Optical Manipulation

Transcription

1 Optical Manipulation of Bose-Einstein Condensates PETER BLAIR BLAKIE A thesis submitted for the degree of Doctor of Philosophy at the University of Otago, Dunedin, New Zealand. July 2001

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3 Revision History Original submitted for examination 6th July, st corrected version (post examination) 6th September, 2001.

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5 Abstract In this thesis we investigate methods for manipulating Bose-Einstein condensates with electromagnetic radiation fields. Condensates are considered in the low temperature limit, where meanfield treatments based on the time-independent and time-dependent Gross-Pitaevskii equations are applicable. We develop numerical methods to solve for eigenstates, and present a novel and efficient algorithm for dynamical evolution in the cylindrical 3D case. We generalise the quantum optics dressed state to the case of a radiatively coupled condensate. A systematic numerical investigation of these stationary states is performed over a wide parameter regime, emphasising the role of different trapping potentials, and collisional effects. We consider how adiabatic transfer can be used to prepare a condensate dressed state, and investigate the use of spatially varying fields for transferring the condensate into a soliton state. The major focus of this thesis is Bragg scattering of Bose-Einstein condensates, a phenomenon based on the interaction between a condensate and a standing light wave field. We derive a Gross-Pitaevskii equation for the evolution of a condensate in the presence of Bragg light fields, which we investigate numerically. Using a linear two state model we consider the noninteracting atomic evolution, derive the condition for Bragg resonance, and obtain a momentum linewidth. We show that under appropriate conditions, Bragg scattering is sensitive to the spatial phase distribution of the initial condensate, allowing preferential scattering from a selected spatial region. This behaviour provides a distinctive signature of the condensate state, and in particular with an appropriate choice of light field parameters, an asymmetric Bragg scattered beam emerges from a vortex state. Finally, we give a detailed theoretical analysis of Bragg spectroscopy, the technique whereby low intensity Bragg scattering is used to probe a condensate. Using many-body Bogoliubov theory in the low intensity limit, the validity of the linearised Gross-Pitaevskii equation is demonstrated. We give a critical discussion of the appropriate observable, and obtain an analytic expression for this, allowing us to quantitatively identify the underlying physical mechanisms. The spectral properties of the condensate are numerically calculated for a wide regime, including for a vortex state.

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7 Acknowledgements I would like to thank my supervisor, Professor Rob Ballagh. His motivation, enthusiasm for physics, and continual support during my PhD has been tremendous. His attention to detail, and dedication to his students is truly amazing. Professor Crispin Gardiner has been a major influence on much of the work presented in this thesis. His talent for spotting holes in arguments and seeing the underlying physics is scary. I thank him for keeping me honest. I have had the pleasure of interacting with numerous brilliant people. Otago theorists (past and present): Dr. David Hutchinson, Dr. Andrea Eschmann, Dr. Ben Caradoc-Davies, Dr. Terry Scott, Andreas Penckwitt, Max Krüeger, Adam Norrie, and Katherine Challis. Otago experimentalists (past and present): Dr. Andrew Wilson, Dr. John Sharpe, Dr. Jos Martin, Dr. Callum McKenzie, Nick Thomas, and Reece Geursen. Special thanks goes to Dr. Peter Manson, Dr. Matthew Davis, Dr. Peter Marzlin, Dr. Sam Morgan, Professor Kieth Burnett, and Professor Peter Zoller for their help, support, and contribution to this work. Life over the last few years wouldn t have been the same without the input from my friends John (a.k.a Dirty-John), Jack, Victor, Malcolm, Paul, and the people and coffee at Governors. I am grateful to my parents, Beverley and William, for their support during my university years.

8 viii To my dearest fianceé and best friend Bee Sian, whose support has been invaluable, even from a distance. I look forward to us finally being together sometime in the near future: Saya amat terharu di atas sokongan dan kasih sayang yang tidak terhingga yang diberikan oleh tunang saya, Bee Sian. Saya amat berharap untuk dapat bersamasama melayari hidup ini dengannya di masa hadapan. This work was supported by the Marsden Fund of New Zealand under contracts PVT603 and PVT902.

9 Dedicated to Albert and Nora Waghorn

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11 Contents 1 Introduction Bose-Einstein condensation Development of Bose-Einstein condensation Recent studies of Bose-Einstein condensates Atom optics with light fields Bragg spectroscopy This work Overview Peer-reviewed publications Background theory Introduction Theory of a uniform condensate Uniform ideal gas Weakly interacting gas Uniform-Bogoliubov theory Theory of a non-uniform condensate Second quantised field theory Bogoliubov theory The ground state The Thomas-Fermi approximation Quasiparticle excitations Time-dependent Gross-Pitaevskii equation Computational units Dimensionless quantities Making the Gross-Pitaevskii equations dimensionless

12 xii CONTENTS Thomas-Fermi limit Atom-radiation interactions Hyperfine structure of 23 Na and 87 Rb Spherical unit vectors The radiation field Electromagnetic interaction Electric-dipole Rabi frequency Magnetic-dipole Rabi frequency Field polarisation choice for magnetically trapped condensates Semi-classical two state model Numerical techniques Introduction Solving nonlinear algebraic equations Newton-Raphson (zero-finding) Optimisation Numerical application Large scale solvers: the conjugate gradient technique Summary Solving for Gross-Pitaevskii eigenstates Notation Real 1D case D results and performance Real 2D case Complex 2D case D results and performance Time-independent Gross-Pitaevskii equation in cylindrical coordinates Cylindrical coordinates Cylindrical wavefunction Discretising the cylindrical Gross-Pitaevskii equation Cylindrical eigenstate results and performance Solution of time-dependent Gross-Pitaevskii equation Review of 2D RK4IP algorithm Time-dependent Gross-Pitaevskii equation for cylindrical systems

13 CONTENTS xiii Numerical evolution of cylindrical wavefunction Numerical implementation Accuracy Dressed states of a Bose-Einstein condensate Introduction Formulation Internal state coupling Hamiltonian Time-independent Gross-Pitaevskii equation Solving for dressed states Gross-Pitaevskii equation in computational units Thomas-Fermi approximation Collisional degeneracy Numerical solution Characteristics of dressed states Identical species Effect of different relative trap potentials Rabi frequency dependence Effect of different relative collisional interactions D dressed states Creating a dark soliton by adiabatic transfer Experimental realisation in rubidium Internal state coupling Dressed state observations Conclusion Bragg scattering of a released condensate Introduction Derivation of the optical potential Adiabatic elimination of excited state Regimes of condensate scattering Raman-Nath diffraction Bragg regime Channeling regime

14 xiv CONTENTS 5.4 Bragg scattering of a released condensate Numerical solutions Analytic treatment of the linear case Noninteracting atoms First order couplings and the two-state model Results and features of the two-state model Rabi oscillations Dispersion curves and resonance-coupling Coupling width and breakdown of two-state model Application to nonlinear case Numerical result Free expansion Nonlinear dispersion curves Momentum spectroscopy Conclusion Spatially selective Bragg scattering Introduction Two state model Two state equations Evolution in a moving frame of reference Formal solution Thomas-Fermi approximate solution Solution validity Steady state beam profiles Pulsed Bragg results: the topological atom-laser Conclusion Theory of coherent Bragg spectroscopy of trapped Bose-Einstein condensates Introduction Low-intensity Bragg scattering theory Many-body field theoretic approach Gross-Pitaevskii equation approach Comparison of approaches

15 CONTENTS xv Quasiparticle occupation Observable of Bragg spectroscopy Spectral response function Dynamic structure factor Quasi-homogeneous approximation to R(q, ω) Homogeneous spectral response Quasi-homogeneous spectral response function Bragg spectroscopy Numerical results for R(q, ω) Scattering beyond the linear regime Long time excitation: energy response Conclusion Conclusion Summary Future work A Deriving the Gross-Pitaevskii equation 221 A.1 Constrained functional minimisation A.1.1 Constrained minimisation A.1.2 Complex variable functions A.1.3 Functionals and functional differentiation A.1.4 Constrained functional minimisation A.2 Derivation of the time-independent Gross-Pitaevskii equation A.3 Derivation of the dressed state equations B Derivative operator 227 B.1 Calculating the finite difference derivative operator B.2 Derivative operator performance B.3 1D analytic derivatives B.4 2D analytic derivatives B.4.1 Real case B.4.2 Complex case B.5 Cylindrical state optimisation B.5.1 Optimality function

16 B.5.2 Analytic derivatives B.6 1D dressed state analytic derivatives C Raman coupling 243 D Bragg results 247 D.1 Raman-Nath time scale D.2 Momentum expansion time constant D.3 Projecting the two state equations (.) D.3.1 Evaluating P n 2 : m 2 ψ D.3.2 Evaluating P n ( V cos(q r ωt)ψ): D.3.3 Evaluating P n ( ψ t ) : D.3.4 Evolution of ψ n : D.4 Bogoliubov conventions

17 Chapter 1 Introduction 1.1 Bose-Einstein condensation Bose-Einstein condensation is a phenomenon in which a single mode of a system becomes macroscopically occupied by atoms, analogous to a highly occupied light mode in an optical cavity of a laser. This ordered collection of atoms, referred to as a Bose-Einstein condensate, has fascinating properties exemplifying the de Broglie nature of matter and providing a remarkable demonstration of the duality principle. The 1995 experimental realisation of a nearly pure condensate that could be manipulated and directly observed, marks the beginning of a major new area of research. Since then, a wide range of novel experiments and theoretical efforts have been concerned with this exciting form of matter wave Development of Bose-Einstein condensation Particles are either bosons, which have integral spin, or fermions, which have halfinteger spin. In a system of identical particles, the interchange of any two particles must leave physical observables unaffected. This imposes some constraints on the state vector for the system - for bosons it must be symmetric with respect to the interchange, while for fermions it must be anti-symmetric. This requirement gives rise to the characteristic behaviour of each type of particle: fermions can have at most one particle in any particular quantum state, this is the well-known Pauli exclusion principle. On the other hand, bosons are gregarious, and can congregate

18 2 Introduction into any state without constraint. In fact, the symmetry of the boson s state vector leads to the remarkable behaviour that the probability of transition into a state grows linearly with the number of bosons in that state. In 1924, the Indian physicist Bose gave a derivation of the Planck distribution based on the idea that photons obey Bose statistics. The following year Einstein hypothesised that atoms would also obey these statistics and, at low temperatures, he predicted a large occupation of the ground motional state, that is a Bose-Einstein condensate would form. Unlike a thermodynamic phase change, where inter-particle interactions are responsible, Bose-Einstein condensation occurs solely because of quantum statistics. In 1937, Kapitza and Allen independently discovered that 4 He exhibited remarkable properties at low temperatures, referred to as superfluidity. The following year London suggested that, because 4 He is a boson, these properties were due to Bose- Einstein condensation. However, superfluid- 4 He is a liquid and the strong interparticle interactions cause the condensate to be largely depleted. Later calculations by Penrose and Onsager showed that at T = 0K the condensate fraction in the superfluid would be only 8% [136], far removed from Einstein s original conception of a pure condensate occurring in an ideal gas. The quest for Bose-Einstein condensation in a weakly interacting gas began with atomic hydrogen in the late 1970 s, but was plagued with difficulties in getting to the condensation point. With the development of laser cooling for alkali atoms, magnetically trapped alkali gases took the lead in what was becoming a race to achieve a gas condensate. The breakthrough came in 1995 when the JILA group [7] reported Bose-Einstein condensation in a gas of 87 Rb atoms, approximately 70 years after Einstein s prediction was made. They were closely followed by the Rice University group with 7 Li, and the MIT group with 23 Na. 1.2 Recent studies of Bose-Einstein condensates As of May 2001, about 26 experimental groups around the world routinely make Bose-Einstein condensates. Most groups work with 87 Rb or 23 Na, however condensates have also been formed with 1 H [69], meta-stable 4 He [145], 7 Li [27], and 85 Rb [48]. These experimental groups, particularly those early on the scene, have been very active and have investigated a wide range of applications. The explosive in-

19 1.2 Recent studies of Bose-Einstein condensates 3 terest in this field has led to a huge number of theoretical works being published, largely driven by experiments. In this section, we give a general overview of the main categories of research carried out over the last six years, with particular emphasis on experimental results. Ground state properties Condensates are usually formed in harmonic traps, though due to interactions between the condensed particles, the ground state wavefunction is very different from a harmonic oscillator state. Experimental groups have concluded an extensive list of ground state properties, including the energy, occupation and critical temperature of formation [60]. The ground state coherence has been examined through measurements of the first [9, 87, 24], and higher order [34, 153] spatial correlation functions. Hau et al. [92] have made detailed measurements of the spatial shape of the condensate, showing excellent agreement with meanfield theory. Due to these, and other experiments, the Gross-Pitaevskii equation [140, 86] is now accepted as an excellent description of condensate behaviour at low temperatures. Excitations Excitations of a many-body system are a central concern in condensed matter physics. For a dilute weakly interacting condensate, the microscopic theory formulated by Bogoliubov in 1947 [25] provided an excellent description at zero [104] temperature. At finite temperatures, the measured shifts in excitation frequencies of the lowest modes [105, 158] have necessitated self consistent treatments of condensate and thermal atoms [99] for an accurate description. A complete manybody theory, which adequately describes the behaviour of all the experimentally measured modes, has yet to be demonstrated, and is a major ongoing theoretical effort (e.g. see [128]). The Ketterle group has measured the speed of Bogoliubov (zero) sound [8], and recently demonstrated the selective excitation of surface modes [133]. The Oxford group has used scissor mode [114] excitations to observe harmonic generation [94] and Beliaev coupling [97] in condensates. Condensate growth Until recently the time scale over which a condensate grows out of the thermal vapour cloud remained unknown. In a major theoretical effort Gardiner et al. (e.g. see [72, 73, 75]) have developed a Quantum Kinetic Theory to provide a unified description of the growth of a weakly interacting condensate over

20 4 Introduction a wide regime. Experiments in this area have been reported by Miesner et al. [123] (also see [71]). Atom lasers The term atom laser is used in analogy to the optical laser to describe a directed, monochromatic beam of coherent matter. The famous MIT experiment [9] showing the interference of two condensates, unequivocally demonstrated the suitability of condensates as a source for an atom laser. A variety of atom lasers have been realised in both pulsed [122, 116] and quasi-continuous output [88, 23, 24], but are generally in closest analogy with Q-switched optical laser (see [12]). As yet no pumping mechanism has been demonstrated to replenish the trapped condensate, and thus the lasing time is limited by the population of the mother condensate. Condensate mixtures In a multiple component condensate, several distinguishable phases coexist. Such a system has been realised by converting a condensate into several different hyperfine states in both magnetic [59, 90, 91, 119, 121] and optical [157] traps. These systems can be described by a vector order parameter (e.g. see [95]), and offer a richer physics than single component condensates [96, 80]. Multiple component condensates have been used to investigate inter-penetrating superfluids and phase separation [91], and tunneling between the spin domains of the different hyperfine components [157]. Recently, a mixture of the bosonic- 7 Li and fermionic- 6 Li has been cooled to the degenerate regime, clearly showing the remarkable differences that exist between these two classes of particles. Vortices and solitons Vortices are the mechanism by which superfluids support circulating flow. Vortices are a major theoretical interest, but until recently, their existence and stability in trapped condensates was uncertain. Using optical techniques, generating a single vortex [120], or a vortex lattice [113, 39, 1], has become a routine experimental procedure. Attention is now turning to characterising vortex dynamics and stability [5, 89]. A dark soliton is a structure which propagates in a nonlinear medium without changing its shape, and has a characteristic phase kink. By using optical fields to imprint this phase kink, dark solitons have been observed [32, 55] in condensates.

21 1.3 Atom optics with light fields 5 Optical manipulation of condensates Light fields have been used for optical trapping [154], to form optical lattices [6], and to impart momentum with Bragg scattering [109] and diffraction [134]. Bragg scattering has been used as an experimental tool to investigate the Talbot effect [53], four-wave mixing [54] of matter waves, superradiant light scattering [101], and matter wave amplification [103, 110]. In set of experiments by the MIT group [160, 155] Bragg scattering was also used to spectroscopically probe a condensate. Utilising the high density and steep refractive index of a cold trapped atoms, Hau et al. [93] were able to slow the speed of light to 17 ms 1 while travelling through a condensate. Atomic collisional properties The scattering length determines the nature of low momentum collisions between atoms. In lithium [27, 77] the scattering length is negative, and only small condensates are stable. Feshbach resonances have been used to manipulate the scattering length, and hence alter the interaction properties of the condensed atoms [159, 146, 48]. Atom interferometers The extended phase coherence of the Bose-Einstein condensate are ideal for atom interferometry. The key part to an interferometer, the beam-splitter, has been realised by converting the condensate internal state [90], or by using Bragg pulses [162] to spatially separate the condensate. 1.3 Atom optics with light fields Atom optics is concerned with techniques for manipulating neutral atoms by using their wave properties. A large portion of research in this area has concentrated on developing devices analogous to those for light optics, such as mirrors, beam splitters, and lenses. The most important application of atom optics so far has been atom-interferometry [14], which has been used in tests of quantum mechanics, and for measuring gravitational and rotational effects (also see [2]). Atom interferometers have the potential to surpass the sensitivity of laser interferometers because atoms have smaller wavelengths, and lower velocities. The availability of Bose-Einstein condensates and atom lasers as the source of highly coherent matter waves offers enhanced scope for the application of atom optics and interferometry.

22 6 Introduction The fields of electron and neutron optics are more mature than atom optics, with electron microscopes dating back to the 1930 s and neutron scattering being used in the 1950 s. These forms of matter-wave optics developed rapidly because of the ease with which electrons are manipulated by electric fields and neutrons by Bragg scattering in crystals. Atom optics largely developed in the last 20 years with the availability of intense tunable lasers and micro-fabrication technology. Neutral atoms are more difficult to manipulate since they are relatively unaffected by static electromagnetic fields and do not penetrate through matter. However, atoms do have several advantages; the insensitivity to static fields is useful for interferometers, atoms are more readily available than neutrons, which require a particle accelerator, and because of their rich internal structure atoms have extra degrees of freedom and are easily probed with light. (a) Laser Standing Wave Atomic Beam Diffracted Beam (b) Laser Standing Wave Bragg Scattered Beam Atomic Beam Transmitted Beam Figure 1.1: Schematic illustration of scattering of atoms from a standing wave of laser light. (a) diffraction (b) Bragg scattering. Bose-Einstein condensates are different from atomic beams, which are currently the most common source of matter waves for atom interferometers. For condensates to have an impact on atom optics, tools to manipulate them need to be developed.

23 1.3 Atom optics with light fields 7 An essential element for an interferometer is a beam splitter. In light optics, a beam splitter amplitude divides a light beam into spatially separated paths. To realise the equivalent element in atom optics it is necessary to impart momentum to the matter wave. In general, a beam splitter will be useful only if it can amplitude divide coherently to provide well defined output beams, clearly separated in momentum, so that they may be used in further optical elements. Condensates can only exist in high vacuum and we are limited to using electromagnetic fields to manipulate them. The most convenient way to impart momentum to a condensate is with photon recoil, of which several techniques have been developed for atomic beams (see [2]). The preferred scheme for condensates is two-photon scattering, whereby an atom instantaneously absorbs and re-emits photons between two fields in a standing wave configuration, and the atom recoils with the difference of the photon momenta. This process can leave the atom in the same internal state (Bragg scattering [109]), or in a different hyperfine state (Raman scattering [88]). Experiments with Bragg scattering have developed over the last 15 years. The 1986 experiment by Pritchard s group [82], demonstrated diffraction 1 of an atomic beam from a standing wave light field (see Fig. 1.1(a)). This was not the first experiment to use light to deflect atoms, however it seems to be the first experiment where the diffraction orders (up to 4th) were clearly discernible. Two years later the Pritchard group demonstrated Bragg scattering [117], which is where a single momentum state is selectively occupied. The Bragg regime of scattering requires a larger region of interaction between the atoms and the standing wave than is needed for diffraction, and scattering is only observed when then beam is incident to the optical fields at certain angles (see Fig. 1.1(b)). In this experiment Pritchard and his group were able to observe up to 4th order scattering (i.e. the scattered and unscattered states differed by 4 photon momentum differences) by carefully selecting the incidence angle. In 1996, the Colorado group [79] used Bragg scattering to make the beam splitters and mirrors in a Mach-Zehnder atom interferometer based on meta-stable neon atomic beams, with fringe contrast of up to 62%. Bragg scattering was applied to condensates by the NIST group [109] in 1999, but since the condensate is initially stationary, they introduced a frequency difference between the laser fields so that the standing wave field (or Bragg grating) was moving relative to the condensate (see [15]). By adjusting the frequency difference, 1 Diffraction is also referred to as Bragg diffraction or normal diffraction in the literature.

24 8 Introduction the condensate was selectively scattering with up to 6 photon momentum differences. In 1999, the Tokyo group [162] constructed a Mach-Zehnder interferometer for Bosecondensed rubidium atoms using Bragg scattering to make the beam splitters and mirrors. Because of the narrow momentum distribution of the condensate, their interferometer exhibited essentially 100% fringe contrast. The theory of many-body atom optics has developed relatively recently. In 1993, a report by Lenz et al. [111] titled nonlinear atom optics considered many-body effects on Bragg scattering. They derived coupled nonlinear Schrödinger equations for the ground and excited states, closely resembling the Gross-Pitaevskii equation, except same-species collisional interactions were ignored. The following year Zhang and Walls [168] also considered a many-body treatment of atoms interacting with a light field, but allowed photon exchange between the particles, giving rise to a long range Kerr-type nonlinearity. Using numerical simulations they demonstrated Bragg-type beam splitting of atomic wavepackets. 1.4 Bragg spectroscopy A significant development in the use of Bragg scattering with condensates, was made by Ketterle s MIT group in a series of experiments [160, 155] using a technique they called Bragg spectroscopy. The essence of that technique was to measure the amount of condensate scattered by a low intensity Bragg grating, as shown in Fig. 1.2(a). The Bragg spectrum was given by the strength of this signal as the frequency difference between the lasers in the Bragg grating was varied. From their measurements, the MIT group was able to infer difficult to observe information about the condensate, such as its momentum width, chemical potential, and excitation spectrum. These experiments were analysed [160, 155, 156] as a measurement of the dynamic structure factor. The dynamic structure factor is most well known as the observable of neutron scattering experiments in superfluid helium [137, 138, 84], and is defined in terms of the Fourier transform of the Density-Density correlation function. In Fig. 1.2(b) we show the setup of a helium scattering experiment: an incoherent thermal neutron beam is inelastically scattered from a helium sample. The dynamic structure factor S(q, ω) is proportional to the intensity of neutrons scattered with the momentum change q and energy change ω from those in the incident beam.

25 1.5 This work 9 (a) Scattered Condensate BEC Bragg Grating (b) Detector Neutron Beam Superfluid He Scattered Neutrons Figure 1.2: Schematic illustration of (a) Bragg spectroscopy and (b) neutron scattering from superfluid helium. 1.5 This work Overview In Chapter 2 we give the theoretical background for this thesis. We begin by reviewing Bogoliubov theory for the uniform and non-uniform cases, and derive the time-dependent and time-independent Gross-Pitaevskii equations. These equations are central to the work in this thesis, and to facilitate their numerical solutions it is convenient to express them in a dimensionless form. We introduce our choice of computational units and give dimensionless forms of important equations and results. Electromagnetic coupling between the internal states of the condensate atoms is of central importance in this work. We review aspects of atomic theory of particular relevance to alkali atoms, and give a detailed calculation of the Rabi frequencies for electric-dipole and magnetic-dipole transitions.

26 10 Introduction The Gross-Pitaevskii equation in time-independent and time-dependent forms is a difficult equation to solve. Much of the work we present in this thesis requires solutions of both these equations for quantitative understanding of the physics we consider, and for these purposes in Chapter 3 we present the numerical methods we have developed. Stationary solutions of the time-independent form of the Gross- Pitaevskii equation were obtained with a novel method based on finite differences and conjugate gradient optimisation. This scheme allows us to accurately solve for highly nonlinear eigenstates, including vortex states. We have also developed an efficient evolution algorithm for the 3D cylindrically symmetric Gross-Pitaevskii equation. In Chapter 4 we begin our investigation of condensate manipulation by looking at the stationary solution behaviour of a two component condensate coupled by a cw radiation field. These states are the condensate generalisation of the quantumoptical dressed states. We consider the properties of these states with a particular emphasis on the spatial effects which arise from trapping and collisional differences between the components. We also investigate the use of spatially varying fields to engineer the condensate into solitonic states using adiabatic passage. The remainder of this thesis is concerned with Bragg scattering in Bose-Einstein condensates. We begin in Chapter 5 by deriving a Gross-Pitaevskii equation for a condensate wavefunction interacting with a Bragg grating. We outline the main categories of scattering that may occur, and indicate the parameter regime for Bragg scattering. We develop a two state model for first order scattering in the noninteracting case, and use this to derive an analytic expression for the momentum transition linewidth. With the aide of numerical simulations we consider how nonlinearity affects Bragg scattering. In Chapter 6 we show that Bragg scattering can be made sensitive to the spatial phase of the condensate, and can preferentially scatter matter from selected spatial regions. An analytic model, which accurately describes this phenomenon, and explains the underlying mechanisms is developed. We present analytic and numerical results demonstrating that the spatially selective nature of Bragg scattering can be used to provide a robust vortex signature. We also show how a topological atom laser can be made by using a pulsed Bragg grating to scatter a sequence of vortices from a vortex state. In Chapter 7 we consider Bragg spectroscopy. A quantum mechanical treat-

27 1.5 This work 11 ment of low intensity Bragg scattering is given, where the condensate atoms are described by Bogoliubov theory, and the Bragg grating is approximated as an optical potential. We demonstrate the equivalence of the quantum result to linearised Gross-Pitaevskii theory and give a critical discussion of the dynamic structure factor, and its applicability to Bragg spectroscopy. Using a quasi-homogeneous approximation and numerical simulations, we identify the dominant physical mechanisms in Bragg spectroscopy. By considering the energy response of a trapped condensate to Bragg scattering and in the long pulse limit, we show that the trapped quasiparticle spectrum can, in principle, be measured directly Peer-reviewed publications Much of the work we report in this thesis has been published in peer-reviewed journals. The work on the dressed states of a Bose-Einstein condensate, which we report in Chapter 4, appeared in Journal of Optics B: Quantum and Semiclassical Optics [21]. Gross-Pitaevskii eigenstates, found using the numerical techniques we have developed (Chapter 3), were extensively used in recent studies of vortex dynamics in Bose-Einstein condensates published as a rapid communication in Physical Review A [36]. Our study of Bragg scattering of a released condensate, which forms Chapter 5, was published in the special issue on coherent matter waves of Journal of Physics B [20]. The spatially selective Bragg scattering results, presented here in Chapter 6, were published in Physical Review Letters [22]. The Bragg spectroscopy work in Chapter 7 is currently in preparation for publication.

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29 Chapter 2 Background theory 2.1 Introduction In this chapter, we review the theory necessary for our investigation of optical manipulation of Bose-Einstein condensation in atomic gases. Our aim is to outline the theory we need in later chapters in a consistent notation so that this thesis may be a self-contained document. We begin by examining condensation in a uniform Bose gas, which is described by a second quantised Hamiltonian. Using Bogoliubov theory we reduce the Hamiltonian to a tractable form which we can simplify by making a quasiparticle transformation. The uniform solution allows us to understand the excitation spectrum of a condensate and to discuss characteristic length and velocity scales. Using field theory we extend our treatment to the non-uniform case, which allows description of experimentally realised condensates. Following the Bogoliubov procedure for the uniform case we approximately diagonalise the Hamiltonian, and in doing so we introduce the Gross-Pitaevskii equations describing the condensate mode, and the Bogoliubov-de Gennes equations describing the quasiparticle excitations. These equations are in general difficult to solve, but the Gross-Pitaevskii equation possesses an approximate solution, known as the Thomas-Fermi solution. In later chapters we consider the numerical solution of the Gross-Pitaevskii equation and it is convenient to introduce our computational units and notation at this point. Finally we look aspects of atomic theory relevant to the radiation coupling of atomic levels.

