C-Field Description of the Quasi-Equilibrium Bose Gas:

Transcription

1 C-Field Description of the Quasi-Equilibrium Bose Gas: Persistent Currents and Kelvon-Induced Vortex Decay Samuel J. Rooney A thesis submitted for the degree of Master of Science at the University of Otago, Dunedin, New Zealand.

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3 Abstract Recent experiments at the University of Arizona have demonstrated superb control and imaging of flattened superfluids revealing a rich landscape of vortex dynamics. Vortex dipoles have been created and their dynamics observed [71], and highly charged persistent currents have been formed through a clever combination of stirring and dissipation [70]. Many ultra-cold gas systems, including these, are in a regime that we may characterize as quasi-equilibrium: despite some external forcing, a large fraction of the gas remains in thermal equilibrium. In this thesis we use the stochastic projected Gross-Pitaevskii equation (SPGPE) to make quantitative predictions of the quasi-equilibrium Bose gas. We quantitatively compare the SPGPE with experiment by modeling the persistent current formation experiment performed at the University of Arizona [70]. We determine all SPGPE parameters for the toroidal system prior to simulation, enabling quantitative modeling of the experiment with no fitting parameters, giving a true test of the SPGPE theory. We find the SPGPE gives quantitative agreement with experiment, accurately predicting the size of the persistent current as well as the decay time of the vortices. We also observe the crucial role that thermal fluctuations have on enabling this agreement, showing the comprehensive SPGPE treatment is necessary to make quantitatively accurate calculations of quasi-equilibrium systems. This is the first quantitative agreement with experimental observations of vortex dynamics obtained with a theory of dissipation from first principles using no fitting parameters. We then systematically study the Kelvin mode excitations on a vortex line in a three dimensional Bose-Einstein condensate using the SPGPE. We give a quantitative measure of the magnitude of vortex bending caused by the activation of Kelvin modes, and find that vortex bending can be suppressed by tightening the confinement along the direction of the vortex line. This leads to a strong suppression of the vortex decay rate as the system enters a regime of two-dimensional vortex dynamics, characterized by a critical oblateness. In our final application of the SPGPE we simulate the decay of a vortex dipole. We model a recent experiment of Neely et al. [71], finding the SPGPE predicts a dipole lifetime consistent with experimental observations. We also show the experiment lies in the twodimensional regime of vortex dynamics, validating our critical oblateness calculation. iii

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5 Acknowledgements First I would like to thank my supervisor, Dr. Ashton Bradley, without whom none of this work would have been possible. His guidance and knowledge that he has imparted on me over my time working with him has been of enormous benefit. I could not have asked for a better supervisor and I thank him greatly for all the opportunities he has given me. Also I would like to thank Assoc. Prof. Blair Blakie who has had significant input into a number of areas of this research. He is always full of great advice, and has always been able to help me in anyway when needed which I am very thankful for. It has been a pleasure collaborating with Assoc. Prof. Brian Anderson throughout the past year, especially during his visit to Otago. I am grateful to him for many helpful discussions, and hope to continue to work with him in the future. Thanks to everyone who made life in room 426 enjoyable, and for many interesting discussions, namely Andrew Wade, Jono Squire, Andrew Martin, and Andrew Sykes. Thanks to Danny Baillie for always being available to help with any computing problems I have had, also thanks to Peter Simpson in the same way. Finally I am grateful for the continual love and support from my parents, my sister, and all my family and friends. To Arezoo especially, thank you for all your love and support you have given me. v

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7 Contents 1 Introduction Overview Research Aims Thesis Outline Papers Arising Background Bose-Einstein Condensation Helium II Dilute-Gas Bose-Einstein Condensation The Gross-Pitaevskii Equation Quantum Vortices The Superfluid Vortex The Quantum Vortex Dipole Finite Temperature Theory C-Field Theory C-Field Theory Introduction Effective Field Theory Projection into the C-Field Region Theory of the Projector Operator Decomposition of the Hamiltonian C-Field Methods Wigner Formalism Overview of Phase-Space Methods vii

8 3.4.2 Coherent States Wigner Representation of a Single Quantum Mode Wigner Representation of the Quantum Field The Truncated Wigner Approximation The Projected Gross-Pitaevskii Equation The Stochastic Projected Gross-Pitaevskii Equation The Master Equation The Stochastic Projected Gross-Pitaevskii Equation Growth and Scattering Processes Grand Canonical Equilibrium Simple Growth SPGPE The Damped Projected Gross-Pitaevskii Equation Numerical Methods Persistent Current Formation Introduction Experimental Description Condensate Formation Procedure Experimental Results Theoretical Description of the System Overview Trapping Potential Harmonic Toroid Approximation Treatment of the Incoherent Region Numerical Procedure Overview Equations of Motion Simulation Parameters Initial States Dynamics DPGPE Treatment Numerical Overview Results SPGPE Treatment viii

9 4.6.1 Numerical Overview Hold 1 Parameters SPGPE Treatment of Hold Effect of Barrier Height Average Number of Vortices Summary Kelvon-Induced Decay of Quantized Vortices Introduction Procedure Physical System Numerical Procedure Results Power Spectra Effect of Geometry: Kelvon Power and Vortex Decay Effect of Temperature: Anomalous Decay in Quasi-Two-Dimensional Systems Summary Vortex Dipole Decay Introduction Procedure Physical System Numerical Procedure Results Decay in a Spherical Condensate Decay in an Oblate Condensate Summary Conclusions Thesis Summary Directions for Future Work References 91 ix

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11 Chapter 1 Introduction 1.1 Overview The achievement of Bose-Einstein condensation in dilute alkali gases in 1995 by Anderson et al. [5], sparked intense experimental and theoretical work in the field of ultra-cold gases. These systems typically consist of 10 6 atoms, and due to the diluteness of the system temperatures of order 10 5 K or less are required to observe the quantum phenomenon of Bose-Einstein condensation. Bose-Einstein condensation presents a unique opportunity for developing ab initio theory that can be directly compared to experiments. The main tool used to theoretically describe Bose-Einstein condensates is the Gross-Pitaevskii equation [44, 78, 23, 76], in which the entire condensate is described by a single wave function. This mean field description of the coherent evolution of the condensate wave function is only strictly valid when all the atoms are in the condensate. Thus in the zero temperature regime, the Gross-Pitaevskii equation has been tremendously successful at giving a quantitative description of experiment. However experiments routinely occur at finite temperature, where the Gross-Pitaevskii equation gives an incomplete description of the spontaneous and incoherent processes induced by a significant thermal cloud. In high temperature Bose-Einstein condensates where the temperature is much larger than the single particle energy, the presence of a large thermal cloud means coupling processes between thermal atoms and the condensate can not be neglected. Accounting for scattering processes between the thermal cloud and condensate leads to a stochastic theory known as the stochastic projected Gross-Pitaevskii equation (SPGPE), which arises from the c-field formalism [12]. The SPGPE formalism has been presented in rigorous detail [37, 17] and 1

12 Chapter 1. Introduction 1.1. Overview provides a non-perturbative quantum treatment of the highly occupied low energy modes of the degenerate Bose gas, not just the condensate itself. In this sense it gives the most complete quantum treatment of the high temperature Bose gas, as other relevant methods consider either only the condensate ground state quantum mechanically, or neglect high energy atoms that form a reservoir in the SPGPE [80]. The ability of the SPGPE to make quantitatively accurate predictions is the main theme of this thesis. The SPGPE has been applied to spontaneous vortex formation during Bose-Einstein condensation [98] showing agreement with experiment, however the damping rate had to be fitted to achieve this. The SPGPE has the potential to make quantitative predictions of finite temperature Bose gas dynamics without fitting any parameters [85]. A recent experiment on the formation of persistent currents performed by Brian Anderson s group at the University of Arizona [70] lies in the high temperature regime, and is extremely well characterized, providing a setting to quantitatively test the SPGPE. As well as being an excellent test of finite temperature field theory, the persistent current experiment makes use of a novel process to form a superfluid flow. A persistent current is formed via thermal damping of vortex dipoles, which are nucleated from moving a bluedetuned laser through the condensate. However vortex dipole dynamics in a dissipative toroidal configuration is a subject that is not well understood theoretically. We have previously applied the SPGPE to the decay of a single vortex [85], finding quantitative predictions for the lifetime of an initially centrally located vortex. The vortex lifetime was found to be shorter as the condensate geometry became more three-dimensional, where it was proposed (but not shown explicitly) that active kelvon modes creating vortex bending lead to increased vortex instability. There has been some investigation into the excitation of kelvon modes in trapped Bose-Einstein condensates [68, 89]. However there is little understanding of the role these excitations have on vortex decay. A recent experiment observing the decay of vortex dipoles by Neely et al. [71] provides a further setting to quantitatively test the SPGPE. In this work they determine the lifetime of a vortex dipole in an oblate Bose-Einstein condensate. In such a trap, vortex bending is suppressed and vortex crossings and reconnections are inhibited, prolonging the dipole lifetime. Determining the vortex dipole lifetime in the experimental regime serves as another test of the SPGPE theory. 2

