the order parameter [3] is depicted Figure 14) The stability of the rotational flow depended on which state the vortex was prepared in: a vortex in the more buoyant F = 1, m F = 1 state, surrounding a core of the F =, m F = +1 com- ponent, lived longer than a vortex prepared in F =, m F = +1 This is related to the more energetically favourable mean field interaction which determines the nature of the station- ary states of the system Interesting vortex dynamics were observed, and further investigated in [33, 34]: in the stable configuration, long lived vortex-pinning was observed, whereas the metastable topological mode resulted in the vortex being dragged through the non-rotating component By using the minimally destructive phase contrast imaging technique [7], the order parameter of the vortex mode was uniquely probed Sequential images were captured as resonant microwave coupling (without the rotating laser beam) was used to transfer the vortex mode between the two spin states Imaging midway between the transfer allowed a form of a phase interferogram to be made, which demonstrated the azimuthal winding of the relative phase Another unique feature of this system was the ability to spatially resolve the vortex in situ The size of a vortex in a single component bec is determined by the heal-

26 1 introduction ing length of the condensate mean field In two-component condensates, there are multiple healing lengths, which we investigate in 5 The mutual repulsion allows the non-rotating component to support a much larger vortex than would be possible with a single component 14 Interferometry with pseudospin- 1 bec Atom interferometry involves splitting a group of atoms into two quantum states which acquire phase differently, coherently combining the two components to produce an interference fringe, and measuring the phase of the resulting fringe The different states of an atom interferometer may be translational/external states (ie position or momentum) or internal states of an atom In analogy to optics, the states are loosely referred to as paths or arms of the interferometer The relative phase acquired between each arm of the interferometer depends on some physical process that affects each state differently, and which we desire to measure The signal-to-noise ratio of a measured interference fringe dictates how well the phase of such a fringe can be determined As a simple example, suppose the phase of an interference fringe is ϕ = E T/ħ, where E is the difference in the energy each state experiences during a known time T during which the states evolve separately The energy E is related to the physical process that we desire to measure, for example the differential effect of an electromagnetic or gravitational field, or their gradient, on the two states Decreasing the uncertainty in the measured phase shift between each path of the interferometer therefore improves the fidelity with which the physical quantity can be measured Bose-Einstein condensates have many properties which make them promising candidates for precision measurement In addition to those they share with ultracold thermal atoms a low energy spread, and isolation from environmental sources of decoherence bec is ideal for atom interferometry due to its large coherence length, and ability to occupy nonclassical many-body states Both of the latter properties are important as they enhance the precision with which an interference fringe, and hence phase shift, can be measured Long range coherence facilitates high phase contrast in matter wave interference experiments, and nonclassical many-body states can be used to modify the quantum-limited uncertainty of a given observable The interactions between atoms in a bec can cause a diffusion of the phase of the conden- sate [35, 36] This is an entirely quantum effect, pertaining to the fundamental uncertainties in observables of the many-body wave function In a two-component bec, interactions change the uncertainty in the projections of the macroscopic and collective pseudospin These can decrease the merit with which an interferometric measurement of phase can be performed, but can also result in an improved interferometric measurement and serve as a means of studying entanglement This will be briefly outlined in 55 Many atom optics experiments

27 14 interferometry with pseudospin- 1 bec 11 to date have focused on extracting a coherent beam of atoms from a condensate, to create an atom laser [37, 38, 39, 4] The relatively low density of the atom laser beam circumvents the deleterious effects of interactions in the condensate Atom lasers draw upon the techniques of coherent manipulation of thermal atomic beams in concert with the interferometric techniques of optical lasers, and have great potential for precision measurement In combination with atom optical elements such as beam splitters, atom lasers offer high sensitivity to sensing rotation and acceleration Experiments have demonstrated the spatial and temporal coherence properties of atom lasers [41, 4, 43], and performed Ramsey interferometry using two internal states of an atom laser propagating in free space [44], to name but a few Rather than extract a coherent matter wave from a condensate, there are many motivations for using trapped sources of cold atoms for interferometry, in particular trapped becs Some benefits of trapped atom interferometry are Confinement of the source: allows manipulation and probing of the system over long evolution times, which has implications for precision measurement In the above example, for a given level of phase noise δϕ (how well an interference fringe can be measured), increasing the interaction time T decreases the uncertainty in the measured energy shift E Large signal (in the language of atom optics, a bright source): has the potential to increase the signal-to-noise ratio which may be bound by both fundamental and technical limits This in turn, also decreases the phase noise δϕ Localisation of atoms: beneficial when the interaction to be measured is position dependent (ie short-range potentials like surface interactions and small objects) This also aids detection generally and cavity enhanced detection in particular (modematching to cavity detection) Also allows controlled coupling to sources of environmental decoherence, when one wishes to study the decoherence mechanisms Size and portability (especially atom chips): Confined atom interferometers have the potential for future technologies including space-based applications Strong interactions: despite being considered a dilute gas, the constituent atoms of a trapped bec can interact strongly enough to exhibit behaviour beyond the realm of single-particle quantum mechanics As noted, whilst these interactions may hinder the coherent phase evolution in a bec interferometer, they can also be used to the advantage of an interferometric measurement The interaction-induced many-body dephasing in a trapped quantum gas can be mitigated in various ways Experiments which involve modifying the interactions using a Feshbach

28 1 introduction resonance to vary the scattering length have reduced the dephasing of Bloch oscillations in a condensate [45, 46], and the spin-projection noise of a one-dimensional Bose gas used in a Ramsey interference experiment has been modified [47] Another form of trapped bec interferometry involves coherently splitting a condensate into two adjacent potential wells [48, 49, 5] In this double-well interferometry, the two paths of the interferometer are different spatial modes occupying each of the two wells In some respects, this system is a pseudospin-1 bec, as the Hamiltonian in a two-mode approximation can be expressed in the form of a spin-1 system [51], and the quantum limited uncertainties of the relative number and relative phase of atoms in each well are related, in many ways, to the corresponding quantities of a bec in a superposition of two internal spin states 7 The many-body dephasing in double-well bec interferometry has been investigated in depth, and in recent experiments, reduced phased diffusion and number squeezing has been observed [5, 53] The many-body effects on bec coherence are important; problems certainly remain to be overcome, and features to be exploited, before the promises of bec interferometry are fulfilled This thesis is chiefly concerned with another effect that changes the way two-component becs can be used for precision measurements: spatial variations of the two-component order parameter Although the order parameter is shared by every atom in the condensate, and is affected by the mean field interaction with every other condensate atom, this is essentially a single-particle effect, as the order parameter obeys a classical field equation, the nonlinear Schrödinger equation (Gross-Pitaevskii equation) of single-particle quantum mechanics Hall et al demonstrated the first interferometric measurement with a pseudospin-1 bec [3] Preparing a superposition of the two-components using a π/-pulse initiated nonequilibrium dynamics of the condensate wave functions, as described in 1 Ramsey interference fringes were observed in a region of space where both components remained overlapped following the strongly damped centre of mass motion, as shown in Fig 15 Before application of the second π/-pulse, the components were vertically offset from each other for a varying length of time As the spatial overlap between the components was poor, a large interference signal was only observed in the overlap region Alternatively, if the total number of atoms in each component was used to compute an interference fringe, it would have been of limited visibility As with the interference of two coherent beams of light, the visibility of matter wave interference fringes is affected by mode matching, how well the spatial modes of each interfering wave overlap Mode matching is important for other types of atom interferometry such as double-well interferometry, and much effort has been devoted to improving the spatial mode of atom lasers Mode matching improves the visibility of interference fringes, 7 In this thesis, we reserve use of term pseudospin-1 bec for condensates composed of different spin states of the same atom; eigenstates with opposite spin projection

29 14 interferometry with pseudospin-1 bec 13 (a) before second π/-pulse 1 optical density (arbitrary units) (b) after second π/-pulse 1 optical density (arbitrary units) (c) vertical position (µm) pulse separation (ms) Figure 15 Adapted from Reference [3]: Ramsey interferometry with pseudospin-1 condensate coupled by microwave π/-pulses (a) density of states 1 F = 1, m F = 1 and F =, m F = +1 following strongly damped centre of mass oscillations ( 45 ms in Fig 1) Although the densities remain fixed, the relative phase between the two components continues to evolve, after which time (b) a second π/-pulse is applied, allowing the relative phase to be measured from the resulting densities (c) At the centre of the region where they overlap, the density of each component oscillates with varying temporal separation of the two pulses (the central density of is plotted) The resulting interference fringe has a frequency which depends on the detuning of the coupling drive from the energy separation of the two spin states and hence via the signal-to-noise ratio the fidelity of a measurement Mode matching requires intensity and phase matching; two waves with identical intensity profiles interfere poorly if their relative phase is spatially inhomogeneous A key result of this thesis regarding the spatial overlap of the wave functions of a two-component bec is that uncommon spatial modes of the components are neither necessary nor sufficient for loss of interferometric visibility (Section 6) As outlined in this work, the relative phase of the two-component order parameter [Eq (1)] can develop spatial inhomogeneity well before the components spatially separate, as illustrated in Fig 16 As spatial inhomogeneity in the relative phase plays a dominant role in degrading the performance of the trapped condensate interferometer, it is desirable to consider wave functions of the states which do not have perfect overlap, but do preserve the uniformity of the relative phase (Fig 16) There are strong connections between two seemingly disparate fields of physics discussed so far, bec interferometry and multi-component superfluidity Let us discuss this from the

30 14 introduction framework of superfluid flow In a single component superfluid, gradients of the phase are manifest as flow [51], from which the quantisation of superfluid circulation follows Likewise, for the two-component order parameter in Eq (11), the superfluid velocity of each component is vi (r) = ħ ϕ i (r) m (15) Gradients in the relative phase of a two-component bec therefore determine the relative velocity of the components The evolution of the relative phase is coupled in a complex way to the relative motion and hence spatial overlap of the two components, as we shall see in Chapter 6 Moreover, the ground state properties and nature of the miscibility in twocomponent condensates are intrinsically manifest in the nature of the interference, which dictates the usefulness of pseudospin- 1 condensates for precision measurement and studies of macroscopic entanglement 15 Outline of this thesis Chapter outlines the framework of two-level systems relevant to two-component bec, including their equivalence to a spin- 1 particle, adiabatic passage and Ramsey interferometry In Chapter 3, we provide a detailed analysis of magnetic dipole coupling amongst the ground states of alkali atoms This coupling is used to prepare, manipulate and interrogate superpositions of the two internal states of a pseudospin- 1 bec Particular attention is paid to the two-photon magnetic dipole coupling between Zeeman levels commonly used in 87 Rb pseudospin- 1 becs Predictions of the multi-photon radiative shifts including effects be- yond the rotating wave approximation are markedly different than previous, simpler treat- ments of these transitions; these shifts are important for interpreting the Ramsey interference signal attained using pseudospin- 1 condensates without ambiguity Chapter 4 describes the apparatus and techniques of the experiments in this thesis The magnetic dipole coupling dis- cussed in Chapter 3 is characterised for microwave and radiofrequency transitions in trapped condensates, and the spatial dependence of the magnetic dipole coupling is considered The necessary elements of two-component bec theory are summarised in Chapter 5: the mean field formalism used extensively throughout this thesis, the nature of excitations and stability of two-component condensates, the ground state wavefunctions and the breakdown of the Thomas-Fermi approximation near the threshold of miscibility, simulating the Gross- Pitaevskii equations, and the many-body theory of two-component bec In the penultimate Chapter, we study the effect of spatial inhomogeneities of the two-component order parameter on trapped condensate interferometry, and investigate how these effects can be mitigated using the techniques of nuclear magnetic resonance

31 15 outline of this thesis a) 5 ni axial position μm Pz Pz axial position μm Φ Π ni b) 4 1 Φ Π T ms T ms Figure 16 Pseudospin- 1 bec interferometry: Comparison of different spatial modes (a) The initial state with densities n i (top) results from applying a π/-pulse to a singlecomponent condensate in the ground state of 1 This is no longer a stationary state of the system, and its relative phase ϕ evolves inhomogeneously (middle) Despite the ideal overlap of the spatial modes, the dephasing causes the visibility of the Ramsey interference fringe to decay (bottom) (b) The densities n i of the two-component ground state, whereby the two components are partially miscible By virtue of this being a stationary state, the relative phase should remain spatially uniform throughout its evolution In practice, inelastic collisions result in decay of atoms from the condensate, changing the mean field potential seen by remaining atoms The initial state follows the time-dependent ground state nearadiabatically, but a small inhomogeneity in the relative phase develops (middle) Despite the seemingly poor overlap of the ground state spatial modes, its uniform relative phase results in a prolonged interferometric visibility, rendering it useful for an interferometric measurement (bottom)

32

33 Chapter Two-level systems Two-level systems capture the elemental quantum mechanics of myriad processes in laser cooling and trapping, magnetic trapping, optical detection, and coherent manipulation of atoms This Chapter outlines the fundamental results about two-level systems applicable to this thesis, and demonstrates how they can be attained using a single formalism The starting point for this analysis is a general two-level system interacting with electromagnetic radiation ( 1) The case of a single two-level system is extended to ensembles in, where the notion of coherence is introduced, using the example of electric dipole transitions in a two-level atom interacting with an optical field The core concepts of radiative forces on atoms are derived in 3 In Section 4 we revert to a general discussion of time dependent interactions in two-level systems, and examine population inversion using adiabatic passage Consideration is given to the time in which population can be inverted using adiabatic passage for a desired transfer efficiency The spin-1 particle, and its equivalence with any general two-level system, is outlined in 5 The notion of a fictitious spin that can be assigned to any twolevel system is discussed, with specific reference to pseudospin-1 condensates The Chapter concludes by applying many of the subsequent results to Ramsey interferometry The basic considerations herein provide a foundation with which to consider Ramsey interference of pseudospin-1 condensates in Chapter 6 The results of this Chapter appear throughout this thesis, and motivate more complicated analyses such as those of Chapter 3, Appendix A, Appendix B, and Appendix C 1 Two-level system formalism This Section describes the behaviour of a two-level system driven by electromagnetic radiation in the semi-classical formalism, in which the electromagnetic field is treated classically; adequate for the purposes of this thesis A complementary approach involves accounting for

34 18 two-level systems the quantised nature of the electromagnetic field (see [54, 55]) The two formalisms are equivalent when there are many photons in the field The semi-classical method is preferred here for its brevity, the instructive implementation of rotating wave approximation it permits,1 as well as the concise way in which radiative forces on atoms can be derived ( 3) Consider two energy levels 1 and, with an energy separation of ħω, interacting with an electromagnetic field of frequency ω = ω +, and arbitrary phase ϕ The two-levels could be electronic levels of an alkali atom with different orbital angular momentum driven at optical frequencies, different spin-states of an electron or nucleus driven by microwave or radiofrequency radiation, or bound levels of a dimer interacting with a radiofrequency field The coupling may be any interaction that causes transitions between the two levels: magnetic- or electric-dipole transitions, for example For now, we assume that the polarisation, amplitude, and frequency of the field have no spatial dependence For convenience in implementing the rotating wave approximation, the coupling term is written as 1 H = ħ Ω cos(ωt ϕ) = ħ Ω i(ωt ϕ) + e i(ωt ϕ) ), (e (1) with Ω > The bare atomic Hamiltonian, of which states 1 and are eigenstates, has matrix representation in the { 1, } basis HA = ħ ω, ω () and the total Hamiltonian in the Schro dinger picture is represented in the { 1, } basis by H= ω + Ω cos(ωt ϕ) ħ Ω cos(ωt ϕ) ω Ω (e i(ωt ϕ) + e i(ωt ϕ) ) ħ ω ħ + = ω Ω(e i(ωt ϕ) + e i(ωt ϕ) ) ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ H (3) V In the last equality, the Hamiltonian is decomposed via H = H + V Transforming to the interaction picture of H facilitates the rotating wave approximation, whereby the time de- pendence of the Hamiltonian is eliminated The desired transformation is achieved with the unitary operator S = e i H t/ħ ; the resulting Hamiltonian in the interaction picture (or 1 The rotating wave approximation in two-level systems with one radiation field (below) is extended to n-level system with m-radiation fields in B Inadequacies of the rotating wave approximation are investigated in 35

35 1 two-level system formalism 19 rotating frame ) is H I = S V S Ω e iϕ (1 + e i(ωt ϕ) ) ħ Ω e iϕ (1 + e i(ωt ϕ) ) (4) We invoke the rotating wave approximation, ignoring terms that oscillate at twice the driving frequency, and hereafter refer to H I as H Then Ω e iϕ ħ, H= Ω e iϕ (5) and we ascribe Ω to be the Rabi frequency, and the detuning Unless otherwise stated, the conventions used in this Section will be used throughout this thesis In most cases, the phase of the coupling term can be chosen arbitrarily without consequence3 The time evolution of a unit-normalised state ψ = c1 1 + c is found in terms of the complex state amplitudes c1 and c for which the Schro dinger equation reads i c 1 (t) = c1 + Ω e iϕ c (6a) i c (t) = Ω e iϕ c1 c (6b) When the system begins in state 1 at t =, the initial condition is c1 () = 1, c () =, and the solutions to Eqs (6) are ΩR t ΩR t sin ( ) i ) ΩR Ω ΩR t c (t) = ie iϕ sin ( ), ΩR c1 (t) = cos ( (7a) (7b) yielding the Rabi solutions [56] c1 (t) = where c (t) = Ω ΩR t = Ωt + cos ( ) ÐÐ cos ( ) ΩR ΩR Ω ΩR t sin ( ) ΩR ΩR = Ω + ÐÐ sin ( = Ωt ), (8a) (8b) (9) 3 The exceptions in this thesis are phase-based Ramsey interferometry, discussed in 6 and 63, and the equivalence between spin- 1 systems and general two-level systems in 5

36 two-level systems is defined as the total Rabi frequency4 ; the frequency at which the state probabilities oscillate for any detuning These formulae indicate that (i) resonant ( = ) Rabi oscillations occur at the Rabi frequency Ω and are of unity amplitude (population oscillates between exclusively in state 1 and exclusively in state ), and (ii) off-resonant ( ) Rabi oscillations occur at the total Rabi frequency ΩR ; they are always faster than resonant Rabi oscillations and are incomplete, with reduced amplitude given by Ω /ΩR The one- and two-photon resonances of magnetic dipole transitions between Zeeman states in the S hyperfine levels of 87 Rb are found using these expressions, as described in 44 The eigenstates of the Hamiltonian in Eq (5) are known as the dressed states of the system, + and [54] Expressed in the { 1, } basis, they are with energies θ θ + = e iϕ/ cos ( ) 1 + e iϕ/ sin ( ), θ θ = e iϕ/ sin ( ) 1 + e iϕ/ cos ( ), E± = ± where tan θ = ħωr, Ω, θ<π (1a) (1b) (11) (1) defines the mixing angle θ The geometric interpretation of this angle will be elucidated in 5 The energies of the states in Eqs (1) are shown as a function of detuning in Fig 1 Far from resonance ( Ω), the mixing angle is close to either or π, and the eigenstates asymptotically approach the bare states The equivalence depends upon the sign of the de- tuning; { ±, } ÐÐÐ { 1, } In this limit, the off-resonant coupling can be thought ± of as perturbing the energies of the bare states 1 and This energy shift is known as the light shift, and can be estimated using the preceding analysis in the rotating frame Taking the difference between the energy of each bare state, and that of the dressed state which it asymptotically approaches, ħ, ±, ħ δe = E, ± δe1 = E± + (13a) (13b) Series expansion in Ω/ gives an expression for the light shift of each state, which to lowest 4 Also known as the generalised Rabi frequency

37 1 two-level system formalism + + E+ 1 1 E ħω E 1 4 Ω Figure 1 Energy eigenvalues of the two-level Hamiltonian in Eq (5) (solid curves) The energies E+ and E in Eq (11) [see Eq (9)] are those of the dressed states + and in Eqs (1) Without coupling (Ω = ), the eigenstates of the system are the bare states 1 and, with energies ±ħ / (dashed curves) The bare state energies cross at = (ω = ω ) With coupling, these curves exhibit an avoided crossing at = (the resonance), and approach the bare state energies when the detuning is far from resonance In this limit, the dressed states approach the bare states, as indicated order is ħω 4 ħω δe = 4 δe1 = (14a) (14b) These expressions are valid for large positive and negative detunings, in spite of the sign specification in Eqs (13) We return to this result in 36, in a study of radiative shifts in 87 Rb 5 S1/, driven by both radiofrequency and microwave fields, and improve upon it in 35, looking at effects beyond the rotating wave approximation In the semiclassical formalism used thus far, we have considered the electromagnetic field to be classical In the fully quantised approach, the quantum nature of the electromagnetic field is considered, and the state of the field can be described by enumerating the number of photons In the combined atom-field Hilbert space, the states i, M denote a system where the atom is in state i and there are M photons in the field While such states are countably infinite (as M is a non-negative integer), the behaviour of the system can be understood by considering a subset of the atom-field states The semiclassical treatment amounts to considering a subset that consists of two states The equivalence between the states of the semi- classical and fully quantised formalisms is 1 1, N and, N in the Schro dinger

38 two-level systems picture; the two states are the different atomic levels with the same number of photons in the field, separated by an energy ħω In the interaction picture [the rotating frame, Eq (4)], the equivalence of states is 1 1, N + 1 and, N, separated by an energy ħ In the rotating wave approximation, the coupling of only two such atom-field states with a given N is considered (single-photon coupling); these are the bare states The dressed states are the eigenstates of the combined atom-field Hilbert subspace under such coupling They do not have a well defined photon number or atomic state, and are equivalent to + and in Eq (1) with energies Eq (11) In Section 35, we investigate the coupling of many atom-field states via multi-photon processes, studying effects beyond the rotating wave approximation for the case of radiofrequency coupling between Zeeman states Density matrix An ensemble of two-level systems can be described by the density operator ρ, that quantifies the degree to which each system shares the same quantum state, ie whether the ensemble is pure or mixed [57] The matrix representation of the density operator is the density matrix ρ, whose diagonal elements are the populations, the ensemble averaged c i, and whose off- diagonal elements represent the amount of coherence between the states The evolution of the density matrix can be used to determine how the populations and coherence of the ensemble evolve in the presence of decay The decay may include relaxation via radiative decay, coupling to a thermal bath, collisional relaxation in a gas, or a dephasing mechanism such as variation of ω throughout the ensemble The evolution of the density operator is governed by ρ i = [H, ρ ] + L ρ, t ħ (15) where L is the Liouvillian operator describing decay [54] In this Section, we separately in- corporate decay of the population5 in state to state 1 (diagonal elements of ρ) with a rate Γ1, and decoherence (decay of off-diagonal elements of ρ) with rate Γ, such that the matrix representation of L ρ is Γ1 ρ Γ ρ1, Γ = Γ ρ1 Γ1 ρ where ρ i j = i ρ j Equation (15) is then ρ 11 (t) = ρ (t) = Ω Im(ρ1 ) + Γ1 ρ i ρ 1 (t) = Ω (ρ ρ11 ) (i + Γ ) ρ1 5 This decay preserves the total population, equivalent to Tr(ρ) = 1 and Tr(Γ) = (16) (17a) (17b)

39 density matrix ss Ρ1 Ρ1 populations time Π Ω time Π Ω Figure (Left) Decay of Rabi oscillations for Γ = Ω/4, = (solid curves), compared with on-resonance Rabi oscillations without decay (dashed curves) (Right) Decay of coherence, ρ1, for Γ = Ω/4, = (solid curves), compared with on-resonance coherence without decay (dashed curves) In this example, the driving field is in phase with the rotating co-ordinates, hence Re(ρ1 ) S x =, and ρ1 = Im(ρ1 ) S y ; the coupling rotates the Bloch vector about the x-axis (see 5) The steady state solution of Eq (19b) is indicated by the black dash-dotted line The steady-state solution of the ensemble can be found by linearising the equation of motion for ρ This is achieved by ordering the elements of the density matrix ρ i j in a column vector ρ and finding the matrix T such that dρ = Tρ dt (18) Finding the steady state solution for ρ amounts to finding the nullspace of T The solutions ρ ss i j must then be normalised such that Tr(ρ) = 1 as the diagonal elements of ρ are the popu- lations In the case of Γ Γ1 = Γ, the steady-state solutions are s 1, 1 + s + δ s i+δ ss ρ1 =, 1 + s + δ Ω where s = is the saturation parameter, and Γ δ= is the detuning in half-linewidths Γ ss ρ = 1 ρ11ss = (19a) (19b) (19c) (19d) These expressions are of great utility; they describe the quintessence of the electric dipole interaction between laser light and a medium composed of two-level atoms

40 4 1 two-level systems Optical resonance of two-level atoms The optical resonance of two-level atoms is due to electric dipole coupling between a ground and excited state of the atom, described by the Hamiltonian H = H A d E(r, t), () where H A is the bare atomic Hamiltonian of which 1 and are eigenstates [Eq ()], d = e R is the electric dipole moment operator, with R the local position of the atomic electron of charge e, and E(r, t) is the electric field of the optical radiation at the centre of mass position of the atom r In order to evaluate radiative forces on atoms in Section 3, we make the spatio-temporal dependence of the light field explicit In the Schro dinger picture: E(r, t) = E(r, t) є ; E(r, t) = 1 E (r)(e iϑ + e iϑ ) ; ϑ ϑ(r, t) = ωt k r, (1a) (1b) (1c) where є = 1 and k is the wave vector6 We take the atomic states 1 and to have well defined and opposite parity As a result, d contains only off-diagonal terms in the { 1, } basis, namely 1 d = d1 Comparison of Eq (1) with Eqs (1) and () indicates the Rabi frequency for electric dipole transitions is given by ħω = d1 E ϕ = k r (a) (b) The decay mechanism is spontaneous emission between the ground [ 1 ] and excited state [ ] In this case, Γ1 = Γ = Γ = τ 1 [58], where Γ is the decay limited linewidth of the optical transition, and τ is the lifetime of excited state The steady state solutions of the density matrix in Eqs (19) can thus be applied The Rabi frequency and saturation parameter can be expressed in terms of atomic parameters and the intensity of the field using the decay rate for spontaneous emission [58] Γ= ω3 d1 є 3πє ħc 3 The intensity of the field is I = 1 cє E, which results in [using Eq (a)] Ω = 3λ3 ΓI σ IΓ = πhc hc/λ (3) (4a) 6 In Eq (1c) we have written the phase dependence of a running plane wave, but many of the results that follow can be used for other optical fields, for example, a standing wave

41 density matrix 5 where σ = 3λ /π is the cross section of the transition with wavelength λ The saturation parameter in Eqs (19) is s = I/Isat, where Isat = πhcγ hcγ ħω3 Γ = = 3λ3 λσ 1πc (5) is the saturation intensity; the intensity at which the resonant scattering rate is half that of its maximum possible value Among the myriad applications of Eqs (19) is the imaging an ultracold atomic gas using closed optical transitions The rate at which each atom scatters photons in the steady state is that of spontaneous emission, which is7 ss γsc = Γ ρ (6) The refractive index of a dilute atomic gas is related to these solutions by the proportionality ss n 1 ρ1 /s, which is independent of intensity of the field in the weak coupling limit s 1: n 1 δ+i 1 + δ (7) This indicates the absorptive and dispersive properties of the medium The absorption and phase shift of light are proportional to imaginary and real parts of Eq (7), respectively In the far off-resonant limit (large δ) the absorption scales as δ, whereas the phase shift decreases as δ 1, motivating the use of diffractive forms of imaging that minimise the num- ber of photons absorbed by a bec; as photon scattering destroys bec Diffraction contrast imaging [59] is one technique allowing the column density of an ultracold atomic gas to be inferred from the diffraction pattern it imparts on off-resonant laser light passing through it In doing so, the technique uses Eq (7) to find the ratio between the absorption, and the phase shift, of light passing through the sample, Im(n 1)/Re(n 1) = δ 1 The author has extended the concepts of this Section to find the refractive index of an ultracold gas of multilevel atoms, excited by one optical field and imaged with another This has been used to image the excited state population of atoms in a magneto-optical trap (mot) using diffraction contrast imaging [6] 7 This can be formally shown by considering the average power absorbed by an atom [54], which is ħω Ω Im(ρ 1 ) In the steady state, photons are absorbed at the same rate which they are scattered, and we have γsc = Ω Im(ρss 1 ) = Γ ρ ss, as seen from Eq (17a) This last equality is a consequence of the Kramers-Kronig relations, which connect the dispersive and absorptive response of the system

42 6 3 two-level systems Radiative forces on atoms We again focus on electric dipole transitions between atomic states 1 and, and consider the case where the intensity of the light field varies in space There are two forces exerted by the light upon an atom, a conservative force: the dipole force, and a dissipative force: the radiation pressure force We now account for the external degree of freedom of the atom moving in the optical field of Eq (1) The Hamiltonian is H = p + H A d E(r, t), m (8) where p is the momentum operator for the centre-of-mass of the atom Application of the Ehrenfest theorem [61] can be used to calculate the average force F on the atom in terms of its internal state and the light field, F= d dt p 1 = (iħ) [p, H ] = H = d E(r, t) (9) = d є E(r, t) In the last step, we have assumed that the variation of E(r, t) across the electronic wave functions of 1 and is negligible The Rabi frequency for electric dipole transitions has spatial dependence through E (r) To evaluate the expectation value in Eq (9), we write the gradient of the electric field in Eq (1) E(r, t) = E iϑ E (e + e iϑ ) + i ϑ(e iϑ e iϑ ) (3) The matrix representation of (d є) E(r, t), after transforming to the interaction picture and making the rotating wave approximation, is (d є) E E ie ϑ d1 є E + ie ϑ Ω iω ϑ ħ = Ω + iω ϑ (31a) (31b)

43 3 radiative forces on atoms 7 The expectation value in Eq (9) can now be evaluated, using the density matrix; A = Tr(ρ A ) The motion of the atom in the field E(r, t) is taken to be adiabatic, in the sense that its internal state equilibrates in a time much shorter than the light field appreciably varies8 The steady-state solutions of the density matrix [Eq (19)] can then be used to describe the internal degrees of freedom of the atom at any position in the field, even if the atom is free to move Returning to Eq (9), we have F = Tr{ρ (d є) E} ss ss = (d1 є) [Re(ρ1 ) E Im(ρ1 )E ϑ] ss ss = ħ Ω Re(ρ1 ) + ħ Ω ϑ Im(ρ1 ) ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ dipole force (3) pressure force The two terms in (3) are the dipole force and the radiation pressure force, respectively 31 Dipole force Inspection of Eq (19b) indicates that far from resonance, we can neglect the second term in Eq (3) The dipole force is Fdipole (r) = ħ Γ δ s(r), s(r) + δ (33) which is conservative in the steady-state regime, with potential9 Vdipole (r) = ħ s(r) ln (1 + ) 1 + δ (34) In the far off-resonance limit, this becomes Vdipole (r) ħ Ω(r) ħ Γ I (r) = 4 8 Isat (35) The first equality above is identical to the light shift derived in Eq (14), illustrating that the dipole potential can be interpreted as a spatially dependent light shift [63] Grimm et al [64] observe that in this limit, which, as they say, is ħ γsc = Γ Vdipole, a direct consequence of the fundamental relation between the absorptive and 8 The author has investigated deviations from this assumption for thermal atoms [6] 9 A spatially independent energy ħ ln(1 + δ ) has been subtracted from the integration of Eq (33) (36)

44 8 two-level systems dispersive response of the oscillator This is another consequence of the Kramers-Kronig relations, as discussed in Their formulation in Reference [64] summarises the calculation of the dipole potential for multilevel atoms 3 Radiation pressure force and magneto-optic trapping The radiation pressure force for a running plane wave [Eq (1c)] is Fpressure (r) = ħk Γ s 1 + s + δ (37) The radiation pressure force is a product of the momentum transferred to the atom from the recoil of a single photon, ħk, with the photon scattering rate, γsc [Eq (6)] This force is responsible for magneto-optic trapping, where orthogonal sets of counter-propagating beams, each with circular polarisation, are used to cool and trap atoms The detuning in a magnetooptic trap is velocity dependent (from the Doppler shift), and position dependent (from the Zeeman shift in a quadrupole magnetic field) In one dimension, the detuning for each of the two beams is given by k (x, v) = k v µ B q k x/ k, ² ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ Doppler (38) Zeeman where B q > is the magnitude of the quadrupole field gradient, > is magnitude of the red-detuning of each beam at zero-field and zero-velocity, and k {k+, k } with k+ k = k and k = k+ = k The Zeeman detuning is proportional to the differential magnetic moment between the optically coupled states, whose magnitude is ħµ = m F g F m F g F µb, (39) where µb is the Bohr magneton, m F and m F are total atomic spin-projection quantum numbers for the ground and excited states, respectively, and g F and g F are the Lande g-factors for the ground and excited states, respectively The sign of the final term in Eq (38) originates from the product of signs of the differential magnetic moment and the local field gradient The polarisation of each beam must be chosen such that it drives a transition with Zeeman detuning of opposite sign to k x; equivalently, (m F g F m F g F ) has the same sign as k x, where x is the unit vector in the x-direction The choice of quantisation axis and quadrupole field orientation results in various conventions for specifying the σ ± character of the counterpropagating beams The above requirements for the atom-field orientation in a mot can be