30 14 Background theory 2.2 Theory of a uniform condensate The standard microscopic theory of a uniform or homogeneous condensate originates from the 1947 paper by Bogoliubov [25]. We briefly review this theory here, with emphasis on important ideas such as excitation behaviour, and characteristic lengths and speeds, which will be useful later in this work Uniform ideal gas We take the ideal gas (noninteracting) system to be enclosed in a cube of sides L and volume V = L 3. The single particle states, subject to periodic boundary conditions, are ζ k (r) = 1 V e ik r, (2.1) where we have labeled the state by its wave vector k, given by k = 2π (n x, n y, n z )/L, where n x, n y and n z are integers (see [144]). These plane wave states have momentum k and their energy is given by where m is the particle mass. ω k = 2 k 2 2m, (2.2) Weakly interacting gas In any physically realisable system the particles will interact with each other. If the system is sufficiently dilute only two body interactions are important, and the Hamiltonian can be written as Ĥ = k ω k â kâk + 1 2V k 1 k 2 k 3 k 4 Ṽ k1 k 3 â k 1 â k 2 â k3 â k4 δ k1 +k 2,k 3 +k 4. (2.3) Here Ṽk is the Fourier transform of the interaction potential between a pair of particles, and the Kronecker-δ ensures momentum conservation in each collision. The two operators â k and â k are the single particle destruction and creation operators and obey the bosonic commutation relations [â k, â k ] = δ k,k (2.4)

31 2.2 Theory of a uniform condensate 15 ] [â k, â k ] = [â k, â k = 0. (2.5) Pseudo-potential approximation For low momentum collisions in a dilute gas, s-wave scattering dominates, and the particle interaction can be simplified to Ṽ k U 0 = 4π 2 a m, (2.6) where a is the s-wave scattering length [51]. This corresponds to making a pseudopotential approximation to the inter-particle potential (in real space), of the form Ṽ (r) = U 0 δ(r). (2.7) This approximation, while very common leads to some unphysical effects, in particular it gives rise to to ultra-violet divergences. However, Morgan [128, 127] has shown that the use of the contact potential (2.7) as an approximation to the twobody T -matrix leads naturally to a high-energy renormalization. The pseudo-potential approximation (2.7) also neglects elastic scattering loss, which will occur between condensate wave packets moving at relative velocities, as discussed by Band et al. [13], and observed by Chikkatur et al. [40]. This effect will be important in atom optical applications where the condensate is often split into moving and stationary parts, however it only causes slight population reduction and does not tend to effect the overall qualitative physics (see [163]). None of these limitations of the pseudo-potential approximation are severe for the cases we consider in this thesis, so we will use (2.7) to approximate the inter-particle potential Uniform-Bogoliubov theory Bogoliubov approximation At low temperatures T T c, where T c is the critical temperature for the onset of Bose-Einstein condensation, N 0 particles occupy the ground state with k = 0. Assuming N 0 is large, adding or subtracting a few particles from this state would not have any appreciable effect on the system. This motivates the Bogoliubov approximation in which the ground state operators are replaced by c-numbers (see

32 16 Background theory [138, 65]), i.e. we write â 0, â 0 N 0. (2.8) Using results (2.6) and (2.8) in the homogeneous Hamiltonian (Eq. (2.3)) and only retaining terms second order in N 0 we have Ĥ ω k â kâk + U 0N 0 (N 0 1) 2V k [( ω k + N ) ] 0U 0 (â kâk + â 2 2 kâ k) k [ ] N0 U (â kâ k + â kâ k ). k 0 (2.9) Bogoliubov Transformation The Hamiltonian (2.9) may be diagonalised by means of the well known Bogoliubov transformation [67, 18] â k = u kˆbk + v kˆb k, (2.10) â k = u kˆb k + v kˆb k, (2.11) where the {ˆb k, ˆb k } are known as quasiparticle operators. The real isotropic coefficients u k and v k are chosen to have the values u 2 k = 1 ( ) ωk + nu 0 + 1, (2.12) 2 ωk B vk 2 = 1 ( ) ωk + nu 0 1, (2.13) 2 ω B k where ω B k is given by the so called Bogoliubov dispersion relation ω B k = ω 2 k + 2nU 0ω k /, (2.14) and n = N 0 /V is the density of the uniform system. The u and v coefficients satisfy the relation u 2 k v 2 k = 1, (2.15)

33 2.2 Theory of a uniform condensate 17 so that the Bogoliubov transformation is canonical, i.e. the new operators satisfy the usual boson commutation rules of the same form as Eqs. (2.4) and (2.5). Applying the Bogoliubov transformation to (2.9) yields the diagonalised Hamiltonian Ĥ B = E 0 (N 0 ) + ω Bˆb k kˆb k, (2.16) k 0 where E 0 (N 0 ) = U 0N 0 (N 0 1) 2V k 0 ω B k v 2 k, (2.17) is the ground state energy. This expression is ultra-violet divergent because we have assumed that the Fourier transform of the inter-particle potential is constant (see the discussion following Eq. (2.7)). Homogeneous quasiparticles In Eq. (2.16) the quasiparticle energies are given by the Bogoliubov dispersion relation (2.14), which has the well known limiting behaviour ω B k { ck, for k 0, ω k + nu 0, for k. (2.18) The long wavelength expression (i.e. quasiparticle excitations of velocity where k 0) corresponds to phonon-like c = nu0 m, (2.19) known as the speed of sound. The high k limit yields a shifted free particle spectrum arising from the Hartree interaction (nu 0 ) with the other particles in the condensate. The crossover between these regimes occurs for ω k nu 0, and the wavelength at which this occurs is called the healing length, given by ξ = 2mnU0 = 1 8πna. (2.20) The healing length has another physical meaning, as the distance over which the condensate, constrained to zero density at one point, recovers to its uniform density. The behaviour of quasiparticle amplitudes, u k and v k is also dependent on the

34 18 Background theory wavelength of excitation. For k 1/ξ, both of these amplitudes are large and can be characterised as u 2 k vk 2 1, (2.21) whereas in the small wavelength regime, where k 1/ξ, we have u 2 k 1 v 2 k 0, (2.22) and the Bogoliubov operators become essentially identical to the single particle operators. 2.3 Theory of a non-uniform condensate In experiments to date, condensation of a dilute Bose gas has occurred within the confines of an external trapping potential and as such a homogeneous theory will be unsatisfactory for quantitative comparison. Here we briefly review non-uniform condensate theory with a particular emphasis on the equations of the condensate mode. We take the trap potential V T to be harmonic in the region where the atoms collect, and we shall write V T (r) = 1 2 m(ω2 xx 2 + ω 2 yy 2 + ω 2 zz 2 ), (2.23) however it is often convenient to leave the potential as a more general time dependent function V T (r, t) to allow for the possibility of dynamically controlling the trap stiffness Second quantised field theory Hamiltonian The usual formulation to extend Bogoliubov theory to the non-uniform case begins with the second quantised Hamiltonian, written in terms of the field operator, ˆΨ(r, t), namely Ĥ = ] dr ˆΨ (r, t) [ 2 2m 2 + V T (r, t) ˆΨ(r, t) (2.24)

35 2.3 Theory of a non-uniform condensate 19 + U 0 2 dr ˆΨ (r, t) ˆΨ (r, t) ˆΨ(r, t) ˆΨ(r, t), where we have taken collisions to be in the s-wave scattering limit. This form of the Hamiltonian can be related to the uniform case (2.3) by noting that the field operator, ˆΨ(r, t), can be written as ˆΨ(r, t) = j ζ j (r)â j, (2.25) where {ζ j (r)} is basis of orthogonal single-particle states and {â j } are the corresponding annihilation operators for these states. equal-time Bose commutation relations [ ˆΨ(r, t), ˆΨ (r, t)] [ ] [ ˆΨ(r, t), ˆΨ(r, t) = ˆΨ (r, t), ˆΨ (r, t)] The field operators satisfy the = δ(r r ), (2.26) = 0, (2.27) where δ(r r ) is the Dirac delta function. Heisenberg equation of motion The Heisenberg equation of motion for the field operator is i ˆΨ t = [ ˆΨ, Ĥ]. (2.28) Utilising the commutation relations (Eqs. (2.26) and (2.27)) the operator evolution equation can be simplified to i ˆΨ ] [ t = 2 2m 2 + V T (r, t) + U ˆΨ 0 (r, t) ˆΨ(r, t) ˆΨ(r, t). (2.29) Bogoliubov theory The Bogoliubov approximation for a non-uniform system is based on the assumption that a single mode is macroscopically occupied and that the field operator can be written as ˆΨ(r, t) Ψ(r, t) + ˆφ(r, t), (2.30)

36 20 Background theory where Ψ(r, t) = ˆΨ(r, t) is the condensate wavefunction and ˆφ(r, t) is the remaining (zero mean) fluctuation operator. Taking the field operator to have a mean value of the form in Eq. (2.30) breaks symmetry with respect to phase. This is not a necessary approximation, and several number conserving approaches have been developed [70, 127] which do not break symmetry, however these treatments yield essentially the same results. The number operator ˆN = dr ˆΨ (r, t) ˆΨ(r, t), (2.31) has the expectation value N = dr Ψ(r, t) 2 + dr ˆφ (r, t) ˆφ(r, t), (2.32) = N 0 + N, (2.33) where N 0 and N are the condensate and non-condensate contributions to the total number of particles. Even at zero temperature N N 0, since interactions will deplete the ground state, however this is a very small effect in dilute condensates [100]. At finite temperatures an additional thermal contribution to the depletion will arise. In this thesis we focus on the zero temperature case where the following approximation should be well satisfied N 0 N. (2.34) In this regime the contribution of ˆφ will always be small and using Eq. (2.30) to simplify the Hamiltonian (2.24) we have Ĥ dr Ψ (r, t) [ 2 2m 2 + V T (r, t) + U ] 0 Ψ(r, t) 2 Ψ(r, t) (2.35) 2 + dr ˆφ (r, t) [ 2 2m 2 + V T (r, t) + U ] 0 Ψ(r, t) 2 Ψ(r, t) 2 + dr Ψ (r, t) [ 2 2m 2 + V T (r, t) + U ] 0 Ψ(r, t) 2 ˆφ(r, t) 2 ] + dr ˆφ(r, t) [ 2 2m 2 + V T (r, t) + 2U 0 Ψ(r, t) 2 ˆφ (r, t)

37 2.3 Theory of a non-uniform condensate 21 + U 0 2 dr [ 4 Ψ(r, t) 2 ˆφ (r, t) ˆφ(r, t) + Ψ(r, t) 2 ˆφ (r, t) ˆφ (r, t) +Ψ (r, t) 2 ˆφ(r, t) ˆφ(r, t) ], where we have only retained terms second order or less in ˆφ. The first line of Eq. (2.35) yields the energy contribution of the condensate wavefunction Ψ(r, t) E[Ψ(r, t)] = dr Ψ (r, t) [ 2 and is referred to as the energy functional. 2m 2 + V T (r, t) + U 0 Ψ(r, t) 2 2 ] Ψ(r, t), (2.36) The ground state When the external potential is time-independent (i.e. V T (r, t) = V T (r)), we expect that a minimum energy stationary state will exist, that is the ground state. Our procedure for finding this is to minimise the energy functional (2.36) subject to the constraint that the condensate wavefunction is normalized to N 0. We provide details in Appendix A.2, where we show that the necessary condition for the ground state is the so called time-independent Gross-Pitaevskii equation [86, 140] µ 0 ψ 0 (r) = ] [ 2 2m 2 + V T (r) + N 0 U 0 ψ 0 (r) 2 ψ 0 (r). (2.37) For convenience, we have chosen ψ 0 (r) to be normalised to unity 1, i.e. we have set Ψ(r) = N 0 ψ 0 (r). (2.38) The quantity µ 0 is the Lagrange multiplier required for the normalisation constraint, but has the physical meaning of the condensate chemical potential, i.e. µ 0 = E. (2.39) N N0 1 We shall write wavefunctions normalised to N 0 as Ψ, and wavefunctions normalised to unity as ψ in this thesis.

38 22 Background theory The Thomas-Fermi approximation When there is a large number of particles in the condensate the kinetic energy contribution to the Gross-Pitaevskii equation (2.37) can be neglected, i.e. µψ 0 (r) = [ V T (r) + N 0 U 0 ψ 0 (r) 2] ψ 0 (r). (2.40) Rearranging this expression we get the Thomas-Fermi density (n TF = N 0 ψ 0 2 ) profile n TF (r) = µ TF V T (r) θ U 0 where we have introduced the unit-step function ( µtf V T (r) U 0 ), (2.41) { 1, x 0, θ(x) = 0, x < 0. (2.42) The value of the Thomas-Fermi chemical potential µ TF is determined by requiring that n TF (2.41) is normalised to N 0. For the case of a axisymmetric trap where only the z-spring constant differs from the other directions, i.e. (ω x, ω y, ω z ) = ω T (1, 1, λ), the Thomas-Fermi chemical potential is ( ) 2 15λN0 U 5 0 (2mω µ TF = T 2 ) 3 5. (2.43) 64π The Thomas-Fermi approximation provides a useful estimate of the condensate behaviour and has proven to be very accurate for large condensates. approximation, the radial and axial extents of the condensate are given by R r = R z = 2µ TF mωt 2 2µ TF Within this, (2.44), (2.45) mλ 2 ωt 2 and peak condensate density at the condensate centre is (from Eq. (2.41)) n p = µ TF U 0. (2.46)

39 2.3 Theory of a non-uniform condensate 23 It is normal to define the healing length as the uniform coherence length (2.20) at trap centre, which is in the Thomas-Fermi approximation ξ = 1 2mµTF. (2.47) Similarly we can estimate the speed of sound at peak density as (also see Eq. (2.19)) c = µtf m. (2.48) Quasiparticle excitations The Bogoliubov transformation (Eqs. (2.10) and (2.11)) can also be generalised to the non-uniform case. We take the condensate to be in the ground state of a static trap, i.e. Ψ(r) = N 0 ψ 0 (r) and assume that ˆφ(r, t) can be expanded as ˆφ(r, t) = j [u j (r)ˆb j (t) + v j (r)ˆb j (t) ], (2.49) where we take the quasiparticle operator time dependence to be ˆb j (t) = ˆb j (0) exp( iω j t), with ω j to be specified. The sufficient condition on the non-uniform quasiparticle amplitudes u j and v j, for the transformation (2.49) to diagonalise the Hamiltonian (2.35) is the so called Bogoliubov-de Gennes equations [76], Lu j (r) + N 0 U 0 ψ 0 (r) 2 v j (r) = ω j u j (r), (2.50) Lv j (r) + N 0 U 0 (ψ 0(r)) 2 u j (r) = ω j v j (r), (2.51) where L = ] [ 2 2m 2 + V T (r) µ 0 + 2N 0 U 0 ψ 0 (r) 2, (2.52) and ω j is the quasiparticle eigenvalue. The normalisation condition on these modes is dr {u i u j v i v j } = δ ij, (2.53) dr {u i v j v i u j } = 0, (2.54)

40 24 Background theory and is obtained by the requirement that the {ˆb j, ˆb j } satisfy the Bose commutation relations (see (2.4) and (2.5)). Substituting (2.30) into the field operator commutation relations (2.26) gives [ ˆφ(r, t), ˆφ (r, t)] [ ] ˆφ(r, t), ˆφ(r, t) = δ(r r ) N 0 ψ 0 (r)ψ0(r ), (2.55) [ = ˆφ (r, t), ˆφ (r, t)] = 0. (2.56) The non-local commutation relation (2.55) arises because ˆφ(r, t) acts in the space orthogonal to the condensate wavefunction ψ 0 (r) [70, 127]. The quasiparticles as defined in Eqs. (2.50) and (2.51) are in general not orthogonal to ψ 0 (r) and must be made so. Morgan et al. [129] describe how this can be done by projecting into the subspace orthogonal to the condensate, i.e. ũ j (r) = u j (r) a j ψ 0 (r), (2.57) ṽ j (r) = v j (r) + a j ψ0(r), (2.58) where a j = dr ψ 0 (r)u j (r) = dr ψ 0 (r)v j (r). (2.59) Time-dependent Gross-Pitaevskii equation For many situations a T = 0K approximation provides an excellent description of the condensate dynamics. We can obtain an evolution equation for the condensate in a systematic manner, beginning from the Bogoliubov Hamiltonian (2.35). Taking the mean of the Heisenberg equation of motion (2.29) gives Ψ(r, t) i t = where we have made use of (2.30). ] [ 2 2m 2 + V T (r, t) Ψ(r, t) + U 0 Ψ(r, t) 2 Ψ(r, t) (2.60) +2U 0 Ψ(r, t) ˆφ (r, t) ˆφ(r, t) + U 0 Ψ (r, t) ˆφ(r, t) ˆφ(r, t) +U 0 ˆφ (r, t) ˆφ(r, t) ˆφ(r, t), The the operator expectations appearing in Eq. (2.60) are: the meanfield of uncondensed particles ˆφ ˆφ ; the anomalous correlations between atoms ˆφ ˆφ ; and ˆφ ˆφ ˆφ represents collisions between thermal particles leading to the deposition of

41 2.4 Computational units 25 an atom into the condensate. In a dilute, weakly interacting gas at T = 0K, these terms involving ˆφ are all small and can be neglected to arrive at the time-dependent Gross-Pitaevskii equation Ψ(r, t) i t = ] [ 2 2m 2 + V T (r, t) Ψ(r, t) + U 0 Ψ(r, t) 2 Ψ(r, t). (2.61) 2.4 Computational units For notational simplicity and numerical convenience we carry out our simulations in dimensionless units. For trapped condensates we follow the choice of Ruprecht et al. [147] for defining distance and time units as Combining these we can define units of momentum r 0 =, 2mω T (2.62) t 0 = 1/ω T. (2.63) p 0 = 2m ω T, (2.64) and energy E 0 = ω T. (2.65) Dimensionless quantities Using the computational units given in Eqs. (2.62)-(2.65), we can introduced dimensionless variables, which we will indicate in this thesis with barred notation, i.e. r = r/r 0, (2.66) t = t/t 0. (2.67) It is worth pointing out explicitly that the wavefunction has units [Length] D/2, where D is the number of spatial dimensions, and we define a dimensionless wavefunction as ψ( r, t) = ψ( rr 0, tt 0 )r D/2 0. (2.68)

42 26 Background theory The chemical potential has units of frequency and in dimensionless form is µ = µ/ω T, (2.69) while the dimensionless potential energy is V ( r, t) = V ( rr 0, tt 0 )/ ω T. (2.70) The harmonic trapping potentials, take the form V T ( x) = x2 4, (2.71) V T ( x, ȳ) = x2 + ȳ 2, 4 (2.72) V T ( r) = x2 + ȳ 2 + λ 2 z 2, 4 (2.73) for the 1, 2 or 3 dimensional cases respectively. In Eq. (2.73) λ is the trap asymmetry parameter, which typically ranges in value from 1/20 2 in the alkali experiments. Finally, in the nonlinear term appearing in Eqs. (2.37) and (2.61), the coefficient U 0 has S.I units of [Jm 3 ] and can be written in dimensionless form as Ū 0 = U 0 / ω T r 3 0. We note that the derivation of U 0 in terms of the scattering length a 0 (2.6) is only valid in three spatial dimensions. Lower dimensional cases can be realised by having the trapping potential sufficiently tight in one (or more) directions along which the condensate motion will be frozen. In these cases Ū0 relates to to the scattering length scaled by the spatial extent of the condensate in the frozen direction (see [57]) Making the Gross-Pitaevskii equations dimensionless In computational units the time-independent Gross-Pitaevskii equation (2.37) is µ ψ 0 ( r) = [ 2 r + V T ( r) + w ψ 0 ( r) 2] ψ0 ( r), (2.74)

43 2.4 Computational units 27 where w = N 0 Ū 0. (2.75) Similarly, the time-dependent Gross-Pitaevskii equation (2.61) is transformed to i ψ( r, t) t = [ 2 r + V T ( r, t) + w ψ( r, t) 2] ψ( r, t). (2.76) where, we note that both the time-dependent and time-independent wavefunctions are normalised to unity, i.e. d r ψ 2 = 1. (2.77) Thomas-Fermi limit For completeness, we present the dimensionless form of the Thomas-Fermi density, chemical potential, speed of sound and healing lengths previously given in S.I units in subsection We also consider the generalisations of the Thomas-Fermi result to the case of one and two spatial dimensions. Thomas-Fermi in 3D The dimensionless Thomas-Fermi density, corresponding to Eq. (2.41) but normalised to unity, is n TF ( r) = µ TF V T ( r) θ ( µ TF w V T ( r) ), (2.78) allowing us to define the Thomas-Fermi wavefunction (assuming real phase) µtf ψ TF ( r) = V T ( r) θ ( µ TF w V T ( r) ). (2.79) The chemical potential (2.43) becomes in dimensionless form µ TF = ( ) 2 15 wλ 5, (2.80) 64π while the healing length and speed of sound (from Eqs. (2.47) and (2.48)) become ξ = 1 µtf, (2.81)

44 28 Background theory c = 2 µ TF. (2.82) We note that since w = N 0 Ū 0, the Thomas-Fermi chemical potential is a monotonically increasing function of the condensate population N 0. Thus as the number of atoms in the condensate is increased (i.e. as the chemical potential increases) the speed of sound increases and the healing length becomes smaller. Thomas-Fermi for 1D and 2D The Thomas-Fermi approximation can be applied to lower dimensional cases, by ignoring the kinetic energy term in the respective time-independent Gross-Pitaevskii equations. The Thomas-Fermi wavefunctions are identical in form to Eq. (2.79), but with the appropriate lower dimensioned trap and chemical potential. The latter are given for 1 and 2 dimensions by µ (1D) TF = µ (2D) TF = ( ) 3 3 w 2, 8 (2.83) ( w ) π (2.84) Vortex states Vortex states are a central interest in Bose-Einstein condensation, and in this work we have studied aspects of their behaviour in two and three dimensional cases. We have been concerned only with central vortices of the form ψ mz ( r) = ϕ mz ( r)e imzφ, (2.85) where φ is the azimuthal angle, and the integer m z is the quantised vortex circulation. In generalising the Thomas-Fermi approximation to these states we can not completely ignore kinetic energy since the centrifugal component of kinetic energy, V c ( r) = m2 z x 2 + ȳ 2, (2.86) associated with the vortex angular momentum is large near the origin. However, by defining an effective potential that includes both the external potential and the

45 2.5 Atom-radiation interactions 29 centrifugal terms (see [96]), i.e. V eff ( r) = V T ( r) + V c ( r), (2.87) we find the expression for ϕ mz, µtf ϕ mz ( r) = V eff ( r) θ ( µ TF w V eff ( r) ). (2.88) Again, the chemical potential is determined by requiring that ψ mz ( r) is normalised to unity. 2.5 Atom-radiation interactions There are many excellent treatments available on atom-radiation interactions, such as Cohen-Tannoudji et al. [45], Corney [47], Allen and Eberly [4], and Loudon [112]. For the purpose of making this thesis self contained we bring together and summarise the material necessary to quantitatively understand the couplings between internal states of condensate atoms Hyperfine structure of 23 Na and 87 Rb In the six years since the first JILA condensate, experiments have been predominantly performed in sodium and rubidium. Here we focus on the internal state couplings for these atoms since, without exception, all atom-optics type experiments with condensates to date have been conducted with these species. Nevertheless, we present the theory in this section in a sufficiently general manner to be easily applicable to any other atomic species with a single valence electron (such as other alkalis or hydrogen). In what follows we use the notation: J = L + S as the sum of spin and orbital angular momentum, while F = I + L + S is the total angular momentum including nuclear spin. The z-component of the total angular momentum is labelled m Fα. Both 23 Na and 87 Rb have nuclear spin I = 3/2 and hence have a nearly identical hyperfine structure which we show in Fig Condensates for these species are made in the ground state hyperfine manifold, 3S 1/2 for sodium and 5S 1/2 for Rubidium.

46 30 Background theory np 3/2 F=3 F=2 F= np 1/2 F= D2 D1 F=2 F= ns 1/2 F= F= m F Figure 2.1: The hyperfine states in the D transitions of 23 Na (3S 3P ) and 87 Rb (5S 5P ). The reverse ordering on the m F states in the lowest two F = 1 manifolds is because the Landé g-factor for these levels is negative.

47 2.5 Atom-radiation interactions Spherical unit vectors To describe the angular momentum properties of the atom-radiation interaction, it is convenient to decompose the vectorial part of the electromagnetic field and the dipole operators in terms of spherical unit vectors, which can be defined ê 1 = 1 2 (ê x + iê y ), (2.89) ê 1 = 1 2 (ê x iê y ), (2.90) ê 0 = ê z. (2.91) These vectors satisfy the property ê q = ( 1) q ê q, (2.92) and the orthogonality relation ê q ê q = δ qq, (2.93) where q = 1, 0, 1. For example, using relation (2.93) we can express an arbitrary vector A by A = +1 q= 1 A q ê q, (2.94) and the dot product between two vectors, A and B, becomes (using Eqs. (2.92) and (2.93)) A B = +1 q= 1 ( 1) q A q B q. (2.95) The radiation field In our development of the evolution equations for atoms interacting with a light field, we assume that the light can be treated classically. We note that Mollow [125] has shown, by applying a canonical transformation, that a coherent state radiation field can be replaced by a c-number field.

48 32 Background theory We confine our attention to plane wave fields of the form E(r, t) = 1 2 E 0(t)e i(k L r ω L t) + c.c, (2.96) B(r, t) = 1 2 B 0(t)e i(k L r ω L t) + c.c, (2.97) where the electric and magnetic field amplitudes, E 0 (t) and B 0 (t), include the polarisation and have a time dependence only to allow the fields to be turned on and off. The wave vector and angular frequency are k L and ω L respectively. Furthermore we assume that the field is purely linearly or circularly polarised, so that we may write these amplitudes as E 0 (t) = E 0 (t)ê p, (2.98) B 0 (t) = B 0 (t)k L ê p, (2.99) where p = 1, 0, 1 for right circular, linear and left circular polarisations (see subsection for polarisation discussion) Electromagnetic interaction The interaction between an atom and an electromagnetic field can be written in the Coulomb gauge in terms of a multipole expansion [112] H I = H ED + H MD + H EQ (2.100) The first term on the right of Eq. (2.100) is the electric-dipole interaction, H ED = d E, (2.101) where d is the electric dipole of the charged particle distribution, and E is the electric field. The second term is the magnetic-dipole interaction H MD = µ B, (2.102)

49 2.5 Atom-radiation interactions 33 where µ = q (L + 2S) /2m is the magnetic dipole and B is the magnetic field, and finally the electric-quadrupole interaction is H QP = Q E, (2.103) where the tensor Q is the quadrupole moment. The dominant term in Eq. (2.100) is the electric dipole interaction, with the magnetic dipole and electric quadrapole interactions being typically of order the fine structure constant smaller (α 1/137). Higher moments than these will be negligible. Here we shall only be interested in situations where the interaction is given by either the electric or magnetic dipole interactions. Which one is important depends on the internal states under consideration and the transition selection rules. Electric-dipole selection rules The selection rules for an electric-dipole transition [43] are L = ±1, (2.104) J = 0, ±1 (0 0), (2.105) m = 1, 0, +1, (2.106) The requirement that L = ±1 is because the electric-dipole operator may only connect states of different parity. The change in magnetic quantum number ( m) can be engineered by making the appropriate choice of field polarisation. Magnetic-dipole selection rules The selection rules for a magnetic-dipole transition [43] are L = 0, (2.107) J = 0, ±1 (0 0), (2.108) m = 1, 0, +1. (2.109) The magnetic dipole requirement for L = 0 is complementary to the electric-dipole selection rule (2.104), so these transitions are mutually exclusive. With reference to the hyperfine structure of rubidium and sodium (Fig. 2.1)

50 34 Background theory we see that all states within the ground state manifold have L = 0, so that only magnetic-dipole transitions are allowed between the states. Electric-dipole coupling is allowed for the D-line transitions from the ground state (3S or 3P ) to the first excited state, 3P or 5P for sodium and rubidium respectively, which has L = 1. Our primary interest is electric dipole coupling, and we consider the Rabi frequency calculation for this in detail in the next subsection. For completeness, we discuss the magnetic dipole coupling, in subsection Electric-dipole Rabi frequency. A two level atom in a resonant light field undergoes cycling between the states at a frequency known as the Rabi frequency. This characterises the strength of the radiation coupling between the states, and our aim here is to develop a quantitative expression of the Rabi frequency for electric-dipole coupling. To begin, we take two internal states of the atom, coupled by an optical electricdipole transition. We label the lower energy state l, and the higher energy state u, and for the rubidium or sodium case, these will be given in general by l = ns 1/2, F l, m Fl, (2.110) and u = np Ju, F u, m Fu, (2.111) where n = 3 or 5 for sodium and rubidium respectively. Using Eqs. (2.95), (2.96) and (2.98) we evaluate the electric dipole interaction (2.101) for a pure polarisation in the spherical basis d E = 1 2 E 0e i(k L r ω L t) d p E 0e i(k L r ω L t) d p. (2.112) The spherical dipole component d p connects particular internal states of the condensate atoms as shown for a simplified F = 0 F = 1 transition in Fig For example, we consider the case of left circularly polarised light (i.e. p = 1 in Eq. (2.112)) propagating along the axis of angular momentum quantisation. The first term on the right of Eq. (2.112) now corresponds to the absorption of a photon and the excitation to the upper m F = 1, state whereas the second term returns the atom to the lower m F = 0 state via emission. In this manner the choice of polarisation of

51 2.5 Atom-radiation interactions 35 m F = -1 m F = 0 m F = 1 d 1 d -1 d 0 d 1 d -1 m F = 0 Figure 2.2: Relationship between the components of the dipole operator and z-component of angular momentum in radiative transitions. the incident radiation can be used to select which states are coupled together. The strength of this coupling is characterised by the Rabi frequency, defined as Ω E = d p (ul)e 0 /, (2.113) where d p (ul) = u d p l, (2.114) is the electric-dipole matrix element between the states involved. In principle, the Rabi frequency could be calculated directly if the electronic wavefunctions are known. In practice however, it is calculated by using experimentally measured excited state life times, i.e. from the Einstein-A coefficients. These Einstein-A coefficients are given for the overall J-level transition, but we can use angular momentum recoupling algebra to find the individual hyperfine contributions. Invoking the Wigner-Eckhart theorem [43], allows us to express the purely geometric part of the matrix element (2.114) in terms of Clebsch-Gordan or 3Jcoefficients as follows d p (ul) = np Ju, F u, m F u d p ns 1/2, F l, m F l, (2.115) = ( F u 1 F l m Fu p m Fl ) (np Ju, F u d ns 1/2, F l ), (2.116)

52 36 Background theory Species, transition Einstein A coefficient Wavelength 23 Na, D1 (3S 1/2 3P 1/2 ) A D1 Na = s 1 23 Na, D2 (3S 1/2 3P 3/2 ) A D2 Na = s 1 87 Rb, D2 (5S 1/2 5P 3/2 ) A D2 Rb = s 1 λ = m λ = m λ = m Table 2.1: Experimentally determined values of Einstein A coefficients for the D-transitions in Sodium and Rubidium (from [132]) where (np 3/2, F u d ns 1/2, F l ) is the reduced matrix element of the electric-dipole moment in the F -basis. The Einstein-A coefficients that are experimentally tabulated relate to J-basis reduced matrix element by [47] A = (2π)3 d 2 3πɛ 0 λ 3, (2.117) where d = (np Ju d ns 1/2 ). (2.118) The reduced matrix elements in F and J are related by a 6J-coefficient (see page 310 in [81]) (np Ju, F u d ns 1/2, F l ) = { J u F u I = 3 2 F l J l = (2F u + 1)(2F l + 1) (np Ju d ns 1/2 ). } (2.119) Now combining Eqs. (2.116) and (2.119) we can relate the spherical component of the hyperfine dipole matrix element to the Einstein-A coefficient as d p (ul) = ( ) { F u 1 F l m F u p m F l J u F u I = 3 2 F l J l = (2F u + 1)(2F l + 1)3πɛ 0 λ 3 A/(2π) 3. } (2.120) In Table 2.1 we show the experimentally measured values of A and λ for sodium and rubidium.