13 Chapter 1. Introduction 1.2. Research Aims 1.2 Research Aims The main aims of this research are as follows: 1. To model the persistent current formation experiment performed at the University of Arizona [70] using the SPGPE. 2. To extend our previous work on the decay of a single vortex [85] to quantitatively determine the effect that condensate geometry has on the suppression of kelvon modes, and how such excitations effect vortex decay. 3. To determine the lifetime of a vortex dipole in an oblate Bose-Einstein condensate. We aim to model a previous experimental setup [71] where they determined the vortex dipole lifetime in an oblate system with ω z /ω r 11, and also investigate the decay of a dipole in a spherical condensate. 1.3 Thesis Outline We begin by giving some background discussion on fundamental aspects of Bose-Einstein condensation, vortices and finite temperature theory in chapter 2. The work in this thesis is based on the c-field treatment of the finite temperature degenerate Bose gas. In chapter 3 we give an overview of this formalism, summarizing the comprehensive review article by Blakie et al. [12]. The main emphasis in this thesis is on making quantitatively accurate calculations of vortex decay in the high temperature regime using the SPGPE [37, 17]. We outline the derivation of the SPGPE, and present the DPGPE which is used to qualitatively investigate purely dissipative vortex dynamics. We then present our results of quantitative calculations of vortex decay using the SPGPE in a range of different systems and regimes, focusing on applications of the SPGPE to quasiequilibrium dynamics. Firstly in chapter 4 we quantitatively model a recent persistent current experiment, performed at the University of Arizona as reported in Tyler Neely s PhD thesis [70], where a persistent current is formed via stirring of a Bose-Einstein condensate. We detail how we apply the SPGPE to a toroidal system, finding all simulation parameters in a physically consistent manner prior to simulation of the dynamical procedure. We first perform DPGPE simulations to give a qualitatively clear picture of the vortex dynamics in this system. This 3

14 Chapter 1. Introduction 1.4. Papers Arising also gives a good comparison with the full SPGPE simulations, highlighting the role fluctuations have on the dynamics. We then model the experiment using the SPGPE, with parameters determined to match the experimental system. We find the SPGPE gives quantitative agreement with the experimental results without any fitted parameters, the most significant result of this thesis. In chapter 5, we present a systematic study of the role that condensate geometry has on vortex decay in a harmonically trapped Bose-Einstein condensate at finite temperature. We investigate how flattening the system from a spherical geometry to a pancake geometry reduces the vortex decay rate by suppressing vortex bending. Such vortex excitations are known as kelvons, and we identify a regime where their suppression leads to effective twodimensional vortex dynamics. We then extend this dimensional investigation to the decay of a vortex dipole in chapter 6. We look at the SPGPE dynamics of a vortex dipole in a pancake condensate where vortex bending is suppressed, modeling the recent experiment by Neely et al. [71]. We measure the lifetime of the vortex dipole, and find this compares well with the experimental results. We then compare this to calculations in a spherical system where we observe a more complex decay mechanism in effect. Finally we conclude and summarize this thesis in chapter Papers Arising A paper based on the results of chapters 5 and 6 has been submitted to Phys. Rev. A, and is available on arxiv [84]. 4

15 Chapter 2 Background 2.1 Bose-Einstein Condensation Bose-Einstein condensation was first theoretically predicted by Einstein in 1925 based on the work of Bose on the statistics of photons. The phenomenon is due to bosons being indistinguishable particles with integral spin, in contrast to fermions having half-odd-integral spin. The wave function for a system of identical bosons must be symmetric under interchange of two particles. Bosons in thermal equilibrium at temperature T obey the Bose-Einstein distribution, n BE (ɛ) = 1 e (ɛ µ)/k BT 1, (2.1) which gives the number of particles in a state with energy ɛ, where the chemical potential µ gives a constraint on the total number of particles in the system. The Bose-Einstein distribution says that at a critical temperature T c, the ground state occupation n 0 can become macroscopic when the chemical potential µ approaches the lowest energy level ɛ 0. This can be achieved by cooling the system while holding the number of atoms constant. When the temperature becomes lower than T c, there is a sudden rush of particles into the the single-particle ground state, and the system has a Bose-Einstein condensate Helium II The earliest example of Bose-Einstein condensation was liquid 4 He, which below the lambda point (T λ = 2.17 K at saturated vapor pressure) becomes a superfluid known as He II with 5

16 Chapter 2. Background 2.1. Bose-Einstein Condensation many remarkable properties [28, 95]. In 1938, London suggested that the superfluid component of He II was of the form of a Bose-Einstein condensate, as at very low temperatures the density of the superfluid component approaches the total density of the entire liquid. The most striking feature of superfluids is that unlike a normal fluid, flow is not dissipated due to the superfluid component having no viscosity. These persistent currents are connected to quantized vortices, which are topological defects whose behavior in 4 He is well known [28, 95]. However as He II is strongly interacting, it prevents a large condensate fraction from forming due to the strong depleting effect of these interactions Dilute-Gas Bose-Einstein Condensation A system has a Bose-Einstein condensate when the ground state becomes macroscopically occupied, which will occur when the thermal de Broglie wavelength becomes comparable to the mean interparticle spacing. This sets the critical temperature for Bose-Einstein condensation of N atoms, T c, and for a harmonically trapped gas this is [23] k B T c = 0.94 h ωn 1/3, (2.2) where ω = (ω x ω y ω z ) 1/3 is the geometric average of the trapping frequencies. Atomic Bose- Einstein condensation was first observed experimentally in dilute atomic gas in 1995 in either rubidium [5], lithium [19] or sodium [24]. The low densities of alkali gases means to observe quantum phenomena in such systems, temperatures of order micro-kelvins or less must be achieved. The observation of Bose-Einstein condensation in dilute gases sparked a massive amount of experimental and theoretical work in the ultra-cold atoms field. The dilute nature of the alkali gases means that interactions can be well controlled (compared with the strongly interacting superfluid helium), so quantum phenomena can be observed on a macroscopic scale. Most important to this work is that the observation of Bose-Einstein condensation in dilute gases provides a setting in which superfluid behavior can be more easily observed than in He II, as the interactions are much weaker in an alkali gas. Hence the condensed gas is an appealing regime in which to study superfluidity The Gross-Pitaevskii Equation Bose-Einstein condensation is theoretically appealing due to the highly Bose-degenerate nature of the system, in which at least some single particle states have an occupation larger 6

17 Chapter 2. Background 2.2. Quantum Vortices than unity. This degeneracy means that valid approximations can be made to the full many body quantum field theory describing all the particles in the system, to create more tractable theories. The Gross-Pitaevskii equation describes the zero-temperature properties of the nonuniform Bose gas [23, 76], and is of the form ψ(x, t) i h t = h2 2m 2 ψ(x, t) + V (x, t)ψ(x, t) + u ψ(x, t) 2 ψ(x, t), (2.3) where V (x, t) is an external potential. The parameter u determines the interparticle interactions and is related to the atomic mass m, and scattering length a s, by u = 4πa s h 2 m. (2.4) The Gross-Pitaevskii equation is a nonlinear Schrödinger equation which gives a mean field description of the macroscopic wave function of the Bose-Einstein condensate. The Gross- Pitaevskii equation is only strictly valid in the zero temperature limit where all atoms are in the condensate. However even at zero temperature interactions lead to quantum depletion of the condensate. Despite this the Gross-Pitaevksii equation gives an excellent description of Bose-Einstein condensation of alkali gases at zero temperature, where the amount of condensate depletion is 0.5% [49]. 2.2 Quantum Vortices The Superfluid Vortex The superfluid behavior of Bose-Einstein condensation, particularly the dynamics of quantized vortices, is a primary feature of this work. Here we give a brief overview of some key properties of vortices in Bose-Einstein condensates. Vortices are topological defects observed in superfluids where the fluid rotates about the vortex at a speed inversely proportional to its distance to the center of the vortex. As a Bose-Einstein condensate is a superfluid, circulation in a Bose-Einstein condensed gas must be quantized. In the mean field description of Bose-Einstein condensation using the Gross- Pitaevskii equation, the single particle wave function represents the condensate. This wave 7