45 4 adiabatic passage 9 described using as few conventions as possible in the following way: Beams propagating towards the quadrupole centre must have circular polarisation which is left handed with respect to the local magnetic field Subsequently, atoms moving away from the quadrupole centre will preferentially scatter pho- tons from the beam which both slows them down (F v < ), and pushes them towards the quadrupole centre (F x < ) This is verified by substituting Eq (38) into Eq (37) for each beam, and taking a series expansion for low velocities k v Γ, and small displacements from the origin µ B q x ħ Γ, yielding F = F+ + F (µ B q x + k v) 4ħk s (1 + s + δ ), (4) where δ = /Γ In this limit the atom undergoes damped simple harmonic motion If the motion were described by Eq (4) alone, there would be no limit to the cooling of atoms achievable with a mot Spontaneous emission, however, results in a randomly oriented force from each photon recoil, limiting the achievable temperature of atoms in a mot Additional techniques are employed to trap and laser cool atoms below the Doppler limit; a discussion of these techniques can be found in Reference [65] 4 Adiabatic passage In the rotating wave approximation ( 1), we eliminated the time-dependence of the Hamiltonian that was fast compared to all other relevant time scales of evolution There are many cases where the Hamiltonian has time-dependent coupling strengths, energy differences, or decay rates, which change slowly compared to the other time scales of the system If the Hamiltonian changes slowly enough, a system that begins in an eigenstate of the initial Hamiltonian will remain in an eigenstate of the time dependent Hamiltonian at any later time This process is known as adiabatic following, a result of the adiabatic theorem (a summary can be found in [66, p 744], and early works are [67, 68, 69]) Many processes in the trapping and manipulation of ultracold gases involve adiabatic following The confinement of atoms with magnetostatic traps or optical dipole traps is contingent on the internal state adiabatically following an eigenstate of the light-atom interaction despite its external motion In the frame of a moving atom in a magnetic trap, the magnetic field amplitude and direction are dynamic For the atom to remain trapped, its magnetic moment must follow the direction of this changing field, and stay in the same eigenstate of spin projection with respect to the time dependent quantisation axis (local magnetic field direction) [56, 7, 71, 7] Similarly, when an ultracold gas is imaged following its ballistic expansion, or when it is transferred to

46 3 two-level systems an optical dipole trap, a magnetic quadrupole trapping field is commonly ramped down, and a uniform bias field is applied The direction of the uniform field is often orthogonal to that of the local magnetic field at the minimum of the antecedent magnetic trap Atoms in an optical dipole trap adiabatically follow the steady-state solution of the optical Bloch equations ( 3) Transitions between eigenstates of the time-dependent Hamiltonian occur when adiabaticity is violated If the eigenstates depend weakly on time near the bounds of some interval, the probability that a transition is made at an intermediate time can be found using asymptotic methods [73] A lauded example of this is coupled two-level system, with a detuning that is linear in time and passes through zero; the two-level crossing problem Remarkably, this problem was solved independently (in quite different contexts and with varied techniques) in 193 by Landau [74, 75, 76], Zener [77], Majorana [78] and Stu ckelberg [79]1 The time de- pendent detuning amounts to the substitution λt in Eq (5), for which the eigenvalues of the dressed states { +, }, Eq (11), are shown in Fig 1 The crossing of uncoupled levels at t = = becomes an avoided crossing when there is a coupling between the two-levels11 Without loss of generality, suppose the system begins in state 1 + for early times, t [as seen by Eqs (1) and (1) with λt] If the subsequent evolution is adiabatic, the system will remain in the + eigenstate for all times Long after the resonance has been crossed, the + state corresponds to a different bare state, ie + for t In the { 1, } basis, the system has undergone a transition from 1 to, a process known as adiabatic passage, which plays a crucial role in several results of this thesis ( 4, 43) If the avoided crossing is approached too fast, the system cannot adiabatically follow the + dressed state, and a superposition of both and + (and hence 1 and ) will exist following the sweep, thereby leading to a nonzero probability of the system remaining in state 1 The probability of traversing the avoided crossing nonadiabatically, and finding the system in state 1 following the sweep is commonly known (despite the efforts of Stu ckelberg and Majorana) as the Landau-Zener probability PLZ = e πγlz ΓLZ = where Ω d, λ= 4λ dt (41a) (41b) The adiabatic limit is then ΓLZ 1 In this example, the Landau-Zener probability corre- sponds to PLZ = c1 (t ) with c1 (t ) = 1 1 Recent reviews of these seminal works can be found in [8, 81] 11 The term avoided refers to the raising of degeneracy of the two dressed states The uniqueness of the eigenvalues can be proved by elementary consideration of the characteristic (or secular ) equation of a Hermitian matrix with non-zero off-diagonal elements

47 adiabatic passage coupling 4 31 Ω populations max c time TΠ 5 1 Figure 3 (Top) Rabi frequency and detuning during the archetypal Landau-Zener sweep linearly chirped detuning, with a finite duration (Bottom) Population transfer during nearadiabatic passage At the end of the sweep, some population remains in state 1 due to imperfect adiabaticity, and the off-resonant Rabi oscillations that occur (inset) The parameters of this sweep are ΓLZ,min = 768 and αmin = 111, chosen to yield the shortest possible passage for an approximate level of fidelity, as measured by the residual population during the final several periods of the off-resonant Rabi oscillations max c1 The duration of this sweep is T = 18Tπ, ie times longer than a resonant π-pulse In practice, the range of the frequency sweep will never be infinite, and it is of experimental interest to consider what this range must be for Eqs (41) to hold More generally, we consider transferring population from state 1 to state, using a detuning sweep of finite duration T, with a given fidelity ξ, such that P1 c1 (T/) < ξ P c (T/) 1 ξ (4a) (4b) We seek to find Pi c i (T/), the probability amplitudes at the end of the sweep In doing so, it is useful to express the sweep rate λ, and duration T in terms of the Landau-Zener parameter ΓLZ, and a parameter α which characterises the range of the sweep α λt/ω1 1 The range of the detuning sweep can then be expressed as [ αω, αω]

48 3 two-level systems Then λ= Ω 8αΓLZ,T= 4ΓLZ Ω (43) The parameterisation of Eq (43) proves appropriate, as the solutions for Pi (α, ΓLZ ) are both analytic, and independent of Ω For example, 1 αγlz F 1/,3/ F,1/ c (T/) = i + 4αΓLZ F 1,3/ F 1/,1/ (44) where F a,b = 1 F1 (a + i ΓLZ, b, iα ΓLZ ) and 1 F1 is the Kummer confluent hypergeometric func- tion [8] A detailed numerical study of the time-dependent solutions13 reveals small oscil- lations in c i (t) at the end of the detuning sweep For the population remaining in state 1, these oscillations are bounded below by the adiabatic limit imposed by Eqs (41), otherwise largely insensitive to ΓLZ, and bounded below by the amplitude of the off-resonant Rabi oscillations that occur at the limits of the detuning sweep, given by Eqs (8) Consequently, we identify two independent criteria which must both be satisfied for Eqs (4) to hold: 1 asymptotic criterion: α αmin, where αmin is the chosen such that the off-resonant Rabi oscillations at the limits of the sweep have amplitude less than ξ adiabatic criterion: ΓLZ ΓLZ,min, where ΓLZ,min is chosen such that the Landau-Zener probability is less than ξ At the limits of the sweep, = α Ω and the amplitude of the off-resonant Rabi coupling is (1 + α ) 1 as determined by Eqs (8) Along with Eqs (41), this yields αmin = ξ 1 1 ΓLZ,min = (π) 1 ln ξ (45a) (45b) For a given ΓLZ and α, the weakest satisfied criterion determines which lower bound best approximates the amplitude of the oscillations in Pi Hence, the transfer will result in a fractional population remaining in the initial state which is bounded below, and well approxi13 Analytic expressions for c i are obtained which depend only on α, ΓLZ and t/tπ where Tπ = π Ω 1 is the duration of a resonant π-pulse For brevity, they are omitted

49 adiabatic passage 33 P1 (α, Γ LZ = ) 1 6 adiabatic limit ( + 99 ) PLZ ( Γ LZ ) it lim P1 (α = 99, Γ LZ ) c ati ab ( + α ) PLZ (Γ LZ = ) it lim ic i ad 1 3 ot pt ym as max c 1 1 max c asymptotic limit Γ LZ Α Figure 4 Analytic approximation to linear adiabatic passage fidelity, Eq (46) At the bounds of the sweep, the populations exhibit small oscillations, whose amplitude is bounded below by the adiabatic criterion and the asymptotic criterion (see text) The larger of these two bounds closely mimics the local maximum of the oscillations, max c1 [taken over several periods near t = T/; see Fig 3] as found from a numeric study of the solutions to Eq (6) with λt Each plot shows the remaining population in the initial state for: (left) fixed adiabaticity ΓLZ = and variable sweep range α, and (right) fixed asymptoticity α = 1 and variable adiabaticity ΓLZ In each case, the approximation of Eq (46) is a near equality, especially when at least one of the criteria in Eqs (45) is either strongly met or violated mated via, ξ(α, ΓLZ ) max {(1 + α ) 1, PLZ (ΓLZ )} (46) The degree to which the above expression holds is illustrated in Fig 4, validating the conclusions presented above One may ask, how rapidly can population be transferred with a given fidelity? Using Eqs (45) and (43), we note that ΩT 8αmin ΓLZ,min 4 1 = ξ 1 ln (ξ 1 ) π (47) Denoting the length of a resonant π-pulse by Tπ, we have ΩTπ = π, and find the ratio of the time taken to transfer population using a linear frequency sweep to that of a resonant π-pulse is T 4 ξ 1 1 ln (ξ 1 ) > 1 Tπ π (48) This shows that the form of adiabatic passage described here always takes longer than a resonant π-pulse The dominant scaling in Eq (48) comes from the asymptotic criterion in Eq (45a) Far from resonance, the detuning can be swept significantly faster than the rate imposed by Eqs (41) and (45b), whilst maintaining adiabatic following Indeed, the rate imposed by Eqs (41) is an upper limit for a given level of adiabaticity, which is most nec-

50 34 two-level systems essary near resonance A vast area of research has focused on the design of optimal coupling pulses for population inversion using adiabatic passage, particularly in the realm of nuclear magnetic resonance (nmr) Many alternatives to a linearly chirped detuning exist [83], including sweeps of the frequency and amplitude of the coupling pulse The simplest improvement to the linearly chirped detuning discussed so far is to ramp the amplitude of the coupling on and off at the beginning and end of the sweep, relaxing the asymptotic criterion Alternative pulses for adiabatic passage are discussed in Appendix A, in which the arguments of this Section are applied and extended Adiabatic passage is sometimes referred to as adiabatic rapid passage The results of this Section would seem to indicate there is nothing rapid about adiabatic passage However, the term rapid refers to a condition separate from adiabatic following, and applies to systems where there is a loss mechanism resulting in a decay of coherence In nmr for example, where adiabatic rapid passage was initially termed adiabatic fast passage, the detuning must be swept fast enough to combat the decay of transverse magnetisation, whilst obeying the adiabatic criterion [84, 85, 86, 87] ³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ d Γ Ω Ω, dt ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ adiabatic (49) rapid where Γ is the rate of decoherence (see 5, 6) Another example for which this additional criterion needs to be met is adiabatic passage via an intermediate level, as in the case of stimulated Raman adiabatic passage (stirap) [88, 89] For optical Raman transitions, the initial and final states are often ground states, and the intermediate state is an optically excited level with some finite lifetime (limited by spontaneous emission) For the passage through the intermediate state to be coherent, the sweep must be faster than the time for spontaneous emission to occur In this thesis, adiabatic passage is performed exclusively using magnetic dipole transitions amongst the 5 S1/ hyperfine ground states of 87 Rb The lifetimes of these states, and decay of transverse magnetisation, are sufficiently long to ignore the effect of decoherence; as such, only the phrase adiabatic passage will be used here 5 Bloch vector formalism The Bloch vector is a powerful way to visualise the dynamics of an ensemble of two-level systems For general two-level systems, it is an algebraic construct with which to discuss Ramsey interferometry or adiabatic following, equivalent to the density matrix presented in the previous Section For actual spins in a magnetic field, it also corresponds to a physical

51 5 bloch vector formalism 35 observable: the ensemble averaged expectation value of the spin operator (or magnetisation) We consider the latter case first, specifically, that of a spin- 1 particle in a magnetic field, before presenting the equivalence between actual spin and the fictitious spin in any two-level system Consider the two-level system of a spin- 1 particle, with spin operator S, and associated magnetic moment µ = γ S The proportionality between the spin and the magnetic moment is the gyrometric ratio, γ = µb g S /ħ, where µb is the Bohr magneton, and g S is the Lande gfactor of the spin- 1 particle In the basis of states whose spin-projection along the z-axis is well defined, ie z and z (the spin-up and spin-down states), the spin operator has matrix representation S ħ σ, where σ = {σx, σ y, σz } is the three-vector of Pauli-spin matrices The interaction of the spin-system with a magnetic field B is described by the Hamiltonian H B = µ B ħ γ Bz B + ib x y B x ib y Bz (5) Since the density operator is Hermitian, the density matrix can be written as ρ = 1 (1 + P σ) ; P R3 = Pz P + ip x y Px ipy 1 Pz (51a) (51b) This representation illustrates that the expectation value of the spin-operator S is proportional to P : ħ Tr(ρ σi ) ħ = [Tr(σi ) + Tr ((P σ) σi )] 4 ħ = Pi Sˆi = We define S = S = ħ P (5) as the Bloch vector, an equivalent representation to the density matrix for describing an ensemble of spin- 1 systems14 Inspection of Eq (51b) shows that the length of the transverse component of the Bloch vector is a measure of the coherence, since S x + S y = ħ Px + Py = ħ ρ1 (53) 14 We will loosely refer to P as the Bloch vector also, and redefine the constant of proportionality as to ensure P is of unit norm, where required (see, for example, 56)

52 36 two-level systems The longitudinal component of the Bloch vector is the difference of the populations in the spin-up and spin-down states along the z-axis, Sz = ħ ħ Pz = (ρ11 ρ ) (54) The equation of motion for the Bloch vector can be determined using the Heisenberg equation for ρ, Eq (15) In the absence of decay, we have ds i ħ d = Tr(ρ σi ) dt dt ħ dρ = Tr ( σi ) dt 1 = Tr ([H, ρ]σi ) i (55) Substituting Eqs (5) and (51) into Eq (55), we arrive at ds = γb S, dt (56) illustrating that the Bloch vector S rotates about the magnetic field B with angular frequency ωl = γ B, the Larmor frequency With decay, the time evolution of the density matrix has the additional term of Eq (16), which leads to an amended equation of motion for the Bloch vector: Γ S x ds = γb S Γ S y dt Γ1 (Sz ħ/) (57) Decay serves to shorten the length of the Bloch vector; it lies on the Bloch sphere, ie S /ħ = P = 1, only for pure ensembles Relaxation returns the Bloch vector to the lowest energy configuration (spin-up in the convention used here) at a rate Γ1, and decoherence serves to shorten the magnitude of the Bloch vector in the x-y plane, at a rate Γ In the context of nmr, we identify T1 = Γ1 1 : relaxation T = Γ 1 : dephasing / decoherence = +, T T1 T (58a) (58b) (58c) and note that Γ Γ1 /, or T1 T, a consequence of the requirement that the Bloch vector

53 5 bloch vector formalism 37 does not increase in length We then have Γ = Γ1 /+ Γ, where Γ1 / is the relaxation induced decoherence rate, and Γ = 1/T is the rate of additional dephasing or decoherence In other words, decoherence is a by-product of relaxation, but can also occur in its absence We now turn to the equivalence between any two-level system coupled by electromagnetic radiation, and a spin- 1 particle in a magnetic field Comparison of the Hamiltonian for each in Eqs (5) and (5), respectively, invites the identification of γ B x Re(Ω e iϕ ), γ B y Im(Ω e iϕ ), γ Bz, or γ Beff = Ω (Ω cos ϕ, Ω sin ϕ, ), (59a) (59b) (59c) (59d) which defines the effective magnetic field in the rotating frame of the two-level system coupled by radiation, and tan θ B, B = Bx + By, Bz 1 z, z (59e) (59f) (59g) Upon making this identification, many of the results for the two-level system described thus far have a direct analogy for spin- 1 particles, and vice versa For example, using the substi- tution in Eqs (59d)-(59f), the eigenstates of the Hamiltonian in Eq (5) are identical to those of the Hamiltonian in Eq (5): θ θ = e iϕ/ cos ( ) z + e iϕ/ sin ( ) z, θ θ = e iϕ/ sin ( ) z + e iϕ/ cos ( ) z, (6a) (6b) so we have the equivalence +, E± = ±, ħωr ħγ B ± (61a) (61b) (61c)

54 38 two-level systems Pure ensembles in such superpositions have Bloch vectors given by cos(ϕ) sin(θ) ħ S = S = sin(ϕ) sin(θ) = S = S cos(θ) (6) This shows the eigenstates of a spin- 1 particle in a magnetic field have spin parallel, and antiparallel, to the magnetic field; and are simply the spin-up and spin-down states with respect to the quantisation axis aligned along B Alternatively, there exists angles θ [, π) and ϕ [, π) such that any two component superposition ψ can be written in the form θ θ ψ = e iϕ/ cos ( ) 1 + e iϕ/ sin ( ) (63) Pure ensembles in such a superposition can be assigned a fictitious spin, with Bloch vector given by the expression in Eq (6) The polar angle of the Bloch vector is related to the relative population, and the azimuthal angle is the relative phase, ie the angles θ and ϕ define a point on the Bloch sphere The equivalence between the two-component superposition in Eq (63) and the Bloch vector in Eq (6) is not unique; the state e πi S x /ħ ψ = ψ also has the Bloch vector in Eq (6) Physically, this is because π rotations of the spinor yield the same expectation value of the spin, but an overall phase factor of the two-component superposition Mathematically, in the language of group theory, this is because the group of unitary, complex matrices, su(), is a double-cover of the group of orthogonal, 3 3 real matrices, so(3), that perform rotations in R3, or su()/{1, 1} is isomorphic to so(3) [9] Mixed ensembles of a general two-level system can also be assigned a Bloch vector for their fictitious spin, using the density matrix mapping in Eqs (51) In this way, one can also cast the equation of motion for S in linear form; where ds = M S + v, dt Γ ħ Γ1 M = Γ Ω ; v = Ω Γ1 (64a) 1 (64b) This is useful, as the steady-state solution for S amounts to solving the linear equation M S =

55 5 bloch vector formalism 39 v For Γ1 = Γ = Γ, this results in s δ ħ 1, Sss = s δ 1+s+δ 1+δ (65) where s and δ are defined in Eqs (19c) and (19d) This is entirely consistent with the results for ρ ss i j in, as seen by substitution of Eq (65) into Eq (51), and comparison with Eqs (19) As a final example to consolidate this equivalence, consider the hybrid situation of a spin- 1 particle in a magnetic field, with spin states coupled by electromagnetic radiation We first choose the z-axis parallel to the magnetic field; the states z and z are then energy eigenstates in the absence of electromagnetic coupling, with energy splitting ħω = γ Bz In the Schro dinger picture, the Bloch vector of an ensemble will precess around the z-axis at the Larmor frequency ωl = ω If we transform to the rotating frame, without coupling the Bloch vector still Larmor precesses about the static magnetic field along the z-axis, but at a frequency ωl =, by way of Eq (59c) In this frame, electromagnetic coupling pulses rotate the actual spin vector about the effective magnetic field Beff in Eq (59d) We are well equipped to apply the results of this Section, and determine how the Bloch vector evolves in certain circumstances 1 Rabi regime: when the applied electromagnetic coupling is strong, such that Ω, Γi In this case, the effective magnetic field in the rotating frame [Eq (59d)] lies in the x-y plane, at an azimuthal angle ϕ from the x-axis The Bloch vector rotates about this direction by an angle θ = Ωt for a coupling pulse of duration t Free evolution: when the electromagnetic coupling drive is off, Ω = and the effective magnetic field is along the z-axis The Bloch vector rotates an angle t about the z-axis of the rotating frame, during free evolution time t The transverse component of the Bloch vector decays exponentially at a rate Γ, and the longitudinal component decays exponentially towards Sz = ħ/ (spin-up) at a rate Γ1 These are related to the decay of coherence, and relaxation to state 1, respectively For the application of a single coupling pulse, we can choose the phase of the coupling pulse arbitrarily In the spin-space of the Bloch vector, this amounts to an arbitrary choice of ori- entation of the x-y plane A convenient choice is to set ϕ = ; the rotating frame is in phase with the electromagnetic coupling, which induces spin rotations about the x-axis What does it mean to change the phase of the electromagnetic coupling in this picture? Such a phase change may be deliberately applied to the electromagnetic field during, or between, the ap-

56 4 two-level systems plication of multiple coupling pulses In the rotating frame with which we transformed to in 1, the axis about which the Bloch vector rotates can be changed at will by varying the phase of the electromagnetic coupling 6 Ramsey interferometry We now study the ensemble behaviour throughout a basic Ramsey interference experiment A schematic of the Ramsey interference sequence is shown in Fig 5 Ψ( t ) = c c = coupling Π i T e i T i e i T i cos( T ) sin ( T ) Π time Figure 5 Schematic of pulse sequence for Ramsey interferometry, and evolution of the two-component spinor in the absence of decoherence A pure ensemble of two-level atoms, initially in state 1, is put into a superposition of both states by a π/-pulse [Eqs (77)] Following the first coupling pulse, this ensemble evolves during time T, at which point it is described by Eqs (78) The coherent fraction of the ensemble has relative phase ϕ = T, which is converted into relative population by the second π/-pulse [Eqs (8)] In the absence of decoherence, the evolution of a pure state can be found by propagating the state vector with the unitary time evolution operator, U (δt) = exp( i H δt/ħ) ψ(t + δt) = U (δt) ψ(t), S(t + δt) = ψ(t) U (δt) σ U (δt) ψ(t) (66a) (66b) (66c) The evolution operator in the frame rotating out of phase with the electromagnetic field by δϕ has matrix representation ΩR δt Ω cos ( ΩR δt ) i sin ( ΩR δt ) ie iδϕ sin ( ) ΩR ΩR, U(δt) = ΩR δt ΩR δt ΩR δt iδϕ Ω ie sin cos + i sin ( ) ( ) ( ) ΩR ΩR (67) which we use to perform spin rotations of the two-component order parameter of a pseudo-

57 6 ramsey interferometry 41 spin- 1 bec in Chapter 615 Particularly useful examples are a π/-pulse in the Rabi regime with phase δϕ ie i δϕ 1 1, U π/ = ie i δϕ 1 (7) a π-pulse in the Rabi regime with phase δϕ and free evolution for a time T ie i δϕ, Uπ = ie i δϕ (71) e i T/ Ufree = e i T/ (7) Colloquially, a π/-pulse can be said to convert relative phase into relative population To illustrate this, consider the application of a π/-pulse to the state in Eq (63) c1,f ie i δϕ e iϕ/ cos(θ/) 1 1 = ie i δϕ c,f 1 e iϕ/ sin(θ/) ² ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ψfinal (73) ψinitial U π/ The normalised population difference after the π/-pulse is c1,f c,f = S = sin(θ) sin(ϕ δϕ) ħ z (74) For states with equal population before the second π/-pulse, θ = π/ and the Bloch vector 15 This operator can also be expressed as U = exp( i Ω σ δt), which rotates the Bloch vector about the unit vector n = (cos δϕ sin δθ, sin δϕ sin δθ, cos δθ) = Ω/ΩR by an angle β = ΩR δt, where tan δθ = Ω/ For a resonant π/-pulse [ =, Ω = ΩR, δθ =, β = π/, n = (cos δϕ, sin δϕ, )], the rotation matrix is cos (δϕ) π ˆ Dn ( ) = sin(δϕ) cos(δϕ) sin(δϕ) sin(δϕ) cos(δϕ) sin (δϕ) cos(δϕ) sin(δϕ) cos(δϕ) (68) Since Pz is usually measured, a π/-pulse probes the transverse spin, or coherence [Eq (53)] For free evolution, the rotation operator is cos( t) sin( t) sin( t) (69) cos( t) 1

58 4 two-level systems lies in the x-y plane After the π/-pulse, the normalised population difference depends only on the initial relative phase ϕ, and the phase shift of the pulse δϕ Alteratively, the action of a π/-pulse on the corresponding Bloch vector in Eq (6) is cos(δϕ) sin(θ) cos(δϕ ϕ) + sin(δϕ) cos(θ) ħ Sfinal = sin(δϕ) sin(θ) cos(δϕ ϕ) cos(δϕ) cos(θ), sin(θ) sin(ϕ δϕ) (75) ie a π/-pulse with a phase shift of δϕ rotates the Bloch about by π/ about the equitorial unit vector n = (cos(δϕ), sin(δϕ), ) The final longitudinal spin projection is the initial component of the spin in the transverse direction perpendicular to n, ( sin(δϕ), cos(δϕ), ) In cases where decoherence or decay is present, the evolution of the density matrix is determined using Eqs (17), and the evolution of the Bloch vector using (57) We consider an initially pure ensemble, with 1 Ψ() =, ħ S() =, 1 (76) ie the spin-up state In the Rabi regime, the first π/-pulse preserves the purity of the ensemble, and results in 1 1 Ψ(t π/ ) =, i ħ S(t π/ ) = 1 (77) A coherence has been established between states 1 and, which are equally populated The Bloch vector has been rotated about the x-axis by π/; it lies entirely in the x-y plane The system then evolves freely, in the absence of electromagnetic coupling, for a time T Decoherence will occur during this time, and the quantum state of the ensemble can no longer be completely described by a two-component state vector In the following, the density matrices and Bloch vectors will be given for the general case in which decay exists, and the state vector will be expressed for the limiting yet insightful case of Γi = After the free

59 6 ramsey interferometry 43 evolution, and immediately prior to the second π/-pulse, the state vector is i T/ 1 e Ψ(t π/ + T) = i e i T/ (for pure ensembles), e Γ T cos( T π ) ħ Γ T π S(t π/ + T) = e sin( T ) 1 e Γ1 T (78a) (78b) The off-diagonals of the density matrix and the transverse components of the Bloch vector have decreased in magnitude, indicating a loss of coherence during the free evolution The diagonals of the density matrix and the longitudinal component of the Bloch vector, signify the partial relaxation to state 1 ; the Bloch vector no longer lies in the x-y plane, since the relaxation has caused a population imbalance The transverse component has also precessed about the z-axis by an angle T, equivalent to a relative phase between state and state 1 of after the free evolution ϕ = T π (79) Finally, the application of the second π/-pulse, phase shifted by δϕ with respect to the first, results in e i δϕ/ sin ( T δϕ ) (for pure ensembles), Ψ(t π/ + T + t π/ ) = i i δϕ/ T δϕ cos ( ) e e Γ T cos(δϕ) cos(ϕ δϕ) + (1 e Γ1 T ) sin(δϕ) ħ S(t π/ + T + t π/ ) = e Γ T sin(δϕ) cos(ϕ δϕ) (1 e Γ1 T ) cos(δϕ), Γ T e sin(ϕ δϕ) (8a) (8b) The longitudinal component of the Bloch vector, indicate that a measurement of either population will yield an interference signal whose phase is offset by δϕ to the relative phase ϕ between the two states prior to the second π/-pulse Accordingly, one can acquire a Ramsey interference fringe in one of three ways; measuring the populations for varying either (i) evolution time T, or (ii) detuning, or (iii) phase shift of the second π/-pulse δϕ There are merits to each, and the preferred technique depends on the system with which Ramsey interference is being performed In the example given here, the amplitude of the Ramsey fringe will decrease in time This decrease is referred to as a loss of interferometric contrast or vis-

60 44 two-level systems ibility, and depends explicitly only upon Γ We shall make use of this result in 6, where measurements of the Ramsey interference signal of a pseudospin-1 bec are presented

61 Chapter 3 Magnetic-dipole coupling in alkali hyperfine ground levels We use magnetic dipole coupling to prepare, manipulate, and interrogate two-component becs The coupling is between the two states of the pseudospin- 1 system, which are amongst the hyperfine ground levels of 87 Rb Such coupling is considered here for alkali atoms in general, as they are most often used to create Bose-Einstein condensates in dilute atomic gases Coupling between these states is ordinarily achieved with magnetic dipole transitions, using a combination of microwave (mw) and radiofrequency (rf) radiation Optical Raman transitions (two laser fields driving electric dipole transitions) can also be used to achieve two-photon coupling between the desired levels, however rf and mw are easily manipulated and not limited by the problems associated with spontaneous emission, beam alignment, spatial uniformity of the electromagnetic field intensity, and the greater technical challenges of the stability and modulation of frequency and phase The coupling strength between two levels determines the time required to prepare a desired superposition, or enact a desired spin rotation In experiments considered in this thesis, and elsewhere, it is often desirable to induce these transitions in the strong coupling regime, where the time taken to manipulate the internal state of the condensate is short with respect to the time scales of other dynamics which we seek to investigate The strong coupling regime allows us to consider the action of magnetic dipole coupling in the context of simple two-level (or multi-level) system, whose internal dynamics are decoupled from the external evolution due to mean field, kinetic and potential energies We seek to couple states whose spin projection differs by two units of angular momenta These states have low differential sensitivity to magnetic fields, allowing uniform coupling of the states in the inhomogeneous magnetic trapping field, but require two-photon transitions for their coupling This Chapter investigates how the strongest two-photon magnetic

62 46 magnetic-dipole coupling in alkali hyperfine ground levels dipole coupling can be achieved for a given experimental circumstance, and identifies which experimental conditions to change for an improvement in coupling strength While the requisite details of magnetic dipole coupling can be found in disparate sources, a united and detailed study of two-photon magnetic dipole transitions between alkali ground states, in the context of pseudospin- 1 condensates, is lacking Previously, mw-rf two- photon transitions in this system have been understood in terms of three-level Raman coupling (Appendix B) Here, we show the important differences and surprising consequences of considering all Zeeman states of the hyperfine ground levels, and their respective coupling As we are measuring energy shifts on the order of 1 Hz with pseudospin- 1 condensates ( 6), it is important to consider the effect the interrogation mechanism has on the energies of the constituent states In 35 we investigate the effects of radiofrequency coupling beyond the rotating wave approximation Inclusion of these effects indicates the locations of magnetic resonances can vary upon application of a coupling field, the long understood Bloch Siegert shift The rf which composes the two-photon transitions is off-resonant, and more relevant here are the shifts of the state energies beyond the rotating wave approximation Furthermore, considerations of these effects are commonly for spin- 1 systems, insufficient for the system we seek to describe Precise knowledge of the magnetic-dipole coupling and associated level shifts are also important for work involving radiofrequency dressed-state potentials used for atom interferometry [91, 5, 9], and simultaneous addressing of multiple transitions amongst atoms trapped in optical lattices, which can act as independent qubits [93] We also study the mechanisms which broaden microwave transitions in a magnetically trapped condensate 31 Hyperfine ground levels The alkali ground levels are n S1/ where n is the principal quantum number distinguishing the atomic species of alkali The total electronic angular momentum has quantum number J = S = 1/, as the single valence electron has no zero orbital angular momentum The total atomic spin can differ by one quanta of angular momentum, owing to the relative orientation of the electronic spin and nuclear spin The ground states are split by the hyperfine interaction H hfs = Ahfs I J = Ahfs (F I J ) (31) This describes the contact interaction between the total electronic spin J = S (as L = for the hyperfine ground levels of all alkali atoms) and the total nuclear spin I The energy eigen- states of this Hamiltonian are I, J, F, m F, where F is the quantum number of total angular

63 3 zeeman states of the hyperfine ground levels momentum F = I + J and I I, J, F, m F = I(I + 1)ħ I, J, F, m F, J I, J, F, m F = J(J + 1)ħ I, J, F, m F, F I, J, F, m F = F(F + 1)ħ I, J, F, m F, F z I, J, F, m F = m F ħ I, J, F, m F 47 (3a) (3b) (3c) (3d) In the hyperfine ground states of a given alkali, J = S = 1 and I are fixed, and F, m F can take the values I J F I + J F m F F, ie F = I ± 1, (33a) (33b) in integer steps As F and m F are the only quantum numbers of the hyperfine interaction which vary for a given alkali species, we denote the hyperfine eigenstates more concisely by F, m F The eigenvalues of the hyperfine interaction Hamiltonian are, using Eqs (31) and (3) I 1 Ahfs ħ F, m F, F = I + ; Ahfs I J F, m F = (I+1) 1 Ahfs ħ F, m F, F = I (34) The hyperfine splitting is then 3 Ehfs = hνhfs = Ahfs (I + 1 )ħ (35) Zeeman states of the hyperfine ground levels The Hamiltonian for both hyperfine and Zeeman interactions in the ground state of alkali atoms is H Zeeman = Ahfs I J µ B (36) The second term in the above gives the interaction of the total magnetic moment of the atom with an external magnetic field B The magnetic moment is given by µ = µb (g J J + g I I ), ħ (37) following the sign convention of [94] In most of this thesis, we take B to be commensurate with the quantisation axis z, the exception is the discussion of spin- 1 particles in 5 and