53 2.5 Atom-radiation interactions 37 To complete the calculation of the Rabi frequency requires the electric field amplitude, which is most conveniently obtained from the measured light intensity, I, through the relationship I = 1 2 ɛ 0cE 2 0. (2.121) From Eqs. (2.113), (2.120) and (2.121), we finally write the hyperfine Rabi frequency as Ω E = ( ) { F u 1 F l m F u p m F l J u F u I = 3 2 F l J l = (2F u + 1)(2F l + 1)3λ 3 IA/(2π) 2 c. } (2.122) Magnetic-dipole Rabi frequency For any two states within the ground state hyperfine manifold, L = 0 and thus electric-dipole transitions are forbidden. However, magnetic-dipole transitions are allowed and we now provide an expression for the Rabi-frequency for this case. We label the two states involved as left (l) and right (r), with l = ns 1/2, F l, m Fl, (2.123) and r = ns 1/2, F r, m Fr. (2.124) In this case, the Rabi frequency is Ω M = r µ B l /, (2.125) which is in fact precisely the form used by Rabi in his study of nuclear magnetic resonance (see [152]). Because the magnetic moment depends purely on the angular properties of the states, i.e. since µ (L+2S), the Rabi frequency (2.125) can be calculated directly. Scott [149] has shown Ω M = g JF α µ B B 0, (2.126)

54 38 Background theory where α = ( F r 1 F l m F r p m F l ) { J r = 1 2 F r I = 3 2 F l J l = } (2.127) (2F r + 1)(2F l + 1)3/2, the Landé g-factor is [ g JF = 1 + J(J + 1) + 3 ] L(L 1) 4, (2.128) 2J(J + 1) (which is g JF = 2 for J = 1/2, L = 0) and p is the field polarisation. Finally, we note that since the B 0 relates to the field intensity as we can write the Rabi-frequency in terms of the intensity I = 1 2µ 0 cb 2 0, (2.129) Ω M = g JF α µ B 2µ0 I/c. (2.130) Field polarisation choice for magnetically trapped condensates The atoms in a magnetically trapped condensate are spin polarised in weak field seeking hyperfine (i.e. m F ) states, where the polarisation direction is determined by the bias field of magnetic trap (see [108]). It is convenient to take this direction to be the angular momentum quantisation axis, since the atoms are then in an initially well characterised state. When the electromagnetic radiation propagates along the positive direction of the quantisation axis we can identify the ê 1, ê 1, and ê 0 components of the electric and magnetic field (2.98) with left circular, right circular and linear polarisations respectively. Thus having radiation purely polarised means that a single transition between hyperfine states can be selected (i.e. one of the transitions in Fig. 2.2). If the propagation and quantisation directions differ, a field polarisation now projects to several spherical basis vectors and more than one transition may occur. In practice, the direction of the bias field in magnetic traps varies over the condensate by, at most, a few percent and so we assume that the

55 2.5 Atom-radiation interactions 39 propagation direction is always parallel to the magnetic bias field Semi-classical two state model In this subsection we derive the approximate rotating wave form of the magneticdipole interaction for a two level atom, which we use in Chapter 4. We assume that only two states, l and r, are coupled by a magnetic-dipole interaction with the light field and we expand the dipole operator relevant to the transition between these states as µ = µ(rl)σ + + µ(lr)σ, (2.131) where µ(rl) = r µ l, (2.132) and we have introduced the raising and lowering operators σ + = r l, (2.133) σ = l r. (2.134) In Eq. (2.131) we have made use of the magnetic-dipole selection rules to set the diagonal i µ i terms to zero. We can now write the magnetic-dipole Hamiltonian (2.102) projected into the { l, r } subspace as H MD ĤMD = 1 2 ei(k L r ω L t) [B 0 (t) µ(rl)σ + + B 0 (t) µ(lr)σ ] + h.c, (2.135) where we have taken the form of the magnetic field given in Eq. (2.97). Spontaneous emission In this thesis we ignore atomic coupling to the light field vacuum states, which amounts to neglecting the effects of spontaneous emission. One way to satisfy this condition is to ensure that the durations of experiments is much less than the excited state lifetime, so that spontaneous emission is negligible. This is well satisfied for magnetic dipole transitions in the ground state manifold (e.g. see Fig. 2.1) where the lifetimes are long compared to condensate lifetimes (see page 182 of [47]). For the cases where we consider electric-dipole coupling, the excited state lifetime is

56 40 Background theory quite short ( 10 7 s) and spontaneous emission from the excited level can not be neglected. However, in these situations we choose the field to be far detuned from resonance so that no appreciable occupation of the excited state occurs. Rotating wave approximation In the rotating frame defined by the unitary transformation operator Û = r r e iω Lt + l l, (2.136) the interaction Hamiltonian (2.135) takes the form Ĥ MD = ÛĤMDÛ + i Û t Û, (2.137) = 1 [ 2 ei(k L r ω L t) B 0 (t) µ(rl)σ + e iω Lt (2.138) ] +B 0 (t) µ(lr)σ e iω Lt + h.c ω L r r. Collecting together terms in Eq. (2.138) we have Ĥ MD = 1 2 eik L r [ B 0 (t) µ(rl)σ + + B 0 (t) µ(lr)σ e i2ω Lt ] (2.139) +h.c ω L r r. The rapidly varying terms exp(±i2ω L t) give rise to small Bloch-Siegert shifts [4], which in the fully quantum mechanical treatment, correspond to energy non-conserving processes such as atomic excitation accompanied by photon emission. Making the rotating wave approximation we ignore these terms and write the interaction as Ĥ MD = Ω M(t) 2 [ e ik L r σ + + e ik L r σ ] ω L r r, (2.140) where Ω M is the Rabi frequency given by (2.125). The interaction Hamiltonian, (2.140), is suitably adapted to the many-body case by writing Ĥ MD = Ω M(t) 2 dr [ ˆΨ r(r, t) ˆΨ ] l (r, t)e ikl r + h.c ω L ˆΨ r (r, t) ˆΨ r (r, t), (2.141) where ˆΨ α is the field operator for the α hyperfine state.

57 Chapter 3 Numerical techniques 3.1 Introduction The Gross-Pitaevskii equation, in its time-independent and time-dependent forms, provides a remarkably good description of condensate behaviour in a wide regime and is a major focus of this thesis. However, this is a nonlinear equation and finding solutions is not particularly straightforward. In order to achieve our goal of exploring new phenomena and obtaining physical insight into them, we are often confronted with verifying analytic approximations by numerical means. A significant portion of the work for this thesis was devoted to developing robust, efficient and generally applicable methods for both forms of the Gross-Pitaevskii equation, which we outline in this chapter. Our scheme for finding stationary solutions of the time-independent Gross-Pitaevskii equation is to use finite differences to approximate the problem as a set of algebraic equations. Rather than solve these equations directly, we instead find that minimising the sum of the residues squared is scalable to a greater range of problems. This involves using the techniques of optimisation, the relatively mature branch of numerical mathematics concerned with function minimisation. In section 3.2 we review the relevant theory of unconstrained optimisation, from which we develop novel methods for finding stationary solutions to the time-independent Gross-Pitaevskii equation. The details of these techniques and performance evaluations are presented in section 3.3.

58 42 Numerical techniques For dynamical studies, we have developed a novel propagation algorithm applicable in the 3D case with cylindrical symmetry. This has considerable memory and speed benefits over full 3D algorithms, and requires little more computation than a 2D technique. We detail this method in section 3.5. Finally we note that in this chapter all physical quantities and equations are considered in computational units (see section 2.4). So for this chapter only we shall suppress the barred notation of these quantities for notational simplicity. 3.2 Solving nonlinear algebraic equations The equations we shall wish to solve can all be written as f(x) = 0, (3.1) where f = [f 1 (X), f 2 (X),..., f n (X)] T, (3.2) is a vector of n equations, and X = [X 1, X 2,..., X m ] T, (3.3) is a vector of m variables, with m n. We will show how to write the timeindependent Gross-Pitaevskii equation in this form using a finite difference approximation in section 3.3. We assume that there always exists at least one solution, X say, for which f(x ) = 0. Because f(x) is in general nonlinear, a direct solution is usually impossible or impractical, and iterative methods must be employed. In these methods an initial guess X (0) is refined by a series of steps X (k+1) = X (k) + p, (3.4) where p is the iterative step. There is a large number of techniques which could be applied to iterate a solution to Eq. (3.1), however there is no general best technique. To make a choice we need to know something about the function f: e.g. is it smooth? how many solutions does it have? etc. In section 3.3 we will find that all the functions we consider are smooth (analytic functions of cubic order), their derivatives can be calculated, and a good

59 3.2 Solving nonlinear algebraic equations 43 initial guess can be provided (e.g. using the Thomas-Fermi solution for X (0) ). The most appropriate methods for this class of functions use Taylor expansions (based on the derivative information) to make a simplified model of the function from which a better estimate of the solution can be found. For small scale problems we consider vector root finding Eq. (3.1) and scalar optimising f 2, and show that the sequence of steps X (k) from both techniques are essentially identical. For large-scale problems, with a large number of variables and equations, neither of the aforementioned iteration schemes will be feasible because of storage and time constraints. However, there are iteration schemes based on optimising f 2 that require minimal storage and eliminate the computationally intensive operations; we choose to use the conjugate gradient technique. An excellent presentation of the numerical techniques we consider here can be found in chapter 4 of Practical Optimization [78]. Also see chapters 9-10 of Numerical Recipes [141]. Generalised derivatives Here we define the generalised derivatives needed to compactly define the multivariate Taylor expansions we will use frequently in this section. For each component function f j (X) of f(x) the gradient vector is defined as g j (X) = f j X 1 f j X 2., (3.5) which points in the direction of steepest increase of f j (X); the Hessian matrix, given by G j (X) = 2 f j X 1 X 1 2 f j X 2 X 1. 2 f j X 1 X f j X 2 X , (3.6) contains the curvature information for f j (X). We will also need the Jacobian matrix of f(x) defined as J(X) = f 1 f 1 X 1 f 2 f 2 X 1. X 2... X , (3.7)

60 44 Numerical techniques which has the individual gradients of the component functions of f as its rows, i.e. J(X) = [g 1 (X), g 2 (X),...] T Newton-Raphson (zero-finding) The Newton-Raphson method is one of the most commonly used algorithms to find the zero of a function. The essence of this technique is that for a small departure p from the current point X (k) the change in f is well approximated by the first-order Taylor expression, i.e. f(x (k) + p) f(x (k) ) + J(X (k) )p. (3.8) The right hand side of Eq. (3.8) can be regarded as a linear model f(x), which can be expected to provide a good description in some neighbourhood about X (k). This model predicts that the location of the zero for f(x (k) + p) occurs where J(X (k) )p = f(x (k) ). (3.9) In principle, it is possible to invert the Jacobian matrix to find p, but instead we leave equation (3.9) in the implicit form since in any numerical implementation, matrix factorisation (e.g. LU factorisation, Cholesky factorisation or the singular value decomposition) would be used to solve for p rather than inversion. This procedure (i.e. solving for p in Eq. (3.9) and then finding the new position according to Eq. (3.4)) is iterated until f(x) = 0 to the desired accuracy Optimisation Shortly we will show how optimisation can be used to solve a nonlinear set of equations. To begin with consider a multi-variate scalar valued function, F (X) say, which we want to minimise. To improve on our current estimate X (k) for the minimum or optimal point (X ), we make a second-order Taylor expansion for a small displacement p F (X (k) + p) F (X (k) ) + g(x (k) ) p pt G(X (k) )p, (3.10)

61 3.2 Solving nonlinear algebraic equations 45 where g and G are the gradient and Hessian matrix of F (X), found by replacing f j (X) by F (X) in Eqs. (3.5) and (3.6) respectively. We assume that this quadratic model provides a good description of F (X) in the neighbourhood of X (k), and use this to estimate the location of the minimum of F (X). The stationary point of (3.10), with respect to changes in p, is G(X (k) )p = g(x (k) ). (3.11) Using this to determine p at each step is known as Newton s method of optimisation. As long as G is positive-definite (i.e. has generalised upward curvature) at X (k) the stationary point will correspond to a minimum. Gauss-Newton optimisation The Gauss-Newton method is an adaption of Newton s method for optimising leastsquared (scalar) functions of the form F (X) = 1 2 n f i (X) 2 = 1 2 f(x) 2. (3.12) i=1 This method is of particular interest to us because the minimum of the function F (X) (due to its sum of squares form) is 0, which occurs only when each of the f i (X) = 0. In this sense zero-finding of f(x) is equivalent to optimising F (X), and the root (X ) and the optimal point (X ) of the two respective procedures will be identical. However, it is possible to have a local minima which does not have F = 0 and hence does not coincide with a root. For the applications of optimisation we are interested in, this difficulty does not arise, because the initial guess X (0) we provide (e.g. Thomas-Fermi solution) is sufficiently close to a minima X where F (X ) = 0. For functions F of the form (3.12) we can explicitly calculate the gradient vector and Hessian matrix g(x (k) ) = J(X (k) ) T f(x (k) ), (3.13) n G(X (k) ) = J(X (k) ) T J(X (k) ) + f i (X (k) )G i (X (k) ), (3.14) where J and G i are defined in equations (3.7) and (3.6) respectively. During optimisation the f i (X) (or residuals) become small as the solution to f(x) = 0 improves. i=1

62 46 Numerical techniques For this reason the Gauss-Newton technique ignores the contribution of the summation term in (3.14) to the Hessian matrix, and the Newton iteration (3.11) for the step to minima is J(X (k) ) T J(X (k) )p = J(X (k) ) T f(x (k) ). (3.15) If J is square and non-singular the Gauss-Newton iterate (3.15) is identical to the Newton-Raphson step (3.9) and we see that no particular advantage in either approach. We have introduced the optimisation formulation of the problem since it is more convenient for the large scale problems we will meet in subsection Numerical application The calculations in this thesis have been carried out using MATLAB [124], a numerical mathematics environment developed by The MathWorks. For some of the problems we tackle in this thesis, it is possible to apply built in optimisation routines from the Optimisation Toolbox add-on to MATLAB. These functions implement the techniques described here, and in addition contain convergence and conditioning checks that switch between a range of methods as the local conditions of the function vary. The fsolve.m routine of the Optimisation Toolbox solves nonlinear sets of equations of the form given in Eq. (3.1) using the least squares technique. Depending on the options supplied, this function uses the Gauss-Newton method or a minor variation of it called the Levenberg-Marquardt method. Typically fsolve.m calls a user written function that returns f(x), and J(X) is calculated approximately by finite differencing f(x), unless provided by another function. Limitations Both Newton zero finding and Gauss-Newton optimisation suffer from the same limitation: the requirement that we construct the Jacobian J(X). As the number of variables n increases, the memory required to store J(X) (which has n 2 elements) becomes unmanageably large. In addition, the factorisation techniques used to solve either Eq. (3.9) or (3.15) become increasingly slow. The number of operations required to perform the commonly used Cholesky factorisation method scales like n 3 /6. In Fig. 3.1 we show how the storage and factorisation of the Jacobian changes

63 3.2 Solving nonlinear algebraic equations 47 Jacobian Storage [MB] (a) 64 MB Factorisation Ops (b) 60 s at 1G op s number of variables Figure 3.1: Scaling of (a) storage requirements and (b) inversion operations of the Jacobian matrix on the number of variables. For comparison the dotted lines indicate in where the Jacobian: (a) would require 64MB of memory (each element stored as a 8-Byte double float); (b) would require 60 seconds to invert on a system performing a billion floating point operations per second. with the number of variables n. The factorisation time usually limits the number of variables to the order of a thousand. For one dimensional Gross-Pitaevskii case, where the number of discretised points (i.e. variables) is of the order of hundreds, the techniques provided by fsolve.m will provide an adequate method for solving the problem. However in two dimensions and above, where the number of discretised points is typically in the range of , we need to consider alternative methods Large scale solvers: the conjugate gradient technique The Newton-Raphson or Gauss-Newton methods could be adapted to large scale problems by using sparse matrix techniques, whereby only the non-zero elements of the Jacobian matrix are stored. However, our ability to solve for the iterative step (p in Eq. (3.9) or (3.15)) using factorisation techniques is very dependent on the structure of the Jacobian matrix. We have instead chosen to use the conjugate

64 48 Numerical techniques gradient technique, which only requires the gradient (which is of order n). We review the application of this method to purely quadratic function and then outline how it is adapted to the general nonlinear case. Quadratic functions We begin with a quadratic function of the general form Φ(X) = c T X XT GX, (3.16) where we have ignored any constant term. We also assume that the Hessian matrix G (which is a constant matrix for the quadratic case) is symmetric and positive definite, so that a minima exists. The gradient of Φ is g(x) = c + GX. (3.17) When the number of variables in X is large, directly solving the Newton equation (3.11) to find the minimum of Φ is a formidable task. Instead, it is advantageous to optimise Φ over a subspace of X. We denote the subspace we will use for the k-th iteration as P k, and consider it to be formed by the k + 1 linearly independent vectors {p 0, p 1,..., p k }. So, for the first iteration (X (0) X (1) ) we minimise on the one dimensional subspace P 0 (i.e. along the line X (0) + t p 0, t R). For the second iteration we minimise over a 2D plane (defined by the vectors {p 0, p 1 }), and so on, increasing the subspace dimension by one at each stage. The key to the conjugate gradient method is how the new vectors, the p i, are generated at each iteration. We will come back to this point shortly. The k-th iteration is thus a k + 1 dimensional minimisation problem, which we write as min Φ(X (k) + P k w), (3.18) where w (a k + 1 element vector that we wish to solve for) is the projection of X X (k) into the P k subspace, and P k = [p 0, p 1,... p k ], is a matrix with the basis vectors as its columns. Expanding Φ in terms of w yields Φ = const. + w T P T k g(x (k) ) wt P T k GP k w. (3.19)

65 3.2 Solving nonlinear algebraic equations 49 Using the Newton formula (3.11), the minima of Φ in the P K subspace is w = (P T k GP k ) 1 P T k g(x (k) ), (3.20) where we have formally inverted the projected Hessian matrix term for later convenience. Using the P k matrix to transform back to our original space (i.e. n-space), we find the new point to be X (k+1) = X (k) P k (Pk T GP k ) 1 Pk T g(x (k) ). (3.21) So far we have not gained anything, since determining w still requires full knowledge (i.e. storage and inversion) of G. However, several points allow us to simplify the computation of X (k). The gradient of Φ at the P k subspace minima (i.e. g(x (k+1) )) will be orthogonal to this subspace. If this were not true, then moving in the direction defined by the projection of the descent direction (i.e. g(x (k+1) )) projected into P k would further minimise the function, in contradiction to the initial premise. In terms of the basis vectors, this property can be written g(x (j) ) T p i = 0, j > i. (3.22) This means that only one column of the matrix Pk T Pk T g(x(k) ), and Eq. (3.21) can be simplified to will contribute to the term X (k+1) = X (k) + γ P k (P T k GP k ) 1 e k, (3.23) where e k is the k-th column of the identity matrix and γ = g T (X (k) ) p k. The second feature we use to drastically simplify subspace minimisation is the defining characteristic of the conjugate gradient technique: if the p k are chosen to be mutually conjugate to the Hessian matrix 1, i.e. p T i Gp j = [ ] p T i Gp j δij, then the matrix product Pk T GP k is diagonal and is easily inverted. This would allow us to simplify Eq. (3.23) to X (k+1) = X (k) + α k p k, (3.24) 1 Two vectors, a and b are conjugate if a b = 0. We say that these two vectors are conjugate to the matrix M if a T Mb = 0.

66 50 Numerical techniques where α k is the constant α k = g(x (k) ) T p k /p T k Gp k. (3.25) The conjugate gradient technique is essentially a method to generate new directions p k conjugate to the Hessian matrix. The procedure to do this at each step is p k = g(x (k) ) + β k 1 p k 1, (3.26) where β k 1 = g(x(k) ) 2 g(x (k 1) ) 2. (3.27) We will not show the details of how these conjugate directions were derived and instead refer the reader to either [78] or [150] for details. Thus, generating the conjugate directions only requires us to evaluate the gradient of Φ. However, α k appearing in Eq. (3.24) still depends on G. This term can be calculated without forming G by considering how g changes along the direction p k. This is the vectorial analogue of the directional derivative, which we formally write as D pk (g(x)) X (k) = [ ] g(x (k) + h p k ) g(x (k) ) lim, h 0 h (3.28) = Gp k. (3.29) We arrive at the last line by making use of Eq. (3.17). The limit to zero is unnecessary in the quadratic case since the curvature is constant, but we retain it for the later non-quadratic extension. In summary we outline the conjugate gradient iterative procedure: Beginning at X (0) with β 0 = 0, and p 0 = g(x (0) ), the steps α k = g(x(k) ) 2 D pk (g), (3.30a) X (k) X (k+1) = X (k) + α k p k : improved minima, (3.30b)

67 3.2 Solving nonlinear algebraic equations 51 β k = g(x(k+1) ) 2 g(x (k) ) 2, (3.30c) p k+1 = g(x (k+1) ) + β k p k : new direction. (3.30d) are cycled through. For each complete iteration of this procedure (except for the first one) only two evaluations of the gradient are required. If exact arithmetic is used, this method will calculate the minima X of the quadratic function (3.16) within n (the number of variables) iterations. Applying the conjugate gradient technique to nonlinear functions The conjugate gradient method is an attractive method for minimising a nonlinear function F (X) when the gradient is readily calculable. We can visual the conjugate gradient algorithm (Eqs. (3.30a)-(3.30d)) as minimising a quadratic model of F (X), obtained by the local derivatives. If the Hessian matrix of F (X) changes rapidly with X, then the quadratic model may become inappropriate and the procedure may no longer decrease F at each step. In many cases it is necessary to determine α k by a line search, or univariate minimisation along the direction p k. This is expensive in terms of function evaluations of F (X) and is to be avoided if possible in large problems. It is usually necessary to check that F (X) decreases at each new position (i.e. at step (3.30b)). If F (X) begins to increase at some step k say, then it is probably necessary to reset the algorithm, by taking β k = 0, p k = g(x (k) ), and begin iterating again. Steepest descent It is interesting to compare the conjugate gradient method against the steepest descent algorithm, which also only requires evaluation of the gradient of the function we wish to minimise. The main difference between the two methods is that steepest descent always minimises along the direction p k = g(x (k) ). This choice is intuitively appealing because this is the immediate direction of maximal decrease. Also, the steepest descent algorithm is more efficient with less computation required for each iteration, since we no longer need to ensure that the directions are conjugate to the Hessian. In detail, the iteration process is identical to the conjugate gradient method (Eqs.(3.30a)-(3.30d)), except the step (3.30c) is unnecessary as β = 0 al-

68 52 Numerical techniques ways. In Fig. 3.2 we compare the conjugate gradient and steepest descent algorithms 4 (a) 2 y (b) 2 y x Figure 3.2: Comparison of paths taken by the (a) conjugate gradient and (b) steepest descent methods to optimise the 2D quadratic function, F (X) = (X 1 ) 2 + 6(X 2 ) 2, whose level curves are shown. The initial (X (0) ) and the optimal (X ) points are indicated with a circle and square respectively. at minimising a two dimensional quadratic function. Neither of these approaches is appropriate for such a low dimensional function (n = 2), but this example does indicate a common pathology of steepest descent. The simple Newton method (Eq. (3.11)) would optimise this problem in a single step. The conjugate gradient method minimises over the one dimensional (steepest descent direction) subspace for the first step; the second step minimises over the two dimensional subspace, which is equal to the dimension of the problem we consider here, so the true minimum is found in two steps. In Fig. 3.2(b) the weakness of the steepest descent algorithm is clearly apparent. This method minimises only along the current direction it is concerned with and often suffers from this form of oscillatory behaviour. After ten iterations, the steepest descent algorithm is a distance of 0.2 from the optimal point X = 0. The level curves of the quadratic function can be thought of as defining a valley, and the optimisation procedure is trying to find the bottom of the valley. As this valley becomes longer and narrower, this characteristic behaviour (see Fig. 3.2(b))

69 3.3 Solving for Gross-Pitaevskii eigenstates 53 of the steepest descent becomes more problematic and convergence is painfully slow. The extra (defining) step in each iteration (i.e. Eq. (3.30c)), which the conjugate gradient algorithm takes to ensure each direction is conjugate to the Hessian matrix, is a small price for this method s far superior convergence properties Summary Having investigated many different algorithms it is worth reviewing our basic findings. Vector root finding with the Newton-Raphson method or optimising using the Gauss-Newton method is a highly efficient way to solve small systems of equations. Both of these algorithms require knowledge of the Jacobian matrix and, as the number of variables and equations increases, these powerful methods rapidly become overtaxed. The conjugate gradient technique is a scheme to optimise a function only requiring evaluations of the gradient. Furthermore, though the conjugate gradient technique is less efficient than the Newton techniques per iteration, the conjugate gradient procedure does not require matrix factorisation and hence the iterations can be evaluated much more rapidly. 3.3 Solving for Gross-Pitaevskii eigenstates Notation We seek a notation that will conveniently describe the variety of situations we consider in this thesis. This includes the 1D case, where the variables are stored in vectors and the operators acting on them are matrices, and 2D and (cylindrically symmetric) 3D cases where the variables are most conveniently stored as matrices and the operators are multi-dimensional arrays. We have chosen to adopt a tensor-like notation since it easily extends to describe arrays of any dimension 2. The number of subscript indices indicates the dimension of the array, e.g. A i is a vector, B ij is a matrix and C ijk is a three dimensional array. 2 Note we do not use the Einstein summation convention, and explicitly indicate when summation is carried out over any index.

70 54 Numerical techniques Real 1D case The one dimensional Gross-Pitaevskii equation (2.74) and associated boundary conditions are [ ] d 2 dx + µ V 2 T (x) w ψ(x) 2 ψ(x) = 0, (3.31) ψ(x = + ) = 0, (3.32) + ψ(x = ) = 0, (3.33) dx ψ(x) 2 1 = 0. (3.34) Notice that every condition of the solution is formalised as an equation. Once these equations are discretised, they form a set of algebraic equations of the form f(x) = 0, and they can be optimised. Typically the trap will be harmonic and of the form V T (x) = x 2 /4. However, this is not essential, so we retain V T as a general function for the time being. We do however restrict our attention to potentials which are increasing sufficiently fast that the asymptotic boundary conditions (Eqs. (3.32)-(3.33)) apply. We also restrict our attention here to the case where ψ is a real function and consider the complex case separately in section ψ V T x 1 x x m Figure 3.3: Representations of a 1D functions on a discretised mesh. A harmonic oscillator ground state wavefunction (solid) and scaled potential (dotted) are shown with their respective discretised representations ψ i (circles) and V i (crosses). To discretise the differential equation (3.31) a suitable basis to approximate ψ must be selected. We use the usual finite difference basis, whereby we sample ψ at m x uniformly spread points over some range of x-values, which are contained in the

71 3.3 Solving for Gross-Pitaevskii eigenstates 55 vector x = [x 1, x 2,..., x mx ] T, (3.35) where the spacing x = x i+1 x i. The wavefunction and trapping potential sampled on this grid are ψ i = ψ(x i ), (3.36) V i = V T (x i ), (3.37) as shown schematically in Fig Using this discretization, the differential equation can be written as a vector of equations m x f i = D (2) ij ψ j + [ µ V i w(ψ i ) 2] ψ i, 1 i m x, (3.38) j=1 where D (2) ij is an operator based on finite differences (see Appendix B.1) for which m x j=1 D (2) ij ψ j ψ (x i ). (3.39) The value of f i (3.38) corresponds to the residual of the differential equation at the grid point location x i. On our uniform grid it is not practical to have points at ± where the exact boundary conditions are evaluated (Eqs. (3.32)-(3.33)). However, as long as the potential increases sufficiently fast on the x-grid chosen, the wavefunction will rapidly decrease towards zero, and evaluating this condition at the grid end points should be sufficient. That is, we assume the boundary conditions are well approximated by ψ(x 1 ) = ψ(x mx ) = 0, which is formalised by requiring the two equations f B1 = ψ 1, (3.40) f B2 = ψ mx, (3.41) to be equal to zero. Finally the normalisation condition (3.34) is well approximated

72 56 Numerical techniques by using the rectangular rule for numerical integration 3 m x f N = ( x) (ψ i ) 2 1. (3.42) i=1 The complete set of equations (3.38)-(3.42) can be expressed as the vector f(x) = 0 of m x + 3 equations. As discussed in section 3.2.2, we replace f(x) by the scalar function F (X), which we minimise, where (see Eq. (3.12)) F ({ψ i, µ}) = 1 2 [ mx (f i ) 2 + i=1 We refer to F as the optimality function. ] 2 (f Bi ) 2 + (f N ) 2. (3.43) All the equations f i are polynomials of the variables {ψ i, µ} up to cubic order. Thus F is also a continuous polynomial function (of up to order 6), and the gradient can be explicitly evaluated (see Appendix B.3) F ψ α = m x i=1 i=1 D (2) iα f i + (µ V α 3w(ψ α ) 2 f α (3.44) + ψ 0 δ α0 + ψ mx δ αmx + f N 2( x)ψ α, mx F µ = ψ i f i, (3.45) i=1 where δ ij is the usual Kronecker δ-symbol. Now that we have the gradient of F with respect to all the variables, we can apply the conjugate gradient technique. If we could survey the complete landscape of F in (3.43), we would see many minima. Some of these will be such that F 0, and will correspond to noded excited states. An essential key to the success of an optimisation technique is related to how close the initial guess we choose is to the desired solution. E.g. an initial guess for the variables X (0) = {ψ (0) i, µ (0) } of say the Thomas-Fermi solution (Eqs. (2.4.3) and (2.83)) will usually be sufficient to ensure we begin optimising in the valley of F near the ground state. To start searching for an excited state instead, it usually suffices to simply modulate the Thomas-Fermi wavefunction (2.83) so that it has the number of nodes we desire for the final state. Generally, we find that in optimising 3 We note that higher order discrete integration methods, such as Simpson s rule, are exactly equivalent to the rectangular rule except for the boundary points where we assume the function is zero anyway.