18 Chapter 2. Background 2.2. Quantum Vortices function can be written in terms of its amplitude and phase by ψ(x, t) = ψ(x, t) e iθ(x,t). (2.5) The superfluid flow velocity of the condensate is defined by v(x, t) = h θ(x, t), (2.6) m thus the phase acts as a velocity potential meaning that the condensate flow is irrotational v = 0. (2.7) From the single-valuedness of the condensate wave function, it follows that the change in phase of the wave function around a closed contour is given by φ = θ dl = 2πl, (2.8) where l = 0, ±1,... is the winding number. The energy of a vortex is proportional to l 2 [33], so a vortex of winding number l > 1 will split into l singly charged vortices. Using the condensate velocity, it thus follows that the condensate circulation is quantized in units of h/m, Γ = v dl = n h m. (2.9) The fact that the condensate flow is irrotational means that the condensate can not support a rigid-body rotation. However a Bose-Einstein condensate can support angular momentum through the presence of vortices, which represent one of the most characteristic features of superfluidity. A vortex is characterized by a line of zero density with a phase circulation of 2πn about this line. Vortices in Bose-Einstein condensation were first observed experimentally in 1999 at JILA by Matthews et al. [67], where a vortex was generated using optical phase imprinting to create a vortex state in the condensate. Since that time a number of different vortex mechanisms have been observed experimentally, from which we list a small number here. Techniques for stirring a Bose-Einstein condensate were developed, leading to the creation of stable vortex lattices at the ENS [66, 65, 64], MIT [2, 1, 81], Oxford [47] and JILA laboratories [46, 30]. Vortex rings have been formed via the decay of dark solitons [3]. The precession of a single vortex has been 8

19 Chapter 2. Background 2.3. Finite Temperature Theory observed [4], where the vortex is seen to spiral out of the condensate due to dissipation. Quantum vortices can support long wavelength traveling waves known as Kelvin waves, which induce vortex bending. The excitation of such kelvon modes in a single vortex has been observed by Bretin et al. [20]. Persistent current formation in Bose-Einstein condensation was observed experimentally by Ryu et al. [86], where quantized rotation was initiated by transferring one unit h of the orbital angular momentum from Laguerre-Gaussian photons to each atom. The spontaneous formation of vortices [98], and formation and decay of vortex dipoles [71] have been observed at the University of Arizona. Freilich et al. have observed the dynamics of single vortices and vortex dipoles in Bose-Einstein condensates [34] The Quantum Vortex Dipole A vortex dipole is made up of a vortex/anti-vortex pair, where the vortex and anti-vortex have opposite circulation. Vortex dipoles play an important role in many areas of superfluidity exhibited in ultra-cold gases. They play a key role in two-dimensional superfluidity, where they arise in the Berezinskii-Kosterlitz-Thouless (BKT) transition [9, 58, 91, 45, 59]. They are also highly prevalent in quantum turbulence [56, 57, 48], as well as in quantum phase transition dynamics [106, 6, 98]. Vortex dipoles have been produced experimentally in an oblate Bose-Einstein condensate by forcing superfluid flow around a repulsive Gaussian obstacle within the condensate [71]. This method is based on the fact that vortex dipoles are nucleated when a superfluid moves past an impurity faster than a critical velocity, above which vortex shedding induces a drag force [35, 100]. Being able to nucleate vortices from stirring a Bose-Einstein condensate with an impurity is key to the investigation into persistent current formation in chapter 4. Despite vortex dipoles being highly prevalent in superfluidity, there is still little known about the decay mechanisms of vortex dipoles. Hence studying the dynamics of vortex dipoles in finite temperature systems is of interest, and their dynamics play an important part in much of the work presented in this thesis. 2.3 Finite Temperature Theory The Gross-Pitaevskii equation gives a good description of Bose-Einstein condensation at zero temperature as the approximation that the entire system can be described by a single wave function is valid. However as the temperature of the system becomes higher, there will be an increasingly significant thermal cloud. The Gross-Pitaevskii equation only deals with 9

20 Chapter 2. Background 2.3. Finite Temperature Theory interparticle interactions from Bose condensed particles, hence it fails to provide an adequate description of a system in which incoherent and spontaneous processes are important. Such processes are present when a sizable thermal cloud exists. The temperature range at which the thermal cloud becomes significant, so that the interaction between the condensate and thermal atoms is important, is classified as the high temperature regime. This occurs when the temperature of the system is significant compared to the single particle energies of the trapping potential. For a harmonic potential with frequency ω as usually found in Bose-Einstein condensed systems, the criterion is hω k B T, (2.10) which generally consists of temperatures in the range 0.6T c < T. In order to treat vortex dynamics in high temperature Bose-Einstein condensation, a theoretical description is required that necessarily includes the physics of spontaneous and incoherent processes. In this work we primarily use a stochastic Gross-Pitaevskii equation based on c-field techniques [12]. We now give a brief introduction into c-field methods C-Field Theory In this thesis, the high temperature Bose-Einstein condensate regime is treated using c-field methods [12]. This formalism utilizes the fact that as temperatures become larger than even the critical temperature, there is still significant population of the lowest energy modes. When this occupation is larger than one quantum the system is highly Bose degenerate and the quantum field can be treated using a classical field. This description still includes quantum features, hence the formalism is named as c-field methods to avoid the perception that it is a classical treatment. C-field methods employ a projector operator which creates a cutoff where the occupation of the energy modes becomes approximately one. This means that all the modes below the cutoff can be treated fully quantum mechanically using a classical field representation. The modes above the cutoff are treated semi-classically and represent the thermal cloud. To account for interactions between the condensate and thermal cloud, a stochastic Gross- Pitaevskii equation can be derived by treating the trapped Bose gas as an open system [12, 37, 17]. The thermal cloud is treated as a thermal reservoir which couples to the condensate atoms to include additional damping and noise terms to the Gross-Pitaevskii treatment. The stochastic projected Gross-Pitaevskii equation has been used to investigate spontaneous 10

21 Chapter 2. Background 2.3. Finite Temperature Theory vortices in the formation of Bose-Einstein condensates [98], modeling the dynamics of the formation of a rotating Bose-Einstein condensate [18], and the decay of a single vortex [85]. We outline the c-field formalism next in chapter 3. 11

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23 Chapter 3 C-Field Theory 3.1 Introduction C-field techniques utilize the fact that Bose-Einstein condensed systems have significant occupation of many modes of the system. The name is given since when a system has high Bose-degeneracy, the highly occupied modes can be described by a classical field instead of using the full many body quantum field theory. In this chapter we give a general outline of the formulation of the stochastic projected Gross-Pitaevskii equation, with the general procedure following that shown in the review article by Blakie et al. [12]. Further details for all C-field techniques can be found in this review [12]. 3.2 Effective Field Theory The theoretical treatment of many systems of a quantum nature in which quantum statistics are important use second-quantized field theory. Any second-quantized description of Bose fields begins with the second-quantized Hamiltonian describing N interacting Bosons Ĥ(t) = d 3 x ˆΨ 1 (x, t)h sp ˆΨ(x, t) + 2 d 3 x d 3 x ˆΨ (x, t) ˆΨ (x, t)u(x x ) ˆΨ(x, t) ˆΨ(x, t), where the single particle Hamiltonian is of the form H sp = h2 2 2m + V (x, t), (3.1) 13

24 Chapter 3. C-Field Theory 3.2. Effective Field Theory and V (x, t) is the external potential. The second-quantized Bosonic field operator, ˆΨ(x, t), annihilates a particle at position x and time t. These field operators obey the Bosonic commutation relations [ ˆΨ(x.t), ˆΨ(x, t) ] = [ ˆΨ (x.t), ˆΨ (x, t) ] = 0 [ ˆΨ(x.t), ˆΨ (x, t) ] = δ(x x ). (3.2) The inter-atomic potential, U(x x ), characterizes two body interactions between Bosons. The magnitude of the interaction potential is set by the effective range parameter r 0 which depends on the relative separation of atoms. Typical temperatures of Bose-Einstein condensation are cold enough (order 1µK) that the de Broglie wavelength of the atoms is much larger than their size. Thus length scales of interest in Bose-Einstein condensed systems are larger than the effective range parameter, which means that most details at the atomic level are not important. Short wavelength modes are eliminated by imposing an energy cutoff E max, that is, the modes of the system are restricted to a low energy subspace L. This cutoff enables the details of small length scales to be ignored, and all scattering between two atoms can be parameterized by the S-wave scattering length. Describing the atomic interactions in this way is known as an effective field theory, and the effective Hamiltonian becomes Ĥ eff = d 3 x ˆψ u (x)h sp ˆψ(x) + 2 d 3 x ˆψ (x) ˆψ (x) ˆψ(x) ˆψ(x), (3.3) where the value of the interaction parameter u is dependent on the S-wave scattering length a s u = 4πa s h 2 m. (3.4) The effective Hamiltonian describes the coarse-grained field operator ˆψ(x) which only contains modes in L, and can be expanded as ˆψ(x) = n L â n φ n (x), (3.5) where the â n obey the Bose commutation relations [â i, â j ] = [ â i, â j] = 0, [âi, â j] = δij, (3.6) and φ n (x) are single-particle eigenstates of H 0 with energy ɛ n ɛ n φ n (x) = H 0 φ n (x), (3.7) 14