64 48 magnetic-dipole coupling in alkali hyperfine ground levels Majorana spin flips in A1 In the presence of the Zeeman interaction, only m F remains a good quantum number at any magnetic field,1 as such we denote the energy eigenstates of H Zeeman the Zeeman states α, m F In general, the Zeeman states will be superpositions of the F, m F states given by α, m F = U αf F, m F (38) F For low magnetic fields B 1 G, the Zeeman states are dominantly composed of a single F, m F state, ie typically, U αi± 1, U αi, so there is little admixture of each Zeeman 1 1 state between the two hyperfine levels Hereafter, we make no distinction between the actual Zeeman states α, m F, and the approximate Zeeman states in low magnetic fields F, m F, and label each α, m F state by the unique F, m F state to which it is equal at zero magnetic field The energies of the Zeeman states are given by the Breit-Rabi equation [95] νhfs g I µb m F B s νhfs 4m F ν F = I ± 1, m F = + ± 1+ x + x, where (I + 1) h I + 1 (g J g I )µb B x=, hνhfs 1, for m F = I 1 and x > 1 ; s= 1, otherwise (39) We use this equation to accurately infer the magnetic field from a spectroscopic measurement of a transition frequency between Zeeman states, and in turn to predict the frequencies at which such transitions will occur for other magnetic fields and/or sets of states 33 Magnetic dipole transition matrix elements Magnetic dipole transitions are induced by the light-atom interaction H rf in the Hamiltonian H = H Zeeman + H rf = Ahfs I J µ B µ Brf cos ωrf t, ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ H Zeeman (31) H rf where H Zeeman is the Zeeman Hamiltonian in Eq (36) We consider the effect of H rf on the zero-field eigenstates of H Zeeman, ie when B = and H Zeeman = H hfs is completely diagonal in the F, m F basis We seek to find the matrix elements F, m F H rf F, m F The matrix 1 m F = m I + m J remains a good quantum number for all magnetic fields, as [H Zeeman, F z ] = 1 1 At G, for example, U αi = 1 U αi± < 1 6 for all Zeeman states α, m F

65 33 magnetic dipole transition matrix elements 49 elements of H rf in the actual Zeeman basis are I+ 1 α, m F H rf α, m F = I+ 1 F F U α U α F, m F H rf F, m F, F =I 1 F=I 1 (311) but we neglect the admixture of Zeeman states between the different hyperfine levels, as discussed in the previous Section Any non-zero component of the oscillating magnetic field in Eq (31) has the potential to induce transitions between the Zeeman states This can be seen by expressing the magnetic dipole operator in terms of the electronic and nuclear angular momentum operators, in Eq (37), and noting that F, m F are not eigenstates of any component of either J or I This leads to non-zero off diagonal matrix elements of H rf of the type in Eq (1), as such H rf can induce radiative coupling between the Zeeman states As g I /g J 1, we neglect transitions induced by the nuclear spin, and consider only the matrix elements of J x, J y and J z in the F, m F basis The transverse components of the electronic angular momentum can be decomposed into the raising and lowering operators3 J x = 1 ( J + + J ), J y = 1 i ( J + J ) (31) The matrix elements of the electronic angular momentum components are readily evaluated in the J, m J basis using [96, p 7]: J, m J J ± J, m J = J(J + 1) m J (m J ± 1) ħδ J J δ m J,m J ±1, J, m J J z J, m J = m J ħδ J J δ m J m J (313a) (313b) To find the matrix elements of J ± and J z in the F, m F = I, J, F, m F basis, we expand I, J, F, m F = m J C(IJF; m F m J, m J ) I, J, m I, m J using the Clebsch-Gordon coefficients with the convention of Rose [96], C( j1 j j; m m, m ) = j1, j, m1, m jm, which gives F, m F J q F, m F = (314) J J C(IJF; m F m J, m J ) I, J, m I, m J J q I, J, m I, m J C(IJF ; m F m J, m J ) m = J m J = J J 3 The raising and lowering operators J ± should not be confused with J ±1 = 1 ( J x ± J y ) = 1 J ±, the spherical representation of J x and J y The former do not transform as a tensor when composed in ( J +, J, J ) with J = J z, whereas the spherical representation J λ = ( J +1, J, J 1 ) does The matrix elements of J λ can be expressed using Eq (313), or alternatively, J, m J J λ J, m J = ( 1) λ J(J + 1)C( j 1 j; m J + λ, λ)ħ with λ =, ±1 This also applies to all other types of angular momenta

66 5 magnetic-dipole coupling in alkali hyperfine ground levels We first use Eq (313a) to find simple expressions for the raising and lowering operators: F, m F J ± F, m F = J 1 1 m J = J+ 1 1 C(IJF; m F m J 1, m J 1)C(IJF; m J, m F m J ) J(J + 1) m J (m J ± 1) ħ = C(I 1 F ; m F ± 1, ± 1 )C(I 1 F; m F ± 1, 1 )ħδ m F,m F ±1 (315a) In the last step, we have taken J = S = 1 which limits the sum over m J to only the m J = 1 term, and implies that m F = m F ± 1 There are two types of allowed transitions in the above; transitions amongst Zeeman states of the same hyperfine level ( F =, m F = ±1), and transitions between Zeeman states of different hyperfine levels ( F = ±1) The former typically occur at radio-frequencies (rf) for alkali atoms in magnetic fields < 1 G, and the latter at microwave (mw) frequencies close to the hyperfine splitting The matrix elements for each type of transition can be expressed in an algebraic form: ( 1)F I 1 F(F + 1) m F (m F ± 1) ħ, I mw : F, mf ± 1 J ± F, mf = ± (I ± m F + 1) 41 ħ, F F = 1 I + 1 rf : F, m F ± 1 J ± F, m F = (315b) (315c) The matrix elements of J z are found by combining Eqs (313b) and (314) for J = 1 Magnetic dipole coupling derived from this component of the electronic angular momentum only re- sults in transitions between Zeeman states of different hyperfine levels, with F = ±1 and m F = : ħ mw : F, m F J z F, mf = [C(I 1 F ; m F 1, 1 )C(I 1 F; mf 1, 1 ) C(I 1 F ; m F + 1, 1 )C(I 1 F; m F + 1, 1 )] (I + 1) 4mF ħ = 4I + (315d) Recalling Eqs (37) and (31), the magnetic dipole transition character depends on which components of the oscillating magnetic field are non-zero, as these dictate, in turn, which of the above electronic angular momentum components will be present in H rf In particular, σ + and σ transitions will only be induced by the transverse component of the oscillating magnetic field B (perpendicular to both the z-axis and the static magnetic field B), and π transitions will be induced by the longitudinal component of the oscillating magnetic field, B

67 33 magnetic dipole transition matrix elements Rabi frequency convention We can now evaluate the matrix elements of H rf in the F, m F basis In doing so, it is useful to define the Rabi frequency for magnetic dipole transitions σ ± transitions : π transitions : µb g F B, ħ µb g F B Ω= ħ Ω= (316a) (316b) With this convention, a radiofrequency field linearly polarised perpendicular to the quantisation axis, and resonant with the Zeeman splitting, will 1 induce rotations of F = F around a direction in the x-y plane by an angle Ω t, and drive Rabi oscillations amongst states of different m F in a given hyperfine level, the slowest of which have a period of π Ω 1, for any value of F or I To illustrate 1, we note that for magnetic dipole transitions between states in the same hyperfine level, the Wigner-Eckart theorem [57, p 41] can be applied, giving µ = µb g F F ħ (317) Using this to find the matrix elements of H rf for the subspace of Zeeman states in a given hyperfine level, F, m F H rf F, m F, yields an identical result to Eq (315b) upon noting that g F ( 1)F I 1/ = gj I + 1 (318) for J = 1 For a radiofrequency field linearly polarised in the x-direction, for example, with amplitude B x, µb g F rwa µb g F B x F x B x cos ωrf t ÐÐ F x, (319) H rf = ħ ħ and hence H rf t = θ F x, θ = Ωt (3) In the above, we have implicitly transformed to the frame rotating at the frequency of the radiofrequency field, and made the rotating wave approximation4 [B x cos ωrf t z B x / in Eq (319)] The unitary time evolution operator then becomes U (t) = e i H rf t/ħ = e iθ F x /ħ D x (θ), (31) 4 This relationship can be intuitively seen from the results of 1, and is derived for polychromatic fields in a multilevel system in B

68 5 magnetic-dipole coupling in alkali hyperfine ground levels the operator for rotations of F around the x-axis by an angle θ = Ω t Examples of are found by solving the Schro dinger equation for a given spinor, as illustrated in 341 The following results are for microwave and radiofrequency fields with linear polarisa- tion, and Rabi-frequency defined according to Eq (316)5 Using Eqs (31) and (315), the only non-zero off-diagonal matrix elements of H rf are ħ Ω F(F + 1) m F (m F ± 1), ħω mw σ ± transitions : F, mf ± 1 H rf F, mf = ± (I ± m F + 1) 41, ħ Ω mw π transitions : F, m F H rf F, m F = (I + 1) 4mF, rf σ ± transitions : F, m F ± 1 H rf F, m F = (3a) (3b) (3c) with F F = 1 34 Single photon radiofrequency transitions The first example of magnetic dipole coupling we consider is that involving single photon transitions amongst Zeeman states of a particular hyperfine level, referred to here as Zeeman transitions As the Zeeman splittings are much less than the hyperfine splitting in low magnetic field, states in different hyperfine levels are not coupled by radiofrequency radiation, and it suffices to consider the Zeeman states of each hyperfine level separately Zeeman transitions are used to outcouple atoms from a magnetic trap, as a mechanism for both forced evaporative cooling and the creation of atom lasers, as well as probing the energy distribution of trapped ultracold atoms in radiofrequency spectroscopy These transitions are also used in the preparation and manipulation of spinor and pseudospinor condensates In the case of the latter, the Zeeman transitions are off-resonant and take part in a two-photon transition in conjunction with an off-resonant microwave, as discussed in 36 We focus on the radiofrequency coupling of Zeeman states by radiation that is linearly polarised perpendicular to the quantisation axis, as this was exclusively used in the experiments of this thesis Later in this Chapter, the procedure outlined in Appendix B to perform the rotating wave approximation for an n-level system coupled by m-radiation fields will be applied to all eight levels of 87 Rb 5 S1/, simultaneously driven by radiofrequency and microwave radiation For now, the Hamiltonian for the subset of Zeeman states in each hyperfine level will be given in the rotating wave approximation without derivation Using the results and conventions of 33, 5 We adopt the convention of Eq (316) for mw transitions as well Although the resulting Rabi frequency differs to that of resonant mw Rabi oscillations, it avoids a polarisation and state-specific definition of the mw Rabi frequency Since g F is the same for F = I ± 1, a single frequency parameter then describes the oscillating magnetic field strength for a given alkali isotope, and the differences in mw coupling strengths are taken into account by Eqs (3b) and (3c)

69 34 single photon radiofrequency transitions 53 h g f F = e g F = +1/ d GHz c F=1 4 b g F = 1/ a mf Figure 31 Relative coupling strengths of magnetic dipole transitions in 87 Rb 5 S1/, if the component of the oscillating field driving each transition were to have the same amplitude For each allowed transition the quantity F, m F H rf F, m F /(ħ Ω/) has been given, indicating the square of the relative coupling strength with respect to the general hyperfine Rabi frequency Ω defined in Eqs (316) the Hamiltonian for rf coupling within each hyperfine level is6 F=1 F= ψ+1 ψ = ψ ; ψ 1 ψ ψ 1 ψ= ψ ; ψ+1 ψ+ Hrf = ħ 1 Ωrf 1 1 Ωrf q Ωrf 1 Ωrf, Ωrf 3 Ωrf 3q Ω rf 3 3, Hrf = ħ Ω 4q Ω rf rf 3 Ω 3q Ω rf rf Ωrf 6 The two sets of Zeeman states have an energy separation of ħ(πνhfs + 3q) with this convention (33a) (33b)

70 54 magnetic-dipole coupling in alkali hyperfine ground levels where the Zeeman states in each hyperfine level have been ordered by ascending energy, and ψ m F = F, m F ψ The Zeeman shift is included up to second order in B, via = ωrf ω, π B khz F = 1 linear Zeeman splitting; ω = π B 737 khz F = linear Zeeman splitting, q = π B 7189 Hz, (34a) (34b) (34c) where B is the magnetic field in Gauss The quadratic Zeeman shift is commonly neglected here, as analytic results for energy eigenvalues and solutions to the Schro dinger equation are obtained with q The exact energies of the Zeeman states [described by the Breit-Rabi equation, Eq (39)] are included when required, and the system is studied numerically The Hamiltonians in Eqs (33) can be used to determine the time dependent populations in each Zeeman state when the rf drives Rabi oscillations, and Landau-Zener transitions, amongst the Zeeman states of a given hyperfine level, as studied experimentally in 41 and Analytic Rabi solutions in a spin-f system A novel way to attain analytic expressions for the transition probabilities in a spin-f system was deduced by Majorana [78], whose treatment can be applied to linearly Zeeman shifted levels coupled via magnetic dipole transitions In a given hyperfine level, the F + 1 state amplitudes are expressed in terms of two variables c1 and c, by making the ansatz ψ m F = F, m F ψ = λ m F c1f+m F cf m F, m F = F,, F (35) Identifying c1 and c as the state amplitudes for a spin- 1 system, we can use the two-level Schro dinger equation Eq (6) to find ψ m F (t) in terms of c i (t) Then, in the case of rf magnetic dipole transitions, the coefficients λ m F given by λm F = (F)!, m F = F,, F (F + m F )!(F m F )! (36a) result in the ψ m F satisfying the Schro dinger equation for the Hamiltonians in Eqs (33) When the system begins in a stretched state, ie F, m F = ±F, there is unity population in one of the effective two-level states [this follows from Eq (35)], and the Rabi solutions for c i (t) in Eqs (7) can be used to find the dynamic Zeeman populations For example, the

71 34 single photon radiofrequency transitions 55 population of each Zeeman state in an F = spinor initially in, is ψ = P4, ψ 1 = ψ = ψ+1 = 4P3 (1 P ) 6P (1 P ) 4P (1 P )3, ψ+ = (1 P )4, (37a) (37b), (37c), (37d) (37e) where Pi = c i (t) in Eq (8) with Ω = Ωrf This analysis can also be used to verify the results of the rf-rabi frequency convention discussed in 331 For example, the longitudinal spin projection during Rabi-oscillations of any spinor is F F z = m F ħ ψ m F m F = F F = m F ħ λm F P1F+m F PF m F m F = F = F ħ (P1 P ) ( = F Sz ) Ωrf + t Ωrf = F ħ 1 sin Ω + rf = F ħ cos(ωrf t) when = (38) The example of resonant rf Rabi oscillations ( = ) in each hyperfine level is shown in Fig 3 to consolidate the results of Radiofrequency dressed-state energies The rf dressed-state energies in each hyperfine level, in the rotating wave approximation and the regime of linear Zeeman shift [q in Eqs (33)], are En = ħ n Ωrf +, n { F,, F}, hence the energy splitting of the rf-dressed states is ħ (39) Ωrf +, in analogy with Eq (11) Each dressed state approaches a unique Zeeman state F, m F for Ωrf ; the dominant state of the admixture depends on the sign of both the detuning and g F The asymptotic behaviour of the rf-dressed states, denoted by n, is { n = F,, n = F } ÐÐÐÐÐ { F, m F = ±F,, F, m F = F } g F ± (33)

72 56 magnetic-dipole coupling in alkali hyperfine ground levels 8 1 F=1 populations populations t Π Ωrf F= 4 6 t Π Ωrf 8 1 Figure 3 Resonant Rabi oscillations amongst spin-1 (left) and spin- (right) Zeeman states In each case, the spinor begins in a stretched state F, m F = ±F, and radiation (typically radiofrequency) drives magnetic dipole transitions between the levels, for which the slowest Rabi period is π/ωrf The analytic solution to these Rabi oscillations can be found using Eqs (35) and (36), an example of which is given for F = in Eq (37) In the absence of coupling, the Zeeman states have energies in the rotating frame EmF = ħ mf gf g F (331) In the off-resonant limit Ωrf, when the rf is tuned far from the Zeeman resonance, the Zeeman states experience an rf-induced radiative shift, which is given (for either sign of the detuning) by δe m F = ħ m F g F Ωrf, g F (33) in direct analogy with dressed state shifts of the two-level system Eq (14) These radiative shifts are the magnetic dipole equivalent of the ac Stark shift induced by electric dipole coupling As such, we refer to energy shifts induced by magnetic dipole coupling as ac Zeeman shifts [97, 98, 99, 1], and reserve the term light shift for the ac Stark shift induced by electric dipole coupling with optical fields Equation (33) indicates that the Zeeman states involved in the two-photon mw-rf transitions studied here share a common rf induced (ac)ac Zeeman shift, as mf gf 1, for 1, 1 and, 1 ; = g F 1, for 1, 1 and, 1 (333) As we shall see in 36, this result is an important consideration when studying the twophoton transitions in 87 Rb 5 S1/

73 35 35 beyond the rotating wave approximation 57 Beyond the rotating wave approximation The efficacy of the rotating wave approximation (rwa) in 1 and 34 relies upon ω Ω The terms in Eq (4) pertaining to rotation at twice the frequency of the electromagnetic field were neglected on this assumption, which becomes less valid in the radiofrequency domain The consequence of neglecting these terms was identified by Bloch and Siegert [11] They realised that the resonance of a spin- 1 system does not occur precisely at ωrf = ω when driven by radiation linearly polarised perpendicular to the quantisation axis The frequency of the resonance slightly decreases; to lowest order the resonance shift is given by δω = Ω, 4ωrf (334) the Bloch-Siegert shift This formula (and higher order treatments of the shift, see [1, 13, 14, 15]) indicates the frequency at which the energy eigenstates become closest, ie the location of the avoided crossing It does not, however, describe the frequency dependent energy shift of the eigenstates, ie the correction to Eqs (11) and (39) Moreover, the Bloch-Siegert shift in systems of spin F > 1 has received little attention, in comparison to the large quantity of literature devoted to the spin- 1 case We focus on corrections to Eqs (11) and (39), and in particular, the shift of the 87 Rb 5 S1/ hyperfine ground levels driven by off-resonant rf, linearly polarised, and perpendicular to, the static field This informs the effect of such shifts on the location of two-photon mw-rf resonances, and the frequency of interference fringes attained when using the two-photon mw-rf coupling for Ramsey interferometry Shifts due to microwave coupling beyond the rwa remain neglected, for their magnitude is negligible Furthermore, the rf does not couple states within different hyperfine levels As such, it suffices to consider effects beyond the rwa in the F = 1 and F = levels separately In evaluating expressions for the energy eigenvalues beyond the rwa, it is helpful to con- sider the radiation field as quantised, incorporating photon number into the Hilbert space States of the combined atom-field system are denoted α, M, where α {a,, h} is an atomic Zeeman state, as prescribed in Fig 31, and M is the number of photons in the field There are innumerable such states, as M {, } In the F = 1 case, the atomic Zeeman states are a, b and c, and the rwa amounts to retaining the combined atom-field states a, N + 1, b, N, and c, N 1 for any N 1, ie the states between which there exists single photon resonances at, or near, ωrf = ω For any given N, this set of states defines what is referred to here as a rwa basis, for which the 3 3 Hamiltonian matrix is given in Eq (33a) The allowed transitions between these states involve absorption or emission of a single σ rf photon of frequency ωrf ω To predict the lowest order Bloch-Siegert

74 58 magnetic-dipole coupling in alkali hyperfine ground levels σ σ+ N+3 N+3 N+ N+ number of photons number of photons N+4 N +1 N N 1 N N 3 mf state N +1 N N 1 N N c b a N 4 mf state 1 1 d e f g h Figure 33 Radiofrequency coupling of combined atom-field states in spin systems with F = 1 (left) and F = (right) rf that is linearly polarised perpendicular to the quantisation axis can drive single-photon σ + and σ transitions (grey lines) For F = 1 [F = ], the single-photon σ [σ + ] transitions are resonant when ωrf ω, and the σ + [σ ] transitions are resonant when ωrf ω An rwa basis is defined by states which are resonantly coupled to each other by single rf photons with frequencies near the Zeeman splitting (shaded regions) Considering two rwa bases in proximity to that of which m F =, N is a member is sufficient to predict the Bloch-Siegert shift and level shifts beyond the rotating wave approximation, to lowest order in Ωrf /ωrf Examples of the multi-photon processes responsible for these shifts are shown, for (i) F = 1: (green lines) a, N + 1 b, N, a three photon coupling which is resonant when ωrf = ω /3, and (blue lines) a, N + 3 c, N 1, a four photon coupling resonant when ωrf = ω /; and (ii) F = : (green lines) e, N + 1 f, N, a three photon coupling which is resonant when ωrf = ω /3, and (blue lines) d, N + 4 h, N, a six photon coupling resonant when ωrf = ω /3, as indicated by Eq (335) shift, one accounts for the coupling of these states to those in two additional rwa bases, { a, N 1, b, N, c, N 3 } and { a, N + 3, b, N +, c, N + 1 } In particular, the a, N 1, b, N, b, N +, and c, N + 1 states are directly coupled to states in the orig- inal rwa basis by single σ + rf photons; however, the resonant condition for such coupling is ωrf = ω The additional states also exhibit coupling amongst themselves, within their respective rwa bases, mediated by σ photons and resonant when ωrf ω The Bloch-

75 beyond the rotating wave approximation rwa correction Hz 35 mf gf = mf gf = +1 +1/ +1/ 1 1/ 1/ frf khz 4 5 Figure 34 Correction to rf-induced ac Zeeman shift of 87 Rb 5 S1/ levels for effects beyond the rotating wave approximation The shift of levels dressed by rf that is detuned far from the Zeeman resonance [ ωrf ω Ωrf ] is given by Eq (33) in the rotating wave approximation This estimate is improved by accounting for multi-photon processes described in this Section, leading to the off-resonant ac Zeeman shift in Eq (336) The correction, the latter term in Eq (336), is plotted here for an rf Rabi frequency of 1 khz, and a 61 khz splitting of the Zeeman states Notably, the correction does not depend on the sign of the detuning, and is commensurate for states that share a common value of m F g F Siegert shift, and energy level shifts beyond the rotating wave approximation, arise due to multi-photon coupling between states in different rwa bases, with a resonant condition different to that of single photon σ + and σ transitions The lowest order process involves the transfer of one unit of angular momentum between an atom and the rf field, via the absorp- tion, or emission, of two σ photons and one σ + photon, ie N = ±3 and m F g F / g F = 1, for example m F =, N m F = ±1, N 3 The resonance condition for such transitions is ωrf = ω /3 Similar higher order multi-photon transitions are resonant when ωrf = g F m F ω, m F > g F N (335) When g F <, such multi-photon transitions have a positive resonant frequency when more σ photons than σ + photons are transferred, and vice-versa for g F > Examples of the multi-photon processes responsible for level shifts and resonance shifts beyond the rotating wave approximation are shown in Fig 33 The three-photon couplings with m F g F / g F = ±1 (green lines in Fig 33) are predominantly responsible for the shifts; higher order multiphoton transitions are vanishingly weak We find the dressed state energies by diagonalising the resulting 9 9 Hamiltonian matrix for the basis consisting of three rwa bases Corrections to the rotating wave approximation

76 6 magnetic-dipole coupling in alkali hyperfine ground levels are then determined by comparison with diagonalisation in the single original rwa basis, ie with the energies given by Eq (39) The process can be generalised to any F: a rwa basis is chosen, containing F + 1 states, and to estimate corrections to the dressed state energies, and the Bloch-Siegert shift of the single-photon resonance locations, an additional two rwa bases are included [a further (F +1) states] Using the Rabi frequency convention of Eqs (316), every single-photon Zeeman resonance is shifted in frequency by an amount given by Eq (334) with Ω = Ωrf, for any integer F When the rf is tuned far from the single-photon Zeeman resonances, Ωrf, the ac Zeeman shift of the Zeeman states is approximated by δe = = Ωrf m F g F ħ Ωrf ( ) g F (ωrf ) m F g F ħ Ωrf ( ) g F ω ωrf ω + ωrf (336) The first term in each of the above equalities is the off-resonant ac Zeeman shift in the rwa [Eq (33)], the second term is the lowest-order correction to the rwa, the so-called counter- rotating term Equation (336) holds for any integer F, and is compared with a numeric evaluation of the dressed state energies in Fig 34 As such, ac Zeeman shifts due from higher- order couplings, beyond the rotating wave approximation, are identical for states 1, 1 and, 1 (states 1, 1 and, 1 also experience the same shift) 36 Two-photon mw-rf transitions in 87 Rb 5 S1/ There are three sets of states in 5 S1/ with similar magnetic moments in low magnetic fields These are states 1, 1 and, 1, states 1, 1 and, 1, and states 1, and, Only the first set can be magnetically trapped, whereas the second set exhibits magnetic and radiofrequency tunable Feshbach resonances near 9 and 18 G, and can be optically trapped These are the two sets of states used for experiments in this thesis The states of either set have a total angular momentum projection which differs by ħ (ie m F = ±), and hence cannot be coupled by a single-photon magnetic dipole transition We use two-photon transitions to prepare and interrogate the requisite superpositions of the two states The two-photon transition occurs via an intermediate state with m F =, either 1, or, Both photons are sufficiently detuned from this state as not to populate it during any coupling pulses or sweeps, a process referred to as adiabatic elimination of the intermediate state This Section describes how adiabatic elimination of the intermediate state is possible and the associated level shifts that can arise The methodology, and elementary consideration of, two-photon coupling in a three-level system is presented in Appendix B, from which two important re-

77 36 two-photon mw-rf transitions in 87 rb 5 s1/ 61 sults are applied here: (i) eliminating the time-dependence from the Hamiltonian for the 87 Rb 5 S1/ system driven simultaneously by radiofrequency and microwave radiation, and (ii) reducing the dynamics of the magnetic dipole coupling amongst all Zeeman states to an effective two-level system While the simple analysis in Appendix B reveals the physics behind realising a two-level system with three coupled levels, it fails to describe two-photon magnetic dipole transitions in 5 S1/ in full A complete treatment of the magnetic dipole coupling amongst all eight levels reveals the differential ac Zeeman shift that might be expected to arise from a mismatch of radiofrequency and microwave coupling strengths is not present, in stark contrast with the three-level result 361 Possible state combinations In both cases, 1, or, can be used as intermediate states, as depicted in Fig 35 and Fig 36 The choice does not change the resonant rf frequency (always νzeeman µb g F B/h), but does determine the appropriate microwave frequency (approximately νhfs ±νzeeman ) The scattering properties are less favourable for trapped atomic samples in state,, which has a higher dipolar relaxation rate than 1, This is only important for very weak continuous coupling of the levels, which is not investigated in this thesis Finally, the choice of interme- diate state determines which polarisation mode drives the composite transitions; to couple states 1, 1 and, 1 requires two σ + transitions are driven, whereas two σ transitions are used to couple states 1, 1 and, 1 This is of no consequence when the coupling fields are linearly polarised perpendicular to the quantisation axis, as both σ + and and σ modes are equally present 36 Coupling strengths and ac Zeeman shifts When the coupling strengths of a two-photon transition differ, the two-photon detuning δ, depicted in Fig 35, Fig 36 and Fig B1, is not necessarily zero on resonance The sum or difference of the photon energies must be offset from the energy difference between the two states to account for off-resonant ac Zeeman shifting of the levels The microwave and radiofrequency coupling strengths of the two-photon transition commonly differ in an experiment, as the fields are derived from different hardware, and transmission of high power radiation is easier to realise at MHz frequencies than at GHz frequencies The resulting differential ac Zeeman one predicts from a three-level analysis [Appendix B, Eq (B1)] can be on the order of khz, easily resolvable from a spectroscopic determination of the resonant two-photon detuning δres (see 44) The author has searched for such an effect in 87 Rb 5 S1/ by comparing spectroscopic measurements of one- and two-photon resonances Results consistent with a differential ac Zeeman shift of the form Eq (B1) have proved

78 6 magnetic-dipole coupling in alkali hyperfine ground levels h Ωrf Ωmw g δ F = f e g F = +1/ d 68 GHz 1 c F=1 δ b g F = 1/ a mf Figure 35 Two-photon coupling of the 1, 1, +1 states using either 1, or, as the intermediate state In either case, the two-photon drive is comprised of a single radiofrequency photon and single microwave photon, whose mean detuning from the intermediate state is The two-photon coupling is not necessarily resonant when the two-photon detuning δ is zero, ie when the photon energies match the energy difference between states 1, 1 and, 1 elusive The resolution to this is found by considering the magnetic dipole coupling of all states in the hyperfine ground levels with the off-resonant microwave and rf radiation By using the same procedure as for the 3 level system to adiabatically eliminate the off-resonant populations, we can attain expressions for the effective two-level couplings Throughout this Section, we use an alternative and concise notation for the Zeeman states, labelling them alphabetically in order of increasing energy (Fig 31, Fig 35 and Fig 36) The complex state amplitudes in the Zeeman basis of a general state ψ are denoted ψ α α ψ, with α {a,, h} For the coupling of c 1, 1 with g, 1 using f, as the intermediate state (Fig 35), the Schro dinger equation in the interaction picture reads 3δ Ωrf ) ψ a (t) + ψb (t), Ω Ωrf rf i ψb (t) = ψ a (t) + (δ )ψb (t) + ψ c (t), i ψ a (t) = ( (337a) (337b)

79 36 two-photon mw-rf transitions in 87 rb 5 s1/ 63 h Ωrf Ωmw g δ F = e g F = +1/ f d 68 GHz c F=1 δ b g F = 1/ a 1 - mf -1 1 Figure 36 Two-photon coupling of the 1, +1, 1 states using either 1, or, as the intermediate state In either case, the two-photon drive is comprised of a single radiofrequency photon and single microwave photon, whose mean detuning from the intermediate state is The two-photon coupling is not necessarily resonant when the two-photon detuning δ is zero, ie when the photon energies match the energy difference between states 1, 1 and, 1 populations 1 8 Δ = Π 5 khz Δ = Π 1 khz Δ = Π 5 khz other 1 time ms 3 4 time ms time ms 1 15 Figure 37 Two-photon Rabi oscillations in 87 Rb 5 S1/ for various intermediate state detunings, with Ωrf = π 1 khz, Ωmw = π 6 khz The leakage is the time average of populations other than those in 1 1, 1 and, 1 ; its dependence on the intermediate state detuning is discussed in 45

80 64 magnetic-dipole coupling in alkali hyperfine ground levels Ωmw Ωrf δ i ψ c (t) = ψb (t) + ψ c (t) + ψ f (t), i ψd (t) = (δ 3 )ψd (t) + Ωrf ψ e (t), 3 δ Ωrf ψ f (t), i ψ e (t) = Ωrf ψd (t) + ( ) ψ e (t) Ωmw i ψ f (t) = ψ c (t) + Ωrf ψ e (t) ψ f (t) + Ωrf ψ g (t), 3 δ i ψ g (t) = Ωrf ψ f (t) ψ g (t) + Ωrf ψ h (t), i ψ h (t) = Ωrf ψ g (t) + ( δ)ψ h (t) (337c) (337d) (337e) (337f) (337g) (337h) We now turn to the problem of adiabatic elimination in the 5 S1/ levels The procedure is similar to that of the three-level Raman transitions (Appendix B) Initially, the only populated state is 1, 1, and we seek to describe the way it couples to state, 1 via a two-photon tran- sition, whose constituent photons are not resonant with any single photon magnetic dipole transitions This requires Ωrf, Ωmw, δ and we make the slow change ansatz ψ α (t) for α c, g By sequential elimination of the state amplitudes ψ a (t), ψb (t), ψd (t), ψ e (t), ψ f (t), and ψ h (t) using Eqs (337) in consecutive order, we can derive an effective two-level Hamiltonian for the reduced state vector ψ c (t) ψeff = ψ g (t) (338) This yields analytic expressions for the two-photon Rabi frequency and effective detuning In comparing these results to those of the three-level analysis in Appendix B, it is helpful to define the variables Ω1 = Ω = Ωmw, (339a) 6 Ωrf (339b) These variables both obey the convention of Eqs (5) and (B5), inasmuch as the coupling term in the interaction picture, upon making the rwa, is written i H j = Ω i / (see Fig 31) We note that in the absence of rf, the frequency of resonant microwave Rabi oscillations between the two states 1, 1 and, would be Ω1, whereas in the absence of the microwave, Ω is not the frequency of any resonant Rabi oscillation induced by the rf ( 331) The two-

81 36 two-photon mw-rf transitions in 87 rb 5 s1/ 65 photon Rabi frequency is Ωeff = = (4 δ)(3 δ) Ωrf 3 Ωmw Ωrf (4 δ)(3 δ) + (3δ/ 11) Ωrf Ω6rf 3Ωrf 11Ω4rf 3 Ωmw Ωrf + + O (1 + [ ]) Ω6 Ω 11Ω4 Ω1 Ω + O (1 + + [ ]) (34) The last approximation represents a series expansion in Ωrf / (or Ω / ) up to fourth-order, neglecting terms of linear or higher order in the two-photon detuning δ/ The requirements of minimally populating any state other than 1, 1 or, 1 (Ωrf, see 45) ensures the above result for Ωeff is very well approximated by the expression for the two-photon Rabi frequency in the three-level case, Eq (B11a) The general expression for the effective detuning is cumbersome by comparison, and for simplicity is expressed here for δ = : 7 eff δ= = = (Ωrf 6 ) [Ωrf (Ωmw + 8Ωrf ) 4Ωmw ] Ωrf Ω4rf Ωmw 3Ωmw Ωrf Ω4rf Ω6 + + O [ ] Ω1 Ω1 Ω Ω4 Ω6 + + O [ ], (341) which is in gross disagreement with the equivalent expression attained using only three levels, Eq (B11b) There is no differential ac Zeeman shift term, and to lowest order the effective detuning is equal to the microwave induced shift only The term corresponding to the rf induced shift is not present because both 1, 1 and, 1 are dressed by radiofrequency, they share a common rf induced shift, given by Eq (33) The predictions of the three- and eight-level analyses are compared in Table 31 for Ωrf = π 8 khz, Ωmw = π 48 khz and = π 35 khz Interestingly, the same combination of rf and mw field amplitudes lead to a different combination of rf and mw coupling strengths depending on which intermediate state is used (see Ω1 and Ω in Table 31), owing to the different magnetic dipole matrix elements of the corresponding single-photon couplings (Fig 31) However, their product remains fixed, in turn leaving the two-photon Rabi frequency unaffected by the choice of intermediate state For a fixed combination of rf and mw field amplitudes, one might conclude from a 7 For a given Ωrf, Ωmw, and, one can find the resonant two-photon detuning δres by solving the general expression for eff = In most cases, however, it suffices to use δres = eff δ=, in analogy with the three-level analysis [see Eqs (B11b) and (B1)]