73 3.3 Solving for Gross-Pitaevskii eigenstates 57 states, features such as the number of nodes of the initial state are more important in determining the final state than having the nodal locations correctly placed. The one dimensional case usually involves a low number of variables (typically < 1000) and is well suited to being solved with Gauss-Newton techniques. For Gauss-Newton techniques (e.g. fsolve.m) the quantity of interest is not the gradient of F but the Jacobian matrix, which we give in Appendix B.3 for completeness D results and performance In previous work [19] we have shown that eigenstate optimisation based on fsolve.m is an extremely accurate and robust technique. We do not aim to reproduce those results here, but instead consider factors affecting the performance and accuracy of the conjugate gradient approach. Even though it is not necessary to use the conjugate gradient technique in the 1D case, this is an important test bed for understanding how to most efficiently use this method. Optimising a system of n-equations near a minima can be visualised as descending into a valley of an n-dimensional function. Our rate of descent is critically dependent on the exact nature of the function, and can be extremely slow when we are in regions where the function is rapidly changing, the valley is extremely long and narrow, or our algorithm suffers from numerical round-off errors. Because of the high dimensionality of the functions we wish to optimise, these factors are difficult to diagnose, and may change subtly with the precise problem being considered. The fsolve.m function is designed to be robust and adapts the methods it uses to overcome some of these difficulties. In contrast, the conjugate gradient technique is a simple monolithic algorithm which attempts to optimise by taking a large number of iterations. Making this process efficient often requires the user to have an intuition for how to cast the problem most appropriately. Fig. 3.4 shows a w = 100 ground state and the Thomas-Fermi solution, which served as the starting point for the optimisation procedure. We examine the convergence of the optimality function to zero in Fig. 3.5(a) as this solution is optimised. A common problem which occurs when optimising is that the normalisation of the eigenstate differs from unity while the other conditions (i.e. the differential equation and boundary conditions) are well satisfied. If the algorithm attempts to adjust the normalisation, the residues on the other equations become large, and so the solution gets trapped in a valley where F 0. Some benefit is to be had from periodically

74 58 Numerical techniques 0.4 ψ x Figure 3.4: Optimised eigenstate (solid) with µ = and Thomas-Fermi initial state (dashed) with µ TF = Nonlinearity is w = (a) 10 1 (b) F 10 5 µ iterations iterations Figure 3.5: Performance comparison of 1D algorithm. (a) The value of the optimality function F (3.43) versus number of iterations: dash-dot line, solution renormalised every 600 steps; solid line, solution renormalised and interpolated to a different grid size every 600 steps. (b) the difference in chemical potential from the value µ = Renormalisation and grid changes every 600 iterations are indicated by vertical dashed lines. Parameters are: w = 100, m x = 401 points. renormalising the solution and restarting the conjugate gradient procedure. We can see in Fig. 3.5(a), where we renormalise the solution (dash-dot) every 600 iterations,

75 3.3 Solving for Gross-Pitaevskii eigenstates 59 that the convergence of F is still relatively slow. A simple way to analyse the accuracy of this eigenstate as it is being optimised is to compare the current estimate of the eigenvalue (which is a variable in our optimisation scheme) against the benchmark value of µ = , found for this state using the fsolve.m algorithm (see dash-dot line in Fig. 3.5(b)). Here we see that though the value of F is slowly improving with iterations, the eigenvalue fails to improve upon 2 decimal places of agreement with the fsolve.m value. Finally, we note that the conjugate gradient algorithm takes about 7 seconds 4 to complete the 2400 iterations of the 401 point solution. Multiple-grid approach The basic approach we outline here to improve the conjugate gradient convergence rate was inspired by the multi-grid methods (e.g. see chapter 19 of [141]) commonly used to solve elliptic partial differential equations. We begin by iteratively solving for the wavefunction with a much sparser discretisation than finally desired. Then, once sufficient accuracy is achieved, we interpolate to a larger, more dense grid and iterate again. This procedure continues until the solution has been found to the desired accuracy on the final full-sized grid. 100 pts 200 pts 300 pts 401 pts 600 iterations 600 iterations 600 iterations 600 iterations Table 3.1: Multiple-grids and iterations used in Fig Total optimisation time 6 seconds. The improvement in performance is dramatic, as shown with the solid lines in Fig In that example we used four grids covered by 100, 200, 300 and 401 points respectively. Beginning on the 100 point grid, 600 conjugate gradient iterations are applied to the discretised wavefunction, then the solution obtained is renormalised and interpolated to the next grid as schematically shown in Table 3.1. In Fig. 3.5(a) we see that the convergence of this procedure is far superior than beginning with the largest size grid and performing the same total number of iterations. Another benefit of this procedure, which becomes more significant in the higher dimensional cases, is that carrying out iterations at smaller grid sizes is significantly faster. For the results in Fig. 3.5, the interpolated solutions (solid line) took a total of 6 seconds 4 The times given in this section are as measured on a AMD K7 (Athlon) 600MHz workstation running MATLAB 5.3 under Linux.

76 60 Numerical techniques including the interpolation steps (compared to 7 seconds for the solution on a single large grid (dash-dot line)). It is worth re-emphasizing that the multiple-grid method has much lower final value of the optimality function and agrees with the fsolve.m calculated eigenvalue to at least 4 decimal places (Fig. 3.5(b)). Solution accuracy Penckwitt has examined our fsolve.m and conjugate gradient algorithms in detail in [135], with a particularly careful analysis of the accuracy of these solutions. His findings show that ensuring F 1 on a reasonably large grid (each dimension covered by at least 100 points say) is usually sufficient for a solution to be quite accurate. The degree of accuracy required depends on the application which the eigenstates are to be used for. Most often the eigenstates we make are used as initial states in dynamical simulations. For this application a final value of F < is usually adequate, though our conjugate gradient and the fsolve.m (when the variable count is sufficiently low for it to be applicable) procedures can easily produce states with orders of magnitude lower F values. Other performance factors In Fig. 3.6 we show how other factors affect the algorithm convergence. To simplify the comparison we use the single-grid method in all cases. Fig. 3.6(a) shows the conjugate gradient algorithm is more efficient at larger values of nonlinearity. To a certain extent this accelerated convergence is because the initial Thomas-Fermi state more closely resembles the final eigenstate. This is counter to the behaviour of other methods for finding eigenstates such as imaginary time propagation and basis set expansions which perform more poorly for high nonlinearities. Fig. 3.6(b) shows that increasing the accuracy of the finite difference approximation to the Laplacian (see Appendix B.1), causes the algorithm to converge more slowly. 500 iterations with the 3 point stencil has decreased the value of F by more than an order of magnitude over the same number of iterations with the 5 or higher point stencils. Finally, in Fig. 3.6(c) we address the issue of how often to reset the conjugate gradient direction to the steepest descent direction. Our results show that increasing the number of steps benefits the overall convergence rate and that in this example choosing the number of steps between resets to be approximately equal to the num-

77 3.3 Solving for Gross-Pitaevskii eigenstates (a) 10 1 (b) F 10 5 F iterations (c) iterations F iterations Figure 3.6: Rate of 1D conjugate gradient convergence. (a) Nonlinearity dependence: (solid) w = 100, (dotted) w = 1, 000, (dashed) w = 10, 000; (b) Accuracy of finite differences: (solid) 3 point stencil, (dotted) 5 point stencil, (dashed) 7 point stencil, (large dots) 9 point stencil; (c) Number of iterations until direction reset: (solid) 1, (dotted) 100, (dashed) 200, (large dots) 400. Other parameters are (unless indicated otherwise above) w = 100, 7 point stencil derivative operator and direction reset every 200 iterations. ber of variables ( 400) seems to be most effective. We note that the solid line in Fig. 3.6(c), where the direction is reset at every step, corresponds to the steepest descent algorithm (see discussion in section 3.2.4), which displays almost 3 orders of magnitude worse performance than the other cases. At the 1000 iteration point, the

78 62 Numerical techniques conjugate gradient techniques are all exhibiting a respectable degree of convergence, whereas the steepest descent has essentially levelled off Real 2D case We remind the reader that in the two dimensional case the Gauss-Newton method is usually overtaxed by the variable count and we must use a large scale technique. The full boundary value problem of the two dimensional Gross-Pitaevskii equation is [ ] 2 x y + µ V 2 T (x, y) w ψ(x, y) 2 ψ(x, y) ψ(x, y) = 0, (3.46) x 2 +y 2 = 0, (3.47) + dx + dy ψ(x, y) 2 1 = 0. (3.48) We are interested in solving this problem on a uniform rectangular mesh defined in the x and y directions by the vectors x = [x 1, x 2,..., x mx ] T, (3.49) y = [y 1, y 2,..., y my ] T, (3.50) where the spacings are x = x i+1 x i and y = y i+1 y i respectively. The wavefunction and trapping potential are most conveniently stored as matrices indexed by the respective coordinates on the grid, i.e. ψ ij = ψ(x i, y j ), (3.51) V ij = V T (x i, y j ), (3.52) and the differential equation (evaluated at each mesh location) is now approximated as the matrix of algebraic equations f ij = [ mx k=1 D (2) ik ψ kj + m y k=1 D (2) jk ψ ik ] + [ µ V ij w(ψ ij ) 2] ψ ij, 1 i m x 1 j m y. (3.53)

79 3.3 Solving for Gross-Pitaevskii eigenstates 63 See Appendix B.1 for details on the Laplacian discretisation where we define the (2) y-direction finite difference operator D jk. In a similar spirit to the 1D case, the 2D boundary conditions are evaluated on the boundary of the rectangular grid by requiring the wavefunction to be zero there, i.e. f B1 j = ψ 1j, (3.54) f B2 j = ψ mxj, (3.55) f B3 i = ψ i1, (3.56) f B4 i = ψ imy, (3.57) where 1 i m x and 1 j m y. In other words, we assume the wavefunction is exponentially small at the edge of the grid. Finally, the normalisation condition (3.48) is approximated by using rectangular integration m x m y f N = ( x y) (ψ ij ) 2 1. (3.58) i=1 Equations (3.53)-(3.58) are solved by minimising the residuals squared (3.12), i.e. The gradient of F is F ψ αβ = F ({ψ ij, µ}) = 1 2 [ mx i=1 mx F µ = D (2) iα f iβ + [ m x j=1 m y m y [ (f ij ) 2 + (fb1 j) 2 + (f B2 j) 2] (3.59) i=1 m x + i=1 m y j=1 j=1 j=1 [ (fb3 i) 2 + (f B4 i) 2] + (f N ) 2 ]. D (2) jβ f αi ] + (µ V αβ 3w(ψ αβ ) 2 )f αβ (3.60) +f B1 βδ 1α + f B2 βδ mxα + f B3 αδ 1β + f B4 1δ my1 + f N 2( x y)ψ αβ, i=1 m y j=1 see Appendix B.4 for details. ψ ij f ij, (3.61)

80 64 Numerical techniques Complex 2D case To solve for complex solutions of the Gross-Pitaevskii equation, we split the wavefunction into real and imaginary parts, according to u ij = R(ψ ij ), (3.62) v ij = I(ψ ij ). (3.63) We discretise Eq. (3.46) as in the real case (i.e. Eq. (3.53)), but take the real and imaginary parts separately to get two sets of real equations f R ij = f I ij = m x k=1 D (2) ik u kj + m y k=1 + [ µ V ij w [ (u ij ) 2 + (v ij ) 2]] u ij, m x k=1 D (2) ik v kj + m y k=1 + [ µ V ij w [ (u ij ) 2 + (v ij ) 2]] v ij. D (2) jk u ik (3.64) D (2) jk v ik (3.65) On the rectangular grid the boundary conditions are analogous to (3.54)-(3.57), except applied to both the real and imaginary parts fb R 1 j = u 1j, (3.66) fb R 2 j = u mxj, (3.67) fb R 3 i = u i1, (3.68) fb R 4 i = u imy, (3.69) fb I 1 j = v 1j, (3.70) fb I 2 j = v mxj, (3.71) fb I 3 i = v i1, (3.72) fb I 4 i = v imy, (3.73) where 1 i m x and 1 j m y. Finally, the normalisation condition (3.58) in terms of u and v is m x m y [ f N = ( x y) (uij ) 2 + (v ij ) 2] 1. (3.74) i=1 j=1

81 3.3 Solving for Gross-Pitaevskii eigenstates 65 The least squares expression is now F ({u ij, v ij, µ}) = [ m x m y [ (f R ij ) 2 + (fij) I 2] + (f N ) 2 (3.75) i=0 j=0 m y j=0 m x [ (f R B1 j) 2 + (f R B 2 j) 2 + (f I B 1 j) 2 + (f I B 2 j) 2] [ (f R B3 i) 2 + (fb R 4 i) 2 + (fb I 3 i) 2 + (fb I 4 i) 2] ]. i=0 The gradient corresponding to this function can be readily evaluated (see Appendix B.4), allowing application of the conjugate gradient technique D results and performance Algorithm performance is a major consideration in the two and higher dimensional cases. The central issue is that in 2D, where the number of variables n is large, each iteration takes a significant amount of time. Furthermore, the number of iterations needed to optimise a function usually scales as the number of variables. We have found that the multiple-grid method introduced in section is essential to allowing the conjugate gradient algorithm to compute a solution in a reasonable time. In Fig. 3.7(a) we present a w = 2500 eigenstate calculated on a point grid. The optimisation time, from a Thomas-Fermi initial state to the final solution was 196 seconds, using the multiple-grid arrangement outlined in Table 3.2. In Fig. 3.7(b) we show how the optimality function decreases throughout pts pts pts 200 iterations 250 iterations 150 iterations pts points pts 5 iterations 10 iterations 50 iterations points points 5 iterations 5 iterations Table 3.2: Multiple-grids and iterations used in Fig Total optimisation time 196 seconds. the optimisation process. Vertical dashed lines indicate the point in the calculation

82 66 Numerical techniques (a) ψ y (b) 0-10 x 20 (c) 10 F 10-5 µ iterations iterations Figure 3.7: Optimising a real w = 2, 500 2D eigenstate. (a) the final eigenstate density; (b) the reduction in the optimality function, F, with iterations; (c) the eigenvalue versus iterations. A multiple-grid approach is used with 8 grids, each new grid is indicated with a vertical dashed line in (b) and (c). when the interpolation to the new grid size is carried out. Finally, in Fig. 3.7(c) we see that the eigenvalue, µ, is accurate to about 2 decimal places by the end of the optimisation stage on the second grid (i.e. after 71 seconds of optimisation). In general the performance of the 2D algorithm is effected in a similar manner to the 1D case: increasing the nonlinearity w and number of steps before reseting the conjugate direction speeds up convergence, whereas increasing the finite difference

83 3.3 Solving for Gross-Pitaevskii eigenstates 67 y (a) λ=1, m z =0 (b) λ=1, m z = y (c) λ=1, m z = x (d) λ=8 1/2, m z = x log 10 density Figure 3.8: Solutions of the 2D Gross-Pitaevskii equation using optimisation. Density plots of (a) real isotropic ground state, (b) m z = 1 isotropic vortex state, (c) m z = 2 isotropic vortex state, (d) an asymmetric m z = 1 vortex state (λ = 8). Parameters: w = 2500 and solutions are on point grids. accuracy slows down convergence. We show results for four different 2D eigenstates in Fig. 3.8(a)-(d) which vary in topological nature and trapping potential asymmetry. Here the initial solutions are given by Eq. (2.88), and the convergence of the optimisation procedure for each case is detailed in Fig The multiple-grid progression for these states is shown in Table 3.3. The ground state in an isotropic trap is shown in Fig. 3.8(a), and was obtained with the real 2D algorithm (see section 3.3.4). The convergence behaviour is illustrated by the solid line in Fig In Figs. 3.8(b) and (c) vortex states of the symmetric trap with m z = 1 and m z = 2 respectively are displayed. These eigenstates have a phase circulation and are accordingly complex, and so were solved for using the complex algorithm (see section 3.3.5). Their convergence rates (Fig. 3.9) are seen to be slower than for the ground

84 68 Numerical techniques 10 0 F iterations Figure 3.9: Convergence of the optimality function for the states shown in Fig. 3.8: solid line, isotropic real state; dashed line, isotropic m z = 1 vortex state; large dots, isotropic m z = 2 vortex; dotted, anisotropic m z = 1 vortex. Multiple-grids as shown in Table 3.3 and other parameters as in Fig state, in part because the fully complex procedure uses twice as many variables as the real case. The asymmetric m z = 1 5 vortex state (d) optimises most slowly, and reaches a final value of only F on the largest grid. In general solving for states with λ 1 is significantly slower and finding an accurate solution requires more iterations and a larger set of multiple-grids. By using 8 grids spanning from and 1000 iterations on each grid a final value of F = was achieved for the state shown in Fig. 3.8(d) points points points 200 iterations 200 iterations 200 iterations points points 200 iterations 200 iterations Table 3.3: Multi-grid parameters used in Fig Total iteration times: real case, 57 seconds; complex case, 187 seconds. 5 We note that m z is the eigenvalue of the angular momentum in 2D. In an asymmetric trap this is not a conserved quantity, however we call this state m z = 1 because it has 2π-phase circulation.

85 3.4 Time-independent Gross-Pitaevskii equation in cylindrical coordinates Time-independent Gross-Pitaevskii equation in cylindrical coordinates Cylindrical coordinates Here we use the usual cylindrical coordinates {r, z, φ} and refer to the directions as the radial (r), axial (z) and azimuthal (φ) respectively. These coordinates are defined to have values lying in the ranges 0 < r <, (3.76) < z <, (3.77) π φ < π, (3.78) so that each point in space has a unique set of values in cylindrical coordinates (except for points lying on the z-axis). Any function which has no dependence on the azimuthal angle is called cylindrically symmetric, and can be written in the form A(r, z) Cylindrical wavefunction When the potential is cylindrically symmetric, i.e. V (r) = V (r, z), then the (dimensionless) Gross-Pitaevskii equation (2.74) can be written as ( 2 µψ(r, z, φ) = r r r + 1 ) 2 r 2 φ + 2 ψ(r, z, φ) (3.79) 2 z 2 + [ V (r, z) + w ψ(r, z, φ) 2] ψ(r, z, φ). The most general class of wavefunction we consider is ψ(r, z, φ) = ϕ mz (r, z)e i(mzφ+ϑ), (3.80) where m z is an integer and ϑ is a constant overall phase that we shall take to be zero. We refer to ϕ mz (r, z) as the cylindrical wavefunction and note that, when m z = 0, we have ψ(r, z, φ) = ψ(r, z), so that the wavefunction is cylindrically symmetric. When m z 0, Eq. (3.80) describes a vortex state, which requires ϕ mz (r, z) 0 as r 0 (see [35]).

86 70 Numerical techniques Substituting (3.80) into Eq. (3.79) we get ( 2 µϕ mz (r, z) = + ) r + 2 z 2 r r [ m 2 z r + V (r, z) + w ϕ 2 m z (r, z) 2 ϕ mz (r, z) (3.81) ] ϕ mz (r, z), which we refer to as the cylindrical time-independent Gross-Pitaevskii equation. For the rest of this section we shall concern ourselves with the solution of Eq. (3.81) Discretising the cylindrical Gross-Pitaevskii equation In Eq. (3.81) ϕ mz depends only on r and z, so this equation is essentially a two dimensional problem. The full boundary value problem we wish to solve is [{ 2 r + 1 } ] r r z + µ ϕ 2 mz (r, z) (3.82) [ ] m 2 z r + V 2 T (r, z) + w ϕ mz (r, z) 2 ϕ mz (r, z) = 0, ϕ mz (r, z) = 0, (3.83) r 2 +z 2 + dz + 0 dr 2πr ϕ mz (r, z) 2 1 = 0, (3.84) and the numerical solution can be implemented in a similar manner to the 2D algorithm in subsection We give the details of the optimality function and analytic derivatives for this case in Appendix B Cylindrical eigenstate results and performance In Fig we show results for cylindrical Gross-Pitaevskii eigenstates. These states were optimised using the same multiple-grid procedure outlined in Table 3.3, but with 400 iterations used on each grid. The average time to work through the complete multiple-grid sequence to the solution on the final point grid was 140 seconds. The convergence rates for the states are shown in Fig. 3.11, and we see that (as noted in the 2D section 3.3.6) states with λ 0 tend to optimise more slowly than states in isotropic trapping potentials.

87 3.4 Time-independent Gross-Pitaevskii equation in cylindrical coordinates (a) λ=1 (b) λ=8 1/2 (c) λ=0.1 (d) λ=1 m z =1 z r r r r log 10 density Figure 3.10: Solutions of the cylindrical Gross-Pitaevskii equation found using optimisation. Density plots of: (a) ground state in isotropic trap, (b) ground state in oblate trap with λ = 8, (c) ground state in prolate trap with λ = 0.1, (d) m z = 1 vortex state in a isotropic trap Parameters: w = 10, 000 and solutions are on point grids. Because these solutions are in 3D, we can easily relate the computational parameters (see section 2.4) to experimental units. For example, if we take the radial trapping frequency to be ω T = 2π 50 Hz, then w = 10, 000 can be identified with about atoms of 87 Rb. The trapping ratios in Fig. 3.10(b) and (c) respectively correspond to typical asymmetries for TOP and Ioffe-Pitchard traps, which are commonly used in experiments (see [165]). In Table 3.4 we give the optimised eigenvalues for the states in Fig to show the effect of varying the trapping potential and the presence of a vortex. (a) λ=1, m z =0 (b) λ= 8, m z =0 (c) λ=0.1, m z =0 (d) λ=1, m z =1 µ Table 3.4: Eigenvalues for the eigenstates in Fig

88 72 Numerical techniques 10 0 F iterations Figure 3.11: Value of optimality function during optimisation of states in Fig (a)- solid, (b)- dashed, (c)-dotted and (d)-large dots. 3.5 Solution of time-dependent Gross-Pitaevskii equation Here, we are interested in the simplification of the full 3D Gross-Pitaevskii equation which occurs when the initial state and external potential have cylindrical symmetry. This class of problem can be simulated by a two dimensional algorithm with a modified evolution operator. An ingenious algorithm, dubbed the RK4IP 6, has been developed at Otago (see [10, 68, 164, 35]) for propagating the nonlinear Schrödinger equation in Cartesian coordinates. This algorithm makes use of Fourier methods to exponentiate the Laplacian operator and transform the system into an interaction picture. We briefly review this method here, and refer the reader to the careful discussion presented by B. Caradoc-Davies in [35]. Caradoc-Davies also extended the RK4IP algorithm to the 3D Cartesian case, however the spatial and momentum resolution of simulations is limited with current computational resources 7 and is unsuitable for investigating atom-optics type experiments. Our interest here is in 6 RK4IP: Runge-Kutta Fourth-order Interaction Picture algorithm. 7 E.g. storage of a wavefunction discretised on a point spatial grid requires about 270MB of storage, and a functioning Runge-Kutta algorithm would require at least 3 wavefunctions to be simultaneously stored.

89 3.5 Solution of time-dependent Gross-Pitaevskii equation 73 how to generalise this procedure to a non-cartesian cylindrical coordinate system where the effective dimensionality of the wavefunctions is 2D, allowing us to achieve much higher spatial and momentum resolution Review of 2D RK4IP algorithm To understand the issues associated with numerically evolving Eq. (3.94) using the RK4IP algorithm, we consider the two dimensional case, where the time-dependent Gross-Pitaevskii equation takes the form ψ(x, y, t) i t ( ) 2 = x + 2 ψ(z, y, t) (3.85) 2 y 2 + [ V T (x, y, t) + w ψ(x, y t) 2] ψ(z, y, t). Following the notation used in [35] we write this as ψ t = i[d 2D + N 2D ]ψ, (3.86) where the terms D 2D = ( ) 2 x + 2, 2 y 2 (3.87) N 2D = V T + w ψ(x, y, t) 2, (3.88) give the diffusive and non-diffusive parts respectively. The key to the stability and accuracy of the RK4IP algorithm is the transformation to the interaction picture defined by ψ I = e +i(t t )D 2D ψ. (3.89) The interaction picture evolution equation is ψ I t = i[n 2D] I ψ I, (3.90) where [N 2D ] I = e +i(t t )D 2D [N 2D ]e i(t t )D 2D (3.91) and the a fourth-order Runge-Kutta algorithm (see [141]) is used to propagate ψ I. Because the Fourier basis functions (i.e. {exp(ik x x), exp(ik y y)}) are eigenfunctions

90 74 Numerical techniques of D 2D, transforming to the interaction picture is easily done in Fourier space and replacing the operators by the square of the Fourier-space coordinates (i.e. the Laplacian eigenvalues, see [35] for details). Computationally, the efficiency of the algorithm derives from the Fast Fourier Transform (e.g. see [141]) which allows the interaction picture transformation (3.89) to be both fast and accurate. Further speed optimisations can be made by moving the origin t of the interaction picture at each step of the Runge-Kutta algorithm to reduce the number of evaluations of (3.89) (see [35]) Time-dependent Gross-Pitaevskii equation for cylindrical systems When the physical system exhibits cylindrical symmetry, the 3D time-dependent Gross-Pitaevskii equation simplifies in a similar manner to the time-independent case (3.81). Assuming that the external potential is cylindrically symmetric, i.e. V T (r, t) = V T (r, z, t), the full evolution equation (2.76) can be written as Ψ(r, z, φ, t) i t ( 2 = r r r + 1 ) 2 r 2 φ + 2 Ψ(r, z, φ, t) (3.92) 2 z 2 + [ V T (r, z, t) + w Ψ(r, z, φ, t) 2] Ψ(r, z, φ, t). If the initial state of the condensate wavefunction is of the form Ψ(r, z, φ, t = 0) = Φ(r, z, t = 0)e imzφ, (3.93) then Φ, which we refer to as the cylindrical wavefunction, evolves according to Φ(r, z, t) i t ( 2 = + ) r + 2 z 2 r r [ m 2 z r + V 2 T (r, z, t) + w Φ(r, z, t) 2 Φ(r, z, t) (3.94) ] Φ(r, z, t). As in the time-independent case (see section 3.4), if m z = 0, Φ describes a cylindrically symmetric wavefunction. If m z 0, our system is an eigenstate of the z-component of angular momentum L z with a non-cylindrically symmetric phase circulation which requires that Φ 0 as r 0. Because the trapping potential has been assumed cylindrically symmetric L z is conserved (see Appendix A.6 of [35]),

91 3.5 Solution of time-dependent Gross-Pitaevskii equation 75 and the m 2 z/r 2 centrifugal barrier term appears in Eq. (3.94) Numerical evolution of cylindrical wavefunction The cylindrical evolution equation (3.94) can be written as where Φ t = i[d cyl + N cyl ]Φ, (3.95) ( 2 D cyl = r + 1 ) 2 r r + 2, (3.96) z 2 and N cyl = m2 z r 2 + V T + w Φ(r, z, t) 2. (3.97) We wish to adapt the RK4IP method to this case, to take advantage of its stability and accuracy. However, this is not straightforward, since the operator D cyl is not diagonal in the Fourier basis. The eigenfunctions of the radial part of D cyl are the zeroth-order Bessel functions of the first kind (J 0 ), which satisfy ( 2 r + 1 ) J 2 0 (kr) = k 2 J 0 (kr). (3.98) r r In a similar manner to the numerical Fast Fourier transform, a transformation to the J 0 basis, known as a Hankel transform, can be made. A quasi-fast Hankel transformation has been developed by Siegman [151], but this requires an exponentially spaced grid in the r-direction, and involves approximations which we found to seriously reduce accuracy. It is also interesting to compare the cylindrically and the spherically symmetric cases, since the Laplacian in spherical coordinates has the term (2/r) / r, similar to the problematic term in D cyl. By using the transformation u = rφ sph, the evolution equation of the spherical wavefunction reduces to i u ] [ t = 2 r + V 2 T (r) + w u 2 u, (3.99) r 2 reducing the diffusive part to being diagonal in the Fourier basis, at the expense of

92 76 Numerical techniques complicating the nonlinear term. Morgan [126] has adapted the 1D RK4IP algorithm to simulate (3.99) by extending r to a symmetric grid with positive and negative values, which requires that the solution is odd, so that u vanishes at the origin and Φ sph is regular there. Morgan found that the form of the nonlinear term in (3.99) caused significant numerical difficulties, but projecting out the even part of the solution (which arises from numerical error) the algorithm accuracy was satisfactory. The subtle difference of a factor of 2 in the (1/r) / r term in the cylindrical and spherical Laplacians means that no such transformation is possible for the cylindrical case. Instead, we have developed a different approach. diffusion operator into two parts We begin by splitting the D cyl = D + D r, (3.100) where ( ) 2 D = r + 2, (3.101) 2 z 2 D r = 1 r r. (3.102) The operator D is clearly diagonal in the Fourier basis ({exp(ik r r),exp(ik z z)}). We now define an interaction picture by Φ I = e +i(t t ) DΦ, (3.103) which is isomorphic to the 2D Cartesian case. The remaining diffusive motion is treated by lumping it together with the nonlinear part of the equation, i.e. Φ I t = i[d r + N cyl ] I Φ I, (3.104) where [D r + N cyl ] I = e +i(t t ) D[D r + N cyl ]e i(t t ) D. (3.105) Considerable care must be taken in evaluating the operator D r in (3.104). Part of the reason for the success of the RK4IP algorithm is that the Fourier method used to calculate D (and its equivalent in the Cartesian case) is highly accurate. The

93 3.5 Solution of time-dependent Gross-Pitaevskii equation 77 feasibility of numerically solving Eq. (3.104) depends on how accurately we can evaluate D r. We are aided in this by the fact that D r is first order. We have found that an 11-point sparse finite difference operator offers similar accuracy, and is in fact faster than the Fast Fourier Transformation (see Appendix B.2) Numerical implementation The numerical transformation to the cylindrical interaction picture (3.103) is not exactly identical to the 2D case, because in the cylindrical wavefunction the r- coordinate is restricted to being positive (see (3.76)). However, the negative r values of a general function, A, could be interpreted as A( r, z, φ) = A( r, z, φ π), (3.106) i.e. the negative r value corresponds to its absolute value rotated 180 azimuthally. Hence for the cylindrically symmetric wavefunction, Φ, when extended onto the domain < r <, we must have Φ( r, z) = Φ(r, z), (3.107) i.e. the function is even with respect to the r coordinate. At the expense of doubling the computational domain, we can simply apply the 2D Fourier technique for evaluating the exp( Dt) factors on the right hand side of Eq. (3.103). It would be of great computational benefit to develop an algorithm which needs only the positive r values, therefore reducing both storage and computational time by a factor of approximately 2. From the definition of the Fourier transformation in one spatial dimension, F (k) = = dx f(x)e ikx (3.108) dx f(x) [cos(kx) + i sin(kx)], (3.109) we see that when f(x) is an even function all the odd-sin terms will be zero and (3.109) can be reduced to F (k) = 2 0 dx f(x) cos(kx). (3.110)

94 78 Numerical techniques Thus a Fast Cosine Transformation could be used on the half grid instead of a Fast Fourier Transformation. Our primary reason for not implementing this is because MATLAB currently does not have a Fast Cosine Transformation Accuracy Since the cylindrical propagation algorithm only differs from the 2D algorithm by the inclusion of the D r operator (3.102), we expect this term to be the primary cause of any additional error. The general validity of RK4IP algorithm in the context of Gross-Pitaevskii propagation is covered in detail by Caradoc-Davies [35]. Here we consider two simple tests to compare the 2D and cylindrical propagation algorithms. For w = 0, the ground state of the isotropic harmonic trap for the Cartesian 2D and cylindrically symmetric 3D cases are the Gaussian functions ψ 2D (x, y) = Φ cyl (r, z) = 1 2π e (x2 +y 2 )/4, (3.111) 1 (2π) 3/2 e (r2 +z2 )/4, (3.112) with eigenvalues µ 2D = 1, and µ cyl = 1.5 respectively. In a trap, these states evolve only by uniformly changing their phase at a rate of µ. Our first test is to take the initial state, ψ i, given by Eq. (3.111) or (3.112) for the 2D or 3D case respectively, and numerically propagate this to give the final state ψ f at time t = t f. The accuracy of this evolution can be evaluated using the expressions developed in [35] for: normalisation error N err = dr ψ f 2 1, (3.113) since the normalisation should be conserved; density magnitude error n err = dr ( ψ f 2 ψ i 2 ), (3.114) is a measure of the change in density profile; and density weighted phase error θ err = dr ψ f 2 Arg(ψ f /ψ i ) µt f, (3.115) to assess the degree to which the phase evolution has deviated from the analytic result.