25 Chapter 3. C-Field Theory 3.3. Projection into the C-Field Region where we have introduced the basis Hamiltonian with the harmonic oscillator potential given by H 0 = h2 2 2m + V HO(x), (3.8) V HO (x) = m 2 [ω2 r(x 2 + y 2 ) + ω 2 zz 2 ]. (3.9) The effective field theory imposes an energy cutoff so the commutation relations of the ˆψ(x) are no longer precise delta functions as in (3.2), but become a coarse-grained delta function [ ˆψ(x), ˆψ (x ) ] = δ L (x x ), (3.10) The Heisenberg equation of motion which describes the evolution of the field operator over the included low energy modes in L can be found easily using the above commutation relations, and is given by i h ˆψ(x) t = d 3 x δ L (x x ) { H sp ˆψ(x ) + u ˆψ (x ) ˆψ(x ) ˆψ(x ) }. (3.11) Understanding the dynamics of any Bose condensed system requires solving this Heisenberg equation of motion (3.11), however in general this is not an easy task. C-field techniques provide a method to solve (3.11) in different regimes. 3.3 Projection into the C-Field Region Theory of the Projector Operator The size of the L region generated from the effective field theory however is very large and computationally difficult to deal with. By further dividing the L region into two regions, classical field methods can be applied to low energy modes which have a high occupation. The system is divided into the following two regions (see figure 3.1): (i) The coherent region (C) consisting of low energy modes with significant occupation. This region contains highly occupied modes which will be treated using a classical field. For temperatures below the transition, this contains the condensate and low energy excitations. 15

26 Chapter 3. C-Field Theory 3.3. Projection into the C-Field Region Energy E max P I cut Beyond s-wave Incoherent Region P C Coherent Region Position Figure 3.1: A schematic showing the coherent region C, incoherent region I, and states eliminated to give the effective field theory. In the stochastic Gross-Pitaevskii equation, the coherent region is treated quantum mechanically in a classical field approximation. The incoherent region is treated semi-classically, and is most simply described as a grand canonical reservoir. (ii) The incoherent region (I) which contains all modes in L which are not in the C region. These modes are weakly occupied, and for many systems are essentially thermalized. The single particle energy ɛ cut defines the coherent and incoherent regions by requiring that the C region corresponds to energies ɛ ɛ cut, and that the I region corresponds to energies ɛ cut < ɛ < E max. The field operator can be restricted to a certain region by using orthogonal projector operators, which are defined for the two regions by P C {F (x)} φ n (x) n C d 3 x φ n(x )F (x ), (3.12) P I {F (x)} φ n (x) n I d 3 x φ n(x )F (x ), (3.13) 16

27 Chapter 3. C-Field Theory 3.3. Projection into the C-Field Region where C = {n : ɛ n ɛ cut } and I = {n : ɛ cut < ɛ n E max }, so L = C + I and P C P I = 0. The quantum field operators for each region can be defined by applying the projector operators on the total field operator ˆψ ˆψ C (x) P C { ˆψ(x)} = n C â n φ n (x), (3.14) ˆψ I (x) P I { ˆψ(x)} = n I â n φ n (x), (3.15) so the total field operator can be written as a sum corresponding to the field operator in both the coherent and incoherent regions ˆψ(x) = ˆψ C (x) + ˆψ I (x). (3.16) An important consequence of the projection is that the commutator of the coherent region field operator is not a pure Dirac-delta function, but is of the form [ ˆψC (x), ˆψ C(x ) ] = δ C (x, x ) (3.17) where δ C (x, x ) is known formally as the kernel of the condensate projector, P C, δ C (x, x ) n C φ n (x)φ n(x ). (3.18) Although the commutator (3.17) is not a pure Dirac-delta function, it acts as one on any function in the projected region. So for any function in the coherent region d 3 x δ C (x, x )ψ C (x ) = ψ C (x). (3.19) Decomposition of the Hamiltonian Using the field operator decomposition (3.16), the effective Hamiltonian (3.3) can be written in terms of ˆψ C and ˆψ I in the form Ĥ eff = ĤC + ĤI + ĤI,C (3.20) 17

28 Chapter 3. C-Field Theory 3.3. Projection into the C-Field Region where ĤC involves only the coherent region field operators, Ĥ I involves only incoherent region field operators and Ĥ I,C = Ĥ(1) I,C + Ĥ(2) I,C + Ĥ(3) I,C (3.21) represents atomic interaction terms involving one, two or three coherent region operators. The terms take the form of Ĥ C = d 3 x ˆψ C(x)H sp ˆψC (x) + u d 3 x 2 ˆψ C(x) ˆψ C(x) ˆψ C (x) ˆψ C (x) (3.22) Ĥ I = d 3 x ˆψ I(x)H sp ˆψI (x) + u d 3 x 2 ˆψ I(x) ˆψ I(x) ˆψ I (x) ˆψ I (x) (3.23) Ĥ (1) I,C = u d 3 x ˆψ I(x) ˆψ I(x) ˆψ I (x) ˆψ C (x) + h.c. (3.24) Ĥ (2) I,C = u d 3 x ˆψ I(x) ˆψ I (x) ˆψ C(x) ˆψ C (x) + h.c. + u d 3 x 2 ˆψ I(x) ˆψ I(x) ˆψ C (x) ˆψ C (x) + h.c. (3.25) Ĥ (3) I,C = u d 3 x ˆψ C(x) ˆψ C(x) ˆψ C (x) ˆψ I (x) + h.c. (3.26) where h.c. denotes the Hermitian-conjugate. Terms arising from the single particle Hamiltonian involving one ˆψ C and one ˆψ I have no contribution as the incoherent region operators have no mean field C-Field Methods There are three c-field methods which differ in the way they treat the different interaction Hamiltonians, which contain field operators of both the coherent and incoherent regions. These three techniques are (i) Projected Gross-Pitaevskii Equation (PGPE) The PGPE is simply the Gross-Pitaevskii equation where evolution is restricted to the coherent region. In this case all cross terms involving the the interaction Hamiltonians are neglected, so the PGPE does not account for any coupling between the coherent and incoherent regions. The PGPE thus treats the coherent region as a microcanonical system of fixed number and energy. (ii) Stochastic Projected Gross-Pitaevskii Equation (SPGPE) The SPGPE retains all interaction terms, hence it accounts for coupling between the coherent and incoherent regions. Such scattering processes are implemented in theory by finding a stochastic differential equation with dissipative and stochastic noise terms. 18

29 Chapter 3. C-Field Theory 3.4. Wigner Formalism Since SPGPE deals with the interaction between the two regions, occupation of the incoherent region can be accounted for. Hence SPGPE is a high temperature theory which is suitable for temperatures much higher than the single particle energy, hω k B T, where there is a significant thermal component. (iii) Truncated Wigner Projected Gross-Pitaevskii Equation (TWPGPE) The TWPGPE is of use when there are modes with low occupation in the coherent region. The population of the incoherent region is negligible meaning the cross terms in the interacting Hamiltonians can be dropped. To model this system containing modes of low occupation quantum mechanically, additional noise must be added to account for quantum fluctuations. These quantum fluctuations can be approximated by sampling the initial Wigner distribution, and leads to the ability to describe non-equilibrium dynamics of Bose-Einstein condensation at low temperatures T T c. Such a system with low occupation of modes is not directly relevant to high temperature systems of interest here, but the truncated Wigner theory is central to the SPGPE. In terms of describing high temperature Bose-Einstein condensates, where hω k B T, the SPGPE will give the most complete account of the system as it treats scattering processes with thermally occupied modes. Since the SPGPE is based on the PGPE, and on the truncated Wigner approximation, it is necessary to consider these in the formulation of SPGPE theory, even though in practice these methods will not be directly used. 3.4 Wigner Formalism Overview of Phase-Space Methods In all c-field methods, the energy cutoff ɛ cut is chosen so all modes in the coherent region have significant occupation. In this case the system is highly quantum degenerate and the full quantum description can be treated by a classical description. Although this Bose degenerate system is being represented by a classical field, the quantum mechanical nature of the system is retained through the interpretation of the classical variables. This is done by mapping the evolution of the quantum density operator to a classical quasi-probability function known as the Wigner function. The key to phase space methods is that the equation of motion for the quasi-probability distribution can be mapped to an equivalent stochastic differential equation, which is much easier to solve than the quantum problem with a large Hilbert space. The background behind phase space methods and the Wigner representation of the quantum 19