82 66 magnetic-dipole coupling in alkali hyperfine ground levels states 1, 1, 1 via, 1, 1, 1 via 1, 1, 1, 1 via, 1, 1, 1 via 1, Ωmw Ωrf Ω1 Ω n=3 Ωeff n=8 Ωeff n=3 eff n=8 eff Table 31 Two-photon Rabi frequency and effective detuning for common state combinations and radiation powers The predictions made using a three-level analysis are compared to those attained by considering all magnetic dipole couplings in the eight levels of 5 S1/ All frequencies are expressed in units of khz All examples correspond to the same radiofrequency amplitude and microwave amplitude In all cases, = π 35 khz and δ = three-level analysis that the choice of intermediate state is still of consequence For instance, choosing the intermediate state which results in the smallest difference between rf and mw coupling strengths may 1 allow a smaller intermediate state detuning for the same amount of population trans- fer to states other than 1, 1 and, 1, yielding higher two-photon Rabi frequencies as per Eq (B11a), and decrease systematic ac Zeeman shifting of the resonance away from δ =, as per Eq (B11b) In fact, neither are true Numerical solution of Eqs (337) confirms the transfer of population into states other than 1, 1 or, 1 is identical for both choices of intermediate state, as discussed in the following Section This debunks 1, on the grounds that leakage does not just happen to the intermediate state, but rather amongst all the Zeeman states Since they are all dressed by the rf, the detuning needs to be such that rf transitions are never driven For a given rf power and initial state, this condition is uniquely defined The dominant origin of the systematic ac Zeeman shift has been identified as the microwave dressing, rather than a differential ac Zeeman shift Bringing the rf and mw couplings closer means increasing the microwave coupling in these examples, which acts to increase the systematic shift of the n=3 n=8 two-photon resonance, debunking The latter is shown in Table 31, whereby eff [ eff ] decreases [increases] when using 1, as the intermediate state We note that for this case, the two-photon Rabi frequency and effective detuning differ slightly from those shown in Equations (341) and (34), in their dependence on Ωrf and Ωmw, as well as Ω1 and Ω Finally, we note that the analysis of this Section has assumed the microwave field contains only the σ + mode, and that this mode off-resonantly couples only states 1, 1 and, In general, each of the σ +, σ and π polarisation modes off-resonantly couple three sets of states

83 36 two-photon mw-rf transitions in 87 rb 5 s1/ 67 These additional off-resonant microwave couplings may slightly modify the two-photon resonance condition expressed in Eq (341) Of these, the couplings which cause the largest modification of the two-photon resonance in Eq (341) are the transitions involving states 1, 1 and, 1 For a linearly polarised microwave field with components B parallel to, and B perpendicular to the quantisation axis, the correction to the effective detuning in Eq (341) is δ eff = = Ωcd Ω + ce 4 cd 4 ce ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ additional c radiative shifts Ω Ωbg ag 4 ag 4 bg ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ additional g radiative shifts 3 µb g F ( ) [B ( ) + B ( )], 8 ħ ω + ω ω + 3ω (34) where α H rf β = ħ Ω αβ /, αβ is the detuning of the microwave field from a particular transition (see Fig 35), and ω is the splitting of adjacent Zeeman states Microwave spec- troscopy of a 1, 1 trapped condensate in 43 reveals that there is comparable power in all modes of the microwave field for our experiments These additional components amount to an additional microwave induced shift of 1 Hz for our experimental conditions 363 Fidelity of the two-photon transition : leakage to other states The simple three-level [Eq (B11a)], and the complete eight-level [Eq (34)] treatment of the two-photon transition both indicate that the effective Rabi frequency is stronger for smaller intermediate state detunings It is important to know how close one can approach the in- termediate state with rf and mw fields of a given strength, without significantly populating states other than the two needed to realise a pseudospin- 1 system The leakage is defined as the time averaged normalised population in states other than 1 or, and has been esti- mated by numerically solving Eqs (337) for various Rabi frequencies and intermediate state detunings In each case, the two-photon detuning (δ in Fig 35) was adjusted to result in two-photon resonance,7 and the time-average taken over one two-photon Rabi period The leakage is well approximated by c i = i 1, Ωeff i 1, π π Ω 1 eff c i (t) dt 6Ωrf + 3Ωmw, 3 (343) which is shown in Fig 38 for 1 = 1, 1 and =, 1, and has been checked for a wide range of, Ωmw, and Ωrf For the two-photon conditions of the Swinburne apparatus (listed in 45), the predicted leakage to other states is approximately 4; despite changing

84 68 magnetic-dipole coupling in alkali hyperfine ground levels 1 Ωrf Π khz leakage 5 Ωrf Π 1 khz Δ Π khz 4 5 Figure 38 Leakage to states other than 1, 1 or, 1 during resonant two-photon Rabi oscillations The leakage, defined as i c,g c i (time averaged over a two-photon Rabi period), is found by numerically solving Eq (337) for Ωmw = π 3 khz, Ωrf = π khz (dots), and Ωmw = π 6 khz, Ωrf = π 1 khz (open circles), for various intermediate state detunings The two-photon detuning δ is chosen to yield the two-photon resonance in each case The curves are analytic estimates to the leakage, given by Eq (343) throughout a two-photon Rabi period, this is similar to the estimated value of population in unwanted states at the end of a π/-pulse This analysis suggests there is nothing to be gained by increasing the rf power when the rf coupling is dominant; for a given threshold of leakage, the intermediate state detuning would need to be increased such that Ωrf / remains fixed In turn, the two-photon Rabi frequency will remain unchanged There is merit in increasing the power in the mw field, however; the mw coupling is not as responsible as the rf coupling for leakage until Ωmw 3 Ωrf For mw powers below this value, there is little increase in the leakage shown in Fig 38, but a proportional increase in the two-photon Rabi frequency, by way of Eq (34) 364 Ramsey interferometry There is a subtle effect of performing Ramsey interferometry using a two-photon transition For Ramsey interferometry with a two-level system driven by a single radiation field, the phase of the interference fringe is ϕ = T when the phase of the coupling field is kept constant [Eq (79)] Consequently, if the coupling field is perfectly resonant ( = ), no fringe will be acquired upon variation of the evolution time T For two-photon transitions, the results of 6 can be used in the context of the effective two-level system formalism namely, the phase of the interference fringe will be ϕ = eff T However, even if the frequencies of the driving fields remain fixed throughout the duration of a Ramsey interference experiment, the effective detuning varies, as it depends upon the amplitudes of the coupling fields; these

85 36 two-photon mw-rf transitions in 87 rb 5 s1/ 69 amplitudes are modulated, naturally, by the nature of the interferometer The phase of the interference fringe depends upon the effective detuning during the evolution time (when the two-photon drive is off), which is generally different to the effective detuning during the π/-pulses, as the atomic energy levels are ac Zeeman shifted [Eqs (341) and (B11b)] This results in behaviour distinct from Ramsey interference using a two-level system coupled by a single-photon transition It is instructive to consider the following examples of Ramsey interferometry using the rf mw transitions considered in this Section: 1 No coupling fields on during evolution time: As the coupling fields are off, Ωrf = Ωmw =, and the effective detuning during the evolution time is eff = δ If the two-photon coupling is resonant, δ = δres (in general), and a Ramsey fringe will be acquired upon variation of the evolution time T Alternatively, if the two-photon detuning is vanishing (δ = ), no Ramsey fringe will be obtained, however the two- photon coupling might not be perfectly resonant Microwave drive on during evolution time: The amplitude of the radiofrequency coupling, Ωrf, is modulated to realise the π/-pulses As Ωrf = during the evolution time, no coupling occurs between the states, however the microwave field continuously shifts one of the states The effective detuning eff is nearly constant throughout the entire experiment [it depends weakly on Ωrf, see Eq (341)], mimicking the constant differential microwave induced shift As such, resonant two-photon coupling will result in no Ramsey fringe, as in the case of single-photon transitions 3 Radiofrequency drive on during evolution time: To first order, this is the same as case 1, as the radiofrequency induced shift (including effects beyond the rwa, 35) experienced by both states is identical This suggests that the differential microwave shift can be measured interferometrically, by modulating the amplitude of the microwave field during the evolution time We have confirmed the qualitative validity of the above three cases, by observing a temporal Ramsey interference fringe for case with a frequency 3 Hz higher than that for cases 1 and 3, consistent with the coupling parameters used for the F = 1, m F = 1 F =, m F = +1 transition ( 36) A more detailed study would involve partial modulation of the microwave amplitude during the evolution time, to confirm the functional dependence of the relative phase evolution8 ϕ=( Ωmw,T + δ) T, (344) 8 This expression neglects the higher-order effects of imperfect microwave polarisation and multiple microwave transitions encapsulated by Eq (34)

86 7 magnetic-dipole coupling in alkali hyperfine ground levels where Ω mw,t is the microwave Rabi frequency during the evolution time Leaving the microwave amplitude fixed during the coupling pulses would ensure the resonance condition is unchanged, eliminating the systematic of imperfect π/-pulses from the measurement

87 Chapter 4 Two-component bec : experiment This chapter describes the experimental techniques and apparatus used to obtain the main results of this thesis The author conducted experiments using the first generation of atom chip apparatus at Swinburne University of Technology in Melbourne, Australia, and in the laboratory of Professor David Hall at Amherst College, Massachusetts, USA Despite myriad differences between the two apparatus, many of the experimental techniques to prepare, control and interrogate two-component becs are common to both laboratories Following a brief description of the Swinburne apparatus in 41, the shared experimental techniques will be summarised without distinction between the two laboratories In cases where it is useful to articulate experimental parameters and hardware, or where an example of a given technique is presented, each laboratory will be denoted by sw (Swinburne atom chip), or ac (Amherst College) Section 3 lays the foundation for discussing the ground state transitions between and amongst hyperfine levels which are used to prepare superpositions necessary for two-component bec experiments This overview precludes a detailed analysis of two-photon magnetic dipole transitions in 87 Rb in 36, which includes new results regarding lights shift in this system The commonly used technique of adiabatic elimination in a three-level system (Appendix B) is extended to the hyperfine ground levels of 87 Rb, and marked differences are elucidated The spectroscopic techniques used to accurately calibrate magnetic fields, and characterise coupling of radiation to a trapped bec are detailed in 43, 44, and 45 Finally, a new and novel imaging technique to simultaneously image both spatial modes of a pseudospinor bec is described in 43 This technique facilitates many of the experimental results of Chapters 6

88 two-component bec : experiment 7 41 Swinburne atom chip The Swinburne atom chip has been used to produce bec since March 5 The atom chip is a hybrid design, with both microwires and a permanent magnetic film for magnetostatic trapping of ultracold atoms [16] The first experiments performed with this apparatus were focused on the preparation and manipulation of bec using permanent magnetic materi- als The origin of fragmentation of a bec in this permanent magnetic film microtrap was identified as inhomogeneity in the magnetisation of the film [17] The spatially corrugated magnetic potential supported a symmetric double-well potential, which was used as a superfluid sensor of field gradients [18] The PhD thesis of Shannon Whitlock [19] provides a detailed description of the apparatus and the above experiments,1 all of which used 87 Rb bec in the F =, mf = + state This Section will focus on the use of this apparatus to prepare 87 Rb condensates in the F = 1, m F = 1 state, and the associated changes implemented since the beginning of Atom chip apparatus The Swinburne atom chip is shown in Fig 41 Atop of the chip are two adjacent glass slides, each of thickness 3 µm One of the substrates supports the permanent magnetic material that can be used to create a magnetostatic trap [16] The other is a covered with only a layer of Cr, used to bond a 1 nm thick layer of gold to the substrate The gold top surface is common to both substrates and serves to reflect the infrared beams of a mirror magnetooptical trap (mmot) The layer of permanent magnetic material is a set of six alternating Cr and Gd1 Tb6 Fe8 Co4 films (of thickness 1 nm and 15 nm, respectively) The latter are magnetised out of the plane of the films, resulting in an effective current of A along the film edge In particular, the effective current along (,, z), in addition to a uniform bias field in the x-direction, creates an Ioffe-Pritchard magnetostatic trap near the chip surface The layered permanent magnetic film did not play a role in the experiments of this thesis; it was neither used, nor did it significantly affect the trapping using the current carrying wires that lay below the glass substrates The current carrying structure is a machined layer of silver foil (5 5 5 mm3 ) that permits currents to flow in U- and Z-shaped paths These current paths, in addition to a uniform magnetic bias field in the x-direction, allow either a mmot or an Ioffe-Pritchard magnetostatic microtrap to be supported near the chip surface [11, 111, 11] Each experimental cycle begins by loading Rb atoms in a mmot, 4 mm be- low the chip surface, from Rb dispensers (saes alkali metal dispensers) pulsed for 1 s The quadrupole field responsible for the initial mmot confinement results from passing 1 A 1 The author contributed to, and co-authored, References [17] and [18]

89 41 swinburne atom chip 73 IZ Irf x y IU IU IZ Irf Irf z Irf 1 mm Figure 41 Current carrying structure of Swinburne atom chip i: a mm3 machined silver foil (scale 9:1) The current paths of (left) the U-wire that supports a compressed mmot to load atoms into the magnetic trap, (centre) the Z-wire that supports a Ioffe-Pritchard magnetostatic trap, and (right) the auxiliary wires used to deliver radiofrequency radiation to trapped atoms, for driving magnetic dipole transitions The Swinburne atom chip i co-ordinate system, and scale of the images, are indicated on the rightmost image The chip is orientated face down, such that gravity is in the positive y-direction (Fig 4) of current through anti-helmholtz coils external to the vacuum system (15 mm diameter polyamide insulated copper wire, 4 turns turn cross section, 6 mm inner coil radius, water cooled), resulting in a field gradient of 89 G cm 1 Upon waiting a further 1 s for recovery of the ultra-high vacuum ( 1 11 Torr), the cloud is transferred within 3 ms to a mmot created solely by the U-wire current on the atom chip and a bias field The subsequent compression heats the cloud from 3 µk to µk This is followed by a polari- sation gradient cooling stage, whereby the trapping light, originally detuned by 3 linewidths below the zero-field resonance, is further detuned to 9 linewidths below resonance During the 1 ms of polarisation gradient cooling the U-wire current is decreased but remains on, confining the cloud in a relaxed quadrupole field At this point, the cloud typically contains 3 18 atoms at 3 µk To spin-polarise the atomic cloud for magnetic trapping, the quadrupole field from the U-wire is switched off, and the atoms are optically pumped into either the F =, m F = + state, or the F = 1, m F = 1 state (see below, 41) This increases the number of magnetically trapped atoms by a factor of 3-5 Approximately 6 17 atoms are then caught in a magnetic trap created by current passed through a Z-wire on the atom chip and a bias field The transfer efficiency is limited by imperfect mode-matching of the initial magnetic trap with the mmot, the finite depth of the magnetic trap, and truncation of the trapping potential by the nearby chip surface The magnetic trap is compressed to increase the elastic collision rate, essential for run- away evaporative cooling The compression is achieved by changing the Z-wire current and

90 two-component bec : experiment 74 bias field as to increase the magnetic quadrupole gradient Evaporative cooling is performed using the standard technique of forced radiofrequency outcoupling [113]; we apply chirped radiofrequency radiation (from MHz to 7 khz) during 9 s The compression, which also moves the trap minimum closer to the chip, is controlled such that the atomic cloud, whose size decreases with the synchronous cooling and compression, is not truncated by the chip surface In this way, we produce almost pure condensates of 3 15 atoms The Z-wire magnetic trap in which condensation occurs typically has radial and axial trap frequencies of 47 Hz and 15 Hz, respectively, for atoms trapped in the F = 1, m F = 1 state, and is 35 µm from the chip surface The trap used for the experiments of Chapter 6 is created by passing 159 A through the Z-wire and applying a 4 G bias field in the negative x-direction A 3 mg field applied in the z-direction, in addition to the z-directed bias field produced by the Z-wire, provides the requisite 33 G field at the potential minimum These parameters result in a relatively weak potential for an atom chip trap, but assist in realising a high degree of cylindrical symmetry On the dimensions of a trapped ultracold cloud, the potential is well approximated as harmonic with measured trap frequencies f ρ = 976() Hz and fz = 1196() Hz for the F = 1, m F = 1 state, which agree with a magnetostatic calculation of the potential to within a few percent This trap has a potential minimum of 33 G at a distance of 56µm from the surface of the atom chip The trap depth of G is not limited by the surface, as can happen with tighter confinement closer to the chip The Swinburne atom chip i apparatus is shown in Fig 4 41 Optical pumping When transferring ultracold atoms from a magneto-optic trap to a magnetic trap, the sample is typically spin-polarised to actively increase the number of atoms in a magnetically trappable state In our system, this is achieved with a brief pulse of optical pumping light (1 3 mw cm intensity, 1 ms duration), near resonant with the F = F = transition of the D line of 87 Rb The optical pumping laser is applied to the atoms in a uni- form magnetic field (nominally 34 G), immediately after switching off the mmot in which they were collected and cooled When creating bec in the, state, the optical pumping pulse is σ + polarised to trans- fer atomic population towards states of increasing m F A repumping beam, tuned to the F = 1 F = transition, is also applied to obviate spontaneous emission to the F = 1 hy- perfine ground level The F = F = optical pumping transition is chosen as to halt Magnetostatic calculations which take into account the exact dimensions of the wires, and gravity, predict the transverse trapping frequencies to be quite close, ω x /ω y 1 1 4

91 41 swinburne atom chip 75 quadrupole coil z x y RF wires U/Z shaped wires MMOT beam optical pumping beam CCD camera atom cloud MMOT beam absorption beam MMOT beam MMOT beam quadrupole coil microwave antenna Figure 4 Adapted from the PhD thesis of Shannon Whitlock [19]: Schematic of the Swinburne atom chip i apparatus and laboratory co-ordinate system Imaging optics, vacuum system, and Helmholtz coils external to the chamber to create bias magnetic fields are not shown the scattering of photons by an atom that has been pumped into the, state, avoiding unnecessary recoil and heating of the cold cloud To magnetically trap atoms in the 1, 1 state, they must be depumped from the F = ground hyperfine level in which they predominantly reside during magneto-optic trapping This can be achieved by merely extinguishing the F = 1 F = repumping light during the final 1 ms of the mmot stage The intense trapping light of the mmot will readily result in decay to the F = 1 hyperfine ground level, but without particular preference to the 1, 1 Zeeman state We find it more effective to actively depump the atoms, preferentially into the 1, 1 state Application of a σ optical pumping beam, in the absence repumping light, can be used for this purpose However, this alone does not result in optimal transfer to the 1, 1 state, as some atoms can be pumped into the, state, which is dark to σ light driving the F = F = transition An elliptically polarised optical pumping beam is used to drive σ and π transitions, allowing the transfer of atomic population to states of decreasing m F,

92 two-component bec : experiment atoms in magnetic trap optical pumping time ms 3 Figure 43 Efficacy of optically pumping atoms into the 1, 1 state, as determined by the number of 1, 1 atoms transferred to a magnetic trap Polarisation gradient cooled atoms from the mmot are optically pumped in a uniform field using an elliptically polarised beam tuned to the F = F = transition, with intensity 3 mw cm The optimal pumping duration is 6 ms, after which, expansion and free fall of the unconfined atoms hinders the efficiency with which they can be caught in a magnetic trap The deliberate transfer of atoms to the 1, 1 state increases the population of magnetically trapped atoms by a factor of 4 compared to a magnetically trapped cloud without optical pumping and improving the subsequent depumping to the 1, 1 state Atoms pumped to the, state can be excited by the π light, to an excited state from which decay to the 1, 1 state is permitted The 1 MHz linewidth of the optical pumping laser ensures the constituent Zeeman shifted transitions between the F = F = levels are all resonant with the pumping light Optimal optical pumping to the, state occurs at a frequency 1 MHz lower than for the 1, 1 state, consistent with the difference in differential Zeeman shifts at 34 G for σ + and σ pumping using the F = F = transition The efficacy of the optical pumping process is shown in Fig 43 for various lengths of exposure of the atoms to the pumping light Several other parameters which influence the pumping process are also optimised, including the frequency, and ellipticity of the polarisation, of the optical pumping light 413 Imaging We use standard absorption imaging to probe the column density of condensates [113] The probe beam used for absorption imaging is directed along the x-axis (tight confinement, Fig 4), yielding information about the axial and radial dependence of the atomic density The cloud is brought 1 mm away from the chip and decompressed prior to imaging; to reduce the effect of the permanent magnetic film on ballistic expansion, and lower the optical depth

93 77 35 measured atom number 14 axial condensate width μm 41 swinburne atom chip repumping time ms 4 5 Figure 44 Increase in the perceived width of a F = 1, m F = 1 condensate due to the recoil from repumping photons A repumping beam on the F = 1 F = transition is applied to a 1, 1 condensate immediately prior to absorption imaging using the F = F = 3 transition The time taken to transfer the entire condensate fraction into the F = level for absorption imaging is 17 ms, at which point the condensate width has almost doubled of the cloud to a level suitable for resonant absorption imaging The probe beam is resonant with the F = F = 3 transition To image condensates comprised of atoms in the F = 1 hyperfine level, we originally employed the commonly used technique of applying a repumping beam on the F = 1 F = transition immediately prior to the absorption probe During the few milliseconds required to repump all the atoms from the F = 1 to the F = level, the recoil experienced by those atoms which have absorbed and emitted a repump photon cause the subsequent absorption image to be blurred This is evident as an increase in the perceived width of an F = 1, m F = 1 condensate for increasing repumping time (Fig 44) While of little consequence when measuring only the number of atoms originally in the F = 1 level, this can be fatal to faithful imaging of a condensate wavefunction, as required in the experiments of Chapter 6 To avert the non-negligible recoil from optical repump photons, we use microwave adiabatic passage to transfer population from the F = 1 to the F = level prior to absorption imaging: outlined in 43 Quantitatively accurate absorption imaging relies upon optimal flat fielding, the sub- traction or division of a reference image of the probe beam, with no atoms present, from the absorption frame, in which the shadow of an atomic sample is case onto the probe Imperfect flat fielding results from spatial inhomogeneity of the probe that is not completely eliminated by the division of the reference and absorption frames A probe beam with high pointing stability and spatial uniformity assist this goal, both of which are improved by fibre

94 two-component bec : experiment 78 coupling the probe to cage mounted optics close to the vacuum chamber We improved our absorption imaging system by fastening a cage mount to a heavy breadboard, with minimal optics downstream from, and in the same cage mount as, the fibre launch; namely, a collimating lens, polarising beam splitting cube, and quarter wave plate The absence of mirrors, and a short beam path (< 1 m) resulted in less inhomogeneity in the reference frame, and better removal of the remaining inhomogeneity upon flat fielding The dominant sources of the remaining technical noise are etaloning caused by the glass cover of the ccd chip, and fringes imparted on the beam by the few optics it passes through, including the imaging lenses (back-to-back achromat doublets), and the vacuum windows Residual fluctuations in the beam direction result in movement, and imperfect subtraction, of these fringes Such technical noise is responsible for not achieving shot-noise limited imaging; we believe the total imaging noise to be roughly twice that of the photon shot noise 414 Computer control The Swinburne atom chip i apparatus is computer controlled using National Instruments hardware and software Timed control of infrared laser, microwave, radiofrequency and magnetostatic fields is achieved using three pci-6733 analog output devices, which each out- put eight high precision (16 bit, ±1 V range, 4 mv accuracy, 1 MS/s update rate) control voltages These are controlled by software written in the LabVIEW programming language by previous members of the laboratory, and contributed to by the author The graphics user interface essentially consists of an array of machine states on a single front panel Each machine state is a set of analog voltages output by the pci-6733 devices, represented by a row of programmable numeric values on the interface The transition between different machine states has an adjustable duration, with a choice of several ramp functions The machine state array of the front panel is converted to a data structure suitable for buffered opera- tion of the pci-6733 devices The output rate of the buffered analog data is largely fixed, we nominally use a step size of 1 µs, synchronised to the 1 MHz internal clock of one of the pci-6733 devices Finally, the control voltages are sent to various pieces of equipment, such as acousto-optic modulator drivers, magnetostatic coil drivers, frequency synthesisers, laser shutter drivers, and rf/mw amplifiers/switches to enact the desired change issued by each control voltage The interface is suitably flexible, but complex timing sequences are difficult to implement (for example, no two control voltages can be ramped simultaneously with a different functional form) Furthermore, the change of a single control voltage at any point of the experiment requires a change of the entire machine state, necessitating the inclusion of a new row on the front panel, where every other control voltage is identical to that of the previous

95 4 magnetic dipole transitions 79 row For typical experimental duty cycles, the fixed update rate results in large arrays to send to each pci-6733 ( 15 8 elements), which is relatively redundant given that the number of unique elements is far less than the array size 415 Microwave and radiofrequency apparatus The 68 GHz microwave radiation is derived from a signal generator (Agilent e857d), pulsed using a fast solid-state switch, then amplified (ma Ltd am , 1 W) and transmitted to the bec using a custom built antenna which resides in air, 1 cm from the condensate The microwave antenna is a helical design, similar to that in Ref [114] It has a measured directional gain of 19 dbi, and consists of 1 turns of mm copper wire wound around a teflon rod The wire is mounted to a copper base plate, which acts as a ground plane, where it is fed with microwave radiation by a bnc connector The forward power, as measured with a reverse-biased directional coupler, is typically 7% A radiofrequency magnetic field can be driven synchronously to the microwaves, using auxiliary wires on the atom chip (Figures 41 and 41) These wires are also used for evapora- tive cooling, albeit with an independent signal generator We source the rf radiation from Stanford Research Systems signal generators (model ds345), and amplify the rf used for two-photon transitions with a ma Ltd W amplifier The radiation resulting from rf current passing through these wires is linearly polarised along the y-axis, perpendicular to the quantisation axis defined by the static magnetic field in the Z-wire trap 4 Magnetic dipole transitions 41 Magnetic dipole Rabi oscillations in a spinor bec We commonly observe rf-rabi oscillations in a trapped spinor bec The coupling of a trapped bec with rf is characterised by performing a Stern-Gerlach experiment after re- leasing the atoms from the trap The rf-rabi frequency is calibrated with respect to the rf-field transmitted by a set of auxiliary-wires on the atom chip (sw, Fig 41) or a loop antenna situated outside the vacuum (ac) During free-fall, we apply a pulsed current through the Z-wire on the chip (sw) or through the quadrupole coils (ac), creating a magnetic field gradient Since the Zeeman states have different magnetic moments, they each acquire a different velocity during the pulsed gradient Further ballistic expansion results in spatial separation of the different spinor bec components This form of state-selective imaging is used to optimise the preparation of the bec in various states prior to addressing the pseudospin- 1 system An example is shown in Fig 45 in which an F = spinor bec is magnetically trapped

96 two-component bec : experiment 8 mf + +1 g 1 B ħ Fz ħ ħ ħ 5 1 time ms 15 Figure 45 rf-rabi oscillations of a F = spinor bec in a magnetic trap (Lower) Longitudinal spin projection, computed using the normalised populations [Eq (41)] and fit using the analytic expression derived in Eq (38) (Upper) Corresponding absorption images using state selective imaging, by application of a magnetic field gradient B in the direction of gravity g (shown) The rf-rabi frequency is π 987(5) khz, and the anti-trapped states (m F < ) can be completely populated without interruption or decay of the coherence The error bars are representative of the uncertainty in measuring the fractional populations Pm F typically 8% for these data with the atom chip, strong rf radiation is applied for several Rabi periods, and the bec is allowed to expand for 1 ms During the expansion, a 3 A current is passed through the Z-wire for 1 ms, resulting in a 3 Gcm 1 field gradient Interestingly, the magnetically untrappable Zeeman states (m F g F ) of the bec are populated during the Rabi oscillations, without interruption or decay of the coherence In fact, for Ωrf t = (n +1)π, the bec is exclusively in the F =, m F = state, which experiences a repulsive force from the magnetic minimum at the trap centre The spinor bec remains trapped however, as the Rabi oscillations are much

97 4 magnetic dipole transitions 81 faster than the trapping frequencies Consequently, the external motion of the bec does not respond adiabatically to the change in its internal state We use the fractional population in each Zeeman state, Pm F = N m F /N, to compute the longitudinal spin projection during the Rabi oscillations F z = m F ħ Pm F mf (41) The simple expression derived in Eq (38) for F z is fit to the spin projection, rather than independently fitting Eqs (37) to the fractional populations In this way, we infer an rf- Rabi frequency of Ωrf = π 987(5) khz, compared with the highest oscillation frequency of the trap f ρ = 4 Hz An elementary calculation of the rf-field amplitude overestimates the Rabi frequency by a factor of, highlighting the advantage of a measurement, rather than a prediction, of the relevant atom-field couplings in this type of experiment A deleterious systematic of the Stern-Gerlach imaging technique is that the various spin components exhibit different cross-sections for optical absorption The effects of optical pumping during absorption imaging of different Zeeman states is discussed in Appendix C For the data presented in Fig 45, the Zeeman populations have been normalised to the total atom number in each shot This does not circumvent the issue of non-uniform detection efficiency, but does mitigate run-to-run fluctuations in the total atom number We note the spatial modes of the imaged spinor components are different, as seen in the absorption images of Fig 45 The same magnetic dipole interaction with the field gradient used to spatially separate the spin components during expansion also distorts their shape (with the exception of the m F = component) This is of little consequence for a mea- surement of net-magnetisation or populations of the spinor bec, but hinders observation of spatial inhomogeneities in the relative phase of a pseudospin- 1 bec, as we will see in Chapter 6 4 Radiofrequency Landau-Zener transitions in a spinor bec Another common technique for state manipulation is the adiabatic passage using magnetic dipole transitions Unlike adiabatic passage with optical Raman transitions, heating and trap loss inherent to spontaneous emission of optical photons is avoided when using rf or mw radiation In this Section, I will discuss radiofrequency Landau-Zener transitions amongst the Zeeman states of a single hyperfine level of a spinor bec These can be used for state preparation, for example the preparation of a 1, 1 condensate from a 1, 1 condensate in an optical trap, or magnetic field insensitive imaging of a single Zeeman component [115] The simplest example of this type of transition is when a single-component bec is exposed to a pulse of chirped rf radiation, swept linearly across the Zeeman resonance An example

98 two-component bec : experiment 8 4 F = atom number ( 1 ) mf sweep rate (MHz / ms) 1 Figure 46 Landau Zener transitions in an F = spinor bec A linearly chirped radiofrequency pulse is applied to a magnetically trapped condensate in the F =, m F = + state, of duration 1 ms for various sweep rates The bec would occupy the F =, m F = state following the sweep, were it slow enough as to be adiabatic Non-adiabatic transitions result from faster sweeps, for which multiple Zeeman states are populated after the sweep (shown here) The population in the F =, m F =, F =, m F = +1 and F =, m F = + states is detected using Stern-Gerlach imaging For fast sweep rates, atoms do not respond to the coupling radiation at all, and remain in the F =, m F = + state The solid curves correspond to the analytic model in Eq (4) of this is shown in Fig 46, whereby chirped rf is applied to a magnetically trapped, condensate (sw) for 1 ms The rate at which the radiofrequency is swept is varied between different experimental realisations, and the resulting Zeeman populations of the F = spinor bec after the sweep are detected with Stern-Gerlach imaging (above) To study this behaviour, we again take a subset of the eight Zeeman states of 5 S1/, in this example the five states with F = The bare states in the rotating frame [ie of Eq (33b)] are degenerate at the Zeeman resonance, and exhibit an avoided crossing there for Ω This can be seen by evaluating the dressed state energies of the (reduced) Hamiltonian in Eq (33b), as shown in Fig 47 At a fixed magnetic field, sweeping the radiofrequency linearly across the Zeeman resonance amounts to the time-dependent detuning = d /dt t For d /dt >, the initial state, is well approximated by the n = dressed state in Eq (33) for Ω <, whose energy is given by the uppermost curve of Fig 47 A superposition of all five Zeeman states is created during the sweep, the rate of which de- termines the nature of the superposition and the Zeeman populations following the sweep, as in the two-level case of 4 If the sweep is slow enough (with respect to the coupling strength Ω), the system adiabatically follows the n = dressed state At the end of the sweep

99 4 magnetic dipole transitions 83 energy khz 5 5 mf Π khz 4 6 Figure 47 Dressed state energies of the F = Zeeman states Each curve is labelled by the dominant Zeeman state of which it is composed at large positive and negative detunings A bec initially in the, state can undergo transitions to other Zeeman states as the frequency of rf radiation is swept across resonance For an adiabatic sweep with d /dt >, the condensate will remain in the dressed state corresponding to the uppermost curve, residing in, following traversal of the avoided crossings If the sweep is non-adiabatic, other dressed states will be populated during the sweep, and population will be distributed across several Zeeman states following the sweep (Fig 46) ( Ω > ), this dressed state is exclusively composed of,, and the spinor has un- dergone a change of state in the Zeeman basis The avoided crossing at = is actually composed of four avoided crossings, each corresponding to a single rf photon transition Vitanov has applied Majorana s mapping between a spin-f system and a spin- 1 sys- tem ( 341) to the problem of Landau-Zener transitions in a general spin system driven by chirped radiofrequency [116]3 Each avoided crossing is considered as an independent two-level system, and the behaviour of the multi-level avoided crossing is treated with conditional probabilities An early detailed analysis of Landau-Zener transitions in this type of multi-level system is given in [117] A sweep of the detuning ( in both the multi-level and effective two-level system) can be treated using the results of adiabatic passage in a two-level system ( 4) The adiabaticity is characterised by the probability of finding the system in its initial state following the sweep, PLZ = e πγlz Eqs (41) If the system begins in the stretched state F, m F = F, the final populations in states F, m F, m F { F,, F} of the multi-level 3 The Rabi frequency convention in Reference [116] is different to that used here, but the results of Eqs (36) and 4 are the same, as the two-level and multi-level Rabi frequencies are both half of that used in this thesis The only consequence of the different convention is the Landau-Zener parameter ΓLZ in 4, which is (consistently) four times smaller here