95 3.5 Solution of time-dependent Gross-Pitaevskii equation 79 Results are given in Table 3.5 for the case of t f = 3. These results indicate as 2D Cylindrical N err n err θ err Table 3.5: Error values for 2D and cylindrical algorithms propagating w = 0 harmonic eigenstates in an isotropic harmonic trap. Other parameters: solutions on computational grid extending from 20 to 20 in each coordinate direction; 500 uniformly spaced time steps taken from t = 0 to t = 3. expected that the cylindrical algorithm is slightly less accurate than the 2D algorithm, but verify that our cylindrical propagator has satisfactory level of accuracy to be of quantitative use (a) (b) 10 5 N err t t Figure 3.12: Normalisation error for cylindrical and 2D propagation algorithms. (a) Propagation of w = 0 (harmonic) ground state for the (solid) cylindrical 3D case, (dotted) 2D case; (b) Nonlinear eigenstate propagation (solid) cylindrical 3D case with w = 10, 000, (dotted) 2D case with w = 500. Other parameters: isotropic trap remains on during simulation; solutions on computational grid extending from (a) 20 to 20 (b) 30 to 30, in each coordinate direction; (a) 4000 and (b) 8000 uniformly spaced time steps taken from t = 0 to t = 24. Caradoc-Davies results [35] show that as long as the algorithm is not suffering from aliasing effects, the normalisation of the eigenstate provides a good indicator

96 80 Numerical techniques of the solution accuracy. This forms a convenient test of our cylindrical algorithm in the nonlinear regime, where no exact eigenstates can be found. In Fig we show the normalisation error (3.113) for the 2D (RK4IP) and cylindrical algorithms for two simulations of duration t = 24. For the linear case shown in Fig. 3.12(a), the cylindrical algorithm performs uniformly worse than the 2D by about 2 orders of magnitude. However, both results have a respectable final error of N err It is difficult to compare 2D and 3D simulations in the nonlinear regime, since the same w value tends to have a higher effective nonlinearity in the 2D case i.e. the eigenvalue and wavefunction density are larger. To have a similar effective nonlinearity, in Fig. 3.12(b) we have chosen to compare a 2D state with w = 500 and µ 2D = 9.0, to a 3D state with w = 10, 000 and µ cyl = The normalisation error for both is several orders of magnitude worse than for the respective w = 0 cases, however the accuracy of the two schemes relative to each other remains almost the same. The final normalisation error of 10 6 is more than satisfactory for the requirements we have of a dynamical algorithm in this thesis.

97 Chapter 4 Dressed states of a Bose-Einstein condensate 4.1 Introduction Experimental work with multiple species Bose-Einstein condensates [130, 154, 121, 91] has motivated theoretical analysis of their wavefunctions and excitations [96, 62, 61, 83, 143, 142, 80]. Also known as spinor or multi-component condensates, these systems consist of atoms in several different trapped hyperfine states. The primary tools used to create and investigate these condensates are external radiation fields. In quantum optics the eigenstates of the full system of the (single) atom plus field are called dressed states (see [45]) and have proved invaluable as calculational and interpretational tools. In this chapter we extend this concept to the radiation-field coupled binary condensate, where two internal states, which we label 1 and 2, are coupled by a cw field. We derive a Gross-Pitaevskii equation, and numerically solve for eigenstates of this system, which we shall call condensate dressed states. The major new feature that occurs in condensate dressed states is the spatial dependence of the wave functions, which can prove significant even when the external field is a uniform plane wave. We explore the properties of the condensate dressed states as a function of system parameters, and present results representing the broad classes of possible behaviour. We begin by considering the most general properties of the condensate dressed

98 82 Dressed states of a Bose-Einstein condensate states. We show that in the simplest case of identical traps for each component and identical collisional interactions, both components have identical spatial behaviour. However, with non identical traps or collisional strengths, the two components may have markedly different spatial character, and we examine the dependence of these shapes on the trap parameters, the collisional parameters, and the external field. We show that the condensate s spatial shapes can be manipulated by changing the external field strength or detuning. For the case of a spatially varying field, we apply our dressed state analysis to the scheme proposed by Dum et al. [58] for adiabatically transferring a ground state into a vortex or soliton state. Finally, we review some preliminary experimental results [118] for dressed states in rubidium. 4.2 Formulation Internal state coupling Here we restrict our attention to the case where the external radiation only couples two internal states ( 1 and 2 ) of the Bose-Einstein condensate atoms. We are interested in the possible stationary configurations of this system, so it is desirable that both states are trapped, since the inclusion of an untrapped state necessitates loss 1. We consider the coupling to be given by a single photon transition between the two levels, as discussed in section However, we show in Appendix C that a Raman system, where the coupling is facilitated by an intermediate state, has an identical form to a single photon transition, and has the advantage that a wider range of states with different trapping properties can be chosen from. For now it will suffice to assume that a coherent process connects the two states directly, as shown in Fig Ballagh et al. [11] have shown for the case of an rf-coupling between internal states, that the loss from the untrapped state is minimal when a large intensity field is used to couple it to a trapped state.

99 4.2 Formulation 83 δ L PSfrag replacements Ω 2 1 Figure 4.1: Two internal states coupled by an external radiation field. The coupling has Rabi-frequency Ω and is detuned from atomic resonance by the angular frequency δ L Hamiltonian The second quantised Hamiltonian for a two component condensate with a coherent coupling is (see Eq. (2.141) and [149]) Ĥ = [ dr ˆΨ ˆΨ 1(r)Ĥ1 1 (r) + ˆΨ ) 2(r) (Ĥ2 δ L ˆΨ2 (r) (4.1) + U ij 2 ˆΨ i (r) ˆΨ j (r) ˆΨ i (r) ˆΨ j (r) i,j Ω 2 ˆΨ 1(r) ˆΨ 2 (r) Ω 2 ˆΨ 2(r) ˆΨ ] 1 (r), where is the single particle Hamiltonian, Ĥ i = 2 2m 2 + V T i (r), (4.2) δ L = ω L ω 0, (4.3) is the detuning of coupling field frequency (ω L ) from atomic resonance, and Ω is the bare Rabi frequency (see section 2.5.3). The quantity U ij represents the collisional interaction strength between atoms in the i and j states respectively, and is of the

100 84 Dressed states of a Bose-Einstein condensate form U ij = 4π 2 a ij m, (4.4) where a ij is the scattering length. We take both trapping potentials, V T i (r), to be harmonic (see 2.23), and related to each other by V 2 (r) = kv 1 (r r 0 ), (4.5) that is, the second component may experience a relative offset r 0 and spring constant k compared to the first component. To a certain extent these factors can be experimentally controlled e.g. by using dynamical effects in the TOP trap (see [59]). The Bose field operators ˆΨ i, obey the equal time commutation relations [ ˆΨi (r), ˆΨ j (r)] = [ ˆΨ i (r), ˆΨ j (r) ] = 0, (4.6) and [ ˆΨi (r), ˆΨ ] j (r ) = δ(r r )δ ij. (4.7) Time-independent Gross-Pitaevskii equation Making a Bogoliubov approximation in a similar manner to section we write each component as ˆΨ i (r) = Ψ i (r) + ˆφ i (r), (4.8) where Ψ i (r, t) is the i-component wavefunction and ˆφ i is a small fluctuation operator. Inserting Eq. (4.8) into the Hamiltonian (4.1), and taking the expectation value gives the energy functional for the radiation-field coupled two component condensate (also see Eq. (2.36)). E[Ψ 1, Ψ 2 ] = [ ) dr Ψ 1(r)Ĥ1Ψ 1 (r) + Ψ 2(r) (Ĥ2 δ L Ψ 2 (r) U ij + 2 Ψ i(r) 2 Ψ j (r) 2 Ω 2 Ψ 2(r)Ψ 1 (r) (4.9) i.j Ω ] 2 Ψ 1(r)Ψ 2 (r). In this expression for the energy, we have ignored terms involving products of fluctuation operators, as these are expected to be small when the condensate is highly

101 4.2 Formulation 85 occupied. We note that the terms in Eq. (4.1) of the form Ω ˆΨ i ˆΨ j, allow species interconversion. That is, an atom in internal state j can convert to i by exchanging energy with the electro-magnetic field, and so the number of particles in each state, given by N i = dr Ψ i (r) 2, (4.10) is not fixed. However, the total number of particles between the two components is constant, i.e. N 0 = N 1 + N 2 = const. (4.11) In an analogous manner to the derivation of the one component Gross-Pitaevskii equation (see section 2.3.3), we can minimise the energy functional (4.9) with respect to the wave functions Ψ 1 and Ψ 2, subject to the constraint of fixed total particle number (4.11). The result of this procedure (see Appendix A.3) is the coupled time-independent Gross-Pitaevskii equations µψ 1 (r) = [Ĥ1 + N 0 U 11 ψ 1 (r) 2 + N 0 U 12 ψ 2 (r) 2 ] ψ 1 (r) (4.12) µψ 2 (r) = Ω 2 ψ 2(r), ] [Ĥ2 δ L + N 0 U 12 ψ 1 (r) 2 + N 0 U 22 ψ 2 (r) 2 ψ 2 (r) (4.13) Ω 2 ψ 1(r). In a similar manner to section we have chosen the stationary states to be normalised according to dr [ ψ 1 (r) 2 + ψ 2 (r) 2] = 1, (4.14) and thus N 0, the total number of particles, appears in Eqs. (4.12) and (4.13). We refer to the solution of Eqs. (4.12) and (4.13) as the condensate dressed states, in analogy with the single atom dressed states of quantum optics [45].

102 86 Dressed states of a Bose-Einstein condensate 4.3 Solving for dressed states In this section we begin by transforming the dressed state equations (4.12) and (4.13) to computational units. To obtain an analytic solution we make a Thomas-Fermi approximation, and introduce the concept of an effective detuning. We shall see that this effective detuning takes into account the trap and collisional effects in addition to the bare field detuning δ L, and allows us to gain insight to the coupled systems behaviour. We show that in certain parameter regimes an explicit solution can be obtained in closed form. Solving the nonlinear dressed state equations numerically is a formidable challenge, and we discuss how the optimisation techniques of chapter 3 can be applied to finding dressed states. We make extensive use of numerical results throughout this chapter to explore the diverse regime of eigenstate solutions Gross-Pitaevskii equation in computational units It is convenient to rewrite the coupled equations (4.12) and (4.13) in terms of our standard computational units introduced in section 2.4. Using barred notation to denote dimensionless quantities, we have µ ψ 1 ( r) = ˆ H1 ψ1 ( r) + [ w 11 ψ 1 ( r) 2 + w 12 ψ 2 ( r) 2] ψ1 ( r) (4.15) Ω 2 ψ 2 ( r), µ ψ 2 ( r) = ( ˆ H2 δ L ) ψ 2 ( r) + [ w 12 ψ 1 ( r) 2 + w 22 ψ 2 ( r) 2] ψ2 ( r) (4.16) Ω 2 ψ 1 ( r), where the dimensionless collisional interaction strengths are w ij = N 0 U ij / ω T r 3 0, (4.17) and other quantities are as defined in section 2.4. In these units, the normalisation condition (4.14) is d r [ ψ 1 ( r) 2 + ψ 2 ( r) 2] = 1. (4.18)

103 4.3 Solving for dressed states Thomas-Fermi approximation We begin by decomposing the dressed state solutions in the form ψ 1 ( r) = c 1 ( r) φ( r), (4.19) ψ 2 ( r) = c 2 ( r) φ( r), (4.20) where we restrict the functions c i and φ to obey the conditions c 1 ( r) 2 + c 2 ( r) 2 = 1, (4.21) d r φ( r) 2 = 1, (4.22) so that the normalisation condition (4.18) is satisfied. Here φ( r) 2 is the total density of both species combined at r, and c i ( r) 2 is the portion of this contributed from the i component. Substituting Eqs. (4.19) and (4.20) into Eqs. (4.15) and (4.16), and making the usual Thomas-Fermi approximation of ignoring the Laplacian term, we arrive at the system of algebraic equations taking the form of a nonlinear eigenvalue problem [ ] [ ] [ ] c 1 ( r) Ḡ( r) Ω/2 c 1 ( r) µ = c 2 ( r) Ω/2 G( r). (4.23) Eff ( r) c 2 ( r) Here Ḡ( r) = V 1 ( r) + [ w 11 c 1 ( r) 2 + w 12 c 2 ( r) 2] φ( r) 2, (4.24) and the term we shall call the effective detuning is Eff ( r) = δ L + V 1 ( r) V 2 ( r) + ( w 11 w 12 ) c 1 ( r) 2 ] φ( r) 2. [ ( w 12 w 22 ) c 2 ( r) 2 (4.25) Solving Eq. (4.23) in general must be done self-consistently, since the coefficient matrix depends on the solution eigenvector {c j }. In the next section we consider a restriction on the collisional terms w ij which permits a solution in closed form.

104 88 Dressed states of a Bose-Einstein condensate Collisional degeneracy Here we consider the case where the collisional interactions are all degenerate, i.e. w 12 = w 11 = w 22, (4.26) and for simplicity we write w = w ij. In this limit Ḡ and Eff reduce to Ḡ( r) = V 1 ( r) + w φ( r) 2, (4.27) Eff ( r) = δ L + V 1 ( r) V 2 ( r). (4.28) Two eigenvalues can be found for Eq. (4.23) and are given by the secular equation µ ± = Ḡ( r, φ) [ ] Eff ( r) ± Eff( r) 2 + Ω 2. (4.29) We have labeled the two possible solutions as µ ±, where µ + µ for all Ω and δ L. Expression (4.29) is not sufficient to determine the eigenvalues, since Ḡ depends on the yet unspecified total density function, φ( r). However, by inverting Eq. (4.29) for the total density, [ Eff ( r) ± Eff ( r) 2 + Ω 2 ] φ( r) 2 = µ ± V 1 ( r) 1 2 w, (4.30) and ensuring this density is normalised to unity, the eigenvalues can be determined. Knowledge of the eigenvalues allows us to solve for the eigenvectors of Eq. (4.27). These are c 1± ( r) = 1 2 (1 ) 1 2 Eff ( r) Eff ( r) 2 + Ω, (4.31) 2 c 2± ( r) = ± 1 c 1± ( r) 2, (4.32) where we have labeled the eigenvectors by ± corresponding to the eigenvalue µ ±. Finally the dressed states ψ 1± ( r), ψ 2± ( r), be found by substituting the results of Eqs. (4.30)-(4.32) back into Eqs. (4.19) and (4.20).

105 4.3 Solving for dressed states Numerical solution The optimisation techniques developed in chapter 3 are readily applied to solve the full coupled equations (4.15) and (4.16). Here we give an overview of the discretisation and functions we optimise in the case of one spatial dimension. Taking a uniform x-grid as in Eq. (3.35), we write the discretised wavefunctions for ψ 1 and (1) (2) ψ 2, as ψ i and ψ i respectively. Discretising Eqs. (4.15) and (4.16), as discussed in section 3.3.2, we have f (1) i = + f (2) i = + m x D (2) ik k=1 [ µ k=1 V (1) i m x D (2) ik ψ (1) k ψ (2) k [ µ + δ L ψ (2) i (4.33) + Ω 2 [ (1) w 11 ( ψ i ) 2 + w 21 ( + Ω 2 V (2) i ]] (2) ψ i ) 2 ψ(1) i, ψ (1) i (4.34) [ (1) w 12 ( ψ i ) 2 + w 22 ( ]] (2) ψ i ) 2 ψ(2) i. The boundary conditions applied to each component wavefunction are f B1 = f B2 = f B3 = f B4 = ψ (1) 1, (4.35) ψ (1) m x, (4.36) ψ (2) 1, (4.37) (2) ψ m x, (4.38) and the normalisation condition (Eq. (4.18)) is m x [ ] (1) f N = ( x) ( ψ i ) 2 (2) + ( ψ i ) 2 1. (4.39) To solve Eqs. (4.33)-(4.39), we minimise the optimality function (1) (2) F ({ ψ i, ψ i, µ}) = 1 2 [ mx i=1 j=1 [ (f (1) i ) 2 + (f (2) i ) 2 ] + ] 4 (f Bi ) 2 + (f N ) 2. (4.40) The analytic derivatives of Eqs. (4.33)-(4.40), necessary to implement a conjugate gradient or Gauss-Newton solution, are given in Appendix B.6. We use the Thomas- Fermi solution (4.29)-(4.32) to provide an estimate of the initial state from which i=1

106 90 Dressed states of a Bose-Einstein condensate our optimisation procedure begins refining. We note that Eqs. (4.33)-(4.39) have ψ (1) been formulated for the case where i and are both real. We have investigated fully complex solutions, but without exception have found that if the Rabi frequency Ω is real (as assumed here) then both solutions are also. ψ (2) i 4.4 Characteristics of dressed states Many properties of the dressed condensate turn out to be analogous to those of the familiar dressed states of quantum optics [45], however a key difference is that the stationary solutions of Eqs. (4.12) and (4.13) have a spatial dependence. As is commonly found in nonlinear phenomena, a diverse range of solutions may exist depending on the parameter regime used. Under certain circumstances many solutions may exist, and in others no solutions exist. We have not conducted a stability analysis, but have used the convergence of the optimisation methods to give an indication of the stability. For cases in this section 1D numerical results are presented to illustrate dressed condensate behaviour. We revert back to S.I units for our equations in the remainder of this chapter Identical species Identical traps In the simplest case, where the trapping potentials are identical (V 1 = V 2 ) and collisional coupling coefficients are all equal, (w ij = w), pairs of dressed states of the form Ψ + = e iϑ (ψ 1+ (r) 1 + ψ 2+ (r) 2 ), (4.41) Ψ = e iϑ (ψ 1 (r) 1 ψ 2 (r) 2 ), (4.42) can always be found, where ϑ is a constant phase factor and ψ 1± and ψ 2± are positive real functions 2. It what follows we shall take ϑ = 0, so that the wavefunctions are real. It is easy to show that the eigenstates Ψ ± can be written in terms of the 2 We take here the case of ground states. This need not be the case if either state is in a topologically excited state, e.g. a vortex state.

107 4.4 Characteristics of dressed states 91 eigenfunction ψ 0 of the uncoupled one component Gross-Pitaevskii equation (2.74) µ 0 ψ 0 (r) = 2 rψ 0 (r) + V 1 (r)ψ 0 (r) + w ψ 0 (r) 2 ψ 0 (r), (4.43) and take the general form Ψ ± = c 1± ψ 0 (r) 1 + c 2± ψ 0 (r) 2, (4.44) with eigenvalues µ ± = µ ( ) δ L ± δ 2L + Ω2, (4.45) where ) 1 c 1± = (1 1 2 δ L, (4.46) 2 δ 2 L + Ω 2 c 2± = ± 1 c 2 1±. (4.47) These amplitudes (c j± ) and eigenvalues (µ ± ) have precisely the same dependence on the field parameters Ω and δ L as for the quantum optics dressed state, and apart from the overall spatial modulation of the ψ 0, these states are completely analogous with the simple two state atom case. Scanning the detuning of the electromagnetic field from the far red (δ L Ω ) through to the far blue (δ L Ω ) and solving for Ψ ± and µ ± at each point reveals that the eigenvalues display an avoided crossing (see Fig. 4.2(a)), similar to that seen in quantum optics. This is associated with a resonance in which the component populations are near equal (typically when δ L Ω ), as shown in Figs. 4.2(b)-(o), where we consider the component wavefunctions of the dressed states for a range of δ L values used in Fig. 4.2(a). On either side of this resonance, as δ L increases, the dressed states approach a single component configuration, i.e. the dressed state is almost entirely in one internal state. We notice that the upper branch wavefunction (Figs. 4.2(b)-(h)) are essentially mirror images of the lower branch solutions (Figs. 4.2(i)-(o)) at the negative value of δ L, except for the phase difference between the components. The centre of mass behaviour of the condensate is an extra degree of freedom which contributes to the richness of the dressed state physics for Bose-Einstein

108 92 Dressed states of a Bose-Einstein condensate (b) (c) (d) (e) (f) (g) (h) ψ x x x x x x x 35 (a) 30 (b) 25 (c) µ [ω T ] 20 (d) µ + (e) (f) (g) (h) 15 (i) (j) (k) µ (l) (m) 10 (n) (o) frag replacements δ L [ω T ] (i) (j) (k) (l) (m) (n) (o) ψ x x x x x x x Figure 4.2: Properties of condensate dressed states as a function of external field frequency. (a) Eigenvalues µ + and µ for upper and lower branches. Dressed state solutions: (b)-(h) upper branch, (i)-(o) lower branch are shown, where the component wavefunctions are: (solid) ψ 1 and (dotted) ψ 2. Parameters are V 1 = V 2, w 11 = w 12 = w 22 = 200w 0 and Ω = 2ω T.

109 4.4 Characteristics of dressed states 93 (b) (c) (d) (e) (f) (g) (h) ψ x x x x x x x 35 (a) 30 (b) 25 (c) (d) µ + µ [ω T ] 20 (e) (f) (g) (h) 15 (i) (j) (k) µ (l) (m) 10 (n) (o) PSfrag replacements δ L [ω T ] (i) (j) (k) (l) (m) (n) (o) ψ x x x x x x x Figure 4.3: Properties of first excited condensate dressed states as a function of external field frequency. (a) Eigenvalues µ + and µ for upper and lower branches. For reference the ground state eigenvalues are shown as a dashed line. Dressed state solutions: (b)-(h) upper branch, (i)-(o) lower branch are shown, where the component wavefunctions are: (solid) ψ 1 and (dotted) ψ 2. Parameters are V 1 = V 2, w 11 = w 12 = w 22 = 200w 0 and Ω = 2ω T.

110 94 Dressed states of a Bose-Einstein condensate condensates. As an example of the greater range of solutions this allows, in Fig. 4.3 we show results for the dressed states when the wavefunction is in its first excited state (also known as a soliton). This state is the 1D analog of a vortex state, and is the first of sequence of topological excitations for the system. Even with this added spatial structure, qualitatively the same physics occurs in this system: the eigenvalues undergo an avoided crossing and a population resonance is observed. In a similar manner to the ground state solutions in Fig. 4.2, we see that the component wavefunctions for the upper branch (Figs. 4.3(b)-(h)) are in phase with each other, whereas those for the lower branch (Figs. 4.3(i)-(o)) are out of phase. For the remainder of this chapter we shall only consider the upper branch (with eigenvalue µ + ). This simplifies discussions, since this branch has the properties: For δ L < 0, component 1 dominates. For δ L > 0, component 2 dominates. Both component wavefunctions are in phase. We note that these comments also apply to Eff defined in Eq. (4.25) Effect of different relative trap potentials The harmonic trapping potentials discussed in Eq. (4.5) take the 1D form V 1 = 1 2 mω2 xx 2, (4.48) and V 2 = k 1 2 mω2 x(x x 0 ) 2, (4.49) which allows the trap for condensate 2 to have a different relative spring constant (k) and an offset centre (x 0 ). No exact analytic solutions are possible in this case, but the representative behaviour is shown in numerical solutions presented in Fig In Fig. 4.4(a) we see that even a small difference in the relative spring constant (5%) can give rise to a significant difference in the component wavefunctions, and as k is further increased a clear phase separation is observed (Figs. 4.4(b)-(c)), whereby one species is excluded from the region where the other species has high density.

111 4.4 Characteristics of dressed states 95 ψ [1/ r0] ψ [1/ r0] ψ [1/ r0] (a) (b) (c) (d) (e) (f) PSfrag replacements x [r 0 ] x [r 0 ] Figure 4.4: Effect of relative spring constant and trap offset on component wavefunctions of dressed states. (solid line) ψ 1, (dotted line) ψ 2. (a) x 0 = 0, k = 0.95, (b) x 0 = 0, k = 0.9, (c) x 0 = 0, k = 0.5. (d) x 0 = 0.2r 0, k = 0 (e) x 0 = 0.5r 0, k = 0 (f) x 0 = 2r 0, k = 0. Parameters are as in Fig. 4.2, except for V 2 (see Eq. (4.49)) and δ L = 2ω T.

112 96 Dressed states of a Bose-Einstein condensate Offsets between the two trapping potentials also cause spatial reshaping of the two components, as can be seen in Fig. 4.4(d)-(f), where x 0 is successively increased. In Fig. 4.4(d), where x 0 is approximately 1% of the condensate size, significant deformation of component 2 is seen, and when x 0 is increased to 1r 0 (i.e. 10% of condensate size) phase separation occurs (Figs. 4.4(e)-(f)). Validity of Thomas-Fermi solution It is worthwhile to compare the results of the degenerate Thomas-Fermi solution developed in section with our numerical results. In Fig. 4.5 we present several examples showing the numerical solutions of the Gross-Pitaevskii equations (4.12) and (4.13) and the Thomas-Fermi solutions for various trap parameters. We see that in general the Thomas-Fermi solutions gives an excellent qualitative estimate of the component wavefunctions. Application of effective detuning The concept of effective detuning developed in Eq. (4.25) affords us an analytic understanding of the structure observed in Fig Substituting the analytic trap forms of Eqs. (4.48) and (4.49) into Eq. (4.25) we have (for the 1D case with degenerate collisional strengths) Eff (x) = δ L mω2 x [ (1 k)x 2 + 2kx 0 x kx 0 ) ]. (4.50) The spatial dependence of the effective detuning is illustrated in Figs. 4.5(g)-(i), which allows us to demonstrate that the effective detuning is a powerful tool for understanding the stationary configurations. In Fig. 4.5(g), where the two solutions are constant scaled ground states, Eff is spatially uniform. For the upper branch solution we are considering, at positive detuning component- 1 is dominant (see discussion at the end of section or Eqs. (4.31) and (4.32)). In Fig. 4.5(h) the difference in trap spring constants gives rise to the quadratic spatial dependence of the effective detuning. As such there are regions where Eff < 0 and the 1 - component dominates and regions where Eff > 0 and the other component is dominant. Similarly in Fig. 4.5(i) we see that the effect of a trap offset is to give a linearly varying effective detuning across the condensate. Once again, in regions where Eff < 0 the 1 -component dominates, while for Eff > 0 the 2 -component

113 4.4 Characteristics of dressed states 97 PSfrag replacements ψ [1/ r0] ψ [1/ r0] ψ [1/ r0] (a) (b) (c) (d) (e) (f) Eff [ωt ] Eff [ωt ] Eff [ωt ] (g) (h) (i) x [r 0 ] x [r 0 ] x [r 0 ] Figure 4.5: Comparison between the numerical (right hand column) and Thomas-Fermi solutions (middle column) for various trap parameters. The solid line is ψ 1 and the dotted line is ψ 2. Numerical solutions are (a) k = 1, x 0 = 0r 0, (b) k = 0.5, x 0 = 0r 0, (c) k = 1, x 0 = 2r 0. In (d)-(f) the corresponding Thomas-Fermi solutions are shown, and in (g)-(i) Eff (x) is shown. Other parameters: all w ij = 500, δ L = 2ω T and Ω = 2ω T.

114 98 Dressed states of a Bose-Einstein condensate ψ1 [1/ r0] 0.2 (a) 0.1 PSfrag replacements ψ2 [1/ r0] (b) Ω [ω T ] 10 Ω [ω T ] x [r 0 ] x [r 0 ] Figure 4.6: Effect of changing Rabi frequency on component wavefunctions: (a) ψ 1, (b) ψ 2. The trapping potentials are V 1 = mω 2 xx 2 /2, V 2 = k mω 2 x(x x 0 ) 2 /2 where k = 0.8 and x 0 = 0.5r 0, the detuning is δ L = 2ω T and the w ij = 500w 0. dominates. We note that zero crossings of Eff correspond to the locations where both components are approximately the same density, we refer to this location as the phase boundary Rabi frequency dependence A scan of the component wavefunction shapes as the external field amplitude Ω is varied is shown in Fig. 4.6, which illustrates the control over the condensate profiles that is afforded by simply altering the strength of the electromagnetic coupling

115 4.4 Characteristics of dressed states 99 field. Two limiting regimes can be seen. At low fields ( Ω δ L ) the components have distinct spatial shapes, determined by different trap potentials and collisional interactions. On the other hand, in the large field regime ( Ω δ L ) Ω becomes the most significant coupling between components and suppresses the spatial differences between the component wavefunctions. In particular, we notice that at large Rabi frequency, phase separation between the condensates is suppressed. This can be understood by noting that when Ω δ L we have (from Eqs. (4.31) and (4.32)) c 1± (r) 1 2, (4.51) c 2± (r) ± 1 2, (4.52) so that both components have identical spatial profiles. In the opposite limit of low intensity, the coupling between the components is dominated by the collisional interaction and the stationary states will be those of a binary condensate [80, 143, 142] Effect of different relative collisional interactions All of the previous results have been given for the case of equal collisional interactions (w 11 = w 12 = w 22 ). Although this is a good approximation to the case of rubidium, larger variations can be expected for other atomic species, and it may even prove possible to manipulate the relative scattering lengths (e.g. see [115]). It has previously been shown that the extent of phase separation of binary condensates in the absence of electromagnetic coupling depends on the relative collisional interactions [96]. Here we find analogous features arising in our dressed state solutions. An analytic solution is not readily available in this collisionally non-degenerate regime, however the concept of effective detuning allows a qualitative consideration of this behaviour. Changing the intra-species collisional strength In Fig. 4.7(a)-(b), we illustrate the effect of increasing w 11 relative to degenerate values of w 12 and w 22. For this case the effective detuning (4.25) is reduced to Eff (r, c 1, c 2, φ) = δ L + V 1 (r) V 2 (r) + (w 11 w 12 ) c 1 (r) 2 φ(r) 2. (4.53)

116 100 Dressed states of a Bose-Einstein condensate ag replacements ψ1 [1/ r0] ψ1 [1/ r0] 0.2 (a) w 11 [w 0 ] (c) w 12 [w 0 ] x [r 0 ] x [r 0 ] ψ2 [1/ r0] ψ2 [1/ r0] 0.2 (b) w 11 [w 0 ] (d) w 12 [w 0 ] x [r 0 ] x [r 0 ] Figure 4.7: Effect on the component wavefunctions of changing collisional strengths: (a)- (b) effect of varying the intra-species strength w 11 on component wavefunctions. (c)-(d) effect of varying the inter-species strength w 12 on component wavefunctions. Parameters: Ω = 2ω T, δ L = 2ω T, V 1 = V 2, and w ij = 500w 0 unless specified otherwise.