30 Chapter 3. C-Field Theory 3.4. Wigner Formalism field has been discussed in detail by Gardiner and Zoller in their book [40]. In this section, we detail the results and methods which are important to the formulation of the SPGPE Coherent States The limit of h 0 corresponds to the approach of the transition from a quantum to classical description of the dynamics of a system. However in some systems, the approach to this limit is not consistent with classical motion. By finding states of minimum uncertainty, α, one will have states which display behavior which is the closest to classical of any quantum state. These states satisfy a α = α α, (3.27) where a is the annihilation operator, α is a complex number, and the states α are known as coherent states. Coherent states are one of the most important tools of studying Bose gases, and are used throughout the Wigner representation of the quantum field Wigner Representation of a Single Quantum Mode If a system has a state vector ψ, t, then the density operator is defined by the outer product ρ(t) = ψ, t ψ, t. (3.28) Phase space methods rely on expressing the density operator in terms of a c-number function of a coherent state variable α. The Wigner representation makes such a connection through the Wigner function, which expresses the density operator on a basis of coherent states. The Wigner function W (α, α ) was introduced by Wigner in 1932 [99], and is defined as the Fourier transform of the symmetrically ordered quantum characteristic function W (α, α ) = 1 π 2 d 2 λexp[ λα + λ α]χ(λ, λ ), (3.29) where the symmetrically ordered quantum characteristic function is given by χ(λ, λ ) = tr { ρexp[λa λ a] }. (3.30) An important fact about the Wigner function is that it exists for any density operator [40], so it is always possible to get a c-number function of α if a density operator is defined. The key to phase space methods is based on the ability to map the action of a quantum operator 20

31 Chapter 3. C-Field Theory 3.4. Wigner Formalism on a density operator to the action of a differential operator on the Wigner function. It is well known that the following operator correspondences exist for the actions of the creation and destruction operators on the density operator [40] ( aρ α + 1 ) W (α, α ) (3.31) 2 α ( a ρ α 1 ) W (α, α ) (3.32) 2 α ( ρa α 1 ) W (α, α ) (3.33) 2 α ( ρa α + 1 ) W (α, α ) (3.34) 2 α These operator correspondences are a crucial aspect of phase space methods in that they can make the equations of motion of the quantum system much simpler, creating an easier path to a solution Wigner Representation of the Quantum Field The properties of the Wigner function in the single mode case extend naturally to the full multimode system needed to describe our system. For a system of M modes in the coherent region, the multimode Wigner function is given by [12] W C (α, α ) = d 2 λ π 2M exp ( λ α α λ ) χ(λ, λ ), (3.35) where α is defined as the vector of mode amplitudes α = [α 0, α 1,, α M 1 ], χ(λ, λ ) is the symmetric quantum characteristic function for the coherent region density operator ρ C, and where d 2 α d 2 α n. (3.36) n C The classical field, ψ C, can be expanded over single-particle energy states using the mode amplitudes ψ C (x) = n C α n φ n (x), (3.37) 21

32 Chapter 3. C-Field Theory 3.4. Wigner Formalism which is the classical field function representing the coherent region. An important result is found by looking at the classical field average which can be found using the fact that moments of the Wigner distribution give symmetrically ordered operator averages. The field density average is given by ˆΨ d 2 α ψ C (x) 2 W C (α, α ) = C(x) ˆΨ C (x) + ˆΨ C (x) ˆΨ C(x), 2 = ˆΨ C(x) ˆΨ C (x) + δ C(x, x). (3.38) 2 This is an important result as there is the presence of vacuum noise in the commutator term δ C (x, x) (from (3.17)), which accounts for half a quantum per mode of noise. This shows that quantum effects are still present in the Wigner representation. As in the single mode case, operator correspondences between the density operator ρ C and the Wigner function can be derived. The multimode calculations can be simplified by defining the projected functional derivative operators δ δψ C (x) φ n(x), (3.39) n C α n δ δψ C(x) φ n (x). (3.40) n C αn Using these projected functional derivatives, the operator correspondences between ρ C and the Wigner function can be easily found ˆψ C (x)ρ C ( ψ C (x) ) δ W δψ C(x) C, (3.41) ˆψ C(x)ρ C ρ C ˆψC (x) ρ C ˆψ C(x) ( ( ( ψ C(x) 1 2 ψ C (x) 1 2 ψ C(x) ) δ W C, (3.42) δψ C (x) ) δ W δψ C(x) C, (3.43) ) δ W C. (3.44) δψ C (x) 22

33 Chapter 3. C-Field Theory 3.4. Wigner Formalism These operator correspondences are used to map the equation of motion of ρ C to an equivalent equation of motion for the Winger function W C The Truncated Wigner Approximation The evolution of the density operator in the coherent region is given by von Neumann s equation i h ˆρ C(t) t = [ Ĥ C, ˆρ C (t) ], (3.45) where we consider no interactions between the coherent and incoherent region. This equation is in general not trivial to solve, so the operator correspondences are used to make a transformation to an evolution equation of the Wigner function. Using the operator correspondences for the multimode Wigner function ( ) leads to the evolution equation W C t ĤC = { iu δ δ d 3 x 4 h δψ C (x) δψ C(x) ψ C(x) δψ C(x) + h.c. + ī } δ h δψ C (x) (H sp + u[ ψ C (x) 2 δ C (x, x)])ψ C (x) + h.c. W C. (3.46) This equation represents the time evolution of the classical field ψ C (x), and is very difficult to solve due to the third-order derivatives. However, this evolution equation becomes manageable if these third-order derivatives are neglected. This approximation is known as the truncated Wigner approximation and is often made in c-field methods. Making the truncated Wigner approximation gives W C t ĤC = d 3 x { } i δ h δψ C (x) (H sp + u[ ψ C (x) 2 δ C (x, x)])ψ C (x) + h.c. W C (3.47) which takes the form of a Fokker-Planck equation with drift due to the presence of the first order derivative, but with no diffusion terms due to the absence of second order derivatives. Making the truncated Wigner approximation is valid in many circumstances [12]. In terms of c-field theory, the approximation is valid when the modes in question have significant population. Hence c-field methods require macroscopic occupation in the coherent region modes so that the truncated Wigner approximation can be made. A general property of Fokker-Planck equations is that they can be mapped to an equivalent stochastic partial differential equation if the diffusion matrix is positive semidefinite [36]. Such a mapping is not possible for the full equation representing the Wigner function 23

34 Chapter 3. C-Field Theory 3.5. The Projected Gross-Pitaevskii Equation evolution (3.46) due to the third order derivatives. Making this mapping gives the following partial differential equation, known as the TWPGPE i h ψ C(x) t ) } = P C {(H sp + u[ ψ C (x) 2 δ C (x, x )] ψ C (x). (3.48) This differential equation is not explicitly stochastic, but noise is introduced by sampling the Wigner function which introduces a random element corresponding to occupation of half a quantum per mode. The TWPGPE (3.48) on its own in this form is not directly applicable to the high temperature regime, but the method behind its derivation is important in terms of how the SPGPE is found. 3.5 The Projected Gross-Pitaevskii Equation The PGPE is derived simply by applying the projector operator (3.14) to the Heisenberg equation of motion (3.11) [27]. As stated, the PGPE neglects all coupling between the coherent and incoherent region, so the resulting equation of motion just includes the Hamiltonian based on the coherent region field operators (3.22). Also as the PGPE is a c-field theory, the mode occupation of the coherent region must be macroscopic. The result is the PGPE i h ψ C(x) t } = P C {L C ψ C (x), (3.49) where L C is the Hamiltonian evolution operator for the C region L C ψ C (x) = ( H sp + u ψ C (x, t) 2) ψ C (x). (3.50) The PGPE is also formally obtained as Hamilton s equation of motion for the c-field Hamiltonian ĤC (3.22) i h ψ C(x) t = δĥc δψ C(x). (3.51) The PGPE gives a microcanonical description of the coherent region, where both the number of particles in the coherent region (N C ) and H C are conserved. As there is no dissipative term in the PGPE, the full complement of coherent region excitations, and their intrinsic noisy effect on the coherent region dynamics are retained. 24

35 Chapter 3. C-Field Theory 3.6. The Stochastic Projected Gross-Pitaevskii Equation 3.6 The Stochastic Projected Gross-Pitaevskii Equation The SPGPE formulation used in this work accounts for interactions between the two regions by treating the incoherent region as fully thermalized, that is, a grand-canonical reservoir at a specific temperature and chemical potential. As interactions between the coherent and incoherent regions are not neglected, all terms from the effective Hamiltonian (3.20) are included in von Neumann s equation describing the evolution of the density operator. The SPGPE can be thought of as an extension of the PGPE to include the interactions between the two regions, and these interactions generate damping and stochastic noise terms in addition to the PGPE. The full derivation of the master equation arising from the effective Hamiltonian (3.20), and subsequent formulation of the stochastic differential equations leading to the SPGPE has been well documented [37, 17]. The full details of these calculations are omitted due to the complexity of the derivations. In this section we outline the key details leading to the formulation of the SPGPE The Master Equation The equation of motion for the density operator for the full system including both the coherent and incoherent regions is given by the von Neumann equation ρ = ī h [ĤC + ĤI + ĤI,C, ρ ]. (3.52) The standard procedure for using phase-space methods for open systems is based on finding a master equation for the reduced system, that is the coherent region density operator, by eliminating the reservoir degrees of freedom [40, 38, 39, 41, 42]. This is done by defining the coherent region density operator as the trace of ρ over the incoherent region degrees of freedom ρ C tr NC {ρ}. (3.53) Since the full effective Hamiltonian (3.20) must be used, a number of different terms arise based on interactions between the coherent and incoherent regions. Accounting for all interactions using the coherent region density operator leads to the full master equation for the reduced system ρ C = ρ C Ham (3.54a) 25