100 two-component bec : experiment 84 system are F+m F Pm F = λm F PLZ (1 PLZ )F m F (4) From the expression for the Landau-Zener parameter ΓLZ = Ω /(4d /dt) Eqs (41), it can be seen that changing the sweep rate will change the distribution of population among the Zeeman states following the sweep The data in Fig 46 has been fit with Eq (4) and no free parameters, since d /dt is precisely controlled and Ω is determined from the above Section Alternatively, this method could be used to calibrate the rf-rabi frequency In conclusion, we have verified the use of an analytic model for Landau-Zener probabilities in a multi-state bowtie system with no independent parameters The application of the single-atom formalism to spinor condensates coupled with radiofrequency has been performed in [118, 119, 1, 11], and the validity of the single-atom Landau-Zener approach to this kind of system is discussed in [1] 43 Microwave outcoupling spectroscopy of a bec in a magnetic microtrap In the previous two Sections, the applied radiation was sufficiently strong that the external motion of the atoms was unaffected by their changing internal state In this Section we study radiative outcoupling, whereby transitions from magnetically trapped states to those which are untrapped results in atoms being expelled from a magnetic trap This relies on the coupling being weak enough for the external atomic motion to respond to the change of internal state Radiofrequency outcoupling spectroscopy of magnetically trapped condensates has been used to probe magnetic field fluctuations of a surface [17], to measure the temperature and quantum phase transition of ultracold clouds trapped using permanent magnetic materials as an alternative to ballistic expansion [13], in matter wave interference [14, 41, 15], and to generate atom lasers [16] Microwave outcoupling has been used as a means to evaporatively cool one atomic species in a multi-species trap [17, 18, 19, 13] As in the case of radiofrequency outcoupling, the lineshape is quite complex and depends on the mean field energy of the condensate, and the gravitational sag it experiences in the magnetic trap [16, 13] Under certain conditions,4 one can recover the distribution of transition frequencies throughout a trapped atomic gas by measuring outcoupling spectrum, the number of atoms remaining in (or lost from) the magnetic trap, as a function of the frequency of the radiation responsible for the outcoupling The outcoupling spectrum can be used, for example, to infer the transition frequency, and in turn the magnetic field, at the location of peak atomic density Here, we study the microwave outcoupling spectra of a magnetically trapped condensate of 15 atoms in the 1, 1 state (sw) All allowed transitions are driven, 4 eg Short pulse length, less than 1% peak outcoupled fraction

101 magnetic dipole transitions energy au f e energy Ec au 4 d c y y 15 f 1 e 5 d 3 σˉ E mw BEC width y y Figure 48 Energy landscape for microwave spectroscopy of a magnetically trapped condensate (Left) Combined Zeeman and gravitational (dotted), and mean field potentials (solid) in the direction of gravity for states c 1, 1, d,, e, 1, and f, (Right) Transition energies for c d, e, and f The energy units are arbitrary, and the ratio between the hyperfine energy and the Zeeman splitting has been reduced for clarity In this case, the gravitational sag is equal to the Thomas-Fermi radius of the condensate, y = ytf, and the broadening of the σ transition is indicated [Eq (44) with m F = 1] ie to the magnetically untrapped,,, 1 and, states, with the microwave field derived from a helical antenna 1 cm from the condensate ( 415) The peak atomic density is located at the potential minimum, which is shifted from the magnetic minimum by the gravitational sag y = g/ωy, where ω y is the trapping frequency in the direction of gravity5 In the regime where the Zeeman shifts are approximately linear with magnetic field, and the differential mean field shifts between the states are negligible,6 the transition energies as a function of vertical position are Emw (y) = E x (y) E c (y) = Ehfs 1 ( m F )(µb B + mωy y ), (43) where B is the value of the magnetic field minimum, x {d, e, f } correspond to the states in the upper hyperfine level coupled to 1, 1 by the microwave (Fig 31), whose energies E c, Ed, E e, and E f consist of the total potential experienced by atoms in each state, includ- 5 For all trapping geometries used in this thesis, the y-axis is in the plane of radial confinement, and ω y = ω ρ 6 We assume the mean field interactions of the states with themselves and each other are identical, in this Section Differential mean field shifts derive from differences in the s-wave scattering lengths of the initial and final states, and shift the resonance by tens of Hz for the densities and states used here

102 two-component bec : experiment 86 ing the mean field interaction Despite the gravitational and mean field shifts, the transition energies (minus the hyperfine splitting) are all approximately integer multiples of the 1, 1, transition energy, as shown in Fig 48 The microwave outcoupling spectra for all allowed transitions are shown in Fig 49 The asymmetric lineshape of each transition can be seen by the deviation of the data from the Lorentzian fits A more detailed study of the lineshape can be found in References [16, 13] The total frequency width of each transition depends on the chemical potential µ, and the ratio between the gravitational sag y, and the vertical Thomas-Fermi radius ytf Emw () Emw (y + ytf ) = µ( m F ) (1 + y /ytf ), y ytf, Emw = Emw (y ytf ) Emw (y + ytf ) = µ( m F ) 4 y /ytf, y > ytf (44) In both cases, gravitational sag can result in the smallest frequency width being several times larger than the chemical potential of the condensate The ratio of the widths is approximately 3 1 for the σ, π and σ + transitions, respectively For the data shown in Fig 49, the trap frequency in the direction of gravity is ω y = π 39 Hz, the chemical potential is µ h 7 khz, resulting in y 53 ytf The frequencies of maximum outcoupling for each transition can be used to compute the magnetic field at the location of peak density in the condensate, y We infer a field of 537() G for the data in Fig 49, and note that this field is higher than the magnetic field minima B ; there is a transition frequency difference of ( m F ) khz between the magnetic field minimum and the potential minimum for these data The peak outcoupled fraction gives an indication of the coupling strength for each transi- tion Using Eqs (316) and Eq (3), the relative components of the microwave field Bmw in the spherical basis can be determined Although the data shown in Fig 49 does not permit a quantitative determination of these ratios, it can be qualitatively seen that there is significant power in each mode of the microwave, despite the helical design of the antenna We attribute this to imperfect alignment of the antenna and the quantisation axis, the near-field nature of the microwave radiation, reflections from the atom chip apparatus, and the simple design of the antenna We will return to the results of this Section in 45, where the effect of microwave transition broadening on the two-photon coupling in a magnetically trapped pseudospin- 1 con- densate is discussed

103 4 magnetic dipole transitions 87 atoms remaining σ 1 π 1 8 fmw fhfs khz σ+ 6 4 Figure 49 Microwave spectroscopy of a magnetically trapped 1, 1 bec Each data point corresponds to the number of trapped atoms remaining following application of a fixed frequency microwave field for 15 ms The σ, π and σ + transitions correspond to microwave outcoupling via the,,, 1 and, states, respectively The relative depth of these features, combined with the microwave coupling strengths of each transition (Fig 31), indicates comparable power in each mode of the microwave field 44 One- and two-photon Rabi spectroscopy We use spectroscopy of one- and two-photon magnetic dipole transitions in a bec to accurately find their transition frequency The non-interferometric measurement of these transitions is most precise when: 1 there is minimal broadening of the transition from spatial inhomogeneity in the transition frequency, and only two states are involved in the transition In many cases, we use spectroscopy of magnetic dipole transitions to calibrate the magnetic field that a condensate experiences whilst trapped, either magnetically or optically The above conditions require both states experience the same external and mean field potentials to within some desired precision For a magnetically trapped condensate, the former requires both states possess similar magnetic moments Optical trapping usually occurs in a uniform magnetic field, and the requirement of common trapping potentials amounts to a negligible differential light shift between the states Any non-zero differential mean field, light shift, or magnetic moment will result in a dynamic change to the external state of the condensate For small differential shifts ( 1 Hz), the time taken for the condensate wave function to change will be much longer than the inverse of any typical Rabi frequency ( 1 ms) In this regime, we can make the single-mode approximation, and the transitions will drive changes in state

104 two-component bec : experiment 88 t TΠ c Δ Ω 4 Figure 41 Rabi spectra of Eq (8b) for pulse duration t {Tπ, Tπ, 3Tπ } The envelope (dashed line) is a Lorentzian with fwhm of fwhm = Ω The maximum population transfer to state does not necessarily occur at zero detuning, as illustrated by the pulse of duration t = Tπ populations well described by the Rabi formulae,7 Eqs (8) Spectroscopic measurements of either population for transitions whose lineshape is described by Eqs (8) will be referred to here as Rabi spectroscopy Coupling pulses with a duration close to that of a resonant π-pulse, Tπ = π Ω 1, are most suitable for Rabi spectroscopy Pulses of duration t = (n 1)Tπ, n Z+ result in the largest population variation over {, }, and hence the largest signal-to-noise ratio for a measurement of either atomic population [or the normalised population difference Pz, Eq (45)] A pulse of length t Tπ results in a spectra similar to that shown in Fig 41 Other notable features of Rabi spectroscopy, when measuring c in Eq (8b) with varying detuning, and fixed duration of an applied coupling pulse, are (see Fig 41): For pulse durations t = (n 1)Tπ, n Z+, the central peak has a full-width at half maximum (fwhm) well approximated by 3Ω/( n) For t = n Tπ, n Z+, the global maxima occurs at Ω/ n with a value well approximated by (1 + n) 1 For any pulse length, the values of all local maxima (other than the central peak) are well approximated by a (1 + /Ω ) 1, ie a Lorentzian with fwhm of Ω An example of Rabi spectroscopy of the 1, 1, transition for a condensate confined in an optical trap (ac) is shown in Figures 41 Rabi spectra such as these are obtained by: 7 By comparison, the outcoupling spectra in 43 were inhomogeneously Zeeman broadened, and the external condensate mode changed significantly during the coupling pulse (by the definition of outcoupling) In 41, the single-mode approximation was only valid for short times ( µs) with strong rf coupling, as multiple states were coupled, and each experienced disparate external potentials

105 4 magnetic dipole transitions 89 1 c 4 Δ Ω π t Ω Figure 411 Navigating the Rabi landscape: population transfer to the upper state of a twolevel system with varying pulse duration t and detuning [Eq (8b)] The transition is characterised by (i) finding a detuning close to resonance and measuring the off-resonant Rabi oscillations as a function of pulse duration (green), (ii) taking a Rabi spectrum for the pulse duration of an off-resonant π-pulse, to precisely locate the resonance (red), and (iii) measuring the resonant Rabi oscillations to directly determine the Rabi frequency, and ensure that unity transfer of population can be achieved (blue) 1 finding an off-resonant π-pulse, the shortest time that results in large transfer ( 7%) of population to state for a fixed frequency near resonance, ie (, T) in the vicinity of (, Tπ ) This requires an estimate of the resonance to within < Ω, followed by a measurement of off-resonant Rabi oscillations at that frequency The duration of the measured π-pulse is always shorter than a resonant π-pulse, giving an upper limit for Ω before the Rabi spectra is measured Following this either atomic population is measured for various detunings, as shown in Fig 41 (or both populations are measured to give the normalised population difference) The lines that these steps correspond to in the parameter space of (, t) are illustrated in Fig 411 Measuring two antinodes either side of the central peak yields excellent agreement with Eqs (8), with high signal-to-noise, constraining the central value of the fitted spectral curve with a remarkable statistical uncertainty for a non-interferometric measurement This form of spectroscopy has been used to calibrate the bias magnetic field in the vicinity of the Feshbach resonance at 91 G between the 1, 1 and, 1 states, as shown in Table 41 and Fig 413

106 two-component bec : experiment 9 6 N detuning khz 4 6 Figure 41 Rabi spectra of the 1, 1, transition for a near resonant π-pulse duration of 48 µs The resonant frequency attained from the fit is (9) GHz, from which the magnetic field can be inferred [using Eq (39)] to be 9871(1) G The Rabi frequency 75(4) khz, slightly above that measured with resonant Rabi oscillations [68() khz] To calibrate a magnetic field using this form of spectroscopy, it is helpful to use singlephoton transitions which are most sensitive to magnetic fields This circumvents systematics that arise from two-photon transitions ( B), and provides optimum sensitivity to variations in the magnetic field For optical trapping, this is best achieved with spectroscopy of the 1, 1, transition 45 Two-photon Rabi oscillations Once the resonance frequencies for a particular two-photon transition are found using Rabi spectroscopy, resonant two-photon Rabi oscillations can be driven between the levels An example of resonant two-photon Rabi oscillations between the 1, 1 and, 1 states of a magnetically trapped bec are shown in Fig 414 The coupling conditions are summarised below: Ωmw = π 6 khz Ωrf = π 1 khz Ω1 = π 85 khz Ω = π 45 khz = π 5 khz Ωeff = π 418 Hz δ = π 7 Hz mw Rabi frequency rf Rabi frequency,, 1 Rabi frequency 1, 1, Rabi frequency intermediate state detuning -photon Rabi frequency -photon detuning at resonance

107 4 magnetic dipole transitions 91 V DAC (Volts) f (GHz) δ f (khz) B (G) δb (G) Table 41 Bias magnetic field calibration using microwave Rabi spectroscopy of the 1, 1, transition, depicted in Fig 413 The errors in the frequency are statistical error from a spectroscopic fit such as that in Fig 41, which results in an uncertainty in the magnetic field determined by inverting the Breit-Rabi equation, Eq (39) These represent a lower bound on the experimental uncertainty; although the precision of a single measurement is below 1 mg, long term drifts of up to several milli-gauss were observed over periods of a few hours The range of magnetic fields depicted here corresponds to a 71 khz range of transition frequencies between 1, 1 and, B G V DAC Volts Figure 413 Bias magnetic field calibration using microwave Rabi spectroscopy of the 1, 1, transition Each point corresponds to the spectroscopic measurement of the resonant frequency at a magnetic field set by the control voltage V DAC ; of the type shown in Figs 41 The magnetic field corresponding to each control voltage is determined by inverting the Breit-Rabi equation, Eq (39) The standard error for each field determination is given in Table 41; the associated error bars are smaller than the size of the data points in this plot The rf-rabi frequency is measured ( 41), as is the -photon Rabi frequency from a fit to the data in Fig 414 The detuning of the rf from the,, 1 transition can be inferred using the magnetic field calibration in concert with the Breit-Rabi equation [Eq (39)], or approximated from the resonant frequency of rf outcoupling spectroscopy The remaining two-photon conditions above can then be inferred from Eqs (34) and (341), in combi-

108 two-component bec : experiment populations pulse length (ms) 8 1 Figure 414 Two-photon Rabi oscillations of a magnetically trapped pseudospin- 1 bec as measured by simultaneous detection of the populations of 1 1, 1 and, 1 Each data point corresponds to a single experimental sequence, with variable duration of the resonant two-photon drive The measured normalised populations are in good agreement with the Rabi formula [Eqs (8)] despite fluctuations in the total number of atoms between each experimental realisation In this scenario, the rf and mw fields were detuned by 5 khz from the intermediate state, The remaining coupling parameters are those listed in the text, for which the magnetic dipole coupling is simulated in Fig 37 nation with the mw frequency that gives the central frequency of the Rabi spectra of the two-photon transition In 43, we saw that the 1, 1, microwave transition of a condensate in a mag- netic trap is broadened due to the spatial extent of the condensate This broadening can be several times larger than the chemical potential, due to gravitational sag of the condensate into regions of higher magnetic field gradient than those near the magnetic field minima This broadening affects the two-photon transition, as the intermediate state detuning becomes non-uniform across the condensate8 This in turn affects the two-photon Rabi frequency Ωeff, and effective detuning eff, whose dependence on is given in Eqs (34) and (341), and which are plotted for typical experimental conditions (sw) in Fig 415 The variation of the two-photon coupling parameters across the condensate can lead to imperfect preparation of macroscopic superpositions The radial density profiles, longitudinal spin projection, and relative phase, following a π/-pulse are shown in Fig 416 For longer coupling pulses, spatial inhomogeneities in either two-photon coupling parameter can decrease the contrast of the two-photon Rabi oscillations, as they ultimately 8 The two-photon detuning δ is unaffected by the broadening of the intermediate state transition, as the shifting of the 1, 1 and, 1 levels is common mode

109 4 magnetic dipole transitions 93 Δ Π khz Ω eff Π Hz Δ eff Π Hz y y μm y y μm 4 4 y y μm Figure 415 Broadening of intermediate state transition: effect on two-photon coupling The 1, 1, transition width is E mw = h 196 khz for a condensate of atoms in a magnetic trap with vertical trap frequency ω z = π 976 Hz [Eq (44)] As a result, the two-photon coupling parameters vary across the vertical extent of the condensate: (left) intermediate state detuning, (middle) two-photon Rabi frequency Ω eff [Eq (34)], and (right) effective detuning eff [Eq (341)], for Ω rf = π 1 khz, Ω mw = π 4 khz, and = π 5 khz at the centre of the condensate The two-photon detuning δ = π 36 Hz has been chosen such that the two-photon transition is resonant at the condensate centre The Thomas-Fermi radius in the vertical direction is z TF = 46 µm, and the gravitational sag is z = 61 µm density leakage 4 4 y y μm Sz y y μm Φ Π mrad y y μm Figure 416 Broadening of intermediate state transition: effect on the pseudospin-1 condensate after a π/-pulse Both the eight-level (solid lines), and effective two-level (dashed lines), Schrödinger equations have been solved for a condensate with the parameters shown in Fig 415 These correspond to the experimental conditions used in the Swinburne experiments (Fig 414, Chapter 6) The intermediate state transition broadening results in a asymmetric density profile following a π/-pulse, and gradients in both the longitudinal spin-projection and relative phase cause spatial dephasing of the condensate wave function Another such effect, which we study in detail in Chapter 6 is the mean field evolution of the condensate We have simulated

110 two-component bec : experiment 94 a resonant π/-pulse being applied to a condensate with atoms, using the techniques in 51 Further discussion of this dephasing mechanism is deferred to later Chapters, but for comparison, we note the predicted relative phase spread owing to the mean field evolution during a typical π/-pulse (Ω = π 418 Hz) is 15 mrad The measured Rabi oscillations can be used to infer a two-photon Rabi frequency as follows The simultaneous detection of the condensate population in both states ( 43) allows the normalised population difference to be measured each time a condensate is created In Section 5, we illustrated the equivalence between the normalised population difference, and the longitudinal projection of the Bloch vector in a spin- 1 system For pseudospinor condensates in this thesis, we use Pz to denote the normalised population difference Pz = N1 N = Nħ 1 Sz N1 + N (45) The right hand side of the final equality above is twice as large as in Eq (54), as the Bloch vector now corresponds to the pseudospin of the condensate, each component of which has longitudinal spin-projection ħ Immediately prior to dual state absorption imaging, the two components of the condensate reside in different states of the F = hyperfine ground level ( 43) The absorption of probe light by atoms in each of these states differs, and results in a relative detection efficiency between each of the two components η= N1,meas N,real, N1,real N,meas (46) where N i,real and N i,meas are the real and measured atom numbers in state i, respectively Similarly, Pz,real = N1,real N,real N1,meas ηn,meas = N1,real + N,real N1,meas + ηn,meas = e t/t c cos(ω t), (47a) (47b) where t c is the characteristic decay time of the oscillations To extract Ω, η, and t c from the measured spin projection (uncorrected for the unequal detection efficiencies), one can use N1,meas N,meas N1,meas + N,meas Pz,real (η + 1) + η 1 = Pz,real (η 1) + η + 1 Pz,meas = (47c) (47d) where Eq (47c) dictates the computation of Pz,meas from the data, which is fit with the model Eq (47d), in conjunction with Eq (47b) Alternatively, the relative detection efficiency can

111 43 dual state imaging 95 be determined separately from the Rabi oscillations of the individual populations The atom number for each component can be fit to expressions similar to those in Eq (8), scaled and with decay, to extract the relative detection efficiency η from comparison of the two amplitudes The latter technique is sensitive to fluctuations of total atom number between different runs of the experiment, as such it does not take full advantage of the dual state detection method 43 Dual state imaging Pseudospin- 1 condensates useful for interferometry and studies of relative phase dynamics are comprised of states whose magnetic moments are similar The energy splitting between the states is then minimally sensitive to magnetic field variations, arising from gradients or spurious fluctuations of the field The low sensitivity to, and inhomogeneous broadening from, magnetic field variations prevents the correspondingly innocuous dephasing Moreover, the states of a magnetically trapped pseudospin- 1 condensate need to have similar magnetic moments for both to share a common external potential, as is desirable for maintaining spatial overlap between the components This renders difficult the simultaneous spatially resolved imaging of both components We present a new imaging technique to simultaneously detect both states of either the 1, 1, +1 or 1, +1, 1 pseudospin- 1 condensates of 87 Rb with uncorrupted spatial resolution This represents the first unambiguous state selective measurement of a pseudospin- 1 condensate that preserves the spatial mode Application of a magnetic field gradient during ballistic expansion is usually used9 to separate and simultaneously detect multiple spin states in a Stern Gerlach like experiment (Fig 45) This is unfeasible with the pseudospin- 1 condensates used in this thesis, unless performed at magnetic fields of a few hundred Gauss, where the differential Zeeman split- ting becomes non-negligible Various methods exist to individually image either the F = 1 or F = populations, using resonant light in combination with optical or microwave repump- ing pulses An alternative method uses a radiofrequency sweep to distribute the 1 and populations among all eight Zeeman sublevels which are then spatially resolved using SternGerlach separation prior to imaging The original 1 and populations are inferred from estimates of how the adiabatic rapid passage distributes population among the sublevels [131] Phase-contrast imaging can also been used to image the difference in populations [5] and has enabled the observation of vortex dynamics [33, 34, 13] The two-photon transitions in 87 Rb (Figures 35 and 36) are such that the microwave field 9 Mixed species multi-component condensates can be imaged with probe light of different frequencies Each optical signal is directed onto a different ccd by polarisation discrimination or different direction of propagation of the probe Alternatively, the exposure to probe light can be done successively for each species This is not practical for same species alkali condensates, as the same optical transition is used to image each component

112 two-component bec : experiment 96 Iz (a) B atom chip (b) 1 g time displacement 1 3 μm Figure 417 Transfer of 1 to the magnetic field insensitive state, is achieved by adiabatically sweeping the atomic resonance in the presence of a fixed frequency microwave pulse (a) Spatial separation is performed by pulsing Iz (see text for description) to induce a Stern-Gerlach force that accelerates away from the atom chip and 1 (b) Absorption imaging using the D (F = F = 3) transition yields a single shot image of a two-component condensate, 1 (above) and (below), where the 1 structure arises during a Ramsey interference experiment described in 6 can be brought into resonance with the, intermediate state by changing the magnetic field by a small amount (corresponding to the intermediate state detuning) This permits transitions to the, intermediate state, whereby spatial separation from the, ±1 state can be achieved with a magnetic field gradient, and the two states can both be resonantly imaged with F = F = 3 light The dual state detection method consists of three stages With the Swinburne appara- tus, the magnetic trap is switched off by reducing the Z-wire current, Iz, to zero in µs The uniform bias field that completes the trapping potential is concurrently ramped off in 3 ms and a fixed frequency microwave field is applied The Zeeman splitting of the mag- netic sublevels decreases, facilitating partial adiabatic passage from 1, 1 to, This pop- ulation transfer method is more robust than a resonant microwave π-pulse, owing to the less stringent requirements of adiabatic passage on the stability of magnetic fields and microwave power The same microwave field is used for the two-photon pulses, obviating the need for a second microwave frequency synthesiser and enabling detection immediately after a two-photon pulse, as in 6 For our initial experiments demonstrating this technique, we transferred 44(1)% of 1, 1 population to, (rapid bias field switching was desired at the expense of adiabaticity) At this point, both spin components of the condensate are spatially overlapped and fall freely under gravity However those in the, state are largely

113 s = 5 s = 1 s = 1 4 N detection efficiency Η 43 dual state imaging 4 (a) 4 6 time (μs) 8 1 (b) N1 Figure 418 (a) The detection efficiency η for dual state absorption imaging as a function of imaging pulse duration, for three typical saturation parameters, s = 5 (solid), s = 1 (dash-dotted) and s = 1 (dashed) (b) Simultaneous measurement of atoms in 1 and after a two-photon coupling pulse showing high correlations for a range of detected bec atom numbers An imaging pulse of 1 µs duration with s = 1 was used insensitive to magnetic fields The second step [Fig 417a] produces spatial separation of the two components This is achieved using a 3 A, 1 ms Iz pulse which induces a Stern-Gerlach force in the direction of gravity to displace such that both components are separated by 3 µm after ms of free fall Finally, an absorption image is taken [Fig 417b] yielding the column density of directly below 1 This image is from a Ramsey interferometry experiment described in 6 and highlights the ability of this dual state imaging method to preserve spatial information about the wave functions This is due to the minuscule recoil velocity (5 1 9 mm/s) associated with microwave photon absorption thus eliminating the image blur associated with optical repumping techniques While conservation of spatial information is crucial for dynamical studies of two-com- ponent bec, the detection method should also determine the atom number accurately We investigate the effects of optical pumping during absorption imaging on η, the relative ef- ficiency of detecting atoms originating in F =, m F = to those starting in The ab- sorption of a σ + polarised probe beam was modelled using the optical Bloch equations [133], including the Zeeman splitting of the ground and excited states (Appendix C) We evaluated the absorption by integrating the total excited-state population (F = 1,, 3 ; m F = F,, F ) during the imaging pulse for a given initial state Results are shown in Fig 418a for differ- ent saturation parameters s = I /168 mw cm of the F =, m F = + F = 3, m F = 3 cycling transition, with I the intensity of the probe beam As the imaging pulse duration

114 two-component bec : experiment 98 atoms in, Tsweep ms 8 1 Figure 419 Microwave Landau-Zener transitions from 1, 1 to, A uniform bias magnetic field was swept 75 G across the 1, 1, resonance while a fixed frequency microwave was applied The duration of the sweep was adjusted, varying the adiabaticity of the avoided crossing traversal A fit to Eqs (41) yields the Rabi frequency of 46(7) khz The microwave used in this instance had lower power than is commonly used to drive 1, 1, transitions (ac, typically we had Ωmw π 7 khz, allowing adiabatic passage in 6 ms) increases, optical pumping acts to populate the F =, m F = + state and more time is spent driving the cycling transition As such, the relative detection efficiency asymptotes to unity and does so faster for higher imaging intensities For our typical imaging conditions, we predict a relative detection efficiency of η = 98 (sw) For experiments performed at Amherst College with optically trapped 1, +1, 1 pseudospin- 1 condensates, some modifications of the above technique were necessary Following extinction of the optical trapping light, the uniform bias magnetic field is ramped linearly1 through the 1, 1, microwave resonance (typically 75 G in 6 ms), realis- ing complete adiabatic passage from 1, 1 to, (characterisation of the adiabatic passage is shown in Fig 419, albeit for a lower than usual microwave power) The bias magnetic field is then rotated into the direction for which separation of the two states is desired, and a burst of current is passed through a set of anti-helmholtz coils, normally used for magnetic trapping The quadrupole field from these coils, in addition to the uniform bias field, results in a field gradient which separates the two components, typically in a horizontal direction11 1 The ramp rate db/dt and intermediate state detuning must have the same sign 11 For separation of the two components in a plane parallel to those of the coils, the combined quadrupole field B Bquad = q (x, y, z) and bias field B = (B,, ) result in a force F = µb q B /(B ) for distances r from the quadrupole centre that satisfy B q r B, and adiabatic following of the non-zero magnetic moment µ This gives a separation of µb q tpulse tpulse x = ( + texp ), (48) m

115 43 dual state imaging (a) 4 μm (b) 4 μm Figure 4 Simultaneous dual-state imaging of 1, +1, 1 pseudospin- 1 bec Following a Ramsey interference experiment ( 6), the optical trap is turned off and population in the 1, 1 component is transferred to the magnetic field insensitive, state; microwave adiabatic passage is achieved by sweeping a magnetic bias field As the two components fall vertically under gravity, they are separated horizontally by pulsing a magnetic field gradient of 8 Gcm 1 for ms The atomic densities in each component are then spatially separated, and can be simultaneously imaged using resonant absorption imaging on the F = F = 3 transition In the above Ramsey interference experiment, the phase of the second π -pulse was varied with respect to the first; the phase shifts used in images (a) and (b) differ by 86π radians by µm The bias magnetic field is then set to have the desired magnitude and orienta- tion for resonant imaging using the F =, m F = + F = 3, m F = 3 cycling transition This requires a rotation of the field by 9, since the quantisation field must now be along the optical axis of imaging, orthogonal to the direction of the field used to spatially separate the states Figures 4 and 41 show images of a 1, +1, 1 pseudospin- 1 condensate acquired using this technique These images were taken following a Ramsey interference experiment, with 15 ms of evolution of the two-component condensate in the optical trap, with variable phase shift of the second π -pulse The different experimental conditions between the two images yielded nearly symmetric density patterns upon interchange of the states, illustrating the ability of this technique to preserve the spatial structure of each state The centre-of-mass of component is also vertically displaced from 1 in these images due to a small component of the Stern-Gerlach force in the vertical direction The Stern-Gerlach force deviates from the direction of the applied bias field because the condensate is not at the centre of the quadrupole field when it is briefly applied; the condensate is falling under gravity during this time This effect can be obviated by applying a larger bias field The dual state imaging method minimises fluctuations in measured relative number that where tpulse is the duration of the pulse and texp is the expansion time following the pulse We typically use parameters tpulse = ms, texp = 6 ms, B q = 8 Gcm 1, and B = G

116 1 two-component bec : experiment arise when detecting individual populations in different experimental realisations This is due to common mode rejection of both probe-laser frequency noise and irreproducibility of total atom number between experimental runs This is demonstrated in Fig 418b which shows a strong correlation between 1 and populations for a wide range of detected total condensate atom number The method allows the relative population to be measured with a sub-percent standard deviation, as opposed to single-component measurements of the absolute population which can typically vary by ±1% This method improves our measurement of phenomena such as Rabi oscillations (Fig 414) and Ramsey fringes (Fig 64) Moreover, it will see important application in atom chip clock research, allowing normalisation of total atom number fluctuations, a practice used to enhance the accuracy of precision fountain atomic clocks [134, 135] It also represents a significant step to achieving quantum-limited detection of the total spin projection in pseudospin-1 condensates

117 MW adiabatic passage Stern-Gerlach kick z (μm) x (μm) 1 Figure 41 Simultaneous dual-state imaging of 1, +1, 1 pseudospinor (see Fig 4) The dotted lines are the result of classical trajectory simulations, terminating at dots located at the centre of each component of the condensate (absorption image to scale) During momentary application of a magnetic field gradient (region labelled Stern-Gerlach kick), state experiences a force which spatially separates it from the magnetic field insensitive state 1 The vector field of the differential force is shown, with the mean arrow length equal to the force of gravity As the kick occurs below the quadrupole centre at x =, z =, the differential force has a small vertical component which accelerates downwards more than gravity alone The Stern-Gerlach kick is applied 6 ms after releasing the condensate from the optical trap (blue ellipse), during which time 1 is transferred from 1, 1 to, using microwave adiabatic passage The quadrupole field of B = B q /(x, y, z) (B,, ) with B q = 8 Gcm 1 and B = 1 G is applied for 4 ms, before another 64 ms of free expansion

118

119 Chapter 5 Two-component bec : theory This chapter outlines the theory of two-component Bose-Einstein condensates relevant to experiments in this thesis We begin with the coupled Gross-Pitaevskii equations (cgpes, 51), which describe the evolution of the two-component order parameter in Eq (11) These classical field equations encompass a rich array of physics, and can be used to predict many features of binary quantum fluids, eg the dependence of ground state wave functions on the nonlinear interactions, and the nature of exotic topological features, such as skyrmions and vortices They also reveal important aspects of using trapped pseudospinor condensates for metrology We include the effects of electromagnetic coupling and losses due to inelastic collisions, and describe the numerical techniques used to simulate the cgpes for this thesis Using the framework of superfluidity, we elucidate the criterion of stability for the spatial coexistence of the two components ( 5) The nature of excitations in two-component condensates and their ground state wave functions, depends on the relative strengths of the two-body collisional interactions that occur between condensate atoms These properties are at the heart of the nonequilibrium dynamics and spatially inhomogeneous relative phase evolution that occur in pseudospinor condensates (Chapter 6) Considerations of a homogenous system lead to a quantitative estimate of the miscibility criterion, and also reveal that multiple speeds of sound exist in interpenetrating superfluids We identify the characteristic time and length scales over which the condensate wave functions respond to a changing potential, due to an external perturbation or the self interaction with their own mean field, and find them distinct to those of a single component superfluid In Section 53, the two-component bec ground states are predicted using the Thomas- Fermi approximation The predicted miscibility threshold in Section 5 is echoed by the Thomas-Fermi approximation for trapped two-component condensates We determine the actual ground state wave functions using the cgpes, and find the threshold is far less abrupt than predicted using the above two methods; the two-components can spatially coexist deep