117 4.4 Characteristics of dressed states 101 Increasing the difference w 11 w 12 (solutions with w 11 > 500w 0 in Fig. 4.7) makes the detuning more positive, thus reducing the population of ψ 1 relative to the collisionally degenerate case. On the other hand, decreasing w 11 w 12 (solutions with w 11 < 500w 0 in Fig. 4.7) makes the effective detuning more negative and enhances the proportion of component- 1. These effects can be understood directly in terms of energy. By increasing w 11 we increase the self energy of component- 1, thus making it less favoured, so that the density of component- 1 is reduced. Changing the inter-species collisional strength In Fig. 4.7(c)-(d), we show the effect of varying the cross coupling term w 12, which mediates the collisional interaction between the two components. Here we consider the case w 11 = w 22 w 12, for which the effective detuning (4.25) becomes Eff (r, c 1, c 2, φ) = δ L + V 1 (r) V 2 (r) + w c 1 (r) 2 ] φ(r) 2, [ w c 2 (r) 2 (4.54) where w = w 12 w jj. When w < 0 (solutions with w 12 > 500w 0 in Fig. 4.7), a competitive interaction occurs in which the larger component in any region is favoured at the expense of the other component, thereby resulting in density differences being enhanced. In Eq. (4.54) this effect can be understood by noting that as the population of 1 increases (i.e. c 1 (r) 2 gets larger) then the detuning will become more negative, hence favouring an even larger c 1 (r) 2 occupation. A similar argument for 2 shows that as this component increases in population the detuning favours further population. Correspondingly, when w 12 is smaller than w 11 and w 22 i.e. w < 0 (curves with w 12 < 500w 0 in Fig. 4.7), it becomes favourable for the components to coexist. This can be understood from Eq. (4.54) because increasing either population alters the detuning so as to favour the other component. Energetically it follows that because the interaction energy between the components is lower than their individual interactions, the system will favour a strong mixture. The influence that w 12 exerts on mixing can be clearly shown by examining how the total populations of the two internal states change as the field frequency is scanned. In Fig. 4.8(a) we see that for the case where all collisional rates are equal (dashed line) the fraction of the total population in state 1 (i.e. N 1 /N 0 )

118 102 Dressed states of a Bose-Einstein condensate (b) (c) (d) (e) (f) (g) (h) (i) ψ x x x x x x x x (a) (j) (b) (k) (c) (d) (e) 0.6 N 1 N (l) (m) (n) (o) (f) (g) (h) (p) (i) (q) rag replacements δ L [ω T ] (j) (k) (l) (m) (n) (o) (p) (q) µ [ω T ψ ] x x x x x x x x Figure 4.8: Behaviour of dressed states with different inter-particle collisional interactions in frequency scans. (a) Population of component 1 (N 1 ). (Solid line) w 12 = 550w 0, (dashed line) w 12 = 500w 0, (dash-dot line) w 12 = 450w 0. Dressed state wavefunctions: (b)-(i) for w 12 = 550w 0, (j)-(q) for w 12 = 450w 0, where the component wavefunctions are: (solid) ψ 1 and (dotted) ψ 2. Other parameters: V 1 = V 2, Ω = 2ω T, and w ii = 500w 0.

119 4.5 3D dressed states 103 changes from 0.3 to 0.7 over the range 1.0ω T < δ L < 1.0ω T. However, to see the same fractional change in N 1 when w 12 is decreased by 10% (dash-dot line), the detuning range has to be extended to 2.0ω T < δ L < 2.0ω T. Notably, about δ L 0 (Figs. 4.8(l)-(o)) the component wavefunctions for this state change slowly with the detuning, showing that reducing w 12 favours mixing of the components. On the other hand, when w 12 is larger than w ii, mixing is unfavourable and one state will usually dominate the other. This effect can be seen with the solid line in Fig. 4.8, where when the frequency scan passes through zero, a very abrupt reversal in the dominant state occurs. We have purposely draw this line discontinuously, as the states to the left (Figs. 4.8(d)-(e)) and to the right of δ L = 0 (Figs. 4.8(f)-(g)) are quite different. In addition, for δ L < Ω, solutions (with w 12 > w ii ) become difficult to solve for numerically, suggesting that the dressed states are unstable in this region D dressed states We are able to solve for dressed states for the cylindrically symmetric 3D case using the conjugate gradient technique outlined in chapter 3. In Fig. 4.9 we give a typical example of a cylindrically symmetric state where the two components experience different trapping potentials. Of particular interest is the phase separation between the two components. It is convenient to identify the phase boundary as the surface at which the two species have equal density, since neither component is completely excluded from the region of the other, due to the occurrence of tunneling. The Thomas-Fermi model (4.31)-(4.32), predicts the components have equal density where the effective detuning is zero. When the collisional interactions are degenerate, the effective detuning takes the form given in Eq. (4.28) and the phase boundary can easily be determined. We show this Thomas-Fermi prediction for the phase boundary in Fig. 4.9(c) and compare it to the actual phase boundary for the full numerical solution. Quite good agreement is observed. 4.6 Creating a dark soliton by adiabatic transfer The major application of condensate dressed states is in manipulating the condensate wavefunction between two hyperfine states in a well defined manner. Condensates

120 104 Dressed states of a Bose-Einstein condensate (a) ψ1 [1/ r 3 0 ] r [r 0 ] z [r 0 ] 20 (b) ψ 2 [1/ r 3 0 ] r [r 0 ] z [r 0 ] 20 (c) 10 z [r0] 0 PSfrag replacements r [r 0 ] Figure 4.9: The component wavefunction and phase boundary for a cylindrically symmetric condensate, with w ij = 10 4 w 0, Ω = 2ω T, δ L = 2ω T, trap offset r 0 = 1.5ẑ r 0, and relative spring constant k = 0.5. (a) ψ 1 component wavefunction. (b) ψ 2 component wavefunction. (c) Analytic phase boundary (dashed), actual phase boundary (solid). The 1/e of total system density ( ψ 1 (r) 2 + ψ 2 (r) 2 ) is shown (dotted) for reference.

121 4.6 Creating a dark soliton by adiabatic transfer 105 (b) (c) (d) (e) ψ x x x x 1 (a) (f) (g) (h) (i) 0.8 N N (e) 0.2 (d) 0 (b) (c) δ L [ω T ] (f) (g) (h) (i) PSfrag replacements ψ µ [ω T ] x x x x Figure 4.10: Dressed states of a spatially varying coupling field, for a range of detunings. (a) Population of ψ 2 component wavefunction. (b)-(i) Dressed state solutions for various δ L values. Component wavefunctions are: ψ 1 (solid) and ψ 2 (dotted). Parameters are V 1 = V 2, w 11 = w 12 = w 22 = 200w 0 and Ω = 2 sin(x/5) ω T.

122 106 Dressed states of a Bose-Einstein condensate are typically prepared in a single hyperfine state with a ground state centre of mass wavefunction. This can be regarded as the dressed eigenstate for δ L Ω (e.g. see Fig. 4.10(b)), where only a single component is significantly occupied. Adiabatic passage, whereby δ L is changed sufficiently slowly that the system remains in an eigenstate (see pages [29]), can be used to bring the condensate to a dressed state with a mixture of hyperfine components. In the next section we reproduce experimental results showing such a transfer for the case of a ground state and spatially uniform coupling fields. In this section we examine the scheme proposed by Dum et al. [58] to engineer the condensate wavefunction into topological states, such as solitons or vortices using a spatially varying coupling field. Using dynamical simulations, they have shown that a ground state condensate in 1 with δ L Ω can be transfered to an excited state in 2 by slowly varying the detuning of the radiation field until δ L Ω. This is a scheme for adiabatic passage, and as the detuning is changed, the condensate will pass through the dressed states (for a spatially varying coupling field), and so our stationary solutions afford us unique insight into this scheme. For the solution branch we are interested in begins with δ L Ω and 1 fully occupied at (upper set of solutions in Fig. 4.2). Then as δ L 0, the second component ψ 2 grows in-phase with ψ 1 where Ω is positive, and out-of-phase in regions where Ω is negative. If the electric field envelope E 0 (r) has a slow sinusoidally variation across the condensate, with a node near the condensate centre, this will cause ψ 2 to develop as a solitonic excitation as we show in Fig These states are dressed by a standing wave coupling field of the form Ω(x) = 2 sin(0.2 x) [ω T ], (4.55) and as the detuning of the optical field is changed from δ L = 16ω T to δ L = 16ω T, the population of the solitonic 2 state (Fig. 4.10(a)) goes from essentially 0 to 1. We note that a sharp peak at the centre of ψ 1 appears near the soliton node when ψ 2 becomes highly occupied, because the more dense ψ 2 component collisionally excludes ψ 1 to this region (where ψ 2 density is low) (see Figs. 4.10(e)- (h)).

123 4.7 Experimental realisation in rubidium F= m F = 1 0 ~2 MHz ~ 6.8 GHz F=1 m F = Figure 4.11: Raman coupling scheme used in rubidium. A microwave field connects the 1 = F = 1, m F = 1 via the intermediate F = 2, m F = 0, to 2 = F = 2, m F = 1 by an rf field. 4.7 Experimental realisation in rubidium-87 Initial binary condensate experiments in rubidium system involved sympathetic cooling of the F = 2, m F = 2 state with the F = 1, m F = 1 state [130]. In a later experiment the JILA group [121] investigated a similar system composed of the F = 2, m F = 1 and F = 1, m F = 1 states. In this latter system, direct radiative coupling between the two states was enabled by using a 2-photon Raman transition (see Appendix C). This coupling is suitable for investigating condensate dressed states, and in this section we review the preliminary experimental results reported by Matthews [118] Internal state coupling The coupling scheme used by the JILA group [121] is shown in Fig A 2- photon transition connects atoms in the F = 1, m F = 1 state with atoms in the F = 2, m F = 1 state, both of which are magnetically trapped. Because both of these states occur in the ground state hyperfine manifold where the L = 0 (orbital angular momentum), the electric-dipole transition is forbidden and the coupling is mediated with a magnetic-dipole transition (see 2.5.6).

124 108 Dressed states of a Bose-Einstein condensate Calculations [106, 33] show that this system is probably only realisable in 87 Rb where a fortuitous cancellation in the inelastic collision rate means that atoms in the F = 2 state do not spontaneously down convert to the F = 1 (the energy release of this process would annihilate the condensate). We also note that in 87 Rb, the scattering lengths and hence collisional interactions (see Eq. (4.4)) are in the ratio {a 11 : a 12 : a 22 } = {1.03 : 1 : 0.97} [46], so we expect that the collisionally degenerate theory in section should be capable of providing a good description of this system. Spatial dependence of coupling Assuming that the envelope of the electro-magnetic field does not vary appreciably over the condensate, which should be true for microwave and rf fields, then the coupling should be spatially independent (see section 4.6). The effect of the photon recoil on the atoms should also be negligible for these field, since the recoil velocity of a Rb atom from a microwave photon is so small that it would take approximately 10 minutes for the recoiling atom to transit the condensate 3. If this coupling was implemented using optical fields, the recoil would be very significant (this recoil is at the heart of the Bragg scattering phenomenon - see chapter 5) Dressed state observations Most experiments using Raman-coupled condensates have focused on dynamics [121, 91, 90] of the two component system which occurs when the light fields are rapidly changed. The main purpose for the coupling in these cases was to control the interconversion of the hyperfine species. The JILA group have created dressed states using adiabatic passage (as outlined in section 4.6, but without spatial variation). The observation of these states was briefly commented on in Matthews et al. [119] and some results were given in Matthews PhD thesis [118], data from which reproduce in Fig and Regrettably the exact parameters for these measurements were not available, and only general trends can be confirmed. In Fig absorption images of the adiabatically prepared dressed condensates for cases of high and low intensity radiation fields are shown. These are in qualitative 3 Here we have taken f 6.8GHz, and assumed a condensate size of approximately 10µm.

125 Experimental realisation in rubidium :9<;=6 4, +3 -/.0, +*! #"%$ µ&%'! (#")$ µ&)' Figure 4.12: Data from Ref. [118] showing effect of Rabi coupling strength on the dressed state profiles. [Left-Upper] absorption image of the spatial profiles of states 1 and 2 in the high intensity regime, [Left-Lower] corresponding cross-section of density profile. [Right-Upper] absorption image of the spatial profiles of states 1 and 2 in the low intensity regime, [Right-Lower] corresponding cross-section of density profile.

126 110 Dressed states of a Bose-Einstein condensate Figure 4.13: Data from Ref. [118] showing stability of the component populations in the dressed state. A 40ms adiabatic passage is used to prepare the condensate in a dressed state with 50% in each component (empty circles) or a dressed state with near 100% (filled circles) in the initially unoccupied internal state. Decay is due to spin exchange collisions of atoms in the F = 2, m F = 1 state. agreement with our results in section (also see Fig. 4.6), and show clearly that the overlap of the two states is controlled by the intensity of the fields (i.e. Rabi frequency). For high intensities the two components essentially sit on top of each other, whereas for low intensities small differences in the trapping potentials and scattering lengths cause the components to separate. Figure 4.13 shows the relative component populations during the adiabatic preparation and subsequent collisional decay of a dressed state. The empty circles show a condensate as the light fields are adiabatically taken from being far detuned to near resonant over a period of 40ms. The fields are then held constant and the component population of the state remains essentially steady. This is the expected behaviour of a dressed state since it is stationary, and is in contrast to the Rabioscillation behaviour which would result if the light field was suddenly (i.e. nonadiabatically) turned on to resonance. The filled circles show how a 40ms adiabatic passage can also be used take the condensate from the F = 1, m F = 1 state to the F = 2, m F = 1 state by passing through resonance. The noticeable decay after the adiabatic passage is complete is due to spin exchange collisions between atoms

127 4.8 Conclusion 111 in the F = 2, m F = 1 states. 4.8 Conclusion We have presented an investigation of condensate dressed states, and explored the similarities and differences to the familiar dressed states of quantum optics. The major new feature that occurs is that the two components of a condensate dressed state may exhibit distinctly different spatial shapes, and we have examined how these shapes depend on the various system parameters. Even when the component wavefunctions have no nodes, and the external field is a uniform intensity plane wave, we have shown that the shape of the component wavefunctions can be manipulated by varying the field parameters. By defining an effective detuning, we have developed an accurate Thomas-Fermi model for the components of the dressed state in the case of degenerate collisional interaction strengths, which allows us to interpret the behaviour features that we observe. Condensate dressed states are a robust concept which we have predominantly illustrated in 1D, but we have also given 3D results. We have used our methods to investigate the results of Dum et al. [58] on adiabatic passage with a spatially varying coupling field as a scheme for engineering the condensate into an excited state. Finally, we have reviewed some preliminary experimental data by the JILA group for condensate dressed states in agreement with our results.

128

129 Chapter 5 Bragg scattering of a released condensate 5.1 Introduction In this chapter we derive a single component Gross-Pitaevskii equation to describe the evolution of a condensate in the presence of an optical potential arising from far detuned laser fields. We then give a general overview of the physics of matter waves interacting with an optical potential, with particular emphasis on the Bragg regime, for which the output state has momentum components only in a single narrow range. A condensate prepared in magnetic trap is initially stationary, has a high degree of coherence, a well defined spatial profile, and usually significant nonlinearity. These features are absent in previous theoretical treatments of Bragg scattering, which was developed to describe atomic beams in terms of non-interacting plane wave atomic states (e.g. see [16, 117, 2, 79, 131]). In order to characterise the effect of nonlinearity and spatial nonuniformity, we have numerically simulated our single component equation in three dimensions and used these results to build an analytic understanding of Bragg scattering. In this chapter we consider the case where the light fields are applied immediately after the condensate is released from a trap. We present a novel treatment of linear Bragg scattering for a released condensate, which conveniently incorporates spatial nonuniformity. This model is based on a partitioned representation of momentum space and under appropriate validity con-

130 114 Bragg scattering of a released condensate Field 1 Field 2 BEC q k 2, k 1, Figure 5.1: Arrangement of fields to form the light grating. ω 2 ω 1 2π ( s 1 ) 2π 10 9 s 1 2π s 1 ω i Table 5.1: Typical optical potential parameters used in experimental Bragg scattering. The detuning is defined in Eq. (5.9) ditions, reduces to a two state system, which allows simple expressions to be found for the dynamical behaviour of momentum wave packets, and the momentum transition linewidth. We determine the validity range over which the linear treatment accurately describes Bragg scattering of nonlinear released condensates, and identify two main mechanisms responsible for the failure of the linear model; meanfield expansion and nonlinear dispersion. 5.2 Derivation of the optical potential We consider as our model a system of atoms in a light field grating provided by two crossed laser beams, as shown in Fig The laser fields are taken to have a spatial extent much larger than the condensate and are treated as plane wave classical fields with wave-vectors and frequencies k i and ω i respectively. For convenience we shall assume the electric fields are of equal intensity so that the total field is given by

131 5.2 Derivation of the optical potential 115 E T (r, t) = 1 2 E 0(t)Λ(r, t), where Λ(r, t) = [e i(k 1 r ω 1 t) + e i(k 2 r ω 2 t) + c.c], (5.1) and the wave vector k i and frequency ω i are related by ω i k i = c. (5.2) Typical orders of magnitude for these and related frequencies are shown in Table 5.1. The atom is modelled as having two internal states g and e with an electric dipole matrix element d = e ˆd g between them (see section 2.5.5). We note that the fields are chosen to couple to the same two internal states g and e, so that both ω 1 and ω 2 will be approximately the same frequency, though there may be a small difference between them of up to 10 5 s 1. We include a time dependence in E 0 (t) to allow for the possibility of pulsed fields, and note that the coupling strength of the two internal states by this field is characterised by the Rabi frequency, Ω 0 (t) = d E 0 (t)/. The meanfield equations for the ground and excited state wavefunctions ψ g and ψ e are the time dependent Gross-Pitaevskii equations (e.g. see Eq. (2.61)) i ψ g t = 2 2m 2 ψ g + V T g (r, t)ψ g 1 2 Ω 0(t)Λ(r, t)ψ e (5.3) + N 0 U gg ψ g 2 ψ g + N 0 U eg ψ e 2 ψ g, i ψ e t = 2 2m 2 ψ e + ω eg ψ e + V T e (r, t)ψ e 1 2 Ω 0(t)Λ(r, t)ψ g (5.4) + N 0 U eg ψ g 2 ψ e + N 0 U ee ψ e 2 ψ eg, where U gg and U ee characterise the intra-species collisional interactions, and U eg the interspecies interaction. The functions V T i represent the trapping potentials for the respective internal states, and we allow a time dependence in order that they can be dynamically altered. We assume the laser frequencies ω i are sufficiently off resonant from the g e transition that the laser field undergoes negligible modification in propagating through the condensate, and furthermore that spontaneous emission (and its effect on condensate coherence) can be ignored. In this large detuning limit

132 116 Bragg scattering of a released condensate it is also permissible to adiabatically eliminate the excited state from Eqs. (5.3) and (5.4), leaving only an equation for ψ g (see the next subsection for details). We have chosen the states ψ g and ψ e to be normalised to unity, i.e. dr [ ψ g (r, t) 2 + ψ e (r, t) 2] = 1, (5.5) thus N 0, the total number of particles, appears in Eqs. (5.3) and (5.4) Adiabatic elimination of excited state Eqs. (5.3) and (5.4) detail the EM field coupling, in the dipole approximation, between the meanfields of the atoms in internal states g and e respectively. By defining a new excited meanfield amplitude ψ e (r, t) = e iω 1t ψ e (r, t), (5.6) the evolutions equations are transformed to i ψ g t = 2 2m 2 ψ g + V T g (r, t)ψ g 1 2 Ω 0(t)Λ(r, t)e iω 1t ψe (5.7) + (N 0 U gg ψ g 2 + N 0 U eg ψ e 2 )ψ g, i ψ e t = 2 2m 2 ψe + V T e (r, t) ψ e 1 2 Ω 0(t)Λ(r, t)e iω1t ψ g (5.8) ψ e + (N 0 U eg ψ g 2 + N 0 U ee ψ e 2 ) ψ e, where we have introduced the detuning = ω 1 ω eg, (5.9) where ω eg is the Bohr frequency. For an initial configuration with the ground state fully occupied, we make the approximations that in the region of significant condensate density the magnitude of ψ e greatly exceeds those of N 0 U ee ψ e 2 ψe, 2 2 ψe /2m, N 0 U eg ψ g 2 ψe and V T e (r, t) ψ e (which can always be arranged by choosing sufficiently large), so that these latter three terms can be dropped from Eq. (5.8) allowing the following approximate formal solution

133 5.2 Derivation of the optical potential 117 ψ e (r, t) + i 2 t 0 { e i (t s) Λ (r, s)e iω 1s } Ω 0(s)ψ g (r, s) ds. (5.10) The adiabatic elimination of the excited state proceeds by noting that while Ω 0 and Ω 0(t) Ω 0 (t), Ω 0(s)ψ g (r, s) is much more slowly varying in time than the terms in the braces in the integrand of Eq. (5.10), which vary at least as fast as e i s. This means the main contribution to the integral arises near the end point, and so Ω 0 (s)ψ g (r, s) may be taken outside the integral as Ω 0 (t)ψ g (r, t), i.e. ψ e (r, t) iω 0(t) 2 ψ g (r, t)e i t t 0 { ds e i s e ik 1 r +e i(k 2 r (ω 2 ω 1 )s) + e i(k 1 r 2ω 1 s) + e i(k 2 r (ω 2 +ω 1 )s) }. (5.11) The rotating wave approximation is now made, in which we neglect the last two rapidly varying terms in Eq. (5.11). The resulting approximate solution to Eq. (5.10) is ψ e (r, t) Ω 0(t) 2 (eik 1 r + e i(k 2 r (ω 2 ω 1 )t) ) ψ g (r, t), (5.12) which can be substituted into Eq. (5.7) to give i ψ g t = 2 2 2m ψ g + Ω 0(t) 2 4 +e i(k 2 r ω 2 t) ( e i(k 1 r ω 1 t) ) Λ(r, t)ψ g + N 0 U gg ψ g 2 ψ g. (5.13) The rotating wave approximation is again be applied to Eq. (5.13), rejecting terms oscillating at optical frequencies. This leads to an effective potential for the ground state of the form Finally, the phase choice Ω 0 (t) 2 [1 + cos(q r ωt)]. (5.14) 2 t ψ = exp( i V (s)ds) ψ g, (5.15) 0

134 118 Bragg scattering of a released condensate removes the D.C term from this potential and yields i ψ t = 2 2m 2 ψ + V T (r, t)ψ + V (t)λ(r, t)ψ + w ψ 2 ψ. (5.16) The wavefunction ψ is the ground state wavefunction in an interaction picture, and for clarity we have written w for N 0 U gg and V T (r, t) for V T g (r, t), and have defined the quantities V (t) = Ω 0(t) 2 2, (5.17) λ(r, t) = cos(q r ω t), (5.18) where q = k 1 k 2, (5.19) ω = ω 1 ω 2. (5.20) Results (5.16)-(5.18) can also be understood by using the dressed states of the (single) atoms in the light fields (e.g. see chapter 7 of [42]). In this approach the internal and external evolution of the atoms is decoupled by introducing an optical potential related to the local eigenvalue of the internal dressed states (see Adams et al. [2] for the standing wave case). In our case the optical potential is V opt (r, t) = V (t) cos(q r ω t), (5.21) identical to the products of Eqs. (5.17) and (5.18). 5.3 Regimes of condensate scattering In this section we classify the broad categories of behaviour that occur in the scattering of atomic waves from an optical potential of the form (5.21), so that we may clearly demarcate the Bragg parameter regime which is the focus of this thesis. The results we present in this section do not require the extended spatial coherence of a condensate and are equally applicable to cases such as atomic beams. Thus for the sake of simplifying the discussion we ignore the nonlinear and trapping potential

135 5.3 Regimes of condensate scattering 119 terms in Eq. (5.16) and take the matter wave to evolve according to i ψ t = 2 2m 2 ψ + V opt (r, t)ψ, (5.22) and refer to ψ simply as the (single particle) wavefunction for the matter wave. We return our attention to condensate case in section 5.5. We take the optical potential amplitude (V (t) in Eq. (5.22)), to define a pulse of average amplitude V p and duration T p, which we shall refer to as the Bragg intensity and pulse length respectively. In this section we show how these quantities determine the regime of scattering Raman-Nath diffraction For short interaction times, where the atomic evolution can be ignored, the Schrödinger equation (5.22) can be reduced to i ψ t = V opt(r, t)ψ. (5.23) This is known as the Raman-Nath approximation [17] (see Appendix D.1), and equation (5.23) has the formal solution ( i ψ(r, t) = ψ(r, 0) exp t 0 ) dt V opt (r, t ). (5.24) For simplicity we assume that both laser fields have the same frequency (i.e. ω = 0) and that the V opt has a constant amplitude V p from time 0 to T p (and zero otherwise). Substituting Eq. (5.21) into Eq. (5.24) and integrating we obtain ψ(r, t) = ψ(r, t=0) exp(iv p sin(q r)t), 0 t T p. (5.25) This result shows that on the time scale where the Raman-Nath approximation is valid, the optical potential simply imprints a phase on the initial spatial wavefunction. A non-zero value of ω causes the optical potential to move, thereby washing out the imprinted phase to some degree. Equation (5.25) is analogous to the optical case of a thin-phase grating in the near field. To evaluate the far-field result it is necessary to evolve the system to some later time, using Eq. (5.22) in the absence of the optical potential. Assuming the system to be free of external forces, the final

136 120 Bragg scattering of a released condensate atom distribution will be determined by the momentum distribution at the conclusion of the optical pulse. Taking the Fourier transformation of Eq. (5.25), assuming that the initial state ψ(r, t = 0) is a plane wave state of zero momentum 1 we find that this momentum distribution is (see [2, 107]) φ(nq, T p ) 2 = J n (V p T p ) 2 n = 0, ±1, ±2,..., (5.26) where φ(k, t) = 1 (2π) 3 2 dr ψ(r, t) e ik r, (5.27) is the momentum space wavefunction, and J n is the n-th order Bessel function. Equation (5.26) predicts that only discrete kicks of momentum are issued to the atoms, and that because J n (x) 2 = J n (x) 2, the diffraction orders are symmetric. It is also interesting to note that the most populated diffraction order is n V p t, so that as the length of time the optical potential is applied increases, more kinetic energy is transferred to the atoms, and the Raman-Nath approximation will no longer be valid. In appendix D.1 we give a simple derivation to show that this approximation will be valid for pulses of duration T p < τ RN, where τ RN = 1 Vp ω q, (5.28) and ω q = q 2 /2m, (5.29) is the atomic recoil frequency. In Fig. 5.2 we show numerical results for the near-field density and momentum distribution for a cylindrically symmetric wavefunction evolving in the presence of an optical potential (we will discuss the numerical simulations we use for solving Eq. (5.16) in section 5.4.1). At t = 0 the light fields are turned on and the wavefunction is in a spherically symmetric ground state, corresponding to a single sharp momentum distribution (see Figs. 5.2(a) and (d)). Since q is in the ẑ direction, we have chosen to show the distribution of the z-component of momentum in the wavefunction, defined as N(k z ) = dk x dk y φ(k) 2 (also see subsection 5.6.1). After being exposed to the optical potential (V opt ) for a duration t = t 0, the density (on the plane 1 In practice this assumption means that ψ(r, t = 0) can be described by a momentum wave packet of mean momentum zero, and spread much less that q.

137 5.3 Regimes of condensate scattering (a) t = 0 0 (d) PSfrag replacements z [r0] z [r0] z [r0] (b) (c) 0-10 x [r 0 ] x [r 0 ] x [r 0 ] t t log 10 N [r0] log 10 N [r0] log 10 N [r0] = = (e) (f) ψ [r0 3 ] k z [r0 1 ] Figure 5.2: Density profiles along the central y = 0 plane and momentum distribution N(k z ) of a condensate evolving in an optical potential. (a)-(c) density profiles; (d)-(f) z-component of momentum distribution shown for a range of times. The approximate momentum distribution from Eq. (5.26) is shown as crosses in Figs (d), (e) and (f). Parameters are: q = 5ẑ /r 0, ω = 25 ω T, V p = 10 ω T and N 0 U 0 = 0.

138 122 Bragg scattering of a released condensate y = 0) and momentum distributions develop as illustrated in Figs. 5.2(b) and (e) respectively. For the parameters used in Fig. 5.2, the Raman-Nath time is τ RN t 0 and as expected Eq. (5.26) is seen to provide a good description for the diffraction order populations. Finally, in Figs. 5.2(c) and (f) we show the system after the optical potential has remained on for t = 0.19 t 0. This is well outside the Raman-Nath regime and the prediction of Eq. (5.26) is now seen to be in significant disagreement with the numerical simulation. We note from Fig. 5.2(f) that the occupation of the zero and the q momentum states is at least an order of magnitude higher than the other states. This preferential selection of the q state occurred because of the choice of frequency difference ω in the optical potential. Outside the Raman-Nath regime, ω no longer has the role of simply washing out the effect of the phase grating. For instance, repeating this numerical simulation with ω = 0, we find that the occupation of both the q and the q states is negligible in comparison to the zero momentum state. We have in fact entered the Bragg regime, where the temporal properties of the optical potential are as important as the spatial properties are in the Raman-Nath regime Bragg regime The Bragg regime is named in analogy to the famous effect discovered by Bragg and Bragg in the scattering of X-rays from a crystal lattice [28]. We devote the rest of this chapter to considering this regime in detail for the case of a released condensate. We begin by presenting a simple energy conservation argument for deriving the Bragg resonance condition for selecting the frequency difference ω to resonantly scatter into a particular momentum state (e.g. see Fig. 5.2(f)). We consider the case of a free atom, initially at rest, which undergoes a sequence of n 2-photon transitions. In each 2-photon transition momentum q and energy ω is transferred from the light field to the centre of mass momentum of the atom. This process will be resonant if the kinetic energy (p 2 /2m) of the atom in its final momentum state (n q) is equal to the energy supplied by the field (n ω), i.e. nω = (nq)2 2m, (5.30) which we refer to as the Bragg resonance condition. This is illustrated graphically in Fig. 5.3 for the case n = 3. Here, the first two transitions to states indicated by grey

139 5.3 Regimes of condensate scattering 123 Energy PSfrag replacements ω q ω q q ω Momentum Figure 5.3: The Bragg resonance condition in an energy-momentum diagram. A multiphoton scattering event must end on the atomic dispersion curve (solid line) to allow overall energy conservation. The black dot at origin indicates the initial state, grey dots show intermediate states, and the upper most black dot indicates the final state for which overall energy is conserved, corresponding to a resonant Bragg transition. dots do not lie on the atomic dispersion curve, so are energetically unfavourable. The third transition allows overall energy conservation, and thus 0 q corresponds to a Bragg resonant transition (see Eq. (5.30)). It is interesting to consider the Bragg scattering process and the resonance condition (5.30) from another point of view. First, we note that the frequency difference ω between the two light fields causes the optical potential (Eq. (5.21)) to move with an effective velocity v opt = ω ˆq, (5.31) q relative to the stationary atom. Making use of Eq. (5.31) we can rewrite the Bragg condition (5.30) as v opt = n q 2m. (5.32) Now consider our system in a reference frame moving at a velocity v rel relative to the laboratory frame, and chosen so that the optical potential is stationary. There are many possible choices of v rel as we simply require that the component of v rel

140 124 Bragg scattering of a released condensate (a) v rel θ v opt (b) Matter Wave Matter Wave v atom sin θ θ v atom PSfrag replacements (c) Incident Matter Wave θ λ atom Bragg scattered Matter Wave Matter Wave Unscattered Figure 5.4: Re-interpreting the moving optical potential and stationary atoms in the traditional Bragg scattering sense. (a) The lab-frame where a stationary atom experiences the optical potential moving at velocity v opt. We also show one possible choice of a relative velocity v rel of a reference frame in which the optical potential would appear stationary. (b) In the reference frame defined by v rel, the atom wavefunction now appears to be moving. (c) In the moving frame the Bragg scattering is analogous to the case of Bragg scattering of X-rays from a crystal lattice.