36 Chapter 3. C-Field Theory 3.6. The Stochastic Projected Gross-Pitaevskii Equation + ρ C growth (3.54b) + ρ C scatt. (3.54c) The first term ρ C Ham corresponds to the term from von Neumann s equation originating from the Hamiltonian H C (3.22) involving just the coherent region field operator. This term is exactly the same as (3.45) used in deriving the TWPGPE. The other two terms, ρ C growth and ρ C scatt, arise from the interaction Hamiltonian (3.21). These terms have the name growth and scattering due to the nature of the interaction processes which are described by the different terms which emerge due to the interaction Hamiltonian The Stochastic Projected Gross-Pitaevskii Equation The full master equation (3.54) is mapped to an evolution equation for the Wigner function using the operator correspondences ( ) as in section where the TWPGPE was derived. Similarly, this leads to a partial differential equation containing third order derivatives. and the truncated Wigner approximation is made to give a Fokker-Planck equation. The mapping of the Fokker-Planck equation to a stochastic differential equation gives the full SPGPE (S)dψ C (x, t) = P C { ī h L Cψ C (x, t)dt (3.55a) + G(x) k B T (µ L C) ψ C (x, t)dt + dw G (x, t) (3.55b) + d 3 x M(x x ) i h j C(x } ) ψ C (x, t)dt + iψ C (x)dw M (x, t) (3.55c) k B T where the (S) indicates that the stochastic differential equation is given in Stratonovich form. The first line of the SPGPE is simply the PGPE (3.49) where there is no representation of coupling between the coherent and incoherent regions. The effect of coupling from the interaction Hamiltonians is introduced into the SPGPE through the growth and scattering terms in the master equation (3.54b, 3.54c), and relate to the second and third lines of the SPGPE respectively. 26

37 Chapter 3. C-Field Theory 3.6. The Stochastic Projected Gross-Pitaevskii Equation q p n I p n I cut cut m C q m C (a) Growth (a) Scattering Figure 3.2: Growth and scattering processes caused by interactions between the coherent and incoherent regions. The growth process (a) occurs when two incoherent region atoms collide, the energy is transferred to one of the atoms, while the other enters the coherent region. The scattering process (b) occurs when a coherent and incoherent region atom collide, leaving no change in either region s population Growth and Scattering Processes Including interactions between the coherent and incoherent regions leads to dissipative and noise terms which can be seen to originate physically from growth and scattering processes (see figure 3.2). Here we give a brief description of these processes. Growth Processes The second line of the SPGPE (3.55b) is responsible for growth in the coherent region by scattering of two thermal atoms in the incoherent region. The µ and T which appear correspond to the temperature and chemical potential of the thermal reservoir containing atoms in the incoherent region. The growth term G(x) represents the collision rate of particles in the incoherent region and acts as a damping term in the equation of motion. The noise associated with the growth process is dw G (x, t), defined by dw G(x, t)dw G (x, t) = 2G(x)δ C (x, x )dt, (3.56) dw G (x, t)dw G (x, t) = dw G(x, t)dw G(x, t) = 0, (3.57) where the noise is complex and additive. 27

38 Chapter 3. C-Field Theory 3.7. Simple Growth SPGPE Scattering Processes The third line of the SPGPE (3.55c) is responsible for scattering processes which conserve the number of atoms in both the coherent and incoherent regions, while providing a mechanism for energy transfer between the two regions. This couples to the divergence of the coherent region current by j C (x) i h { } [ ψ 2m C(x)]ψ C (x) ψc(x) ψ C (x). (3.58) In the scattering process, the noise is real and defined by dw M (x, t)dw M (x, t) = 2M(x x )dt, (3.59) where M(x x ) is the scattering rate function Grand Canonical Equilibrium The SPGPE provides a grand canonical description of the coherent region, parameterized by the temperature and chemical potential of the incoherent region. Irrespective of the form of G(x) and M(x), in equilibrium the SPGPE samples microstates with probability P (ψ C ) exp where the grand canonical Hamiltonian, K C, is given by ( ) KC, (3.60) k B T K C H C µn C, (3.61) where the c-field energy (H C ) is given by (3.22), and the c-field number is given by N C = d 3 x ψ C (x) 2. (3.62) 3.7 Simple Growth SPGPE The scattering process term from the full SPGPE (3.55c) involves the collision of a coherent and incoherent region atom resulting in no change in the population of the coherent region. This is in contrast to the growth term (3.55b) which describes the dominant collision process 28

39 Chapter 3. C-Field Theory 3.8. The Damped Projected Gross-Pitaevskii Equation resulting in Bose-Einstein condensation. This means it is reasonable to expect that the scattering term will be less important than the growth term in describing the condensate dynamics. It seems reasonable physically to neglect the scattering term, but it is also an appealing approximation as the scattering term is very difficult to implement numerically [12]. Neglecting the scattering term leads to the simple growth SPGPE (S)dψ C (x, t) = P C { ī h L Cψ C (x, t)dt + G(x) } k B T (µ L C) ψ C (x, t)dt + dw γ (x, t), (3.63) where dw G(x, t)dw G (x, t) = 2G(x)δ C (x, x )dt, (3.64) The noise term in the simple growth SPGPE is additive, so the numerical implementation of this equation is only slightly more complicated than the PGPE. We use the simple growth SPGPE (3.63) throughout this thesis, so we will refer to this equation as simply the SPGPE from now. The growth rate function (G(x)) can be explicitly calculated in near equilibrium situations, and has a simple spatially constant form G(x) = γ, which is described in more detail in section This means the SPGPE provides an a priori theory of the quasi-equilibrium Bose gas. Thus we can make quantitative calculations with a theory of damping from first principles. We use the SPGPE throughout this thesis, where we use this equation to make quantitative calculations of vortex decay in high temperature Bose-Einstein condensates. 3.8 The Damped Projected Gross-Pitaevskii Equation A damped PGPE (DPGPE) is obtained by neglecting the noise term in the simple growth SPGPE (3.63), and is of the form i h ψ C(x, t) t { = P C L C ψ C (x, t) i hγ } k B T (L C µ) ψ C (x, t). (3.65) 29

40 Chapter 3. C-Field Theory 3.9. Numerical Methods Evolution via the DPGPE (3.65) minimizes the grand canonical Hamiltonian K C (3.61), where dk C dt = 2γ k B T d 3 x P C {(µ L C )ψ C (x)} 2, (3.66) which means thermal fluctuations are damped out, and the equilibrium solution is the zero temperature PGPE ground state satisfying P C L C ψ C = µψ C. (3.67) The absence of noise means the DPGPE will not give quantitatively accurate calculations of vortex decay in high temperature Bose-Einstein condensation since it can not account for thermal fluctuations. However it can be useful to give a phenomenological picture of dissipative dynamics, which is often conceptually clearer as SPGPE dynamics is often masked by the large thermal fluctuations. It also gives a clear comparison with the SPGPE to help highlight the effect that thermal fluctuations have on dynamics. To date several numerical studies of vortex nucleation and dynamics have been based on a phenomenological damped Gross-Pitaevskii equation [74, 75, 96, 54]. The description was first introduced by Tsubota et al. [96], where their approach followed the damped Gross-Pitaevskii equation of Choi et al. [21] for a thermal cloud in equilibrium at a rotation frequency Ω. This equation of motion was also obatained by Kasamatsu et al. [54] from the generalized finite temperature Gross-Pitaevskii equation of Zaremba, Nikuni and Griffin [104]. 3.9 Numerical Methods The numerical methods behind the simple growth SPGPE (3.63) are not trivial and years of research has been spent finding a method of solution. The methods used to solve (3.63) are based on the numerical solution to the PGPE (3.49), as the only difference between these two equations is the damping and additive noise which are numerically easy to deal with. The theory behind the PGPE numerical methods of solution were formulated by Blakie and Davis [11, 13], and we use an adaptive Runge-Kutta method in the interaction picture [15, 26] to solve (3.63) numerically. We refer the reader to these references [15, 26, 11, 13], as the precise details of the numerical implementation are not a key detail of the new research in this thesis. 30