120 two-component bec : theory 14 into a regime where immiscibility is expected Kinetic energy plays a significant role at the boundary where the components overlap in space, which is neglected in the Thomas-Fermi approximation Accordingly, the Thomas-Fermi approximation breaks down at the interface between miscible and immiscible quantum fluids for the scattering lengths considered here Finally, an equivalent order parameter, the macroscopic Bloch vector, is introduced in 56 We use this extensively in Chapter 6 to depict the evolution of the local pseudospin in the condensate during interferometry experiments 51 Coupled Gross-Pitaevskii equations We consider a two-component Bose-Einstein condensate (bec) described by the order parameters Ψ1 and Ψ, coupled by a single (semi-classical) radiation field, with the inclusion of inter- and intra-state many body loss processes The wave functions are normalised such that Ψi d3 r = N i is the number of atoms in state i, hence Ψi = n i is the density of each state In their most basic form (without loss or electromagnetic coupling terms), these equations were introduced in [136, 137] The inclusion of electromagnetic coupling terms has appeared in [118] (multi-component condensates) and [138, 139, 14, 141, 14, 143] (specifically pseudospin- 1 condensates) Addition of nonlinear terms to describe many-body loss processes was used in [144], and applied to a binary condensate system of 87 Rb in [8] Combining all of the above features we arrive at iħ iħ ħ ħ ħω Ψ1 = [ + V1 + g11 Ψ1 + g1 Ψ iħ Γ1 + Ψ, ] Ψ1 + t m Ψ ħ ħ ħω = [ + V + g Ψ + g1 Ψ1 iħ Γ Ψ1 ] Ψ + t m (51a) (51b) where m is the mass of 87 Rb, Vi is the trapping potential experienced by component i and g i j = 4πħ a i j /m are the mean field interaction parameters The dominant two- and three- body loss processes are described by the terms Γ1 = (γ111 n1 + γ1 n )/, and Γ = (γ n + γ1 n1 )/, (5a) (5b) with loss rates γ i j measured and described in Refs [145, 8] The origin of the loss terms is apparent from the following phenomenological treatment The decay of the condensate densities are well described by the rate equations [145, 146] ( dn1 ) = γ111 n13 γ1 n1 n, dt loss (53a)

121 51 coupled gross-pitaevskii equations 15 ( dn dt ) = γ n γ 1 n 1 n (53b) loss The dominant loss processes are three-body recombination of atoms in state 1 (γ 111 ), and spin relaxation between two atoms; one in each state (γ 1 ), or both atoms in state (γ ) The loss terms in Eqs Eq (51) can be obtained by expressing the order parameters as functions of the densities Ψ i = n i e iϕ i and using Eqs (53), ( dψ 1 dt ) = Ψ 1 ( dn 1 loss n 1 dt ) = 1 Ψ 1 loss ( dψ dt ) = Ψ ( dn loss n dt ) loss= 1 ( γ 111 n1 3 γ 1 n 1 n ) = Γ 1 n 1 Ψ 1, (54a) Ψ ( γ n γ 1 n 1 n ) = Γ n Ψ (54b) The origin of the two-body inelastic decay terms can alternatively be gleaned by noting that the s-wave scattering lengths a 1 and a are, in general, complex 1 The imaginary parts of these scattering lengths are related to two body inelastic scattering processes in which the outgoing collision products are not the same spin states of the entrance channel These are mechanisms of decay, as such collisions are exothermic and the atoms subsequently escape the trap The decay terms in Eqs (51) are equivalent to those attained by rewriting the equations without the loss made explicit (Γ i ), but including the imaginary parts of the complex scattering lengths (a i j Re a i j + i Im a i j ) Before the realisation of two overlapping condensates (in states 1, 1 and,, [4]), large inelastic collision rates were expected between the different spin states, leading to the rapid decay of multi-component condensates with even partial miscibility In reality, condensate decay due to inelastic spin exchange collisions is a hinderance, but not fatal, to performing nontrivial experiments with multi-component condensates over several seconds The remarkable likeness of the singlet and triplet scattering lengths for 87 Rb was later found to explain the suppression of certain deleterious s-wave collisions, for any Zeeman states [147] Two-body inelastic collisions do, however, make it difficult to prepare two-component condensates of 87 Rb in their ground state wave functions using thermal mixtures of the spin states Inter- and intra-state two-body decay overwhelms the elastic collisions necessary for rethermalisation during efficient evaporative cooling, preventing simultaneous condensation of the two spin states Consequently, two-component condensates are prepared by evaporating a thermal cloud of 87 Rb atoms in a Zeeman state of the F = 1 hyperfine level, and subsequent radiative coupling to a Zeeman state in the F = hyperfine level This commonly results in condensate wave functions different to that of the two-component ground state, as we shall see in Chapter 6 1 With the equations written in the form of Eqs (51), the terms a i j refer to the real part of the scattering lengths

122 two-component bec : theory 16 5 Stability, miscibility and excitations In this Section, we consider the conditions under which a two-component condensate with overlapping wave functions is stable Such stability indicates how miscible the components of the condensate are when thought of as a binary superfluid Other important and related properties of the condensate are its response to external perturbation, the excitation spectra, and its ground state wave functions The following approaches can be used to determine these properties 1 Bogoliubov approximation (this Section) gives dispersion relations of the homogeneous system The stability criterion follows from requirement of real plane-wave energy Thomas-Fermi approximation (next Section) poorly predicts ground states, but also leads to the critical value of a1 = a11 a for component coexistence This was the grounds upon which the critical value was first derived in the context of dilute alkali gas binary condensates [148] 3 Many-body (second-quantised) methods, eg random-phase approximation [149] give excitation spectra which account for effects beyond the mean field approximation of the cgpes [15, 149] We consider an initially homogenous two-component condensate using Eqs (51) with Vi = Γi = Ω = = The steady state solutions are spatially homogeneous Ψi, (r, t) = n i e i µ i t/ħ with µ i the chemical potential of each state/species µi = n j gi j j (55) To parse the single particle excitation spectra, we linearise Eq (51) using the substitution Ψi (r, t) = Ψi, (r, t)+ e i µ i t/ħ δψi (r, t) This yields, neglecting terms higher than linear order in δψi and δψi, iħ (δψi ) ħ = (δψi ) + g i j n i n j (δψj + δψj ) t m j (56) Seeking plane-wave solutions of the form δψi = u i,k e i(k r єt/ħ), (57a) The nature of collective excitations of trapped binary condensates is more complex, but closely related to the results presented in this Section The mode spectra have been determined for two-component condensates in spherical and cylindrical geometries by Graham and Walls [151], using a combination of methods 1 and

123 5 stability, miscibility and excitations 17 δψi = v i,k e i(k r єt/ħ) (57b) results in a linear system of four equations for u1,k, v1,k, u,k, v,k, the Bogoliubov-de Gennes equations for the two-component homogeneous system [є ħ k ħ k ] uk,i = [ є ] vk,i = g i j n i n j (uk, j + vk, j ) m m j (58) These equations are block diagonal in the basis comprised of the sums and differences of uk,i and vk,i, in which the equations can be written g11 n1 g1 n1 n g1 n1 n uk,1 ± vk,1 ħ k mє uk,1 ± vk,1 = [1 ( ) ] ħ k uk, ± vk, g n uk, ± vk, m (59) The dispersion relation is then found by solving the above eigenvalue equation, giving єk = ħ k ħ k + η± ), ( m m where η± are the eigenvalues of the matrix g11 n1 G= g1 n1 n g1 n1 n g n (51) (511) The spectrum contains two physical branches,3 one for each eigenvalue of G The negative branch (with eigenvalue η ) has eigenvector (uk,1 vk,1, uk, vk, ) and corresponds to out of phase relative motion of the two components (for repulsive interactions) The positive branch (with eigenvalue η+ ) has eigenvector (uk,1 + vk,1, uk, + vk, ) and corresponds to in phase os- cillations of the densities of the two components4 The dispersion relation of Eq (51) has an identical form to the Bogoliubov dispersion of a dilute single-component Bose gas with the equivalence η± n i g ii = mc i, where c i is the hydrodynamic speed of sound for excitations of a single-component condensate5 Equivalently, there are multiple sound velocities associated with two-component excitations, as first remarked upon for the envisioned super- fluid 4 He 6 He mixture [153]6 Indeed, in the long wavelength limit of k, we have the 3 There are four branches in total; for if є k is a solution, so too is є k We only consider physical solutions, for which Re(є k ) 4 The nature of the relative motion of the excitations can be seen by performing the excitation analysis with a hydrodynamic formalism, with Ψi = n i + δn i e i ϕ i, as in Reference [15] 5 The two-component analysis reduces to this case with n =, for example 6 Originally anticipated as an achievable bosonic superfluid mixture, the short half-life of 6 He ( 8 s, [154]) rendered its superfluidity infeasible [155, 156]

124 two-component bec : theory 18 excitation velocities c± = mc± ω k = (є k /ħ) k = η± /m For equal densities n i = n/ n ) = η± = (g11 + g ± (g11 g ) + 4g1 4 πħ n ) = (a11 + a ± (a11 a ) + 4a1 m (51) The dispersion relationship encapsulated by Eqs (51) and (51) has been derived elsewhere by various means [157, 151, 158] The stability criterion for the homogeneous system (identified in the early works [159, 16]) requires єk R k, equivalent to η± > ; imaginary energies (or excitation frequencies) are a signature of instability [161, 157, 16] Should this inequality be violated, the excitations δψi and δψi in Eq (57) would grow exponentially For repulsive interactions, the inequality can only be violated by the negative branch7 and in the case of uniform and equal densities, this amounts to a1 a11 a (513) This stability criterion [148] has also been elucidated in the context of 3 He 4 He mixtures [163] and Bose-condensation of two internal states of hydrogen [164], and can alternatively be derived by minimising the total energy of the free-space system [158, 165, 166] When Eq (513) is violated, the mutual repulsion is strong enough for the components to spatially separate The unstable excitations of the negative branch grow at a rate Im(єk )/ħ; the fastest growing modes can be found by minimising єk in Eq (51) with respect to k, which gives ħkf = c m = η m It is readily confirmed that this mode corresponds to a global and negative minima of єk = m c 4 = η, for η = η < As noted by Timmermans [158], one can then surmise that a uniform homogeneous mixture will break up into domains of size π/kf = h/( m c ) on a time scale ħ/єk = ħ/(m c ) = ħ η 1 This analysis motivates the definition of the healing scales of the two-component system The healing length of a single-component condensate, ξ i,heal = ħ ħ = mn i g ii mc i (514) is the minimum length scale over which density fluctuations can occur [51] Equivalently, it is the length over which the wave function of a uniform condensate with repulsive interactions heals at an infinite potential barrier; over which the density goes from zero to its peak, equilibrium value at such a barrier The temporal equivalent to this quantity, the healing time, is mentioned relatively seldom Deuar and Drummond [167] give an expression for the 7 For attractive interactions, the excitations of the positive branch can violate the inequality, indicating the instability of such condensates [157]

125 5 stability, miscibility and excitations 19 healing time of a single-component condensate based on dimensional analysis t i,heal = mξi,heal ħ = ħ ħ = n i g ii mc i (515) This is approximately the shortest time for the condensate density to heal (in this case, come to equilibrium) after an infinitely large disturbance or perturbation These quantities inadequately describe the scales over which the two-component condensate wave functions change due to immiscibility Indeed, the single-component healing scales for condensates used in this thesis are of order ξ i,heal 1 µm; much less than the length scale over which the immiscibility of the components is manifest, and t i,heal 1 ms; much faster than the time taken for initially overlapping components to spatially separate The sound velocity of the two-com- ponent system relevant to spatial separation is c, and the healing scales obtained from the above hydrodynamic analysis are in direct analogy with the single-component results under the substitution c i c in Eqs (514) and (515) In summary, they are two-component healing length two-component healing time h, m c ħ = m c c ξheal = c theal (516a) (516b) The analysis of Ao and Chui [165] contains a similar result; they define penetration depths Λ i as the length scales over which each component penetrates into the other Their result Λi = [ g1 1] g11 g 1/ ξ i,heal (517) is equivalent to that of Eq (516a) when the intra-state scattering lengths are almost commensurate Then ξ i,heal c ) = ( c ) = ( ξheal ci g g (g11 g ) + 4g1 11 g ii g1 (g11 g ) g11 + g [1 + ] g ii 4g1 g ii g1 1 (for g11 g ), g11 g = 1/ ie ξ i,heal =( ), Λi c ξheal = Λi (518)

126 two-component bec : theory 11 c Ξheal (μm) bg a1 a c t heal (s) a1 a11a 1 5 c c (dashed, Figure 51 Two-component healing length ξheal (solid, red) and healing time theal 87 green) for a Rb pseudospinor condensate in the regime of immiscibility, a1 > a11 a In this plot, the best current estimates of a11 = 14 a and a = 95 a are used, and the inter-state scattering length is allowed to vary fractionally from the background value bg a1 = 9766 a [8] The top axis indicates the remarkable proximity of the background scattering lengths to the miscibility/stability criterion of (513) The total atomic number density is n = 1 m 3, for which the single-component healing lengths are ξ1,heal ξ,heal 3 µm and the single-component healing times are t1,heal t,heal 1 4 s If the scattering lengths are in the unstable, immiscible regime but also similar, ie a1 a11 a (regime of weakly segregated phases [165]) the two-component healing length can be much larger than the single-component healing length, as seen in Eq (518) This is the case c c for 87 Rb, for which ξheal and theal are plotted in Fig 51 with a ii fixed and a i j allowed to vary [in the immiscible regime to which Eqs (516) apply] from its background value The best known values of the scattering lengths in the 1, 1, +1 pseudospinor sys- tem of 87 Rb are such that the stability criterion of Eq (513) is just satisfied, as a1 /a11 a 1 = In this regard, two important things warrant emphasis Firstly, the three scat- tering lengths are not known well enough to irrefutably say which side of the miscibility regime the system lies (in the absence of tuning any of the scattering lengths with a Feshbach resonance) All scattering lengths are known from scattering calculations, and comparisons of observed condensate nonequilibrium dynamics with simulations of the coupled Gross-Pitaevskii equations (Reference [8] and Reference [7] therein), to a precision of 1a Small uncertainties in the background scattering lengths leads to a shift in a1 /a11 a

127 53 of thomas-fermi approximation δa/a > 1 3 ; uncertainty of the background scattering lengths8 alone therefore encompasses a significant region of Fig 51 Secondly, the inequality of Eq (513) does not represent a sharp boundary of miscibility; there exist situations where Eq (513) is violated yet the components still exhibit significant spatial overlap, as we shall see for trapped condensates in the following Section º Having established the conditions under which two-component condensates are able to coexist in space, we now consider the nature of their co-existence or spatial separation; the nature of the ground states of trapped two-component becs 53 Thomas-Fermi approximation The stability of the homogeneous system, described in the previous Section, is instructive in estimating the time and length scales of evolution for an initially homogeneous two-component condensate in the immiscible regime It is tempting to think that the Thomas-Fermi approximation, in which the kinetic energy terms in Eqs (51) are ignored, may give similar quantitative insight into the nature of the two-component ground states Indeed, the Thomas-Fermi approximation has been of great use in describing the ground state of single- component becs [169] To study the ground states of the cgpes in the Thomas-Fermi approximation, we write Eqs (51) with Ψi Ψi (r, t) = e i µ i t/ħ n i (r) Γi = Ω= = (negligible kinetic energy) (ground state of i with chemical potential µ i ) (no particle loss) (no electromagnetic coupling) The ground state densities n i = Ψi satisfy the equations µ1 = V1 + g11 n1 + g1 n, µ = V + g n + g1 n1 (519a) (519b) Finding the solution to these equations is more difficult than the case of a single-component, for which the method will now be outlined Setting n in Eqs (519) and dropping 8 As recently as, these scattering lengths were reported as a 11 = 19a, a = 956a, and a 1 = 98a (Reference [168] and Reference [6] therein), in disagreement with the values used here by 5a With δa = 5a, the value of a 1 /a 11 a is only certain to within 1

128 two-component bec : theory 11 the subscripts denoting state 1, we have the single component Thomas-Fermi ground state density n(r) = g 1 max (µ V (r), ) (5) To find the ground state density profile for a given atom number N, we integrate Eq (5) over all space to find the dependence of the atom number on the chemical potential ie N(µ) = n d3 r This gives 8π 3/ µ 5/ 16 λπµ 5/ N= =, ( ) 15 mω g 15g ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ real units (51) natural units with ω = (ωρ ωz )1/3 the geometric mean of the trapping frequencies Inverting this relation- ship to obtain µ(n) yields the desired result; the ground state density for a given number of atoms: where a = n(r) = ħω 15Na /5 ρ z ( ) max (1, ), a ρtf ztf (5) ħ/mω The radial and axial Thomas-Fermi widths of the condensate, ρtf and ztf, are defined via µ = 1 mωρ ρtf = 1 mωz ztf In the single-component case, the Thomas- Fermi approximation is most accurate when the kinetic energy Ekin is much less than the in- teraction energy Eint, ie Eint /Ekin Na/a 1 The condensate ground state wave function is then well described by Eq (5) except at its edge, where the energetic cost of a discontinuous derivative is averted by quantum diffusion; the kinetic energy is non-negligible at the edge, and in reality the wave function decays exponentially in space For the two-component case, inspection of Eq (519) reveals that in addition to the mean field interaction with itself, each component experiences an effective potential comprised of the external confinement Vi and the mean field of the other component These effective po- tentials are not hard to the mean field of the individual component they confine; they are both malleable and coupled to the mean field of the component they confine As such, the effects of quantum pressure are far greater in two-component condensates, and the ThomasFermi approximation inadequately predicts features of two-component ground states This has been remarked upon by Esry et al [148], and also by Pu and Bigelow [17, 171] In spite of this, application of the Thomas-Fermi approximation to two-component condensates motivated much of the theoretical work providing the initial, core understanding of component separation in binary condensates [136, 158, 17] Of the works that have employed the Thomas-Fermi approximation or evaluated its efficacy, many have done so for relative interaction strengths deep in either regime of miscibility; not representative of the weakly segregated regime present in 87 Rb pseudospinor condensates

129 53 thomas-fermi approximation 113 (a11 a, and a a1 a11 ) In the remainder of this Section, I will outline the method used to find two-component ground states in the Thomas-Fermi approximation, and demonstrate the shortcomings of this approximation by comparison with ground-state solutions of the cgpes I will restrict the discussion to cases where m1 = m (obviously true for two different spin states of the same atom), and ω1 = ω The latter is a good approximation for optically trapped Zeeman states of 87 Rb 5 S1/, where the differential light shift between the levels is negligible, and magnetostatic trapping of the 1, 1 and, 1 states at fields near 33 G In a top trap, however, the antiparallel angular momenta of states with the same magnetic moment results in slightly different trap frequencies [173, 8] The generalisation of the procedure for a single-component condensate to a binary quan- tum fluid amounts to finding the density profiles n i (r) as functions of the particle numbers N i To do this for a given set of three interaction strengths g i j we must 1 Express the densities of the binary mixture as functions of µ i for regions: where the components coexist, occupied exclusively by one component, or where there is zero density Determine the boundaries of these regions for a given µ i 3 Integrate the piecewise defined densities over all space and use N i = n i d3 r to obtain the number of atoms as a function of µ i 4 Invert the relation N i (µ1, µ ) to obtain the µ i (N1, N ), which then gives the densities n i (r) as a function of N i by way of 1 Of these, step 1 is trivial, step is rather subtle and depends very strongly on the relative values of the interaction parameters, whereas steps 3 and 4 are algebraically demanding We perform step 1 by solving for n i in Eqs (519) for (a) non-zero n1 and n, (b) n1, n =, (c) n1 =, n, and (d) n1 = n = This gives coexisting 1 and only 1 only vacuum n1(a) = g µ 1 g1 µ ; g11 g g1 n1(b) = g11 1 µ 1 n1(c) = n1(d) = ; ; ; n(a) = g11 µ g1 µ 1, g11 g g1 n(b) =, 1 n(c) = g µ, n(d) =, (53a) (53b) (53c) (53d) where µ i µ i Vi To simplify the analysis, we assume spherical symmetry (ω ρ = ωz ) and use natural units,9 such that Vi = r / There are six types of density configurations allowed for harmonic confinement Each can be classified by the subset of Eqs (53) which describe 9 See 54 for example

130 type 1 type r ab r ac r ac r ab 1 r bd 1 1 r cd MISCIBLE REGIME 1 type 3 type 4 r ab r ac 1 r bd 1 r cd 1 type 5 type 6 IMMISCIBLE REGIME r s 1 r bd r s 1 r cd Figure 5 Possible density configurations for two-component Thomas-Fermi ground states The illustrated configurations are two-dimensional projections of a condensate trapped in a harmonic potential with spherical symmetry The density in each region is described by Eqs (53), the radii by Eqs (54), aside from r s which is found by minimising the Thomas-Fermi energy functional in Eq (56)

131 53 thomas-fermi approximation 115 the density, in order of increasing r: Type ii -coexisting- 1 -vacuum : (c)-(a)-(b)-(d) Type iii 1 -coexisting- -vacuum : (b)-(a)-(c)-(d) coexisting- 1 -vacuum : (a)-(b)-(d) Type iv coexisting- -vacuum : (a)-(c)-(d) Type i - 1 -vacuum : (c)-(b)-(d) Type v 1 - -vacuum : (b)-(c)-(d) Type vi Scattering lengths which permit miscibility [Eq (513)] result in Thomas-Fermi ground states of type i iv; immiscible cases are of type v or type vi1 Fig 5 illustrates these possible configurations 531 Miscible regime : a1 a11 a Density configurations of Type i and ii have a region of coexistence whose boundary is described by the following radii: Á Á À µ 1 rab = Á Á Á À µ 1 rac = Á µ1 µ, g 1 g111 (54a) µ1 µ g 1 g1 (54b) The above radii are obtained by setting n = (for rab ), and n1 = (for rac ) in Eq (53a) For types iii and iv, the boundary of the coexisting region is described by Eq (54a) or Eq (54b), respectively The outermost non-zero density vanishes at rbd = rcd = µ1 (Types i, iii, and v), (54c) µ (Types ii, iv, and vi) (54d) To find N i (µ1, µ ), one performs the appropriate piecewise-integration For example, rac < rab < rbd in Type i, indicating the density configuration of the components is such that resides within a shell of state 1, with some co-existing region between rac and rab The atom numbers for type i obey Ni = rac (c) n i d3 r + 1 We determine this property a posteriori rab rac (a) n i d3 r + rbd rab (b) n i d3 r (55)

132 two-component bec : theory 116 Determining which type of density configuration describes the ground state of a two-component system with a given N i and g i j is quite involved Ho and Shenoy [136] developed a procedure involving the phase space defined by ( µ 1, µ ) This method predicts which density configurations might be possible; it does not, however, predict a unique configuration for a given N i and g i j Our results indicate the type of density configuration cannot be predicted g11 g, g11 /g1, and g1 /g, alone, as suggested by Trippenbach et al [17] For exfrom g1 ample, a transition from type iii to type i can occur upon varying g1 whilst g < g1 < g11 In practice, a self-consistent approach proves satisfactory, ie calculate N i (µ1, µ ) for all types i-iv, then eliminate those cases for which any of the following criteria are not met: N i (µ1, µ ) = N i permits real solutions for µ1 and µ The boundary radii corresponding to a given type must be real and physically consis- tent for the solutions µ i (N1, N ), eg rac < rab < rbd for type i; Fig 5 provides an illustration of this condition The resulting piecewise-defined density profiles must be non-negative for all radii Should multiple density configurations remain following this elimination procedure, the unique ground state will have the lowest energy, as determined by the Gross-Pitaevskii energy functional in the Thomas-Fermi approximation εtf = ij 1 [δ i j R + g i j n j ] n i d3 r (56) Expressing the chemical potentials as µ µ1 = β µ simplifies the task of inverting N i (µ1, µ ), as integrals of the form Eq (55) reduce to N i = µ i 1+d/ f i (β) for dimension d,11 where f i (β) are cumbersome, yet analytic expressions This greatly constrains the values of µ i when inverting N i (µ1, µ ), and indicates much of the behaviour of Thomas-Fermi ground states depends only on g i j and the relative atom number 53 Immiscible regime : a1 > a11 a For the Thomas-Fermi ground states in the immiscible regime, we take an approach similar to Trippenbach et al [17] Thomas-Fermi ground states with completely separated wave functions are of types v and vi Using type v as an example, the wave functions in this 1+d/ 11 The scaling of the atom numbers with the chemical potentials, N i µ i case, is the same as the single-component

133 53 thomas-fermi approximation regime are of the form n(c) i (r), r rs ; (b) n i (r) = n i (r), rs < r rtf ; rtf < r,, 117 (57) where rs is the separation radius, and rtf is the Thomas-Fermi radius defined by either Eq (54c) (type v) or Eq (54d) (type vi) To find the Thomas-Fermi ground state wave functions, we 1 use the normalisation condition N i = n i d3 r to find µ i (N1, N, rs ), substitute this result into Eq (56), the total energy in the Thomas-Fermi approximation, and 3 minimise εtf with respect to rs with the constraint that rtf (µ i (N1, N, rs )) > In the remainder of this Section, we study Thomas-Fermi ground states where g1 is varied In the immiscible regime, the above procedure is independent of g1, as there is no spatial overlap between the wave functions; all Thomas-Fermi ground states with g ii fixed are the same for g1 > g11 g 533 Ground states of 87 Rb pseudospin- 1 bec We now implement the results of this Section, using the Thomas-Fermi approximation to compute two-component ground states, and compare them to those found by numerical solution of the cgpes A relevant example to current and future experiments with 87 Rb pseudospin- 1 condensates is the variation of a1 about its background value, across the threshold of miscibility For the F = 1, m F = +1 and F =, m F = 1 states, the inter-state scattering length a1 can be tuned using a mixed-spin-channel Feshbach resonance [174, 175, 176, 177] The tunability of this scattering length is limited by the enhanced collisional losses in the vicinity of the resonance We consider the range of a1 over which the losses are not im- practicably severe;1 the largest tunability we infer is such that a1 varies by ±15% about its background value, which is the range used for the present analysis Furthermore, we take the background scattering lengths a i j of the F = 1, m F = +1 and F =, m F = 1 states to be those of the F = 1, m F = 1 and F =, m F = +1 states We find the ground states for N1 = N = atoms, and take ω/π = 4848 Hz, corresponding to the geometric mean of the trapping frequencies used for the experiments described in Chapter 6 The results are 1 Using the characterisation of the Feshbach resonance between the F = 1, m F = +1 and F =, m F = 1 states in Reference [177], we infer the inter-state two body loss rate to be γ 1 = cm3 at magnetic fields 19 mg either side of the resonance, for which a 1 is tuned ±15% from its background value This is 33 times greater than the corresponding loss rate for the F = 1, m F = 1 and F =, m F = +1 states [8])

134 two-component bec : theory 118 a1 /a1 a1 /a11 a bg a11 /a1 a1 /a > < > < > > > > < < > < < < < Type µ1 µ εtf εgp c ξheal iii iii iii i i v v v Table 51 Energy and length scale comparison for the two-component ground states in Fig 53 The chemical potentials µ i, and energies of Eq (56), are shown for the ThomasFermi approximation in units of ħω The character of the Thomas-Fermi ground states can not be classified using a1 /a11 a, a11 /a1, and a1 /a alone, as suggested in Reference [17] (see text) The healing length (shown in µm) has been calculated using Eq (516a); the total density used here corresponds to the radial position where the cgpe simulated densities are equal (n = cm 3 at r 6 µm) The two-component healing length is not well defined for cases where the overlapping homogeneous system is stable [Eq (513)] The Thomas-Fermi radius is 96 µm in all cases (including single-component) shown in Fig 53, and are separated into the two regimes of the miscibility criterion The ground state of the two-component system is important to pseudospin- 1 bec experiments, as the dissimilarity of the initially prepared wave functions to the ground state dictates the resulting nonequilibrium dynamics The nature of these dynamics affects, among other things, Ramsey interferometry of pseudospin- 1 bec Table 51 contains a quantitative summary of the ground states depicted in Fig 53 In all cases, we find µ1 > µ, consistent with the density configuration implied by Eqs (54), ie a ring of state 1 surrounding a shell of state (types i and v), or overlapping com- ponents with 1 extending to larger radii than (type iii) The first two circumstances in Fig 53 show qualitatively good agreement between the Thomas-Fermi approximation and the Gross-Pitaevskii ground states For a1 = 97 a1 = a, the density of state 1 exhibits bg a relatively uniform density near the origin in both Thomas-Fermi, and cgpe ground states Figure 53 (facing page) Two-component ground states as found using the ThomasFermi approximation (dashed lines), or simulations of the coupled Gross-Pitaevskii equations (solid lines), across the threshold of miscibility The density has units of number of atoms per harmonic oscillator unit volume, or a 3 Ψ

135 a 1 a 11 a 7 density MISCIBLE REGIME bg a 1 = 85 a r µm 1 density IMMISCIBLE REGIME bg a 1 = 1 a r µm 1 a 1 a 11 a 1 bg a 1 = 97 a 1 bg a 1 = 11 a density r µm density r µm 1 1 bg a 1 = 9875a 1 bg a 1 = 15 a density density r µm r µm bg a 1 = 99 a 1 bg a 1 = 115 a 1 98 density density r µm r µm

136 1 two-component bec : theory This can be understood by the effective potential experienced by component 1, it has no curvature for g 1 = g, as is seen by the independence of n (a) 1 in Eq (53a) on r As a 1 is increased, and the threshold of miscibility is approached from below, it can be seen that 1 the Thomas-Fermi approximation becomes decreasingly valid confirmed by comparison with the Gross-Pitaevskii ground states, and no single density configuration is unique to the ratios a 11 /a 1 and a 1 /a, and vice versa For example, the transition from a type iii to a type i Thomas-Fermi density configuration occurs for a 1 = 9875 a bg 1, when a < a 1 < a 11 In particular, this transition is predicted to occur when r ac =, implying [via Eq (54b)] which, in combination with the vanishing relative number, β = µ 1 µ = g 1 g, (58) N 1 (µ 1, µ ) N (µ 1, µ ) = N 1 (µ, β) N (µ, β) =, (59) yields an equality dependent only upon g i j For ground states predicted using the cgpes, this transition does not occur until a 1 = 15 a bg 1 ; within the immiscible regime Comparison with the Gross-Pitaevskii ground states highlights the breakdown of the Thomas-Fermi approximation, and indicates the criterion for miscibility in 5 does not represent a sharp boundary of component separation when the threshold of Eq (513) is crossed As the interstate scattering length is increased further, above the threshold of miscibility, the mutual repulsion becomes dominant and the Thomas-Fermi approximation improves The separation radius r s ( 53), whilst not a boundary of complete component separation, accurately mimics the radius of equal density of the two partially overlapping components In addition, the healing lengths for the immiscible cases have been computed using Eq (516a) For a 1 = 11 a bg 1 the healing length of 85 µm is larger than the Thomas-Fermi radius r TF = 96 µm, consistent with the Gross-Pitaevskii ground state exhibiting no region of spatial separation For the final two cases of Fig 53, the calculated healing length approximately coincides with the depth to which the components overlap The comparison of this Section relied upon the numerical solution of the coupled Gross- Pitaevskii equations The core understanding of the experimental results in the following Chapters has derived from simulating these equations The rest of this Chapter is devoted to outlining the methods used to simulate the evolution of pseudospin-1 bec with Eqs (51) in

137 54 simulation of the coupled gross-pitaevskii equations 11 this thesis 54 Simulation of the coupled Gross-Pitaevskii equations To solve Eqs (51) numerically, we exploit the cylindrical symmetry of our experimental ge- ometry, and assume the wave function takes the form: Ψ(r) e i m φ Ψm (ρ, z), where {ρ, z, φ} are cylindrical co-ordinates and m is an integer Since the ground state of the system has m =, we exclusively solve for this case and assume the time evolution does not break this symmetry We use the discrete Hankel-Fourier transform (dhft) to solve the cgpes, and have adapted the procedure used in [178] for multi-component systems Essentially, the technique facilitates the solution of second-order partial differential equations in cylindrical coordinates using spectral methods, where the Fourier transform fails due to divergence of the Laplacian for zero radial momentum Implementation of the Hankel transform in a discrete variable representation is greatly simplified by sampling the fields at zeros of the mth order Bessel function of the first kind, rather than using a Cartesian grid The mth order Hankel transform of a function h(ρ) is defined as Hm {h(ρ)} = h (k ρ ) = h(ρ)j m (k ρ ρ)ρ dρ, (53a) where J m (z) is the mth order Bessel function of the first kind The inverse Hankel transform of h (k ρ ) is H 1 m { h (k ρ )} = h(ρ) = h (k ρ )J m (k ρ ρ)k ρ dk ρ (53b) By performing the Hankel transform over the radial co-ordinate ρ, and the Fourier transform on the axial co-ordinate z, we have the combined Hankel-Fourier transform13 Gm {ψ(ρ, z)} = (π) 1/ = ψ (k ρ, kz ) e ik z z ψ(ρ, z)j m (k ρ ρ)ρ dρ dz The inverse Fourier-Hankel transform of ψ (k ρ, kz ) is 1 {ψ (k ρ, kz )} = (π) 1/ Gm = ψ(ρ, z) e ik z z ψ (k ρ, kz )J m (k ρ ρ)k ρ dk ρ dkz (531a) (531b) 13 The normalisation in Eq (53a) and Eq (531a) is such that both transforms obey Parseval s theorem For example, if {h(ρ, z), h (k ρ, k z )} and {g(ρ, z), g (k ρ, k z )} are transform pairs of the Hankel-Fourier transform, then h(ρ, z) g(ρ, z) ρ dρ dz = h (k ρ, k z ) g (k ρ, k z ) k ρ dk ρ dk z