141 5.3 Regimes of condensate scattering 125 parallel to the direction of q exactly cancels the optical potential motion i.e. v opt = v rel sin(θ), (5.33) where θ is the angle between the velocities v rel and v opt (see Fig. 5.4(a)). In this new reference frame, the atom is now no longer at rest, but moves at a velocity v atom = v rel (Fig. 5.4(b)). From Eqs. (5.32) and (5.33) we can rewrite the Bragg resonance condition as nλ atom = 2d opt sin(θ), (5.34) where λ atom = 2π /mv atom is the de Broglie expression for the wavelength associated with a moving particle, and d opt = 2π/q is the spacing between the intensity peaks of V opt. This condition (5.34) is identical to the famous Bragg result [28], however here it is for the complementary situation of matter waves scattering off a light crystal defined by the optical potential intensity peaks. For the case of X-rays, Bragg scattering only occurs in thick crystals where multiple reflections can arise. This is equivalent to requiring that the optical potential is exposed to the atoms for a duration T p > τ Bragg, where τ Bragg = 1 ω q, (5.35) so that the wavefunction is traversed by approximately a wavelength of the moving optical potential. This condition ensures that the frequency spread about ω in the optical potential due to finite pulse time, is less than the recoil frequency (ω q ). The effect of this frequency spread can be visualised as introducing vertical error bars on the transitions points shown in Fig We note that when the frequency spread is greater than ω q diffraction occurs, whereby the frequency components present are able to satisfy the Bragg resonance condition (5.30) for a number of n values and many output momentum states will be populated. Similarly, if V opt is not formed from planes waves, but from light fields with a finite spatial extent l, then there will be an associated spread in q values of order 1/l. In typical condensate experiments, l 10 3 /q so finite spatial effects are negligible. It is also necessary that the optical potential height is small enough that the atoms can traverse it and hence interact with the many light crystal planes. This

142 126 Bragg scattering of a released condensate V p PSfrag replacements V Bragg Channeling Bragg 0 Raman-Nath τ Bragg T p Figure 5.5: Parameter regimes for the different regimes of matter wave interaction with optical gratings. condition places an upper limit on the optical potential intensity V p V Bragg, where V Bragg = ω q, (5.36) (e.g. see [107]) Channeling regime For completeness, we note that another regime known as channeling [148, 107] exists, with features distinct from both the Raman-Nath and Bragg cases. Channeling occurs when the atoms are exposed to a large amplitude optical potential for a long time. This regime was so named because the energies of the atoms are well bounded by the optical potential and the atoms simply channel along the optical potential valleys. The characteristics of channeling atoms are very sensitive to how rapidly the optical potential is switched on [107], and in certain cases can resemble Bragg scattering. In Fig. 5.5 we summarise the parameter regimes associated with the different categories of matter wave scattering.

143 5.4 Bragg scattering of a released condensate Bragg scattering of a released condensate For the remainder of this chapter we will consider the evolution of a condensate released from a harmonic trap at t = 0, and immediately subjected to a Bragg pulse. As discussed in the previous section, to be in the Bragg regime we require the pulse duration to be T p > 1/ω q, and the optical potential intensity to satisfy V p ω q Numerical solutions The primary equation we use to investigate Bragg scattering of Bose-Einstein condensates is the Gross-Pitaevskii equation (5.16) derived in section 5.2. This equation is in general difficult to solve because of the nonlinear term and in order to explore the types of behaviour that can occur, we have obtained numerical solutions for a range of parameters. Our solutions are in two or three dimensions, with the three dimensional case restricted to the cylindrical symmetry (see section 3.5). These simulations could in principal be carried out with a general 3D algorithm such as developed by Caradoc-Davies [35], however the size of spatial grids required to accurately represent the optical potential and the condensate wavefunction over some practical time duration places significant constraints on systems which can be simulated. Our restriction in the 3D case to cylindrical symmetry requires q to be in the axial direction (i.e. q = q ˆk z ). In the next two chapters we will meet cases where the condensate states we wish to examine are not compatible with the cylindrical symmetry constraint, and so we revert to 2D simulations. As an initial example of Bragg scattering in a released condensate we present in Fig. 5.6 a sequence of images which show the density of a condensate evolving in the presence of a light grating, and illustrate the complexity that can typically occur. During this simulation we can see fringes developing in the density profile, which is characteristic of interference between a stationary and a moving wavepacket. As time progresses the spatial distribution extends in the z direction at a much faster rate than spreading occurs in the perpendicular directions, and the density distribution becomes less uniform (see Fig. 5.6(d)) with distinct regions of high and low density apparent. For convenience our numerical solutions are given in dimensionless units as described in section 2.4.

144 128 Bragg scattering of a released condensate 20 (a) t = 0 (b) t = 0.21 z [r0] z [r0] ψ 2 [r 3 0] (c) t = 0.43 (d) t = 0.65 PSfrag replacements x [r 0 ] x [r 0 ] Figure 5.6: Probability density (in the x-z plane) of a condensate evolving in the presence of a light grating. The condensate is initially in the ground state of a spherical harmonic trap, and at time t = 0, the trap is switched off and a cw-light grating applied. Parameters are w = 2500 w 0, q = 14/r 0 in the z-direction, ω = 196 ω T and V p = 20 ω T.

145 5.5 Analytic treatment of the linear case Analytic treatment of the linear case It is evident that the simplified treatment of Bragg scattering of condensates encompassed in Eq. (5.16) can give rise to quite complex behaviour. Here we derive an analytic treatment of the model that will provide a simple understanding of the behaviour over a wide parameter regime. We begin by transforming the equation into momentum space, where the momentum space wavefunction φ(k, t) is defined in Eq. (5.27). Henceforth, for the sake of notational clarity, we will suppress the explicit time dependence of the momentum wavefunction, and simply write φ(k), for φ(k, t). Straightforward manipulation of Eq. (5.16) leads to i φ(k) t = 2 k 2 V (t) φ(k) + [φ(k + q)e iωt + φ(k q)e iωt ] (5.37) 2m 2 w + dk (2π) 3 1 dk 2 φ(k 1 )φ ( k 2 )φ(k k 1 k 2 ), and we can see that the effect of the light grating is to linearly couple states of momentum k to neighbouring states k ± q. It is useful to partition momentum space to incorporate this regularity, in an analogous way that Brillouin zones are used in solid state physics to reflect the periodicity of the crystal lattice. Choosing the q in k-space to be along the z direction, we divide the ˆk z axis into intervals of length q, and label them by the integer n, such that the centre of the n th interval (along the ˆk z axis) is at k z = nq, (and n = 0, ±1, ±2, ±3,...). The wavefunction φ(k) can now be re-expressed as a set of wavefunctions φ n (k), each defined only within the n-th interval of the partitioned k-space, i.e. φ n (k) φ(k) {k : (n 1 2 ) q < k z (n + 1 ) q }. (5.38) 2 Within each interval, the momentum vector is uniquely defined by its offset from the central k value, and thus we finally replace the wavefunction φ(k) by the set of partitioned wavefunctions φ n (κ) = φ n (k) e inωt, (5.39) where κ = k nq, i.e. the domain of φ n is all values of κ which have a ˆk z component in the range [ q /2, q /2]. We have included phases in the definition in Eq. (5.39), for later convenience. The utility of these new wavefunctions arises because φ n (κ)

146 130 Bragg scattering of a released condensate and φ n+1 (κ) represent the full momentum wavefunction at adjacent positions in k-space separated by exactly one momentum kick q. In terms of this new set of wavefunctions, Eq. (5.37) can be written as the set of equations i φ n(κ) t = ω n (κ)φ n (κ) V (t)[φ n 1(κ) + φ n+1 (κ)] (5.40) w + dκ (2π) 3 1 dκ 2 φ i (κ 1 )φ j(κ 2 )φ n i j (κ κ 1 κ 2 ), i,j where ω n (κ) = κ + nq 2 2m nω. (5.41) Noninteracting atoms We derive an approximate analytic solution for Eq. (5.40) by considering the case where collisional interactions in the BEC are negligible, i.e. w = 0. We shall find that this allows us to make a good representation of the full equation for a large regime of interest for condensates. Putting w = 0 allows us to write the evolution equation (5.40) as a linear system: i t. φ n 1 (κ) φ n (κ) φ n+1 (κ). = ω n 1 (κ) V (t) V (t) 0 2 ω n (κ) V (t) 0 2 V (t) ω 2 n+1 (κ) φ n 1 (κ) φ n (κ) φ n+1 (κ). (5.42) First order couplings and the two-state model The matrix in Eq. (5.42) displays the couplings between the partitioned momentum wavefunctions. For a given momentum κ, the effectiveness of the coupling between φ n (κ) and φ n+1 (κ), is determined by the size of the coupling V p relative to the momentum detuning, n (κ) = ω n+1 (κ) ω n (κ). (5.43) In particular if (for some κ r ), n (κ r ) = 0, then the corresponding transition of a particle from momentum state κ r + nq κ r + (n + 1)q is resonant. Noting

147 5.5 Analytic treatment of the linear case 131 that the momentum kick given to the atoms is q, (the momentum difference of the photons from the two light fields) and that n (κ) = [(2n + 1)q + 2κ] q ω, (5.44) 2m then the resonance condition n (κ r ) = 0 is recognised as a Bragg condition for the process: that is both momentum and energy are conserved. The typical initial condition for Eq. (5.42) is a localised momentum wavepacket. If q is significantly greater than the width of the wavepacket, the initial profile is almost completely contained within a single momentum partition interval, and thus we can take the initial momentum wavefunction to have only one of the partitioned wavefunctions φ 0 (κ) to be non-zero. It is easy to see from Eq. (5.44) that if one of the j (κ) is very small, then all the others are larger by order q 2. For example, if 0 (κ r ) = 0 (a first order Bragg resonance), then n (κ r ) = m [ n q 2 ]. (5.45) Thus for the case 0 (κ r ) = 0, if the neighbouring detunings ( ±1 (κ r )) are much larger than V over the entire κ-domain (see also section 5.6.3), we can make the secular approximation [41] that only the coupling φ 0 φ 1 is significant and the evolution equation can be reduced to: i t [ φ 0 (κ) φ 1 (κ) ] = [ ω 0 (κ) V (t) 2 V (t) 2 ω 0 (κ) + 0 (κ) ] [ φ 0 (κ) φ 1 (κ) ]. (5.46) This equation represents a continuous set of two-state Rabi problems, where the members of the set are labelled by the variable κ. The solution is easily found in terms of the eigenvectors of the coefficient matrix in Eq. (5.46) (e.g. see [44]). Separable Wavefunction The complete Bragg scattering problem in the linear case (w = 0) is described by Eq. (5.42), a system of equations in which the momentum argument κ is a three dimensional vector. This three dimensional character remains even when the equations can be reduced to a set of two-state problems, as in the previous section. It is possible, however, to find a solution for the full linear case which simplifies the geometrical character. If we assume the partitioned wavefunction is separable, i.e.

148 132 Bragg scattering of a released condensate φ n (κ) = ξ(κ x )ζ(κ y )Φ n (κ z ), (5.47) and we take the momentum kick to be in the z direction, then Eq. (5.42) transforms to a set of independent equations, i ξ(κ x) t i ζ(κ y) t i Φ n(κ z ) t = 2 κ 2 x 2m ξ(κ x), (5.48) = 2 κ 2 y 2m ζ(κ y), (5.49) [ ] (κz + nq) 2 = nω Φ n (κ z ) (5.50) 2m V (t) [Φ n 1(κ z ) + Φ n+1 (κ z )]. The wavefunction Φ n (κ z ) obeys an equation identical to Eq. (5.42), except that the momentum argument is now simply the scalar κ z. The wavefunctions ξ(κ x ) and ζ(κ y ) describing the κ x and κ y momentum behaviour are freely evolving one dimensional wavepackets, and for example with a gaussian initial condition have a well known analytic solution (e.g. [44]). Reduction of the system of equations for Φ n to a two-state system would proceed exactly as in the previous section. We have chosen, in that derivation, to retain the slightly more general form φ n (κ), because it also includes the case of non-separable wavefunctions, and we will retain the general form φ n (κ) in the remaining sections of this chapter for the same reason. We emphasize however that our results can be transformed to the somewhat easier separable case by simply making the transformation φ n (κ) Φ n (κ z ). 5.6 Results and features of the two-state model In this section we present the characteristic features of the two state model and compare its behaviour to the numerical solutions of Eq. (5.16) for the case of w = 0. We then discuss the parameter regimes for which the two-state model provides a good description of the full w = 0 behaviour.

149 5.6 Results and features of the two-state model Rabi oscillations The two state system of Eq. (5.46) will Rabi oscillate between the φ 0 and φ 1 wavepackets in accordance with previous theory and experiments [117, 14]. The frequency at which each momentum state oscillates between the two wavepackets is momentum dependent, and is given by the generalized Rabi frequency, defined as Ω (κ) = V (κ) 2. (5.51) This oscillatory behaviour is clearly evident in Fig. 5.7(a)-(c), where we present the time evolution of the momentum wavepackets from the two-state model, for a case similar to Fig. 5.6, but with w = 0. In the linear case the behaviour of the momentum oscillations for a given value of kkis independent of the k x and components, and thus the cycling behaviour of φ(k) 2 is the same as for the k y z-component of the momentum density of the wavefunction, i.e. N(k z ) = dk x dk y φ(k) 2. (5.52) Hence in the w = 0 case we have chosen to visualize the momentum wavefunction dynamics in terms of N, rather than φ(k) 2. As with all examples presented in sections , the condensate is prepared in an eigenstate of a spherical harmonic trap and at time t = 0 the trap is switched off and the light grating is applied. The initial state is thus gaussian (since w = 0), and we have also chosen the detuning at the centre of the φ 0 wavepacket to be zero (i.e. 0 (0) = 0 ). In Fig. 5.7(b), where t = π/ω (0), the population has transferred almost entirely to the φ 1 wavepacket. The κ dependence of the Rabi cycling becomes evident at later times as the difference in the Rabi periods accumulates, an effect which can be seen in (Fig. 5.7(c)) where the cycling at the outer edges of the momentum packets now significantly leads the cycling at κ = 0 giving rise to a pronounced central dip in φ 1. The full numerical solution of Eq. (5.16) for these parameters is presented in Fig. 5.7(d)-(f), and clearly confirms the validity of the two-state model in this case Dispersion curves and resonance-coupling The two-state model represented by Eq. (5.46) was obtained by assuming that only one of the couplings in the matrix of Eq. (5.42) was important. Here we present a

150 134 Bragg scattering of a released condensate (a) t = 0.01 (d) t = 0.01 N [r0] N [r0] N [r0] (b) (c) t t =0.06 = 0.52 (e) (f) t t = 0.06 = 0.52 PSfrag replacements k z [r0 1 ] k z [r0 1 ] Figure 5.7: Evolution of the momentum distribution for the two-state solution with q z = 14/r 0, ω = 196 ω T, and V p = 50 ω T. The boundary for the partitioned wavefunctions is indicated by the vertical dotted line. The wavepacket centred about k z = 0 corresponds to φ 0 and has an initial gaussian profile. The wavepacket to the right of the dotted line corresponds to φ 1 (but shifted by k z = 14/r 0 ). Figures (a)-(c): the analytic two-state solution. Figures (d)-(f): full numerical solution of Eq. (5.16) for w = 0.

151 5.6 Results and features of the two-state model (a) ω 2 ωn [ωt ] ωn [ωt ] PSfrag replacements ω [ω T ] ω 1 ω 1 ω κ [r0 1 ] (b) ω 1 ω 2 ω 0 ω κ [r0 1 ] Figure 5.8: Free particle dispersion curves, with first order resonance indicated by circle. (a) q z = 14/r 0 and ω = 196 ω T, (b) q z = 14/r 0 and ω = 420 ω T. simple means of visualising the possible couplings, and determining the validity and manner of breakdown of the two-state approximation. We first construct a set of dispersion-type curves by plotting the frequencies ω n (κ) against κ for a given choice of q and ω, as illustrated in Fig. 5.8(a). The important feature of this graph is the vertical separation of the curves: the coupling between φ n and φ n+1 becomes appreciable only when n (κ) is small, i.e. when the difference between ω n (κ) and ω n+1 (κ) is small. We need only consider (κ) in the case of κ in the z direction (the direction of q), because the perpendicular components of κ simply offset all the ω curves by the same amount. We will therefore

152 136 Bragg scattering of a released condensate simply write κ z and k z for the momentum arguments in the remainder of the chapter. On the plots, the first order Bragg resonance condition ( n (κ z ) = 0) appears as a crossing of the ω n and ω n+1 curves, and we have highlighted this point (for n = 0) with a small circle. Appreciable coupling will only occur in the vicinity of this point. In Fig. 5.8(a) we have chosen the light grating parameters (q and ω) so that the crossing is at κ z = 0, and thus wavepacket momentum components near κ z = 0 will undergo Rabi cycling. In Fig. 5.8(b), the grating parameter choice puts the Bragg resonance at κ z = 6/r 0, and the response for a wavepacket centered at κ z = 0 depends on the coupling width, which we discuss in the following subsection. Fig. 5.8(b) also shows a crossing of the ω 0 and ω 2 curves near κ z = 0. This is a second order resonance: φ 0 can link to φ 2 via the intermediate state φ 1, and the overall process of φ 0 φ 2 conserves energy and momentum. However the transition from φ 0 φ 1 (and likewise φ 1 φ 2 ) is detuned by the amount 0 = 36ω T at this point, and thus in order that the transition φ 0 φ 2 may proceed, V p must be sufficiently large for φ 1 to be seeded with population and so mediate the resonant second order process. This of course generalises to higher order couplings Coupling width and breakdown of two-state model Power broadened momentum line width As is well known from the Rabi model, the probability of transition from state 1 to state 2 is given by the expression P 12 = V 2 p V 2 p + 0 (κ z ) 2. (5.53) As a function of κ z, this is maximum at the first order Bragg resonance 0 (κ z ) = 0 and falls to half its value at 0 (κ z ) = V p, the power broadened width. Momentum components in φ 0 and φ 1 (or more generally φ n and φ n+1 ) will be significantly coupled only if the separation of the ω curves is less than V p. Converting this to the corresponding κ z width, we find that the momentum width of the transition (i.e. the range of κ z about the Bragg resonance point for which the probability of momentum transfer exceeds 1/2) is σ (V ) q = 2mV p q z. (5.54)

153 5.6 Results and features of the two-state model 137 Temporal broadened momentum line width The Rabi expression (5.53) is valid for the case of a cw monochromatic coupling. For a Bragg pulse of duration T p, about ω there will be a frequency spread of width σ ω π/t p, (5.55) giving rise to a temporal momentum width σ (T ) q = πm q z T p. (5.56) In the Bragg regime of scattering, the maximum size of the temporal width is limited to σ q < q/2 by the requirement T p > ω 1 q (5.35). Total momentum line width The power width (5.54) and temporal width (5.56) should be added in quadrature to give to total linewidth, i.e. σ q = ( ) 2 ( σ q (V ) + σ (T ) q ) 2. For the rest of this section we shall be interested in the large intensity limit where V p ω q and the temporal effect (5.56) can be neglected (i.e. σ q σ (V ) q ). The cycling behaviour of the whole wavepacket depends on the relative size of the momentum width of the initial state, σ p, compared to σ q. If σ q > σ p the whole packet will cycle, as shown in Fig In practice this can be achieved using a sufficiently large laser intensity (i.e. large V p ) to broaden the transition width σ q. It is then possible to apply a π-pulse so that the initial momentum wavepacket (φ 0 ) is transferred entirely to the adjacent location in momentum space (φ 1 ), and then the whole wavepacket receives a quantised momentum change. A π/2 pulse on the other hand acts as a 50/50 beam splitter. For the case of σ q σ p, the Bragg process will couple out only a fraction of the initial state, as we discuss further in Sec Two state validity From the dispersion curves in Fig. 5.8 it is apparent that the validity of the twostate model requires that over the momentum width σ p of the initial wavepacket,

154 138 Bragg scattering of a released condensate 0.75 N [r0] PSfrag replacements k z [r 1 0 ] Figure 5.9: Numerical simulation of Bragg scattering for large V p, showing probability distribution of z-component of momentum at time t = 0.29/ω T. The dotted lines indicate the boundaries between the momentum partitions. Except for V p = 100 ω T, other parameters are as in Fig only the ω 1 curve is within a distance V p of ω 0. If on the other hand V p is sufficiently large that ω 1 and ω 0, or ω 2 and ω 1 are separated by less than (or of order V p ), then additional couplings will occur, and the two-state description will fail. We illustrate this in Fig. 5.9, where the parameters are the same as in Fig. 5.7, except that V p has been increased, to a value of approximately 25% of the ω 1 to ω 2 separation. By the time t = 0.29/ω T, as shown in Fig. 5.9, the additional couplings have resulted in appreciable population transfer to the φ 1 and φ 2 components. Of course coupling to additional φ n will occur if V p is further increased. In view of this discussion we can formulate a condition that the behaviour of a scattered wavepacket will be twostate in momentum space: the minimum detuning to subsequent dispersion curves (i.e. ±1 ) over the momentum range of the wavepacket must be much greater than V p, i.e. [ q 2 q σ ] p V p. (5.57) m 2 It is worth emphasizing that for the case of a first order Bragg resonance the accuracy of the two state model improves as q increases in magnitude.

155 5.7 Application to nonlinear case Application to nonlinear case In this section we compare our analytic results obtained in the linear regime (w = 0) to the case of simulations with nonzero values of w, and characterise the regime where the linear analysis accurately describes the nonlinear case Numerical result We first demonstrate that the main characteristics contained in the two-state analysis still occur in the nonlinear case. We consider as an illustration the density evolution case presented in Fig. 5.6, where w = 2500 w 0, and we plot in Fig. 5.10(a) the wavefunction amplitude along the symmetry axis, corresponding to Fig. 5.10(b). The corresponding momentum wavefunction, obtained by numerical Fourier transform, is shown in Fig. 5.10(b), and reveals two distinct packets. On the same figure we have drawn for comparison a dashed line representing the momentum wavefunction that would arise from the w = 0 two-state case. The agreement is close, and furthermore as time progresses the momentum wavepackets of the w = 2500 w 0 case oscillate at a frequency approximately equal to the frequency V p predicted by the two state model. The momentum distribution enables us to understand the fringe structure that developed in Fig. 5.10(a). We illustrate this by constructing separately the individual spatial wavepackets ψ 0 and ψ 1 corresponding to the momentum packets φ 0 and φ 1 respectively. The wavefunction ψ 0 (shown as a solid line in Fig. 5.10(c)), is of course essentially stationary (apart from the effects of expansion), while the wavefunction ψ 1 (shown as a dotted line) has a mean momentum of k z = q, and moves to the right at a speed of q/m. This packet accordingly has a steep, approximately linear, phase gradient across it, so that superposition of ψ 0 and ψ 1 results in the observed interference fringes Free expansion One of the significant new effects in nonlinear Bragg scattering is that the self repulsion in the free expansion of the condensate causes the momentum wavepackets to expand. In order to clearly demonstrate this effect, it is convenient to isolate the contribution to momentum changes that arise simply from the Bragg kicks. We

156 140 Bragg scattering of a released condensate ψ [1/ r 3 0] (a) φn [ r 3 0] (b) z [r 0 ] 0.1 PSfrag replacements ψn [1/ r 3 0] k z [r0 1 ] 0.06 (c) z [r 0 ] Figure 5.10: Spatial and momentum profiles for the numerical solution presented in Fig. 5.6 at t = 0.21/ω T. (a) Spatial wavefunction amplitude along the z-axis. (b) Momentum wavefunction amplitudes. Solid line corresponds to case in Fig. 5.6(a). Dashed line shows the w = 0 two-state case, for comparison. (c) Amplitudes of ψ 0 (solid) and ψ 1 (dotted) spatial wavefunctions corresponding to φ 0 and φ 1 of Fig. 5.6(b), respectively.

157 5.7 Application to nonlinear case 141 achieve this by defining a momentum density ρ T (κ z ) = dκ x dκ y φ i (κ) 2, (5.58) i which gives the total occupation of all momentum states that could be reached from an initial state of momentum κ by an integral number of momentum kicks (n q). In the linear case, where Bragg scattering is the only mechanism for changing momentum states, ρ T (κ z ) is time independent. In a nonlinear condensate the effect of the ballistic expansion alone can be seen in Fig. 5.11, where the dashed line shows in successive frames, the momentum expansion of a w = 2500 w 0 cylindrically symmetric condensate freely evolving after release from a trap. The solid line shows the evolution of the same condensate in the presence of Bragg scattering. One sees that most of the momentum reshaping of ρ T (κ z ) is due to the repulsive expansion which occurs on a time scale τ E = 7 3 µ 4, (5.59) where µ is the chemical potential (see Appendix D.2 for the derivation of (5.59)). For times t < τ E (see Table 5.2), we can neglect the effects of ballistic expansion, and the effects of the condensate nonlinearity are confined to modifications of the resonance conditions, which we consider in the next section Nonlinear dispersion curves The other significant modification arising from condensate nonlinearity occurs in the Bragg resonance conditions, and the dispersion curve concept provides a useful framework for analysing this effect. In the linear case, the diagonalisation of the Hamiltonian to determine the energy eigenvalues and their momentum dependence is trivial, and leads to the dispersion curves defined by Eq. (5.41), and illustrated in Fig In the nonlinear case diagonalisation is rather less straightforward: for small population transfer from the ground state the appropriate dispersion curves are given (for the case of Bragg scattering in a trap) by the Bogoliubov dispersion relation (2.14), however for large population transfer the dispersion curves have not been investigated. Nevertheless as w increases from zero we can estimate that the dispersion curves will be shifted by an amount of order µ, the chemical potential. In our analysis of the linear case in the previous section, Bragg scattering was

158 142 Bragg scattering of a released condensate 1.5 (a) t = 0 PSfrag replacements ρt [r0] ρt [r0] ρt [r0] (b) (c) t t = 0.49 = κ [r0 1 ] Figure 5.11: Comparison between a purely ballistically expanding condensate (dashed line) and a Bragg scattered, untrapped condensate (solid line). In each case w = w 0. The light field parameters for the Bragg scattered state are V p = 5ω T, ω = 36ω T and q = 36/r 0.

159 5.7 Application to nonlinear case 143 shown to occur where the ω n and ω n+1 curves are separated by less than the power broadened width V of the laser interaction. Thus if the laser parameters ω and q are chosen so that ω 0 and ω 1 cross in the w = 0 case, we can expect our linear analysis to still apply when w 0, provided the shift of the dispersion curves, µ, is less that the transition width. The regime where the results from section 5.6 can be applied in the nonlinear case is therefore µ < V p, (5.60) and we list in Table 5.2 the values of µ at the instant of release from the trap for a range of condensates. We note that if V p becomes sufficiently large, Bragg scattering will evolve into quantum channeling (see Fig. 5.5). A critical test of condition (5.60) can be made by considering condensate evolution in the spatial picture, and the comparison between the linear and nonlinear cases is facilitated by separating the spatial wavefunctions into their ψ 0 and ψ 1 constituents (corresponding respectively to φ 0 and φ 1 ). We have plotted a temporal progression of these wavefunctions in Fig for the w = 0 case (Figs. 5.12(a)-(c)) and for w = 2500 w 0 (Figs. 5.12(d)- (f)). For the latter case we have µ = 8.26 ω T and V p = 10ω T, so that condition (5.60) is satisfied. It is evident that there is very good agreement between the linear and nonlinear cases in all the qualitative features, with the main difference perhaps being the smoothing of the sharpest features in the nonlinear case. Finally we illustrate in Fig how the linear analysis progressively fails as µ approaches and then exceeds V p. We plot for the range of w values given in Table 5.2 the time evolution of, P 0, the total population in the 0 th momentum domain, which is defined according to q/2 P n = dk x dk y dκ z φ n (κ) 2. (5.61) q/2 The linear case is shown as a solid line, and is in close agreement with the w = 250w 0 case, and in reasonable agreement with the w = 2500w 0 case. However for the w = 25000w 0 case, where µ significantly exceeds V, significant disagreement occurs after the first half cycle. We note that by using P 0 rather that φ 0 (κ z ) as our basis of comparison, we have minimised the discrepancy that would occur simply from the repulsive expansion.

160 144 Bragg scattering of a released condensate ψn [1/ r 3 0] ψn [1/ r 3 0] ψn [1/ r 3 0] (a) (b) (c) t = 0.19 t = 0.29 t = 0.39 t = 0.19 t = 0.29 t = 0.39 PSfrag replacements z [r 0 ] z [r 0 ] Figure 5.12: Comparison of Bragg scattered spatial wavefunctions in linear and nonlinear condensates. The solid curve represents ψ 0 and the dotted curve ψ 1. (a)-(c): w = 0. (d)-(f): w = 2500 w 0. Light field parameters are V p = 10ω T, ω = 196ω T, q z = 14/r 0.