41 Chapter 4 Persistent Current Formation 4.1 Introduction A Bose-Einstein condensate trapped in an external potential which monotonically increases from its global minimum, cannot support stable vortices in the absence of an externally imposed rotation [82]. However a multiply connected geometry can enable persistence of superfluid flow. The stability of macroscopic persistent currents in toroidal Bose-Einstein condensates has been well studied [52, 8, 50, 72, 73]. In this chapter we model the persistent current formation experiment performed in Brian Anderson s laboratory at the University of Arizona [70]. In this experiment, a multiply connected toroidal geometry is created by using a blue-detuned laser beam propagating along the axial direction through the center of the trap [70]. In this case quantized circulation is supported around the repulsive barrier, and persists by vortices pinning to this blue-detuned laser beam. Such vortex pinning is stable [14, 60] due to the energy cost of moving a vortex from the zero density pinning site into the atomic fluid. Such vortex pinning has been seen experimentally in Bose Einstein condensation [97], and vortex pining from localized impurities has been studied theoretically [25]. In this chapter we use the SPGPE of section 3.7 to quantitatively model the persistent current formation experiment [70]. In brief, a multiply charged persistent current is formed in a toroidal Bose-Einstein condensate by stirring the condensate with a blue-detuned laser. This stirring nucleates many vortices, whose subsequent thermal damping leads to a persistent current. Vortices decay to the boundary of the condensate, as well as to the center of the toroid where they become stabilized by pinning to the blue-detuned laser beam. A persistent current exists when such stabilization of vortices occurs. These experiments are 31

42 Chapter 4. Persistent Current Formation 4.2. Experimental Description performed in the high temperature regime, with T 0.9T c, so are well suited to the SPGPE theory. As well as being a good test of the SPGPE, the work in this chapter is also important as the physics of persistent current formation is interesting in its own right. In a superfluid, the frictionless flow allows persistent circulation in a multiply connected geometry such as a hollow toroid [61]. Thus the superfluid behavior of Bose-Einstein condensation can be clearly tested through the sustainability of superfluid flow in a condensate. Pinning of multiple quanta to the central barrier in a toroidal Bose-Einstein condensate was first shown experimentally by Ryu et al. [86], where quantized rotation was initiated by transferring one unit h of the orbital angular momentum from Laguerre-Gaussian photons to each atom. The Arizona experiment is unique in that a thermal cloud is used to drive the formation of a persistent current from the decay of vortex dipoles. The blue-detuned laser beam is rotated through the condensate to nucleate vortex dipoles creating a turbulent state. The vortices then decay to either the condensate boundary, or become pinned to the blue-detuned laser beam leading to the formation of a persistent current. We first give a description of the experiment as detailed in Tyler Neely s PhD thesis [70], before outlining the procedure we use to quantitatively model the experimental system. We detail how we determine all SPGPE parameters, including the damping rate, for a toroidal Bose-Einstein condensate using experimentally measured parameters. Thus we can model the Arizona experiment without fitting any simulation parameters. We also model the experimental procedure using the damped projected Gross-Pitaevskii equation (DPGPE), to give a clear phenomenological picture of the underlying mechanisms behind persistent current formation. We compare the SPGPE with the DPGPE, and experimental results, and show the significant effect that thermal fluctuations have on the decay rate of vortices and the quantitative accuracy of the SPGPE calculations. Our results show that the SPGPE accurately describes the physical features of the experiment; it accurately predicts the size of persistent current formed, and the timescale of vortex decay observed in the experiment. 4.2 Experimental Description Here we describe the persistent current experiment performed in Brian Anderson s laboratory at the University of Arizona. Full experimental details can be found in Tyler Neely s PhD thesis [70]. Only details relevant to the theoretical procedure are described here. Un- 32

43 Figure 5.1: In-situ images of the BEC in the toroidal trap. (a) Horizontal phasecontrast image, corresponding to the time-step where vortex pairs are generated, but without the blue-detuned beam enabled. (b) Vertical absorption image at the Chapter same 4. time-step, Persistent Current with theformation blue-detuned beam enabled. (c) Vertical 4.2. Experimental absorptiondescription image after additional cooling. Figure 5.2: Scheme used to create a superflow. (a) RF sequence. (b) Timing sequence (see text). (c) Diagram of the spin sequence, shown relative to the harmonic Figure 4.1: Schematic describing the experimental procedure used to create a persistent current. (a) RF evaporation scheme, leading to a high temperature Bose-Einstein condensate trap center. The harmonic trap center moves in a circle of r =5.7(0.2) µm over 333 with T/T ms in c 0.9. (b) Hold sequence timing. (c) Diagram of the spin sequence, relative to response to the magnetic push coils (see text). Thus, in the trap rest-frame, the harmonic trap center (not to scale). Figure reproduced from [70]. the blue-detuned beam (represented by the solid gray circle) appears to circle about its initial position, as diagramed in zoomed in portion of the figure (not to scale). Subsequent to the spin sequence the harmonic trap is held stationary, keeping the blue-detuned beam centered on the harmonic trap. certainties in experimental measurements are shown in parenthesis. the barrier height was in the range of U 0 1.4µ 0 to U 0 1.8µ 0. As expected, this barrier height produced a clear hole in the middle of the cloud; in-situ images are Condensate Formation given in Fig. 5.1(b,c), where a flat, annular cloud is visible. Also different in this experiment compared to the methods of Chapter 4 was the We begin by describing the properties of the Bose-Einstein condensate formed in the Arizona RF evaporation scheme; the scheme is represented in Fig. 5.2(a). The initial jump of laboratory. A Bose-Einstein condensate is created in a time-averaged orbiting potential the RF from 4 MHz to 3.61 MHz formed a BEC with a considerable thermal fraction magnetic trap with radial and axial trapping frequencies of (ω r, ω z ) = 2π (8, 90) Hz. A Bose- Einstein condensate in a toroidal geometry is achieved by focusing an axially propagating 660-nm blue-detuned laser through the middle of the harmonic trap. The laser beam has a 1/e 2 radius of 23(3)µm. The Bose-Einstein condensate was formed through forced radiofrequency (RF) evaporation in the combined magnetic and optical potential with the bluedetuned beam turned on to create a toroidal condensate. The RF evaporation scheme is shown in figure 4.1(a), which leads to the formation of a Bose-Einstein condensate with a large thermal fraction (T/T c 0.9, with T c 116 nk). After 6 seconds of evaporation the RF jumps out to a value which does not induce further evaporation, giving the system time to equilibrate. The laser intensity was chosen to give a barrier height V 0, in the range 131 hω r V hω r. As the chemical potential of the condensate was µ 90 hω r, the barrier produces a hole in the middle of the condensate creating a toroidal geometry. 33

44 Chapter 4. Persistent Current Formation 4.2. Experimental Description Sequence Step Condensate ( 10 6 ) Thermal ( 10 6 ) T (nk T c (nk) End of 6 s jump 0.97(0.19) 2.08(0.42) 103(15) 116(17) At spin 0.95(0.19) 2.15(0.43) 104(16) 118(18) End of Hold (0.17) 0.82(0.16) 75(11) 96(14) 5 s of Hold (0.17) 0.19(0.10) 47(7) 82(12) Table 4.1: Atom number and system temperature at different stages of the spin sequence. Table reproduced from [70] Procedure After forming the toroidal Bose-Einstein condensate a persistent current is formed by moving the blue-detuned laser relative to the harmonic trapping potential, to generate vortex dipoles. The harmonic trap center was made to move circularly in the radial plane by creating a bias magnetic field whose direction rotates. The trap center rotated in a circle of diameter 5.7(0.2) µm for a single rotation at 3 Hz, so the rotation lasts 333 ms (see figure 4.1(c)). From here on we describe the motion in the frame of the harmonic potential, so the obstacle is seen to rotate relative to the trap. Following the spin procedure, the beam obstacle is kept at the center of the harmonic trap. After an additional hold at 4 MHz for s (Hold 1 in figure 4.1(b)), the RF dropped to 3.5 MHz (Hold 2 in figure 4.1(b)). This removed a large fraction of the thermal cloud, cooling the condensate to T/T c 0.6, with T c 82 nk after 5 s of hold 2. Table 4.1 gives an overview of the atom number and temperature of the system during the experimental sequence Experimental Results The movement of the blue-detuned beam in the spin sequence leads to the creation of many vortices (see figure 4.2 at t h = 0 ms). As mentioned in section 2.2.2, movement of the beam exceeding the critical velocity of the condensate results in vortex dipole nucleation [71]. The critical velocity at the center of the condensate was measured to be 170 µm/s, while the circular motion of the beam was estimated to have a velocity of 430 µm/s. The large beam size means the edge of the beam will be closer to the edge of the condensate where the local critical velocity is reduced, hence the stirring sequence nucleates many vortex dipoles. Figure 4.2 shows axial absorption images where expansion is made immediately after turning the blue-detuned beam off. The hold time, t h, corresponds to the amount of time after the end of the spin sequence. The two main results of this experiment can be seen in 34