138 two-component bec : theory 1 The pseudo-momentum k (k ρ, kz ) is the eigenvalue of the zeroth-order Hankel-Fourier transform applied to the Laplacian in axisymmetric cylindrical co-ordinates: G { ψ(ρ, z)} = G { 1 d dψ d ψ (ρ ) + } ρ dρ dρ dz = (k ρ + kz ) G {ψ(ρ, z)} = k ψ (k ρ, kz ), (53) where k k A proof can be found in Reference [179] The Hankel and Fourier transform in (531a) can be performed separately, and in either order, and we use a different discrete-variable algorithm to implement them along ρ and z, respectively We first consider the Hankel transform along ρ There are many discrete-variable implementations of the Hankel transform A useful algorithm for iterative processes and arbitrary fields is the quasi-discrete Hankel transform [18, 181] (qdht), which obeys the discrete form of Parseval s theorem By sampling the field at zeroes of a Bessel function, the Hankel transform and radial integration can both be performed using matrix multiplication The technique has been successfully used to solve optical propagation and scattering problems, as well as the excitation spectrum of a bec with dipolar interactions [178] The following outline of the algorithm mirrors that of Reference [178] One first chooses a radial size R, and pseudo-momentum K, such that h(ρ) and h (k ρ ) sufficiently vanish at the extents of both sampled co-ordinate space and sampled momentum space As in the case of the fast Fourier transform (fft), this choice dictates the size of the radial basis N ρ, as h(ρ) and h (k ρ ) are sampled at ρ j = K 1 α m j, k i = R 1 α mi, j = 1,, N, i = 1,, N, (533a) (533b) where α mi is the i th zero of the mth order Bessel function of the first kind, J m (ρ) Alterna- tively, for a given range of co-ordinate space R, the number of sampled points N determines the maximum pseudo-momentum K = R/α m,n+1 A scaled pair of transform vectors are introduced R h(ρ j ), J m+1 (α m j ) K H i = h (k i ), J m+1 (α mi ) Hj = (534a) (534b)

139 54 simulation of the coupled gross-pitaevskii equations 13 with which the transform is performed by the matrix multiplication N H i = Ti j H j, j=1 where Ti j = J m (α mi α m j /S), J m+1 (α mi ) J m+1 (α m j ) S (535) (536) is a real, N N symmetric transformation matrix and S = KR = α m,n+1 Substituting Eqs (534) and (536) into Eq (535), we have the direct expression for the qdht h (k i ) = α m j α mi N h(ρ j ) Jm ( ) K j=1 J m+1 (α m j ) S (537) During the course of any simulation of condensate evolution or ground state evaluation, integration in cylindrical co-ordinates is often required The tools of the qdht algorithm provide an efficient, accurate approximation to the radial integral h(ρ)ρ dρ 1 N 1 N h(ρ ) = Fi, i K i=1 J m+1 (α mi ) SK i=1 J m+1 (α mi ) (538) which is exact for band limited fields (ie those fields h(ρ) for which K exists such that h (k) = for k > K) and R, N We use the fft to perform the axial part of the transform in Eq (531a), with uniform sampling of the field along z Another advantage of using the Hankel transform is the ease with which the column den- sities of a two-component bec can be computed from a given field sampled in cylindrical co-ordinates These column densities are required to compare results of a cgpe simula- tion with experimentally measured absorption images of a two-component bec Suppose a density n(ρ, z) has been computed as a result of a simulation The column density is n (y, z) = n( x + y, z) dx, (539) where x is the propagation direction of the light used to absorption image the bec If the field is sampled cylindrically at {ρ i, z j }, this is computationally demanding, as the integral in Eq (539) needs to be evaluated for every y The need to perform such a process is obviated by identifying Eq (539) as an Abel transform, A[h(x, y)](y) = h(x, y) dx, (54) and using the slice-projection theorem This relates the Abel transform to the Fourier trans-

140 two-component bec : theory 14 form and the Hankel transform A[h] = F 1 H[h] (541) Since use of the above techniques requires both of these transforms, the requisite algorithms are already at hand, and the Abel transform can be computed merely by performing the discrete version of Eq (541) We now return to solving the cgpes using the dhft; recasting Eqs (51) in the natural units of the harmonic oscillator The mapping of variables is: ρ ρ = l ρ 1 ρ, radial length z z = lz 1 z, axial length t t = ωρ t, time energy (54a) (54b) 1 E E = (ħω ρ ) E, Ψ Ψ = wave function 1/ l ρ lz Ψ (54c) (54d) (54e) Dividing Eqs (51) by ħω ρ and using the above substitution, we arrive at the cgpes for the fields Ψi in cylindrical co-ordinates (ρ, z ): i i with Ψ1 Ω = [ + V1 + g11 Ψ1 + g1 Ψ i Γ1 ] Ψ1 + Ψ, t Ψ Ω Ψ1 i Γ ] Ψ + Ψ1 Ψ, Ψ + g1 = [ + V + g t = ( λ ), ρ ρ ρ z 1 (ρ + λ 1 z ), ωρ λ=, ωz V1 = V = V = g i j 4πa i j m3/ ω1/ z = = gi j, 5/ lz ħ Γ1 = (γ111 Ψ1 4 + γ1 Ψ )/, Γ = (γ1 Ψ1 + γ Ψ )/, γ111, γ111 = ω ρ l ρ4 lz (543a) (543b) (543c) (543d) (543e) (543f) (543g) (543h) (543i)

141 54 simulation of the coupled gross-pitaevskii equations 15 γi j, ω ρ l ρ lz Ω Ω =, ωρ = ωρ γ i j = (543j) (543k) (543l) The wave function is scaled as in (54e) to preserve the normalisation Ψi d3 r = N i with r = {ρ, z }14 All variables are hereafter expressed without primes, and assumed to be in natural units These cgpes can be expressed in vector form as i Ψ =[ 1 + (V + G n i L n i L3 n ) 1 + H ]Ψ, t ± ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ D differential operator (544a) N nonlinear operator where v M represents row-wise multiplication of a vector v with a matrix M, ie (v M)i j = v i M i j, and V1 V= V H = Ψ1 ; n= Ψ g11 ; G= g1 g1, g 1 γ1 1 γ111 1 Ω ; L = ; L3 = Ω γ γ 1 We then evaluate the evolution after an infinitesimal time increment δt using Ψ(t + δt) = e i δt(d +N ) Ψ(t) e i δt D e i δt N Ψ(t), where 1, N = (V + G n i L n i L3 n ) 1 + H (545a) D = (545b) (545c) 14 With this rescaling, the Thomas-Fermi relationship of Eq (519) still holds in natural units, and Ψi is the atomic number density in units of atoms per unit harmonic oscillator volume

142 two-component bec : theory 16 Differentiation is performed in k-space: e i δt D Ψ = G 1 {exp [ i δt (k + λ 1 kz )] G {Ψ}}, ρ (546) and application of e i δt N amounts to exponentiating the matrix N for each point in co-ordinate space When using the fft in reduced dimensions, the approximation made in Eq (545a) can be improved using the split-step method; using e i δt(d +N ) e i δt N / e i δt D e i δt N / (547) This can result in a marked improvement of the numerical solution for a similar computational time We find that for the discrete Hankel-Fourier transform, the split-step method is no more accurate than Eq (545a) when N contains terms nonlinear in the field The matrix representation illuminates the only mechanism for inter-state conversion in the cgpes electromagnetic coupling; H is the only operator in matrix representation with non-zero off-diagonal elements The equations for each component are, of course, still coupled; Eq (545c) shows that each diagonal term of N depends upon both order parameters These terms, however, are exclusively density dependent; they depend only on the modulus of the wave functions As such, the evolution of each state does not explicitly depend on the relative phase between them 541 Gross-Pitaevskii ground state computation To find two-component Gross-Pitaevskii ground state wave functions, such as those in Fig 53 and in the following Chapters, we propagate an initial field forward in imaginary time This involves: 1 Choosing initial wave functions for each state with the desired normalisation These need not be good estimates for the two-component ground state; the single-component Thomas-Fermi ground state, weighted so that Ψi d3 r = N i suffices Use δt i δt in Eq (544a) with H = (no electromagnetic coupling), and L i = (no particle losses) 3 Propagate the initial field, renormalising at every time step (propagation in imaginary time is not unitary) until the Gross-Pitaevskii energy converges to some desired precision

143 55 many-body theory of two-component bec 17 This procedure finds the lowest energy state, as it is the slowest to decay in imaginary time The expression for the energy, which is minimised by the ground state, is (in natural units) εgp = = d3 r Ψ [ (V + G n) 1]Ψ i d r Ψ 3 i + Vi + j The field is propagated in imaginary time until 55 gi j Ψj Ψi (548) dεgp < 1 7 N dt (549) Many-body theory of two-component bec The physics of spin-squeezing and entanglement are ancillary to this thesis In this Section, however, we digress briefly and examine the many-body formalism of two-component condensates, relevant to spin squeezing and entanglement In the second-quantised formalism, the many-body Hamiltonian of a pseudospinor bec with s-wave collisional interactions is H = ij gi j d r ψ (r) i H j ψ (r) + d r ψ (r)ψ (r)ψ (r)ψ (r), 3 i j ij 3 i j j i (55) where i, j {1, } and H is the Hamiltonian for a single atom with two internal states H = [ ħ ħ + Vext ] 1 + Ω σ m (551) The coupling of the two-levels via an external field is represented by the final term in (551), where σ = {σ x, σ y, σ z } is the three-vector of Pauli-spin operators, and Ω = {Re(Ω), Im(Ω), } is the effective magnetic field in the rotating wave approximation, with Rabi frequency Ω and atom-field detuning 15 We limit our discussion to symmetric trapping of each internal state, ie both states experience the same external potential Vext The bosonic field operators ψ i (r) and ψ i (r) create and destroy (respectively) a particle at position r in state i, and satisfy the commutation relations [ψ i (r), ψ j (r )] = δ i j δ(r r ) (55) 15 When the coupling is via a multi-photon transition, Ω and represent the effective two-level coupling parameters Ωeff and eff (see 36 and Appendix B)

144 two-component bec : theory 18 The contact interaction is described by the parameters g i j = 4πħ a i j /m, where a i j are the s-wave inter- and intra-state scattering lengths We define operators a i = d r ψ (r)ψ (r), 3 i i (553) which create a particle in state i with wave function ψ i (r), normalised as d3 r ψ i (r) = 1 The pair of operators conjugate to a i are a i = 3 d r δ(r r)ψ (r ) = ψ i (r), i (554) yielding the commutation relations [ a i, a j ] = δ i j r ψ i = δ i j ψ i (r) (555) A complete orthonormal basis for the many-body Hilbert space is formed by the set of states for which exactly N i bosons are in state i with wave function ψ i (r) These are the Fock states (or fragmented states ) (a ) (a ) vac N1 ψ1, N ψ = 1 N1! N! N1 N (556) The action of the bosonic field operators on the Fock states is ψ i (r) N1 ψ1, N ψ = ψ i (r) N1 ψ1, N ψ = N i + 1 ψ i (r) N1 + δ i1 ψ1, N + δ i ψ, N i ψ i (r) N1 δ i1 ψ1, N δ i ψ, N i N1 ψ1, N ψ = N i N1 ψ1, N ψ, (557a) (557b) (557c) where N i = d3 r ψ i (r)ψ i (r) is the number operator for state i A coherent spin state16 (css) of this system is defined as Ψ CSS (N!) 1/ (c1 a 1 + c a ) vac, N (558) where c1 + c = 1 This state describes a condensate in which every atom shares the same superposition (of both internal and external states) In the spirit of single-particle spinor states, it is useful to express the complex state amplitudes in Eq (558) as c1 = cos(θ/)e iϕ/ 16 Also known as a relative gp state, relative phase state, or binomial state

145 55 many-body theory of two-component bec 19 and c = sin(θ/)e iϕ/ The many-body spin operators S µ = d r (ψ 3 1 ψ ) σ µ ψ 1, µ {x, y, z} ψ (559) act on coherent spin states to give the mean spin S = ( S x, S y, S z ) = N/ (cos ϕ sin θ, sin ϕ sin θ, cos θ) Moreover, the variance of the spin in directions orthogonal to the mean spin is S = (56) N/ With the number operators defined as N i = d it can be seen that S z /N = cos θ = ( N1 N )/N, the mean relative atom number of a coherent spin state In the 3 r ψ i (r)ψ i (r), Fock basis, the coherent states have admixture of the Fock states described by N Ψ CSS = ( N 1 = 1 N! ) c1n1 cn N1 ψ1, N ψ N1!N! (561) The number of particles in each internal state of any nontrivial coherent spin state17 is not well defined, and for N 1 the number of atoms in state i has a Poissonian distribution with mean value N i = N i = c i N For coherent spin states with θ π/, the width of the binomial distributions are of order N; equivalently, the variance of mean relative number (N1 N )/N = S /N = N 1/ A final note regarding properties of coherent states; they form an overcomplete quasi-orthogonal basis The spin expectation values of coherent spin states facilitates their denotation N ; θ, ϕ ; ψ1, ψ, of which there are an infinitude The thesis of Ashhab [18, Appendix A] derives the result of quasi-orthonormality, N ; θ, ϕ ; ψ1, ψ N ; θ, ϕ ; ψ1, ψ 1 for ϕ ϕ N 1/, and shows that any Fock state (and hence any many-body state of the system) can be expanded in terms of coherent states Most second quantised treatments of pseudospinor becs [7, 183] proceed by assuming that a condensate in a single internal state i is described by a Fock state,18 eg Eq (556) with N1 = N, N =, and ψ1 (r) = ψ (r) (usually the ground state of a 1 bec) Such a state is also a coherent spin state, described by Eq (558) (trivially since c =, for example) The application of an electromagnetic coupling pulse, short compared to the timescales of other dynamics, is assumed to yield a coherent spin state19 taking the form of Eq (558) with ψ1 (r) = ψ (r) = ψ (r), and c1, c found by solving Eq (6) Equivalently, short coupling pulses are assumed to map coherent states to coherent states, performing the same spin rotation to the 17 ie c i, i = 1, 18 The consequences of these assumptions are yet to be gleaned from theoretical investigations [184] 19 At the time of publication, the author is unaware of any second-quantised treatment which proves that coherent spin states are closed under the operation of strong coupling

146 two-component bec : theory 13 macroscopic superposition that one would expect for a single-atom The expectation value of the number of atoms in each state is conserved under evolu- tion in the absence of, and subsequent to, inter-state coupling (Ω = ) As such, it is useful to express the coherent states of Eq (558) in the Fock basis [Eq (561)], and use Eqs (55) and (557) to determine the way in which the Fock states evolve; the evolution of the initial coherent spin state can then be evaluated in terms of that of the Fock states Sinatra and Castin [7] have comprehensively elucidated this formalism, and among their key results are: the Fock states evolve as N1 ψ1,init, N ψ,init Ð e ia(n1,n ;t)/ħ N1 ψ1 (N1, N ; r, t), N ψ (N1, N ; r, t) (56) where ψ i (N1, N ; r, t) obey a set of coupled Gross-Pitaevskii equations ψ i iħ = H + (N j δ i j )g i j ψ j ψ i t j (563) with initial conditions ψ i (N1, N ; r, t = ) = ψ i,init (r), and the complex phase factor A(N1, N ; t) solves da g ii = N i (N i 1) dt i d r ψ 3 i 4 N1 N g1 d r ψ ψ 3 1 (564) As a coherent spin state is composed of a distribution of each N i, its evolution depends on a range of ψ i (N1, N ; r, t) and A(N1, N ; t) Li et al [183] give a concise synopsis of the techniques introduced in Reference [7] which predict the essential many-body dynamics relevant to spin-squeezing in this system, whereby entanglement of the internal and spatial modes of the system can result in reduced variances for certain projections of the macro- scopic spin Notably, the dynamic phase modulus technique relies solely upon the solution of as few as five sets of coupled Gross-Pitaevskii equations This technique assumes that the wave functions ψ i (N1, N ; r, t) have moduli independent of (N1, N ), and phase linearly de- pendent on (N1, N ) about (N 1, N ), and has been shown to agree well with predictions based on simulating many thousands of sets of coupled Gross-Pitaevskii equations for the distributions of N i given in Eq (561) The experimental results of this thesis deal exclusively with bulk condensates (large This can be seen by showing the number operators N i = d3 r ψ i (r)ψ i (r) commute with the Hamiltonian in Eq (55) In 51, we incorporate the effects of collisional decay which do not preserve particle number These processes are ignored in the second-quantised formalism presented here

147 56 bloch vector density 131 N) The mean field formalism of 51 predicts the evolution of the expectation values of the spatially dependent densities and relative phase, and the macroscopic spin vector, to good approximation By using a single set of coupled gp equations for a two-component order parameter, we neglect the effects of quantum noise (of order N 1/ ), and take the single set of classical fields (ie those that obey single-particle quantum mechanics) to describe the desired expectation values1 We assume that our initial state is a coherent state, whereby all atoms are in the same superposition of 1 and For large N, the distribution of Fock states that comprise a coherent state becomes extremely narrow, suggesting it may be sufficient to consider the evolution of the single most dominant Fock state with N i = N i 56 Bloch vector density Just as any two-component spinor can be represented by a Bloch vector, an equivalent order parameter to the two-component complex pseudospinor Ψ(r) is the Bloch vector density, or spin density, a real three-vector p(r) Its components are defined here as Ψ (r)σ µ Ψ(r) p µ (r) =, µ {x, y, z} Ψ (r)ψ(r) (565) With this definition, the spin density has a position-independent unit-norm, and the following position dependent properties: the transverse angle ϕ(r) of the Bloch vector density is the relative phase between Ψ1 (r) and Ψ (r), ie tan θ(r) = p x (r)/p y (r) the longitudinal angle θ(r) is the related to the normalised relative density via cos θ(r) = n1 (r) n (r) = pz (r) n1 (r) + n (r) (566) The total spin is the spatially integrated spin density, weighted with the total density n(r) = n1 (r) + n (r) = Ψ (r)ψ(r): Pµ = (N/) 1 S µ = = Ψ (r)σ Ψ(r) d r µ n(r) p µ (r) d3 r n(r) d3 r 3 (567a) (567b) 1 The ansatz that quantum noise plays a negligible role in the evolution of the bulk condensates considered in this thesis is shown to be appropriate in Chapter 6, where classical effects / single-particle dephasing far outweigh the effects of phase diffusion This is due not only to large atom numbers, but the similarity of the inter- and intra-state scattering lengths A similar vector spin density is used, and its equation of motion derived, in Reference [164]

148 two-component bec : theory 13 For example,3 S z N1 N n(r)pz (r) d3 r Pz = = = N N n(r) d3 r (568) Although Pz { 1, 1}, the length of P can be less than unity Spatial inhomogeneity in the relative phase ϕ(r) or the relative density cos θ(r) cause a decrease of the transverse spin component As such, we identify a net relative phase ϕ, net polar angle θ, and transverse spin projection P defined via Equivalently, P cos ϕ P = P sin ϕ cos θ P tan ϕ (569a) Px + Py, (569b) Px, and Py (569c) cos θ Pz (569d) An important result, which may seem intuitively obvious, concerns rotations of the pseudospin density A spatially independent rotation of p(r) enacts the same rotation upon P To prove this, we express consider any rotation operator R acting on p(r) as a linear transformation, which can be expressed as a matrix with elements R µα The Bloch vector density after the rotation is p µ (r) = α R µα p α (r), and the total spin after the rotation is Pµ = N 1 R α 3 µα p α n(r) d r = R µα (N 1 α p n(r) d r) = R α 3 µα Pα (57) In 5, we saw that free evolution and coupling result in rotation of the Bloch vector The evolution of p(r) under these processes causes the same rotation to P, which we will make use of in the next Chapter For elongated condensates, it is useful to consider the spin projection along the direction of weak confinement (the axial direction, z) The distribution of longitudinal spin projection along the axial direction can be defined in terms of the linear densities n i (z) = πρ n(ρ, z) dρ: pz (z) = n1 (z) n (z) n1 (z) + n (z) (571) 3 Note that ħ (not ħ/) is present in the pre-factor, since the composite states of the pseudospin- 1 condensates we consider have S z j = ±( 1) j ħ j

149 56 bloch vector density 133 As in the case of the spin density p µ (r), the axial distribution of longitudinal spin projection is related to the total spin by weighting with the corresponding atomic density n(z) = n1 (z)+ n (z): Pz = n(z)pz (z) dz n(z) dz (57) Although the quantity pz defined in Eq (571) can be used to find the total spin in Eq (57), it is not obvious that pz (z) is physically meaningful in terms of a spin field This depends on how uniform the spin density is in the radial direction A necessary condition for radial spin uniformity is the uniformity of relative phase in the radial direction This, and how the linear spin density can be used to recover the relative phase, is discussed further in Section 65 and Appendix D We use this representation in Chapter 6 to picture the spatially dependent relative phase evolution of a pseudospin- 1 condensate

150

151 Chapter 6 Relative phase evolution of a two-component bec This chapter describes the spatiotemporal evolution of the density and relative phase of a two-component Bose-Einstein condensate, discussing the results of [9] in detail Section 61 is a brief overview of previous relative phase measurements in two-component quantum gases, supplementing the historical review of pseudospin- 1 condensates in Chapter 1 Observations of spatially inhomogeneous phase evolution, and the effect of this evolution on interferometry of a two-component bec are discussed in 6 Analytic estimates of the rel- ative phase evolution with regard to cold collision shifts are discussed in 63 Techniques of reversing the phase inhomogeneity are discussed in 64 Finally, in 65, we propose a combination of the dual state imaging method and Ramsey interferometry to directly image the spatial inhomogeneity and quantum fluctuations of relative phase 61 Historical development Knowledge of the phase of matter waves is crucially important in interferometry, entanglement, and precision measurement Pertinent to all of these areas is the pseudospin- 1 conden- sate, which can be realised with trapped neutral atoms in a superposition of two hyperfine ground states The states most commonly used in experiments to date are the magnetically trappable F = 1, m F = 1 and F =, m F = +1 states of 87 Rb [, 3, 5, 185, 8] The differ- ential Zeeman shift of these states has a local minimum at a magnetic field of 33 G, and is the least sensitive to magnetic field variations of all pairs of states in 87 Rb 5 S1/ above 138 G More recently, the optically trappable F = 1, m F = +1 and F =, m F = 1 states have garnered attention as a pseudospin- 1 system, as the two-body collisional interaction between the different states is tunable at low magnetic fields Early experiments using the former set

152 136 relative phase evolution of a two-component bec of states have studied: spatial separation of the components [], and interferometric detection of relative phase for a region where both components remained overlapped following strongly damped centre of mass motion [3], the behaviour of a phase winding throughout a two-component condensate driven by continuous electromagnetic coupling [5], the creation of novel vortex states using spatially dependent coupling of the two levels [4], spin excitations in an uncondensed ultracold gas [168, 186] and spin domain growth for mixtures of condensed and uncondensed atoms [185], other vortex experiments [33, 34], and interference between two vortex lattices comprised of each component [13], long-lived ringlike excitations of the binary condensate system [8] The dynamical instability of the latter is accompanied by spatially dependent relative phase dynamics, which we investigate here In particular, we consider the temporal decay of the interference signal obtained with a Ramsey-like measurement of the pseudospin-1 condensate The mechanism of phase diffusion for a two-component quantum gas has been recently studied [47], whereby the evolution of a coherent spin state results in the decay of Ramsey visibility [7] For the close inter- and intrastate interaction strengths in our system, phase diffusion is negligible [7] Rather, we consider mean field driven spatial inhomogeneities of the relative phase, which also act to decrease the interferometric contrast, even before the components have spatially separated This is relevant for proposals to squeeze the macroscopic pseudospin in two-component quantum degenerate gases [187, 188, 189, 19], which are based on using the mean field interaction as a source of entanglement, and development of a trapped atomic clock using an atom chip [191, 134] 6 Ramsey interferometry with a pseudospin-1 bec We use Ramsey interferometry to study the relative phase evolution of a two-component condensate The nonequilibrium dynamics observed in this system [8] are accompanied by spatially dependent dynamics of the relative phase We describe the evolution of the system using a pseudospinor formalism, whereby the internal and external states of the condensate

153 6 ramsey interferometry with a pseudospin- 1 bec Ψ ( r, t) Ψ ( r, t) = n (r ) coupling Π i n (r ) T 137 n ( r ) e iϕ ( r) n ( r ) e iϕ ( r) Π n' ( r ) e iϕ' ( r ) n' ( r ) e iϕ' ( r ) time Figure 61 Schematic of pulse sequence for Ramsey interferometry of a pseudospin- 1 bec, and evolution of its order parameter The condensate, initially in the ground state of 1, with density n (r), is put into a superposition of both spin states by a π/-pulse [Eq (63)]; the wave functions of each state have the form of n (r) Following the first coupling pulse, this state evolves during time T to a state described by Eq (64a) The relative phase of this state, ϕ(r) = ϕ (r) ϕ1 (r), is converted into relative population, locally, by the second π/-pulse This state is described by Eq (64b), and is related to the pseudospinor before the second pulse by Eq (64) are represented by a two-component order parameter [as in Eq (11) and 54] Ψ1 (r, t) n1 (r, t) e iϕ1 (r,t) =, Ψ(r, t) Ψ (r, t) n (r, t) e iϕ (r,t) (61) where n1 and n are the atomic densities of each state in the condensate We begin with a condensate which exclusively occupies state 1, and in the ground state of the combined mean field and external potentials, with density n (r) The corresponding pseudospinor representation is 1 Ψ(r, ) = n (r), (6) Application of a π/-pulse (of length t π/ ) prepares the two-component superposition1 1 1 n (r) Ψ(r, t π/ ) = i (63) This is no longer the ground state of the two-component system, as the mean field interaction between component 1 with component, and component with itself is different to that of component 1 alone This is due to a slight difference in the s-wave scattering lengths 1 This assumes the strong coupling limit, as discussed in 56 The densities of the initial superposition state in Eq (63), and the two-component ground state for the present parameters, are shown in Fig 16

154 138 relative phase evolution of a two-component bec a11 = 14 a, a = 95 a, and a1 = 9766 a, where a is the Bohr radius [8] Through the state inter-conversion, we modify the mean field energy of the system by 1% for our experimental parameters and with a condensate with atoms3 This drives hundreds of milliseconds of weakly damped collective excitations and coherent relative phase evolution The system is allowed to evolve for a time T, after which we observe the condensate without or with the application of a second π/-pulse: n1 (r) e iϕ1 (r) Ψ(r, t π/ + T) = n (r) e iϕ (r) n1 (r) e iϕ 1 (r) Ψ(r, t π/ + T + t π/ ) = n (r) e iϕ (r) ; (64a) (64b) where primes denote the densities and phases after the optional second pulse In this Chapter, we take the spatially dependent relative phase ϕ(r) ϕ (r) ϕ1 (r) (65) to be the relative phase prior to the second π/-pulse, which we seek to measure In the frame rotating at the angular frequency of ωmw ± ωrf,4 this phase evolves to ϕ(r) = T + ϕmf (r, T) π, (66) where is the effective detuning of the two-photon drive during the evolution time 5 T, and ϕmf is the spatially dependent phase whose evolution is driven by the mean field In Fig 6, the simulated densities of each component are shown for various evolution times following preparation of the initial superposition in Eq (6) The relative phase begins to vary across the axial dimension of the condensate This relative phase modulation is associated with a spatial separation of the components along the axial direction, consistent with the relative velocity of the components given by vi (r) = ħ ϕ i (r) m (67) 3 Interestingly, this amounts to a decrease in energy compared to the initial state in Eq (6), as computed using Eq (548) The superposition state in Eq (63) is still out of equilibrium however, as its energy exceeds that of the two-component ground state for equally populated components 4 The sign in the linear combination of the two radiation frequencies depends on the transition being used 5 In general, this is not the same as the effective detuning during the coupling pulses, due to the nature of the two-photon transition This leads to unique features of Ramsey interferometry performed using states coupled via a two-photon transition, as discussed in 364

155 6 ramsey interferometry with a pseudospin- 1 bec ms 45 ms ms 1 Φ Π Π 5 5 axial position (µm) Figure 6 Simulated column densities n i = n i dx and relative phase ϕ(ρ =, z) [see Eqs (64) and (65)] for various evolution times of the initial state in Eq (63) The initially uniform phase exhibits inhomogeneous evolution (blue, solid), accompanied by density excitations in the axial direction z An offset of 14 π rad s 1 has been subtracted from the relative phase for clarity ( = π 7 Hz) The analytic estimate of the axial relative phase evolution (red, dashed), discussed in 63, is proximate to the cgpe simulated relative phase during the first 1 ms The aspect ratio of the density images has been scaled to mimic the shape of the condensate after ms of ballistic expansion, when the condensate is wider in the direction that it was more tightly confined This is evident in Fig 6 at 9 ms, where state 1 has expanded and become wider than state, which is compressed along the axial direction In the absence of loss, the moments z n i d3 r would continue to oscillate out of phase, with occupying a larger volume than 1 at ms For a condensate with atoms, atomic loss dampens the collective oscillations on this time scale With fewer atoms, we have observed the full oscillatory behaviour of the collective excitation (the consequence of this will be discussed in 64) The relative phase remains relatively uniform in the radial direction, owing to the large energetic cost of a phase gradient along this axis The radial trapping frequency f ρ = 976 Hz, is typically an order of magnitude larger than the energy of the collective excitation

156 relative phase evolution of a two-component bec 14 Applying a second π/-pulse with a phase shift of δϕ yields a Ramsey interference fringe upon variation of, T, or δϕ, and measurement of the population or density of either state We present measurements of the density of state for various evolution times T, followed by a second π/-pulse The absorption imaging process ( 43) results in a measurement of the column densities n i of each state [Eq (D)], and n is shown in Fig 63 We observe striking agreement between these results and those of cgpe simulations The only free pa- rameter used in the simulation is the detuning, in Eq (51)6 In particular, the three distinct density maxima at 4 and 115 ms, corresponding to antinodes of the state wave function, are represented accurately by the simulation This prominent density modulation along the z-axis (axial direction, weak confinement) reveals the nonuniform relative phase variation along this direction This is enabled by the second π/-pulse, which converts spatial variations of the relative phase into spatial variations of the atomic density of each state ( 56) In the absence of the second pulse, while density modulation still exists, it is of lower contrast (Fig 6, 9 ms is the most structure we observe from simulations) We attribute residual differences in the precise spatial structure of the measured and simulated macroscopic wave functions to the finite resolution of our imaging system, photon recoil blurring during the imaging pulse, imperfect cylindrical symmetry of the trap, spatial inhomogeneity of the coupling pulses ( 45), and interaction of the condensate with an undetectable yet significant thermal component for longer times The repeatability of the observed interference fringes (both run-to-run and day-to-day) is indicative of the well-defined initial-state preparation and phase evolution in this system 61 Net interference signal We quantify the Ramsey interference by first using the net longitudinal pseudospin projection after the second π/-pulse, scaled to vary as the normalised population difference [Eq (568)] Pz = (N1 N ) /N, (68) where N i = n i d3 r, (i = 1, ) and N is the total atom number A typical Ramsey signal obtained by varying the evolution time T is shown in Fig 64 The Ramsey fringes observed in the time domain are consistent with the spatial evolution of a coherent two-component order parameter, simulated with the cgpes Using the Bloch vector density formalism of 56, in concert with the results of Section 6, we arrive at the following equalities for the 6 The external potential is well characterised, as in 41, and typically has trapping frequencies f ρ = 976() Hz and f z = 1196() Hz

157 6 ramsey interferometry with a pseudospin- 1 bec y μm z Figure 63 Column density of state as a function of Ramsey interferometry time (inset for each image in milliseconds) Application of the second π/-pulse locally converts the relative phase into relative population (Top) Experimental: single shot absorption images, taken after a ms fall time, reveal the mean field driven phase evolution (Bottom) Theoretical: column density plots obtained by solving the coupled Gross-Pitaevskii equations show overall agreement with experimental results We observe similar agreement for state 1

158 relative phase evolution of a two-component bec 14 1 Pz time ms Figure 64 The Ramsey interference signal with a pseudospin bec of Rb atoms, where the effective detuning of the two-photon field during the evolution time is = π 41 Hz The points are experimental measurements and the solid line is calculated using the cgpe theory, with an additional exponential decay due to decoherence The 8 Hz oscillation in Pz is due to the combined effect of the detuning and a net mean field driven phase which evolves at 13 Hz The envelope (dashed) indicates the instantaneous fringe visibility, P (T) in Eq (69) The decay of P is predominantly due to the spatially dependent mean field relative phase evolution interference signal Pz = N 1 = N 1 = N 1 = N 1 = N 1 z p (r) n(r) d r (cos δϕ p (r) sin δϕ p (r)) n(r) d r n(r) sin [θ(r)] sin [ϕ(r) δϕ] d r n(r) 1 p (r) sin [ϕ(r) δϕ] d r n (r)n (r) sin [ϕ(r) δϕ] d r Im {e ψ (r)ψ (r) d r} = N 1 3 y 3 x 3 1 i δϕ 3 z 3 1 = P (T) sin [ ϕ(t) δϕ], 3 (69a) (69b) (69c) (69d) (69e) (69f) (69g) where P is the fringe visibility, equal to the net transverse pseudospin magnitude immedi- ately prior to the second π/-pulse The integrand in Eq (69e) suggests that high contrast interference requires both good overlap of the wave functions [uniformity of n1 (r)n (r)] and