161 5.7 Application to nonlinear case P PSfrag replacements t [1/ω T ] Figure 5.13: Temporal evolution of the population of the 0 th momentum domain, for different condensate nonlinearities. w = 0 solid, w = 250w 0 dashed, w = 2 500w 0 dash-dot, and w = w 0 dotted. Light field parameters are as in Fig. (5.12). w [w 0 ] µ [ ω T ] τ E [1/ω T ] Table 5.2: Chemical potentials and expansion time constants for the eigenstates used in section

162 146 Bragg scattering of a released condensate 5.8 Momentum spectroscopy The NIST and MIT groups recently reported experiments [109], [160] where Bragg scattering was used to directly measure the momentum composition of condensates. The formalism set out in this chapter provides an appropriate framework for the description of those results. Bragg scattering can be used to selectively couple out a portion of the momentum states from a wavepacket, provided the momentum width σ q of the Bragg transition is less than the momentum width σ p of the initial wavepacket. The width σ q is controlled primarily by the laser intensity and pulse duration, and it is easy to see from Eqs. (5.54) and (5.56) that for the purposes of momentum spectroscopy, we require that V p < q 4m σ p, (5.62) T p > 2πm. q σ p (5.63) We illustrate such a case in Fig where a long duration, low intensity, high momentum coupling causes a narrow fraction of the initial momentum distribution to be out-coupled. In the experiments reported by the NIST group, momentum selectivity was obtained by allowing the condensate to ballistically expand before the application of the light grating, so that the self repulsion broadens the momentum distribution to be larger than the Bragg line width σ q. The MIT group achieve precise momentum selectivity by using very low intensity lasers, which allows them to analyse the narrow momentum distribution of a trapped condensate. We examine that regime and those experiments in chapter Conclusion In this chapter, we have presented a meanfield formalism for Bragg scattering in released condensates. Starting from the two-component Gross-Pitaevskii equation driven by a far detuned classical light field, we obtained a one-component equation which provides a framework for treating the scattering of trapped or untrapped condensates from an arbitrary light-field grating. We have concentrated on the Bragg

163 5.9 Conclusion N [r0] 0.4 PSfrag replacements k z [r 1 0 ] Figure 5.14: Momentum spectroscopy. The momentum wavefunction along the symmetry axis of the condensate is shown at t = 0.5/ω T. The out coupled state (centred at k z = 20/r 0 ) is much narrower than the initial packet which is centred at k z = 0. Parameters are w = 2500w 0, V p = 5ω T, q = 20/r 0 and ω = 400ω T. regime, and solved the one component equation in a number of situations using three dimensional cylindrically symmetric numerical simulations. For the case of Bragg scattering from a condensate released from a trap, the dominating behaviour is the oscillation of the atomic momentum components between an initial value k i and a shifted value k i + q. We presented an alternative derivation of linear Bragg scattering based on a partitioned representation of the momentum wavefunction and incorporating spatial nonuniformity, and used this to analyse the full nonlinear system. We showed that this linear model gives an accurate description of the nonlinear system within a well defined validity regime. Dispersion curves in the partitioned momentum representation facilitated visualisation of the possible couplings that may occur, including first and higher order Bragg resonances. We also derived analytic expressions for the the momentum transition linewidth, and used these to define the regime of momentum spectroscopy.

164

165 Chapter 6 Spatially selective Bragg scattering 6.1 Introduction A major interest in condensates is that their macroscopic wavefunctions may be engineered into specific spatial structures. For example, the vortex state, which has been recently observed [120, 113], has a phase circulation of 2π about a vortex core. The most direct means for observing such a phase distribution is by interference with a separate well characterised matter field (e.g. [26, 120, 102, 38]), but this may not always be experimentally convenient. In this chapter we show that under appropriate conditions Bragg scattering is sensitive to the spatial phase distribution of the initial condensate and therefore allows preferential scattering from a selected spatial region. In the case of a vortex, this gives a distinctive signature which we illustrate in Fig There the density distribution of a trapped condensate vortex state is shown following application of a Bragg pulse chosen according to criteria developed later in this chapter. In Fig. 6.1(a) the streaming output (the Bragg scattered component) emerges from only one side of the initial condensate, and gives an asymmetric density pattern which should be detectable with current experimental technology. In Fig. 6.1(b) where the frequency difference of the laser fields has been changed, the streaming output field has a density node that is an order of magnitude wider than the vortex core itself (i.e. the healing length in

166 150 Spatially Selective Bragg Scattering the centre of the original condensate). In the following we develop a treatment of such spatially selective Bragg scattering which explains the behaviour in terms of the underlying spatial phase sensitive mechanism and we give analytic solutions appropriate to 3D condensates. 30 (a) (b) 20 y [r0] 10 0 PSfrag replacements x [r 0 ] x [r 0 ] log 10 ψ 2 [r0 2 ] Figure 6.1: Spatial density of a 2D vortex state at time t = 0.6t 0 after excitation by a Bragg pulse with frequency detuning (a) ω = 216ω T and (b) ω = 203ω T (ω T is the harmonic trapping frequency). The trap remains on at all times. Other parameters are V p = 0.2ω T, q = 14/x 0 and w = 500w 0. The dashed line denotes a suitable position for measuring the output beam profile (see text).

167 6.2 Two state model Two state model We treat the process of Bragg scattering from a condensate using the Gross-Pitaevskii equation (5.16), which we reproduce here for the readers convenience, i ψ ] [ t = 2 2m 2 + V T (r) + w ψ 2 ψ + V (t) cos(q r ωt)ψ, (6.1) q and ω represent the two-photon momentum and energy difference. We restrict our attention here to scattering in the Bragg regime, which limits the pulse length to T p > ωq 1 (5.35) and the intensity to V p ω q (5.36). In order to obtain spatially selective Bragg scattering it is also necessary that V p is sufficiently small that the total amount of condensate scattered in time T p is small compared to the unscattered amount. In addition, since we perform the Bragg scattering while the trapping potential remains on, we assume the Bragg pulse length (and the time of observation) is shorter than a quarter trap period, to avoid the trap forces significantly altering the momentum of the scattered beam. These validity conditions will be quantified in section Two state equations Equation (6.1) can be directly numerically simulated, as for example we have shown in the two dimensional case of Fig We have also obtained an analytic solution, which provides detailed insight into the underlying physics and enables us to provide quantitative calculations over a wide range of parameters in the three dimensional case. For the analytic solution, we assume that the recoil momentum q of the Bragg grating is much larger than the momentum width of the initial condensate, σ p, (which is centred about zero momentum), and also assume that the detuning ω is close to the Bragg resonance (5.30). This means the scattered wavepacket is well separated in momentum space from the initial state and a slowly varying envelope approximation can be made [163] so that we write ψ(r, t) = ψ 0 (r, t) + ψ 1 (r, t) e i(q r ωt). (6.2) In Eq. (6.2) ψ 1 is the scattered wavepacket while ψ 0, which we call the mother condensate, represents a condensate with a momentum wavepacket centered on zero. At t = 0, ψ 0 is exactly the initial condensate which we take to be an eigenstate of

168 152 Spatially Selective Bragg Scattering the time-independent Gross-Pitaevskii equation (2.38). Substituting Eq. (6.2) into Eq. (6.1) and projecting into orthogonal regions of momentum space, we obtain to first order in ψ 1 the coupled equations i t ψ 0 = i t ψ 1 = [ ] 2 2m 2 + V T + w ψ 0 2 V (t) ψ 0 + ψ 1, (6.3) 2 [ ] 2 2m 2 δ i Q + V T + 2w ψ 0 2 ψ 1 (6.4) V (t) + ψ 0, 2 where Q = q/m is the velocity of the scattered atoms, δ = ω ω q and we have neglected the rapidly oscillating term ψ 2 0ψ 1. We show details of the projection procedure used to obtain Eqs. (6.3) and (6.4) for the linear case in Appendix D Evolution in a moving frame of reference Equation (6.4) is simplified considerably in the reference frame moving at velocity Q, as defined by the coordinate transformation r(r, t) = r + Q(τ t), (6.5) where we denote all moving frame quantities with a tilde, and the constant τ is the time when the origins of the moving and stationary frames coincide. Any function can be transformed to the moving frame according to Ã( r, t) = A(r( r, t), t), (6.6) where r( r, t) = r Q(τ t), (6.7) from Eq. (6.5). The equation for ψ 1 (6.4) simplifies in the moving frame by virtue of the derivative operator transformation ( ) ( ( ) ) Q r +, (6.8) t r t r r r, (6.9)

169 6.2 Two state model 153 so that Eq. (6.4) becomes i t ψ 1 = ] [ 2 2m 2 r δ + ṼT + 2w ψ 0 2 V (t) ψ 1 + ψ 0, (6.10) 2 where / t is evaluated at constant r. The solution we develop for ψ 1 in the next subsection only requires knowledge of ψ 0 in the lab frame, so we do not bother transforming Eq. (6.3) into the moving frame Formal solution We ignore the Laplacian in Eq. (6.10) which describes the momentum diffusion about the central momentum ( q) of the ψ 1 packet, and will be small on the time scales we consider. Equation (6.10) can now be formally solved to give t ψ 1 ( r, t) = i ds e i K( r,t,s) V (s) 2 ψ 0 ( r, s), (6.11) 0 where K( r, t, s) = (s t)δ + 1 t ds [ṼT ( r) + 2w ψ 0 ( r) ]. 2 (6.12) s We can express this solution in the laboratory frame by using the inverse coordinate transformation (6.7). It is convenient to choose τ (in Eq. (6.7)) to be equal to the observation time t. With this choice the origins of the moving and laboratory frames coincide at t, i.e. r t = r t and thus ψ 1 (r, t) = ψ 1 (r, t), and we have (from Eqs. (6.6), (6.7), (6.11) and (6.12)) ψ 1 (r, t) = i 2 where t ds e ik(r,t,s) V (s)ψ 0 (r + Q(s t), s), (6.13) 0 t K(r, t, s) = (s t)δ + 1 s +2w ψ 0 (r + Q(s t), s ) 2 ]. ds [V T (r + Q(s t)) (6.14) We use Fig. 6.2 to illustrate the formation of the scattered state according to Eq. (6.13) as follows. As the scattered packet moves across the mother condensate, the amplitude ψ 1 is built up from the sum of contributions coupled in from successive

170 154 Spatially Selective Bragg Scattering r s Q(s t) (a) At time s scattered + V (s)ψ 0 (r + Q(s t)) mother frag replacements Propagator e ik(r,t,s) r t scattered + (b) At later time t V (t)ψ 0 (r) mother Figure 6.2: Physical picture of analytic solution (6.13). (a) at time s atoms are Bragg scattered from the mother condensate ψ 0 into the scattered state ψ 1. (b) at a later time t the system has evolved according to the propagator, and the scattered state has moved to interact with a new part of the mother condensate. points along the mother condensate. The contribution coupled in at time s from a position r s on the mother condensate moves to the position r t = r s + Q(t s) at time t with the propagator exp( ik(r, t, s)). The scattered state at r and t will be appreciable only if the contributions from all earlier times constructively interfere on average. Static mother condensate Our interest is in the scattered state ψ 1 which is assumed to be small, so that we can neglect the scattering from ψ 1 back to ψ 0 (i.e. the term V ψ 1 in Eq. (6.3)). This

171 6.2 Two state model 155 means that ψ 0 then evolves according to the usual Gross-Pitaevskii equation (2.61), and typically we choose ψ 0 to be an eigenstate of that equation (i.e. a solution of Eq. (2.37)). In that case we may write ψ 0 (r, t) in terms of amplitude and phase as ψ 0 (r, t) = A 0 (r)e i(s 0(r) µ 0 t), (6.15) where A 0 and S 0 are real functions and µ 0 is the eigenvalue. Eqs. (6.13) and (6.14) become ψ 1 (r, t) = i 2 and t ds e iθ(r,t,s) V (s)a 0 (r + Q(s t)), (6.16) 0 t Θ(r, t, s) = (t s)δ µ 0 s 1 s ds [V T (r + Q(s t)) (6.17) +2w [A 0 (r + Q(s t), s )] 2 ] + S 0 (r + Q(s t)). Equation (6.16) is useful because the V (s)a 0 term appearing in the integrand is real, and any interference effects must arise from Θ. Thus the condition for an appreciable scattered state to form is that Θ(r, t, s) is slowly varying over some region of the mother condensate Thomas-Fermi approximate solution The formal solution given by Eqs. (6.16) and (6.17) is readily applicable to cases where we possess a solution for the ψ 0 eigenstate in analytic (e.g. w = 0) or numerical form. The main interest for us here is Bragg scattering of condensates in the Thomas-Fermi regime, and in this subsection we further specialise our analytic results to this case. To begin, we note that the Thomas-Fermi density profile is given by where θ is the unit step function (2.42), and A 0 (r) 2 = [µ 0 V eff (r)] θ (µ 0 V eff (r)), (6.18) w V eff (r) = V T (r) + 2 m 2 z 2m(x 2 + y 2 ), (6.19) is the effective potential including the centrifugal potential for the case of a vortex of circulation m z (see Eqs. (2.41), (2.43) and (2.87)-(2.88)). The phase of this state

172 156 Spatially Selective Bragg Scattering is taken to be S 0 (r) = m z φ, where φ is the azimuthal angle. Using (6.18), Eqs. (6.16)-(6.17) simplify in the Thomas-Fermi regime to ψ 1 ψ TF (r, t) = i 2 where t 0 ds e iθ TF(r,t,s) V (s)a 0 (r + Q(s t)), (6.20) t Θ Θ TF (r, t, s) t(δ µ 0 ) w ds [A 0 (r + Q(s t))] 2 (6.21) s sδ + S 0 (r + Q(s t)). The condition for stationary phase dθ(r, t, s)/ds = 0 then gives the generalised Bragg resonance condition [ w δ [A 0(R)] 2 + R S 0 (R) Q] R=r+Q(s t). (6.22) If Θ has sufficiently large spatial curvature, one particular time s may dominate the stationary phase contribution. However for the cases of a ground or vortex initial state, Θ varies sufficiently slowly that most contributions from along the line R = r + Q(s t) on the initial condensate can be in phase. For an initial ground state R S 0 (R) = 0 and Eq. (6.22) reduces in the linear case to the usual Bragg resonance condition δ = 0, while for the nonlinear case, δ = {w ψ 0 (R) 2 / } in which the braces indicate that the precise value of the shift is obtained by a suitable average along the path R 1. For a general initial state the second term in Eq. (6.22), which can be interpreted as a Doppler condition, must be included Solution validity The analytic solutions we presented in Eqs. (6.17) and (6.16), and Eqs. (6.21) and (6.20), have been derived from the full Gross-Pitaevskii equation (6.1) by making a series of approximations illustrated in Table 6.1. To clarify the conditions under which we expect these to be an accurate representation of the full solution we revisit and analyse each assumption in more detail. 1 This shift, suitably averaged over all paths through the condensate corresponds to the shift measured in [160, 155]. The Thomas Fermi values for 2D and 3D are 2µ 0 /3 and 4µ 0 /7 respectively (see [160]). Also see chapter 7.

173 6.2 Two state model 157 Gross-Pitaevskii Equation (6.1) projection onto ψ 0, ψ 1 and neglect nonlinear terms in ψ 1 Two state solution:eqs. (6.3) and (6.4) neglect 2 Formal solution: Eqs. (6.13) and (6.14) neglect back coupling to ψ 0 (i.e. V ψ 1 ) Approximate solution: Eqs. (6.16) and (6.17) Thomas-Fermi approximation Thomas-Fermi approximate solution: Eqs. (6.20) and (6.21) Table 6.1: Summary of approximations and solutions derived for spatially selective Bragg scattering. (1) Two state decomposition The decomposition ψ = ψ 0 +ψ 1 exp(i[q r ωt]), and the subsequent projection into regions of momentum space to yield Eqs. (6.3) and (6.4) relies on ψ 0 and ψ 1 being well localised momentum wavepackets with a width much smaller than q. For a ground state condensate, the momentum width is approximately the inverse of its radius, so we require where R TF is the Thomas-Fermi radius (2.44). q > 1 R TF, (6.23) Condition (6.23) is not strict enough to ensure the two state equations (6.3) and (6.4) will accurately represent the condensate behaviour. The problem lies with the inability of the two state decomposition (6.2) to accommodate the low momentum collective excitations of the condensate where the positive frequency component of the excitation is significant, i.e. v (r)e iωt (see section 2.3.5). This term is fastrotating relative to ψ 0 and ψ 1 (6.2), and is not included in Eqs. (6.3)-(6.4), limiting our description to single particle excitations [50]. The v-amplitudes only have an appreciable effect at low momentum excitations, and will be negligible if we limit

174 158 Spatially Selective Bragg Scattering our consideration to excitations with q > 1 ξ, (6.24) where ξ is the healing length (2.47). In the Thomas-Fermi regime 1/ξ (6.24) is usually much larger than 1/R TF (6.23), except in highly asymmetric traps (e.g. see [155]). (2) Nonlinear terms Expanding the nonlinear term w ψ 2 in Eq. (6.1) with the two state decomposition (6.2) yields w ψ 2 ψ = ( w ψ w ψ 1 2) ψ 0 (6.25) + ( w ψ w ψ 0 2) ψ 1 e i(q r ωt) +w(ψ 0 ) 2 ψ 1e i(q r ωt) + wψ 0(ψ 1 ) 2 e 2i(q r ωt). If we assume that ω is chosen to be Bragg resonant (e.g see Eq. (5.3)) for the momentum transition 0 q, then we expect that the two terms on the last line of Eq. (6.25) can be ignored by making a rotating wave approximation. Rewriting the original Gross-Pitaevskii Eq. (6.1) with the two state decomposition (6.2) (so that ψ 0 and ψ 1 and slowly varying), simplifying the nonlinear term with the rotating wave approximation and projecting to the two orthogonal regions of momentum space (see Appendix D.3), we obtain i t ψ 0 = i t ψ 1 = [ 2 ] 2m 2 + V T + w ψ w ψ 1 2 V (t) ψ 0 + ψ 1, (6.26) 2 [ 2 2m 2 δ i Q + V T + w ψ 1 2 (6.27) +2w ψ 0 2 ] ψ 1 + V (t) ψ 0. 2 To arrive at the Eqs. (6.3) and (6.4) which are linear in ψ 1, it is necessary to ignore the ψ 1 2 terms in Eqs. (6.26) and (6.27). For this to be a good approximation we require that the bracketed factors in Eqs. (6.26) and (6.27) are not changed

175 6.2 Two state model 159 significantly by dropping these terms. This will be true when w ψ 1 2 ( V T + w ψ 0 2), (6.28) which is (in the Thomas-Fermi limit ) ψ 1 n p, (6.29) where n p is the peak density (2.46). We now find an upper bound on the average intensity V p so that the ψ 1 component will not develop sufficient amplitude during the pulse (of duration T p ) to exceed condition (6.29). Using Eq. (6.16) we have ψ 1 (r, T p ) = Tp 0 Tp From Eqs. (6.29) and (6.31), and taking A 0 = n p we have 0 ds e iθ(r,t,s) V p A 0 (r + Q(s t)), (6.30) ds V p A 0 (r + Q(s t)). (6.31) V p T p 2. (6.32) This condition is typically too strict, since the matter wave ψ 1 continuously moves at velocity Q, and only interacts with the peak density region of the mother condensate for a short time. (3) Back coupling V ψ 1 The back coupling term is responsible for scattering atoms from ψ 1 back into the stationary wavepacket ψ 0. By neglecting this we can take the mother condensate to be stationary, yielding the formal solution given in Eqs. (6.16) and (6.17). A similar argument to the forward coupling calculation (6.30)-(6.31) can be applied to show that the amount scattered into ψ 0 is V p T p ψ 1, and as long as condition (6.32) is satisfied (ensuring ψ 1 has low density) then back scattering will be small.

176 160 Spatially Selective Bragg Scattering (4) Neglecting 2 It is difficult to give a general argument for the validity regime of ignoring the Laplacian, needed to arrive at the formal solution (6.11) in the moving frame. In Fig. 6.3 we give supportive evidence by comparing numerical solutions of: the full Gross-Pitaevskii equation (6.1); the two state equations (6.3)-(6.4) in full; the two state Eqs. (6.3)-(6.4) and neglecting 2 and V ψ 1 ; the analytic solution (6.16)- (6.17). We note first, by comparing Figs. 6.3(a) and (b), that the two state model y [r0] 20 (a) (b) 10 0 (c) (d) log 10 ψ 2 [r 0 2 ] g replacements x [r 0 ] x [r 0 ] x [r 0 ] x [r 0 ] Figure 6.3: Density comparison of solutions at t = 0.5/ω T to (a) full Gross-Pitaevskii Eq. (6.1); (b) solution of two state Eqs. (6.3) and (6.4); (c) two state Eqs. (6.3) and (6.4) neglecting 2 and the back coupling (V ψ 1 ) term; and (d) the analytic solution (6.16). Two dimensional simulation with w = 500 w 0 central vortex state, q = 14/r 0, V p = 1ω T, ω = 175ω T. provides an excellent description of the full Gross-Pitaevskii solution. Comparing Figs. 6.3(a) and (c) we see that neglecting 2 does have an observable effect on the scattered component, however the general behaviour is well represented. We point out that some barely noticeable interference fringes near the region y = 0 in Figs. 6.3(a) and (b) are absent from Fig. 6.3(c). These fringes arise from the back scattering from ψ 1 back to ψ 0. The analytic solution Fig. 6.3(d) is seen to be in excellent agreement with Fig. 6.3(c). An upper bound on the validity of neglecting the Laplacian can be given by considering the effect of the trap on the scattered state. We note first that atoms moving fast enough (e.g. at Q) will undergo (approximately) simple harmonic motion in the trap with a period T = 2π/ω T, while an atom scattered at t = 0 will come to rest a quarter trap period later. Consider the wavefunction ψ 1 evolving according

177 6.3 Steady state beam profiles 161 to Eq. (6.4). As this matter wave moves into regions of higher potential it develops spatial phase gradients, and these in turn affect the centre of mass velocity though the Laplacian operator. In our formal solution (6.13)-(6.14) these phase gradients arise, however by neglecting 2 the atomic motion continues unaltered. Thus we take a quarter trap period as a convenient upper bound on the Bragg pulse length for which the formal solution (6.13)-(6.14) will break down, i.e. T p < π/ω T. (6.33) Validity conditions summary To conclude this section we summarise the parameter regime where we expect our solution (6.13) to be valid. From Eqs. (6.23) and (6.24) we have a lower bound on q: { 1 q > max, 1 } ; (6.34) R TF ξ the joint condition (6.32) on Bragg intensity and duration, for us to neglect ψ 1 nonlinearity and ignore back scattering to ψ 0 is V p T p 2; (6.35) Finally, to be in the Bragg regime, but avoid the deceleration of the scattered wave packet by the trap potential, we need to limit the pulse length to the interval 1 < T p < π. (6.36) ω q ω T 6.3 Steady state beam profiles The full spatial solution for the scattered state contains a great deal of information as can be seen in Figs. 6.1 and 6.3. However those solutions share certain characteristics, as we illustrate in Fig There we see that the leading edge of the scattered density has a characteristic crescent shape matching the front edge of the mother condensate (see A in Fig. 6.4). Behind this feature the spatially varying structures appear in the scattered beam (see B). These two parts of the scattered state are easy to understand from the analytic solution (6.13): on a short time scale the action of

178 162 Spatially Selective Bragg Scattering A B C PSfrag replacements 2R D Figure 6.4: Structure of the Bragg scattered matter wave. (A) the spatial insensitive scattered leading edge. (B) the spatially selected region of the scattered state. The large arrows (C) and (D) show the trajectories of (A) and (B) respectively, and the small dashed arrows indicate the regions of the mother condensate where these were formed. The condensate size 2R is shown, and the solid dashed line denotes a suitable position for measuring the output beam profile (see text). the Bragg pulse is to merely copy the mother condensate into the moving frame as ψ 1 V (t)ψ 0 δt. (6.37) The front edge (scattered at t = 0) immediately moves out of the space occupied by the mother condensate, and this is essentially just a shifted image of the edge region of the initial condensate. We indicate this schematically in Fig. 6.4 with the large arrow (C) indicating the trajectory of this part of the scattered state, and the two dashed arrows indicate the region of the mother condensate from which it was formed. Conversely the spatially selected part of the beam (B) is formed from matter scattered over a large region of the mother condensate, indicated by the eight small dashed arrows adjacent to the arrow D. We also note that if the Bragg pulse is applied constantly throughout the time it takes for the scattered beam to traverse the mother condensate, the cross section of the emerging beam becomes constant. We shall refer to this stationary cross section

179 6.3 Steady state beam profiles 163 refer as the steady state beam profile. In the 2D simulation of Fig. 6.1 for example, the steady-state density profile could be measured along the dashed line. The condition for achieving a steady state profile is that the optical potential must be applied for a duration T p > τ s where τ s is the time for the moving frame to completely traverse the mother condensate i.e. τ s = 2R Q, (6.38) with R being the radial extent of the condensate (see Fig. 6.4). The variation in the steady-state beam profile with Bragg frequency (ω) provides a compact characterisation of the response of the condensate to the Bragg pulse, as can be seen from the distinctive behaviour of the scattered matter wave for the two frequencies chosen in Fig For definiteness, we shall take the case where q is in the y direction and define the steady-state density profile of the scattered state to be D q (x, z, ω) = ψ 1 (x, y = R, z, τ S ) 2, (6.39) where the distance R is sufficiently large that the density of the mother condensate in the plane (x, y = R, z) is negligible (i.e. D q ψ 2 ). For strongly interacting condensates it is suitable to use R R T F, where R T F is the Thomas-Fermi radius (in the xy-plane). Since experiments would usually measure the density projected along the z axis (the line of viewing), we define the projected steady-state density profile D q (x, ω) = dz D q (x, z, ω) (6.40) and will use this to characterise the results of the Bragg scattering. We begin by considering the case of two spatial dimensions, which contains the essence of the physics. Fig. 6.5 shows D q (x, ω) calculated from the full 2D numerical solution of Eq. (6.1), for cases where the initial state is (a) a noninteracting ground state (b) a condensate ground state, (c) a condensate vortex state. In Fig. 6.5 (d)-(f) we provide the comparison to the analytic solutions for the same cases, and it is apparent that agreement is very good. We note that in the analytic solutions a Thomas Fermi approximation is made for ψ 0 (see Eqs. (6.20)-(6.21)) The results for the noninteracting ground state (Fig. 6.5 (a) or (d)) show, as expected, that the scattered beam is greatest (i.e. Bragg scattering is resonant)

180 164 Spatially Selective Bragg Scattering x [r0] x [r0] x [r0] 5 (a) (d) (b) (e) (c) (f) PSfrag replacements ω [ω T ] ω [ω T ] D q [r 2 0 ] Figure 6.5: Steady-state Bragg scattered density profiles in 2D for (a) noninteracting ground state, (b) condensate ground state (w = 500w 0 ), (c) condensate vortex state (w = 500w 0, m z = 1). Measurements are made at t = 0.6t 0 for a Bragg field with V p = 1ω T, q = 14/x 0. Frames (d)-(f) give the analytic solution of Eqs. (6.15)-(6.16) for the case corresponding to (a), and Eqs. (6.18)-(6.19) for the cases corresponding to (b)-(c). Dotted lines, free particle resonant frequency (ω q = q 2 /2m); dashed lines, 2D nonlinear shifted frequency (ω nl = ω q + 2µ 0 /3) when ω = ω q (i.e. δ = 0). The frequency width of D q (x, ω) for this case can estimated 2 using a simple Fourier analysis on the integral in Eq. (6.13), to be D Q/R, where R is the size of the noninteracting state. In Fig. 6.5(b) (or (e)) the effects of the condensate nonlinearity appear. At the centre of the scattered profile (x = 0) the resonant frequency for Bragg scattering has been shifted by 0.84µ 2 Here we ignore the effect of the trap potential in Eq. (6.15) so that we may write ψ 1 ds exp(iδs)vp A 0 (r + q(t s)).

181 6.4 Pulsed Bragg results: the topological atom-laser 165 which is close to the value of {w ψ 0 (R) 2 / } averaged along the centre line of the mother condensate (= 4µ 0 /5). The shift at the spatial edges of the scattered beam (x = ±R T F ) is less, because the mother condensate has lower average density along the appropriate lines R = r + Q(s t), and so D q (x, ω) has a crescent shape. The frequency width D (at x = 0) is smaller than for the noninteracting ground state case (a), due mainly to the increased spatial width R T F of the mother condensate. The most significant result is Fig. 6.5(c), namely the vortex signature. At the resonant frequency (indicated by the dashed line), the scattered density profile is essentially spatially symmetric (see also Fig. 6.1(b)), but at other frequencies the scattering is spatially asymmetric. This asymmetry, which we emphasize arises from the spatial phase asymmetry of the vortex, is robust, being present for a wide range of frequencies. The density node at the centre of the beam (x = 0) at the resonant frequency also arises from the phase asymmetry: as the scattered wavepacket passes over the vortex core, the contributions from the mother condensate change phase sharply by π and thus cancel. Our results can also be extended into three dimensions by using the analytic solution Eq. (6.13). In Fig. 6.6 we present the behaviour of D q (x, ω) from a vortex condensate of Rubidium atoms in both oblate and prolate traps. The features discussed in the previous paragraphs are unchanged in three dimensions. 6.4 Pulsed Bragg results: the topological atomlaser Although we have concentrated on the steady state density profile our analytic solution contains other interesting results. For example, in Fig. 6.7 we show that the scattered state is itself a vortex when the Bragg pulse is of length T p τ s, and that a sequence of such Bragg pulses produces a sequence of vortices streaming out from the mother condensate. This is a realisation of a topological atom-laser, where the output beam has a well characterised non-plane phase. In 6.7(a)-(c) we present a sequence of images showing the condensate evolution obtained by solving the Gross-Pitaevskii equation with the Bragg fields pulsed as shown in Fig For comparison we show in Fig. 6.7(d) the analytic solution result (6.16) corresponding to Fig. 6.7(c). We note that parameters for this regime are outside the rigorous validity regime of the analytic model; i.e. dt V 2.4, and the mother condensate

182 166 Spatially Selective Bragg Scattering 10 (a) (b) 5 x [µm] 0-5 PSfrag replacements [khz] ω/2π [khz] ω/2π D q [cm 2 ] 10 9 Figure 6.6: Projected steady-state Bragg scattered density profiles from a condensate of Rb 87 atoms in an m z = 1 central vortex in (a) oblate trap with ω T = 2π 50Hz, aspect ratio λ = 8, µ ω T ; (b) prolate trap with ω T = 2π 100Hz, λ = 1/10, µ 0 9.5ω T. Dotted lines, free particle resonant frequency (ω q = 2π 15kHz); dashed lines, 3D nonlinear shifted frequency (ω nl = ω q +4µ 0 /7). Bragg field (V p =2π 50Hz) provided by counter-propagating lasers of approximately 780rmnm. is 18% depleted at t = 0.8/ω T. Several noticeable differences exist between Figs. 6.7(c) and (d). In the full numerical solution, the effect of 2 arising from the trap forces, causes the outer-most vortex to decelerate, a feature not present in the analytic solution. Also the spatial fringes appearing in the scattered beams 6.7(a)- (c), are due to scattering from ψ 1 to ψ 0, which is enhanced by the large Bragg intensity V used. 6.5 Conclusion We have shown that under appropriate conditions Bragg scattering is sensitive to the spatial phase dependence of the initial matter field state, and therefore allows

183 6.5 Conclusion t (a) = 0.4 t (b) = 0.6 t (c) = 0.8 t (d) = y [r0] PSfrag replacements x [r 0 ] x [r 0 ] x [r 0 ] x [r 0 ] log 10 ψ [r 2 0 ] Figure 6.7: A topological laser formed from pulsing on the Bragg fields on a 2D vortex state. (a)-(c) temporal sequence of images showing the full Gross-Pitaevskii solution (6.1). (d) analytic solution (6.16) corresponding to (d). Parameters are w = 500w 0, q = 28/r 0 and ω = 793ω T. V (t) [ωt ] PSfrag replacements t [ω T ] Figure 6.8: Intensity pulses used in Fig preferential scattering from a selected spatial region. We have developed an analytic model which accurately describes this phenomenon and explains the underlying mechanisms. When applied to a vortex state, a robust signature is obtained, for both oblate and prolate traps. We have shown how the pulsed Bragg scattering of