159 6 ramsey interferometry with a pseudospin- 1 bec 143 a homogeneous relative phase [uniformity of ϕ(r)] The decay of Ramsey contrast we observe is primarily due to the spatial dependence of the mean field driven phase Furthermore, this decay happens significantly faster than the spatial separation of the components We emphasise that uncommon spatial modes of the components are neither necessary nor sufficient for loss of interferometric visibility Consider, for example, the ground state of the two-com- ponent system, whose density consists of a central feature of state, enclosed by two lobes of state 1 (This state is shown for the trapping potential used here, and N = atoms in Fig 16b) While the spatial overlap seems poor, the initial Ramsey visibility is 8% In the absence of losses, no inhomogeneous relative phase evolution occurs for the ground state, and the visibility would not decay In reality, atomic loss from each component leads to a changing mean field interaction, resulting in time-dependent ground state wave functions Our simulations suggest that the condensate follows the time-dependent ground state nearadiabatically Imperfect response of the condensate wave functions to the time-dependent mean field interaction results in a small inhomogeneity of the relative phase, and a subsequent yet slow decay of the interferometric contrast (Fig 16b) Decoherence contributes secondarily to the observed decay of Ramsey visibility Simu- lations of the cgpes yield the mean field limited visibility, VGP (T) [P (T) in Eq (69g)], consistently higher than the visibility we measure When fitting a simulated interference fringe to the experimental data (Fig 64), we impose an additional exponential decay on the visibility, ie V(T) = VGP (T) e T/tc [cf Eq (8b)] to account for decoherence and dephasing from technical sources and quantum phase diffusion We infer a decoherence time of 37 ms for the data in Fig 64 The net relative phase of the condensate prior to the second π/-pulse [the phase of the Ramsey interference signal, Eq (69g)] is equal to [cf Eq (79)] ϕ(t) = ϕmf (T) + T π, (61) where ϕmf (T) is the net mean field phase which drives the precession of the net pseudospin As indicated by Eq (61), a non-zero fringe frequency may result from varying the evolution time T, even when the effective detuning is zero Despite the complex nature of ϕmf (r, T), n i (r), and a loss of atoms from the condensate, the net mean field phase ϕmf (T) remains linear in time for T 1 ms (see Fig 61) The frequency of the mean field phase evolution decreases slightly, as the number of atoms in the condensate decays, leading to a decrease, or increase in the fringe frequency, depending upon the sign of For the data presented here, however, the chirp is less than one might expect from intuitive reasoning based on the decreasing mean field interaction We return to this peculiarity in Section 63, where an estimate of the net mean field phase is presented

160 relative phase evolution of a two-component bec Local spin projection The relative phase evolution of a pseudospin- 1 condensate can also be analysed using the local spin projection We observe an axially dependent Ramsey interference signal, by mea- suring the local longitudinal spin projection pz (z), introduced in Eq (571) An absorp- tion image of the condensate yields the column densities n i (y, z) [Eq (D)], which can be integrated along y to obtain the line densities n i (z) [Eq (D1)] used to compute pz = (n1 n )/(n1 + n ) The integration of density inherent to absorption imaging, and the subsequent integration of the absorption image along y does not corrupt the Ramsey in- terference signal, because the relative phase is almost uniform along the radial dimension, as previously noted An example is shown in Fig 65, along with the local spin projection obtained from a simulation of the cgpes This presentation of the data provides a direct way to visualise the axially dependent evolution of the relative phase Lines of constant pz in Fig 65 can loosely be thought of as wavefronts of relative phase,7 whose curvature increases with the phase variation across the condensate Horizontal cross sections of this plot (fixed z) correspond to local Ramsey interference fringes We obtain an analytic estimate for the position dependent frequency of the Ramsey interference fringes in the next Section When the net Ramsey interference signal is fit using the results of a cgpe simulation, there is ambiguity as to the value of In Fig 64 for example, two values of the detuning yield identical evolution of the net relative phase, and hence the same Ramsey interference curve These values, = π 41 Hz and = π 145 Hz, can be distinguished by their effect on the spatially dependent spin projection following a Ramsey interference experiment The simulated axially dependent spin projection is shown for both in Fig 65 Finally, we note that the mean of the two detunings for which the simulated net Ramsey signal matches the experimental data can be used to infer the contribution of ϕmf to ϕ In this case we have d ϕmf π ( ) = π 13 Hz dt (611) Phase and frequency domains The relative phase evolution can also be probed using Ramsey interferometry in the phase and frequency domains So far we have discussed temporal Ramsey interference, whereby the evolution time T is varied between the π/-pulses This yields a complex interference sig- nal, from which it is difficult to determine the instantaneous net relative phase ϕ(t) and the interference visibility V(T) [Eq (69)] By fixing T, and varying the phase of the second 7 While the densities remain similar, the polar angle of the local spin is θ(r) π, and then p z (r) sin ϕ(r) The axial distribution of spin projection, measured with absorption imaging is similarly related to the relative phase p z (z) sin ϕ(z) under certain circumstances discussed in Appendix D

161 6 ramsey interferometry with a pseudospin- 1 bec 145 Pz 1 (a) z µm 1 1 (b) 3 z µm 1 1 (c) 3 z µm time ms Figure 65 Axially dependent spin projection during Ramsey interferometry of a pseudospin- 1 bec In each plot, the local spin projection pz = (n1 n )/(n1 + n ), computed using the line densities n i (z) = n(z) = πρ n(ρ, z) dρ [Eq (571)] (a) data, whereby line densities are obtained by once-integrating the column densities, (b) cgpe simulation with /π = 41 Hz, and (c) cgpe simulation with /π = 145 Hz Both detunings yield identical Ramsey fringes when averaging over the entire condensate (Fig 68), but the sign of the detuning affects the curvature of the local spin-projection in the t z plane (see text)

162 relative phase evolution of a two-component bec ϕ 1 Pz Π Π δϕ 3Π Π Figure 66 Ramsey interferometry in the phase domain After a fixed evolution time T = ms, the second π/-pulse is applied with a variable phase shift δϕ This results in an interference signal given by Eq (69) when averaged over the entire condensate Such a measurement yields the instantaneous fringe visibility V(T) = 76(1), and the net relative phase ϕ(t) = 86(1)π The raw absorption images for two phase shifts are shown (inset, δϕ π and δϕ π) The densities of both states are mirrored between these different π/-pulses, whose phase shift differed by π π/-pulse δϕ, or the detuning, one can obtain a Ramsey signal in Pz which is sinusoidal and provides the values of ϕ(t) and V(T) at a particular evolution time, thereby measur- ing the instantaneous net transverse pseudospin of the condensate This is useful to quantify the mean field dephasing and decoherence of the system An example of phase shift interferometry is shown for ms of evolution in Fig 66 Interferometry in the phase domain has the advantage that it does not affect the evolution of the system, as might varying the detuning For the spin-echo experiments in Section 64, phase-based interferometry is the only way in which the net transverse pseudospin can be probed, as the phase shifts induced by an off resonant coupling drive are annulled by the nature of the pulse sequence Spatially resolved Ramsey interferometry in the phase domain is a form of phase tomography of the condensate Applying the second π/-pulse with various phase shifts amounts to rotating the local spin projection p(r) around various angles in the x y plane By imaging the longitudinal spin projection pz (r) for multiple rotation angles, one can then reconstruct p(r) before the π/-pulse Absorption imaging may corrupt this reconstruction, as the imaging process inherently integrates along one dimension This is not a problem in elongated trapping geometries, where the dominant relative phase evolution occurs along the axial direction (Appendix D) For cylindrically symmetric potentials that are less elongated, the Radon transform might be used to recover the full three-dimensional dependence of pz (r) [19, 193] We introduce another form of interferometric phase imaging in 65

163 63 collisional shift analysis of relative phase Collisional shift analysis of relative phase A qualitative way of interpreting the phase evolution of a pseudospinor condensate is via the density dependent collisional shift, the same shift that limits atomic clocks, such as fountain clocks which use cold thermal atoms The cold-collision shift in a trapped ultracold gas of 87 Rb has been measured using Ramsey interferometry [194, 168], by spatial binning of the Ramsey signal The detunings inferred from the local Ramsey interference signal provided a measure of the spatially dependent collision shift; deconvolving this spatial dependence with that of the density indicated the density dependence of the collisional shift This density dependence is ν1 = ħ (α a n + α1 a1 n1 α11 a11 n1 α1 a1 n ), m (61) where α i j, i, j {1, } depends upon the exchange symmetry of the collision partners For Bose-condensed atoms of the same species, α ii = 1, whereas α ii = for noncondensed, indistinguishable bosons [195, 196] There has been a long standing ambiguity in the literature about the coefficient for unlike collision parters (sometimes referred to as the factors of controversy); the value of α1 [148, 149, 197, 198, 199, 194, ] In particular, the effect of coherence on α1 led to a paradox involving the Ramsey interference of a de- cohering ensemble of thermal atoms prepared in a superposition of states 1 and In Reference [194], it was verified that α1 = for coherent two-component superpositions of noncondensed 87 Rb atoms Furthermore, it was suggested that should α1 = 1 for incoher- ent two-component mixtures, the collisional shift of a decohering sample of thermal atoms would be dynamic, manifest as a chirp in the frequency of Ramsey fringes in the temporal domain [194, 1] Such a chirp was not observed, and this seeming paradox has been resolved [,, 3, 4, 5] These factors are also relevant for the interaction terms in the many-body Hamiltonian Eq (55), and the nonlinear mean field terms in the coupled gpes Eqs (563) and (51), and consequently, all the results that follow from these equations such as those in 5 Less has been written about the value of α1 for condensed bosons, but should α ii = 1 for a condensate (as demonstrated in [194]), our observations of mean field dynamics and Ramsey interference are only consistent if α1 = 1 for 87 Rb atoms in a coherent pseudospinor condensate Throughout this thesis I have taken α i j = 1 for all i, j {1, } In Reference [185], it is suggested that the mean field relative phase evolution is deter- mined by8 dϕmf = π ν1 dt (613) 8 This is equivalent to the Thomas-Fermi approximation with the ansatz of static densities, ie it can be derived by setting, Ψi = n i e i ϕ i and n i = in Eq (51)

164 148 relative phase evolution of a two-component bec This relationship has motivated several proposals to obviate the collisional shift in a pseudospin-1 condensate, in the hope of mitigating the effect it has on Ramsey interferometry Kadio and Band [6] suggest preparing the two-component system with the relative population difference which makes ν 1 vanish in Eq (61), n n 1 = α 1a 1 α 11 a 11 α 1 a 1 α a (614) This population imbalance could be controlled using a coupling pulse of a specific duration, however, as noted by Kadio and Band, should α 11 = α = α 1 (as proposed here), the background values of the scattering lengths in the 1, 1, +1 pseudospinor condensate of 87 Rb result a negative expression for the requisite population ratio in Eq (614), n /n 1 = 13 Lewandowski, et al demonstrated the cancellation of the cold-collision shift by using the quadratic differential Zeeman shift of the two levels in a magnetically trapped thermal cloud of 87 Rb atoms Rosenbusch has also investigated the dephasing mechanisms relevant to ultracold magnetically trapped atoms [134] We propose that Eq (613) does not describe the evolution of relative phase of a pseudospin condensate To verify this claim, consider the two-component ground state, as discussed in 53 and 541 By definition, the two-component ground state is a stationary state of the system, with a relative phase that is spatially uniform, and which remains so under free evolution The value of ν 1 for a two-component ground state is not, in general, spatially uniform 9 Were the relative phase to develop spatial inhomogeneity by way of Eq (613), the spatial modes of each state would not be stationary, in direct contradiction with the nature of the ground state Moreover, for the initial state prepared using a π/-pulse in Eq (63), we have seen there is very little spatial inhomogeneity of the relative phase in the radial direction The phase is almost uniform radially, as the energy of the excitation inherent to the state in Eq (63) is less than the radial trapping frequency On these grounds we emphasise that density inhomogeneity alone does not induce relative phase inhomogeneity Despite the quantitative limitation of Eq (613), it can be used to estimate how the relative phase evolution scales with density and atom number, by making two phenomenological amendments The first is the static single mode approximation, ie the densities of each component are commensurate and stationary This ignores the relative motion of each component that results from gradients in the relative phase [Eq (15)] To attain analytic results, we approximate the density of the initial state in Eq (63) with that of the single-component Thomas-Fermi ground state [Eq (5)] The second amendment is to assume a priori that Eq (613) only describes the relative phase evolution along the axial direction To this end, 9 For the ground state plotted in Fig 16b, ν 1 varies by 4 Hz across the condensate

165 63 collisional shift analysis of relative phase 149 frequency Hz axial position µm 4 Figure 67 Axially dependent Ramsey fringe frequency of the data in Fig 65 A cross section (fixed z) of the local spin projection in Fig 65 yields a local Ramsey interference fringe, whose frequency is fit with the analytic estimate in Eq (617) For these data, the detuning used was = π 41 Hz (blue, dashed), which in combination with the axially dependent mean field phase, results in the net relative phase evolution at a frequency of 8 Hz (red, dashed), shown in Fig 64 The mean error of the fitted frequencies is 33() Hz we replace the densities n i in Eq (61) with the radially averaged densities πρ n i (ρ, z) dρ n i (z) = πρ n i (ρ, z) dρ (615) This enforces the relative phase to be fixed in the radial direction, whilst accounting for the different density across the radial dimension of the condensate at each axial position The radially averaged density for a single-component Thomas-Fermi ground state is n(z) = ntf z [1 ( ) ], 3 ztf (616) where ztf is the axial Thomas-Fermi radius of the single-component ground state, and ntf is the peak total density in the Thomas-Fermi approximation1 The radially averaged densities of the initial two-component superposition in Eq (63) are approximated by n i (z) = n(z) /, which gives dϕmf πħ = (a11 a ) n(z) dt m z 4πħ = (a11 a ) ntf [1 ( ) ], 3m ztf 1 This can be confirmed by substituting Eq (5) into Eq (615) (617)

166 relative phase evolution of a two-component bec 15 Assuming linear evolution of the local relative phase (valid for short times), we have11 ϕmf (z) = ϕ [1 ( ϕ = z ) ], where ztf (618a) 4πħ (a11 a )ntf T 3m (618b) is the range of relative phase across the condensate at time T The net interference signal is attained by averaging over the entire condensate [cf Eq (69f)], Pz (ϕ ) = P (ϕ ) sin ϕ(ϕ ), where P e i ϕ = N 1 = 15 n(z)e 5/ 64ϕ i(ϕmf (z) π/) (619a) dz (619b) { [3 4ϕ (ϕ + i)] e i(ϕ +π/4) erf (e iπ/4 ϕ ) (3i + ϕ ) ϕ } The visibility and net relative phase are the modulus and phase of the above expression, respectively, although it is difficult to ascertain how the phase depends on experimental parameters To estimate how the net relative phase scales with atom number, time, and trap frequencies, we make use of the results of 56 As the length of P is unchanged by a uniform local rotation of p(r), we choose a rotation about the z-axis which would result in Px = prior to the second π/-pulse The angle of rotation is precisely π ϕmf, which can be found by solving Px = N 1 cos [ϕ mf (z) ϕmf + π/] n(z) dz = (6) The resulting analytic expression for the net mean field phase scales approximately linearly with ϕ, and we use d ϕmf 6 = dϕ ϕ = 7 (61) to estimate the rate of change of the net mean field phase d ϕmf d ϕmf dϕ = dt dϕ dt 8πħ = (a11 a ) ntf 7m (6) 11 A superposition composed of an unequal number of atoms in each state can be accounted for by an additional term (a 11 + a a 1 )Pz in Eq (617), which we ignore as it is near vanishing for the scattering lengths considered here

167 63 collisional shift analysis of relative phase 151 With reference to the experimental conditions considered in this Chapter, the net relative phase scales as /5 fρ d ϕmf N fz = π ( ) ( ) dt Hz 976 Hz 4/5 133 Hz (63) These analytic estimates can be compared with the measured relative phase evolution and with the simulated relative phase using the cgpes To this end we make the following comparisons 1 Axially dependent Ramsey fringe frequency An empirical analysis of the Ramsey interferometry data in Fig 65 can be performed, similar to that of Reference [168] We observe an axially dependent Ramsey interference signal, by measuring the local longitudinal spin projection pz (z), introduced in Eq (571)1 In doing so, we attain a fringe frequency for each axial position along the condensate, plotted in Fig 67, that is consistent with Eq (617) Axially dependent relative phase evolution The analytic estimate of the relative phase evolution along the axial coordinate, Eqs (618), can be compared with results of the cgpe simulations In Fig 6, the simulated relative phase evolution (blue) is compared with the analytic estimate (red, dashed), and they are in good agreement during the first 1 ms of evolution 3 The net relative phase and visibility calculated from the cgpes agree with Eqs (619) to within % during the first 5 ms of evolution, at which point the range of relative phase across the condensate is ϕ = 3π/ The linear phase approximation of Eq (63) is a closer estimate still of the net mean field phase, which contributes 13 Hz to the frequency of the Ramsey fringe in Fig 64 Curiously, the above comparisons are most favourable when decay of the condensate is not accounted for in the analytic estimate One could include the time dependence of N (computed from the cgpes, or measured experimentally) in Eq (617) for the mean field phase evolution, which depends on the peak total density ntf N /5 After 1 ms, for example, when 68% of atoms remain in the condensate, the net relative phase is predicted to evolve at a rate 86% of its original value Such a chirp is not observed in experiments or cgpe simulations (which do account for losses) with the experimental conditions considered thus far This result does not hold for lower atom numbers, however, and we conclude that the predictive power of the analytic model presented in this Section is limited 1 An absorption image of the condensate yields the column densities n i (y, z) [Eq (D)], which can be integrated along y to obtain the line densities n i (z) (D1) used to compute pz (z)

168 relative phase evolution of a two-component bec Rephasing the nonlinear dynamics of a pseudospin- 1 bec We demonstrate the rephasing of spatially inhomogeneous relative phase evolution in a trapped two-component Bose-Einstein condensate Refocusing of the mean-field dephasing is achieved using the archetypal spin-echo sequence of radiofrequency and microwave pulses which couple the two-states of the pseudospin- 1 system The rephasing by a single echo pulse is imper- fect due to decay of the condensate and nonlinear dynamics of its spatial modes, the latter of which can suppressed using a Carr-Purcell sequence of multiple π-pulses 641 Spin Echo Hahn s celebrated result of rephasing the inhomogeneous broadening in nuclear magnetic resonance nmr [7] has been applied to myriad systems, including superconducting qubits [8], trapped ions [9, 1, 11], ultracold thermal atoms [1, 13], nonlinear optics [14] and ultrafast spectroscopy [15, 16] Strong interactions in spatially inhomogeneous systems, like that of a trapped pseudospin Rb condensate, have long been considered deleterious to the application of these systems to interferometry Proposals exist to counteract the mean-field dynamics in trapped condensate interferometers [6] The spin-echo technique has been applied to reverse manybody phase diffusion in two-component quantum gases [47], and it has been suggested as a means to probe magnetic dipole-dipole interactions in a spinor condensate [17] The prospect of squeezing the pseudospin of such systems makes use of these strong interactions to prepare non-classical spin states Whilst our experimental configuration may not be a candidate for observing spin-squeezing, the investigation of single-particle (mean-field) dephasing, and how it can be circumvented, is relevant to these studies Untwisting of the spatially inhomogeneous relative phase has been observed in a pseu- dospinor bec whose constituent states were continuously coupled by a driving field [5] Here we observe another form of rephasing of the condensate pseudospin, due to the spin echo sequence of pulses which actively reverse the mean-field inhomogeneities in the bec A temporal demonstration of the echo on the net pseudospin is shown in Fig 68 The experimental conditions are identical to those in Fig 64, for which the decaying Ramsey interference fringe is reproduced here for comparison At 3 ms, a π-pulse of 14 ms duration is applied, with the same phase as the π/-pulse that created the initial superposition We then allow the system to evolve for various times after the π-pulse, and observe the net longitudinal spin projection after a second π/-pulse The resulting interference fringe exhibits a recurrence of visibility, indicating the two-component order parameter has rephased The contrast of the revived fringe is greatest at approximately 6 ms, as expected from the archetypal spin-echo sequence

169 64 rephasing the nonlinear dynamics of a pseudospin- 1 bec Pz time ms Figure 68 Temporally resolved spin echo of a pseudospin 87 Rb bec A π-pulse applied at 3 ms (grey bar) partially reverses the nonlinear dynamics, and mean field dephasing of the condensate A revival of the interferometric contrast (red), characteristic of a spin echo, can be seen at 6 ms, with respect to the Ramsey interference signal without the π-pulse (blue, Fig 64) The points are experimental measurements and the solid line is calculated using the cgpe theory, with an additional exponential decay due to decoherence The observed spin-echo is described extremely well by simulations of the cgpes As in the case of Ramsey interferometry, 61, the simulated interference fringe must be supplemented by an additional exponential decay to attain agreement with the observed fringe visibility Whilst the mean-field dephasing is the dominant mechanism for the loss of interferometric contrast, an additional exponential decay is required to account for decoherence, viz V(t) = e t/t c VGP (t), where VGP (t) is the theoretically predicted Ramsey fringe visibil- ity The additional decay required to attain agreement between the observed and simulated fringes is significantly longer for the spin-echo experiment (t c 15 seconds) This suggests there is another form of dephasing, in addition to the mean-field dephasing, which is at least partially reversed by the echo pulse sequence It is unclear whether this additional dephasing is due to many-body phase diffusion, of a technical origin, or both A striking illustration of the spin-echo is shown in Fig 69, where the local spin-projec- tion following the second π/-pulse is shown The spatially resolved dephasing in Fig 65 is reversed by the π-pulse applied at 3 ms, directly observable here by a spatially uniform longitudinal spin projection after a π/-pulse applied at 6 ms Wavefronts of relative phase in the t z plane develop curvature during the first 3 ms of evolution, and are then refocused by the π-pulse After this rephasing, the local pseudospin begins to dephase along the condensate once more, similar to the inhomogeneity that occurs following the initial state

170 relative phase evolution of a two-component bec 154 Pz 1 (a) π pulse z µm 1 (b) 3 1 π pulse z µm time ms Figure 69 Axial spin projection during spin-echo of a pseudospin- 1 bec In each plot, the local spin projection pz (n1 n )/(n1 + n ), computed using the line densities n i (z) = n(z) = πρ n(ρ, z) dρ (a) data, whereby line densities are obtained by once-integrating the column densities, (b) cgpe simulation with /π = 41 Hz A π-pulse applied at 3 ms results in rephasing of the spatially inhomogeneous phase, shown in Fig 65 Optimal rephasing occurs at 6 ms, indicated by a uniform phase of the Ramsey fringe across the condensate preparation The efficacy of the echo is limited, ie the rephasing is imperfect, due to irreversibility of the nonlinear spatial dynamics, and decay of atoms from the condensate We investigate the nature and limitations of the symmetric π/ T/ π T/ π/ pulse sequence, and supplement this pathology of the spin echo with a regular Ramsey interferometry sequence, π/ T π/ For each T, we perform interferometry in the phase domain (Section 63) to determine the instantaneous net relative phase and fringe visibility The measured fringe visibility with and without a π-pulse applied at T/ are shown in Fig 61 for up to 3 ms

171 64 rephasing the nonlinear dynamics of a pseudospin- 1 bec (a) 1 (b) 6 Φ Π visibility T ms T ms 15 Figure 61 Net transverse spin projection (fringe visibility) and relative phase during spin-echo (red) and conventional Ramsey interferometry (blue) experiments, probed using phase-shift interferometry (Section 63) All spin-echo data corresponds to the symmetric pulse sequence, in which the π-pulse was applied at T/ (a) The fringe visibility improves dramatically for the spin-echo sequence, but is limited by the imperfect rephasing of the nonlinear mean field dynamics The mean field limited visibility [VGP (T), dashed] is inferred from the cgpe simulations The data is in qualitative agreement with the results of the simulations once decoherence has been accounted for via V(T) = VGP (T) e T/tc (solid; see 61) The ratio between the echo and Ramsey visibilities is 5 at 15 ms (b) The net relative phase for the Ramsey sequence is ϕ(t) = ϕmf (T) + T π/, and we infer a detuning of 178(1) Hz for these data The net relative phase of the echo signal is echo ϕ(t) = ϕecho mf (T) + π/, where ϕmf (T) is the residual mean field phase not cancelled by the echo sequence of evolution For short times, the spin echo results in a significant return of interferometric contrast, whereas near 3 ms, the echo has little to no effect This would remain true in the absence of any dephasing or decoherence mechanism other than the mean field dynam- ics, shown by the results of the cgpe simulations with no additional decay (dashed lines in Fig 61) This indicates the rephasing is, in part, limited by the mean field; the rephasing that occurs after the π-pulse is different to the dephasing that occurs before the π-pulse This asymmetry is due to structure in the spatial modes and atomic decay, causing the condensate to experience a different mean field interaction during the second half of the pulse sequence The experimental spin-echo observation is in qualitative agreement with the theoretical predictions Both exhibit a maximum in the ratio of the echo and Ramsey visibility of 5 at approximately 15 ms There remains some discrepancy due to the finite duration of the π-pulse, and the possibility that the pulses used in these data were not optimal, both due to

172 156 relative phase evolution of a two-component bec imperfect pulse area13 and spatially inhomogeneous coupling, discussed in 45 The imperfect rephasing following the π-pulse is accompanied by a nontrivial net phase shift In an ideal spin-echo experiment, the net phase following the evolution time (at t = t π/ + T/ + t π + T/) is ϕ = ±π/, depending on the phase of the π-pulse We observe a different net relative phase due to the asymmetric evolution before and after the π-pulse, which depends upon the evolution time Were the coupling pulses imperfect, a net relative phase shift other than ±π/ could result, however it would be oscillatory with the evolution time Moreover, the net phase shift we observe is consistent with mean-field simulations of the spin-echo experiment We extract the net relative phase using the fringe attained from phase-based interferometry, shown in Fig 61 The decrease in the net relative phase is a result of less mean field phase being acquired during the second half of the evolution time Echo interferometry offers the possibility of measuring phase shifts of interest which can be modulated in time Any spatially uniform differential energy shift of states 1 and that remains constant throughout the experiment will be cancelled by the echo, and no fringe will result in the frequency or time domains To acquire a fringe with the echo experiment, the phase shift must be asymmetric about the π-pulse A simple example of modulation which would result in a fringe is an instantaneous change of the detuning following the π- pulse The utility of such a measurement requires that Vecho (T) > VRamsey (T/) The echo interferometry probes only that phase acquired during half of the total evolution time The same phase shift could be measured using Ramsey interferometry with an evolution time T/, and for an improvement in the signal to noise ratio, the visibility of the echo signal at T must be greater than that of the aforementioned Ramsey signal In any case, the spurious net mean field shift must be considered when interpreting phase shifts associated with echo interferometry of pseudospin condensates The efficacy of the spin echo can be improved by the application of multiple π-pulses during the evolution time, as shown in Fig 611 We measure an almost fourfold increase in the interferometric visibility after ms of evolution using four π-pulses in the Carr-Purcell sequence [18], whereby the first of n π-pulses is applied at (n) 1 T, and subsequent π- pulses are separated by a delay of n 1 T In the context of nmr, the spatially inhomogeneous evolution of the order parameter of the pseudospin- 1 condensate is analogous to diffusion, which hinders measurement of the transverse relaxation time As in archetypal multi-pulse echo experiments, the technique presented here mitigates the effect of these inhomogeneities on a coherence measurement of the condensate 13 We have observed long term drifts in the microwave and radiofrequency power over the course of the data acquisition Each point in Fig 69 amounts to 13 runs of the experiment ( 1 minutes)

173 65 phase imaging : interferometric reconstruction 157 visbility at ms Π pulses 3 4 Figure 611 Spin echo using Carr-Purcell pulse sequence of multiple π-pulses Application of four π-pulses during ms of evolution results in a fourfold increase in the fringe visibility 65 Phase imaging : interferometric reconstruction In this system, imaging the relative phase amounts to resolving the transverse component of the local Bloch-vector corresponding to the pseudospinor in Eq (61) For a sample of ultracold uncondensed 87 Rb atoms, this has been achieved in Reference [186] There, the Ramsey interference signal was binned in both space and time, recovering a phase spectrogram that illustrated collective spin excitations driven by exchange scattering This requires the repetition of many experimental cycles and gives an averaged relative phase with reduced temporal resolution We propose a technique to spatially resolve the relative phase of the pseudospin- 1 condensate from two dual state absorption images and present theoretical results that investi- gate the method s applicability and limitations Performing an identical experiment with and without the presence of the second π/-pulse permits four column densities n = n dx to be imaged, where n {n1, n, n1, n } and the densities are defined in Eqs (64) The densities can be used to recover the relative phase ϕ(r) using n (r) n1 (r) = n1 (r)n (r) sin(ϕ(r)) (64) There is little variation of the relative phase along the radial direction, and the densities of both components have similar radial dependence This is a consequence of the tight radial confinement, as the energy of the system is not sufficient for significant radial excitations to be induced As such, the radially integrated densities, the line densities n = n πρ dρ =

174 relative phase evolution of a two-component bec 158 _ n_1 n _ n1` _ n` line density (a) Φ Π (b) Π axial position Figure 61 Simulation of interferometric phase retrieval method (a) Two absorption images are captured for each evolution time, yielding the line densities immediately prior to, and after the second π/-pulse: n1, n, and n 1, n respectively (b) Point-by-point arithmetic is used [Eq (65)] to extract the relative phase profile along the axial direction n dy, are representative of the relative phase evolution, such that n (z) n 1 (z) = n1 (z)n (z) sin(ϕ(z)) (65) defines a phase ϕ(z) that is a good estimate of the phase along the centre of the condensate ϕ(ρ =, z)14 We have confirmed this by evaluating Eq (65) with line densities attained from simulations, as shown in Fig 61 For 9 ms of evolution, where the range of relative phase is π across the condensate, the agreement is within 78 mrad In more elongated geometries or where tunable miscibility can be realised, radial excitations are further suppressed and the approximation improves In these cases there is increased spin locking along the imaging axis and the integration of density inherent to absorption imaging corrupts the phase signal less The requirements of this technique are more demanding than those of the experimental investigation presented in 6 For successful implementation of Eq (65) to point-bypoint arithmetic on multiple absorption images, we are currently pursuing improved detec14 For a more detailed discussion of this, see Appendix D

175 66 future directions 159 tion methods with higher signal-to-noise ratio, spatial resolution, and dynamic range This technique may be useful for investigations of spin squeezing, in the context of recent proposals which suggest using the nonlinear interaction to reduce the spin projection noise below the standard quantum limit [187, 188, 189, 19] It is interesting to consider the spatial dependence of the relative phase fluctuations of a macroscopic pseudospinor This technique may allow such quantum noise to be studied with spatial resolution 66 Future directions To conclude this Chapter, a brief outlook is given on future directions of the experiments related to this thesis 661 Self rephasing We have experimentally observed self-rephasing with smaller condensates of 5 14 atoms, whereby the nonequilibrium dynamics responsible for decay of interferometric contrast can also result in multiple revivals When coincident with this self-rephasing, the π-pulse of the spin-echo sequence is most efficient, resulting in extended preservation of the transverse pseudospin The long-lived collective spin dynamics, probed using phased-based Ramsey interferometry, suggest coherence times in excess of 1 second The simulated self-rephasing is depicted spatiotemporally, using the local spin density (cf Figs 65 and 69), in Fig 613 Whilst the net relative phase evolves linearly at the centre of the condensate, the rate of change of relative phase at axial displaced positions oscillates about the mean frequency This is consistent with the cyclic behaviour of the collective oscillations which the condensate is undergoing, and which are damped in larger condensates that do not exhibit the rephasing The rephasing results in a reversal of the concavity of the relative phase along z, accompanied by a change in the relative velocity between the components [cf Eq (67)] This type of rephasing also amounts to rich behaviour of the order parameter, similar to that observed by Matthews, et al [5] The energetic cost of a phase winding along the condensate is countered by the spatial dynamics, and the order parameter explores a large region of su() to untwist itself [3] 66 Wave function engineering Spatial control of the state coupling, using off-resonant optical light shifts, or near field microwave sources, could enable nontrivial topological initial superpositions to be prepared, as well as pseudospin- 1 condensates in their ground state This would be advantageous for interferometry, as discussed in 14 and this Chapter

176 relative phase evolution of a two-component bec 16 Pz z µm time ms 1 Figure 613 Self rephasing of a pseudospin- 1 condensate with 3 14 atoms (simulated) A detuning of = π Hz was used in these simulations to resolve the spatial dependence of the relative phase evolution The relative phase of the condensate initially uniform become inhomogeneous during the initial evolution, as in Fig 65 The energetic cost of a phase winding along the condensate is tempered by the mean field dynamics which change the concavity of the relative phase in the t z plane, akin to the effect observed in the spinecho experiment (Fig 69) The repeatedly uniform relative phase results in a revivals of the interferometric contrast at 3 ms, 65 ms and 9 ms 663 Spin squeezing The nonlinearity responsible for spin squeezing is proportional to the following integrals of the spatial modes of the condensate χ a11 ψ (r) d r + a ψ (r) d r a ψ (r) ψ (r) d r 1 1/ where ψ i (r) = N i (66) Ψi (r) When the spatial modes of the condensate are stationary and common to each state, χ a11 + a a1, which is vanishing for the background scattering lengths of the F = 1, m F = 1 and F =, m F = +1 states of 87 Rb Li et al have proposed controlling the nonlinearity by separating the wave functions using state-dependent potentials Following creation of an overlapping superposition, the states are forcefully separated in space, increasing the value of χ and offering the possibility of squeezing the condensate pseudospin In this work, since the wave functions evolve due to the mean field dynamics on their own, the nonlinearity responsible for spin-squeezing might become significant during this evolution Moreover, the self-rephasing of the condensate plays the role of spatial