45 Chapter 4. Persistent Current Formation 4.2. Experimental Description 83 Figure 4.2: Absorption images, where expansion is made immediately after turning off the Figure 5.3: blue-detuned Superflow laser beam. formation The holdprocess; time (t h ) corresponds images are to the acquired time afterwith the spin axial sequence absorption imaging, is completed. expanding Figure immediately reproduced from after [70]. the blue-detuned beam ramp-down. Imaging just after the spin (t h = 0 ms), rotating the trap leads to the creation of a turbulent initial state (first image). The image scale is identical for each image. With increasing hold time t h an apparent superflow is established (indicated by the central dip figure 4.2; (i) the formation of a persistent current from vortices pinning to the repulsive light beam, and (ii) decay of free vortices due to damping induced by the thermal cloud. in the middle of the BEC) as vortices leave the system. After 7 s of hold (last The persistent current is created from the method of the dipole nucleation process. Since image), a large density dip is visible, suggesting the presence of multiple vortices the obstacle is moved in a circular path, a particular sign of vortex (co-rotating) will be (see text) at the center of the BEC and the blue-detuned beam location. created closer to the center of the condensate inducing a net angular momentum on the system. The vortices near the blue-detuned beam can be localized to this pinning potential. experimentally As the holdintime the is increased, work ofthe Tung large et holeal. in figure [68]. 4.2 Eventually, shows the formation with enough of a persistent hold time, current. As the blue-detuned beam is ramped off in these images, the density dip represents an apparent superflow is established around the blue-detuned beam in the center the presence of multiple quanta of circulation due to the size of the hole compared to the free of the cloud, vortices. and The free unpinned vortices aresimultaneously seen to leave the system leavedue the to thermal system. damping. The presence A quantitative measure of the number of quanta retained in the system is shown in figure of the superflow is indicated by the large dark holes in the images of Fig. 5.3; since 4.3. Individual singly charged vortices were resolved from the multiple quanta pinned to the these images beam byare performing takenexpansion, after the afterbeam an extraramp 3 s of hold off, time they hadare elapsed notafter due turning to the hole in the BEC created by the blue-detuned beam. With the ramp-off of the blue-detuned 35 beam there is no active suppression of the density in the middle of the cloud, and

46 Chapter 4. Persistent Current Formation 4.3. Theoretical Description of the System Figure 4.3: Black boxes show the average number of vortices, after 3 s of hold time after Figure 5.5: Evidence of current persistence, counting vortices. Black boxes represent ramp-down of the toroidal beam to resolve individual vortices. These states after the beam average number of vortex cores over 10 runs per point, allowing 3 s of hold time ramp-down after are ramp-down shown in of the the toroidal inset. Grey beam circles to resolve correspond individual to cores the (examples average number inset). of free vorticeserror in thebars trap, represent where statistical the beamuncertainty is rampedrather down than and kept counting off after uncertainty. t h = An s. The white triangles exponential showfitextra to thespontaneously data gives a lifetime created of 31(4) vortices. Grey Figure circlesreproduced show the lifetime from [70]. of vortices released from the beam immediately after creating a superflow. An exponential fit gives the lifetime of these free vortices of 15(1) s. White triangles show extraneously generated vortices (see text) with the blue-detuned beam held on but no initial spin of the cloud. blue-detuned beam off. Without the pinning potential, the multi-charge vortex is unstable and breaks into single cores. The extra hold time allows this to occur, and hence individual vortices are visible. The absorption images in the inset of figure 4.3 show this separation into single cores, where the number of cores observed varies from 3 to 5. The square boxes in figure 4.3 show the number of vortices pinned to the potential barrier, while the grey circles correspond to counting the free vortices in the trap after they have turned off the blue-detuned laser after s of hold time. The number of vortices pinned drops as the hold time increases, due to magnetic field drift and beam misalignment. However this will only be significant for long hold times, meaning the number of quanta pinned in the toroid will be an interesting value to compare quantitatively with theory. 36

47 Chapter 4. Persistent Current Formation 4.3. Theoretical Description of the System 4.3 Theoretical Description of the System Overview Here we outline our theoretical description of a toroidal Bose-Einstein condensate. A key feature of the SPGPE is its description in terms of the reservoir temperature and chemical potential, and the cutoff energy. While the SPGPE itself is relatively straightforward, determining these parameters requires some care. We use a harmonic toroid approximation [16, 7] to give a Thomas-Fermi description of the toroidal condensate [7]. This description provides our starting point for modeling the system using the SPGPE, enabling accurate parameter estimation. We then describe how we treat the incoherent region, determine the growth rate and other parameters, and numerically solve the SPGPE Trapping Potential The harmonic Gaussian potential is given by V (x, t) = V HO (x) + V G (x, t), (4.1) where V HO (x) is the harmonic oscillator potential given by (3.9), and V G represents the Gaussian potential given by [ [(x x(t)) 2 + (y ȳ(t)) 2 ] ] V G = V 0 exp, (4.2) where V 0 is the intensity, σ 0 is the width, and ( x, ȳ) sets the center of the Gaussian potential. To match the parameters of the experiment, we set (ω r, ω z ) = 2π (8, 90) Hz. The potential (4.1) is shown as a function of (x, y) = r in figure 4.4 for ( x, ȳ) = (0, 0), σ = 23 µm/ 2, and V 0 = 131 hω r = 58 h ω that correspond to the appropriate experimental parameters, where we have introduced the geometric mean of the trapping frequencies, ω = (ω x ω y ω z ) 1/3. σ Harmonic Toroid Approximation Here we summarize previous work on toroidal condensates which is relevant to our description of the system [16, 7]. The harmonic Gaussian potential can be simplified by approximating the potential at the minimum of the toroidal trap as harmonic. This is known as the harmonic toroid approximation. Firstly we define some useful quantities for describing the toroidal 37

48 Chapter 4. Persistent Current Formation 4.3. Theoretical Description of the System potential. The energy V σ 1 2 mw2 rσ 2 0, (4.3) represents the potential energy of an atom at r = σ 0 in the harmonic potential. Rewriting this in terms of the radial harmonic oscillator length a r = h/mω r, V σ becomes V σ = hω r 2 ( σ0 a r ) 2, (4.4) so V σ expresses the area of the Gaussian in length and energy units of the radial trap. The minimum of the toroid has a value V m, located at z = 0 and r = r m, where V m = V σ [1 + ln(v 0 /V σ )], (4.5) and r m = σ 2 0 ln(v 0 /V σ ). (4.6) Thus the harmonic toroid is given by V HT (x) m 2 [ω2 T (r r m ) 2 + ω 2 zz 2 ] + V m, (4.7) which is a parabolic potential centralized about r = r m. The harmonic trapping frequency about the minimum of the toroid (ω T ) is found by expanding the radial potential about r = r m, giving ω T 2ωr r m σ 0 = ω r 2 ln(v 0 /V σ ). (4.8) The toroidal regime is realized for V 0 /V σ > 1, and for the experimental parameters we have V 0 /V σ 14. Thomas-Fermi Description The harmonic toroid (4.7) is completely characterized by r m, ω z, and ω T. The Thomas-Fermi particle density can be written as n 0 (x) = µ ( ) r rm 2 ( z [1 U 0 R T R Z ) 2 ], (4.9) where this is positive, and zero elsewhere. The chemical potential is measured relative to V m, and R T = 2µ/mωT 2 and R Z = 2µ/mωz 2 are the Thomas-Fermi radii. The chemical 38

49 Chapter 4. Persistent Current Formation 4.3. Theoretical Description of the System potential can be found in the Thomas-Fermi regime, and has the form µ HT (N 0 ) = h ω HT 8 a N 0 π ā HT where we have introduced the geometric mean frequency of the toroid 1 2, (4.10) ω 5 HT ω K ω 2 zω 2 T, (4.11) where hω K h 2 /2mπrm 2 is the energy scale associated with azimuthal motion around the toroid. ā HT = h/m ω HT is the geometric mean length scale of the toroid Treatment of the Incoherent Region In the SPGPE theory of the high temperature Bose gas, the incoherent region consists of modes with low occupation, with energy greater than the specified cutoff energy ɛ cut. The treatment of the incoherent region for a harmonically trapped Bose gas has been described previously [12, 18, 85]. For the case of an extra potential as in this chapter, we must be careful that it can be represented by our chosen basis-states, single-particle eigenstates for the harmonic oscillator potential V HO (see (3.8)). In a semiclassical picture, we can define a local cutoff wave number via the expression from which we find λ c (x) h 2 K 2 c (x) 2m + V HO(x) = ɛ cut, (4.12) 2π K c (x) = 2π h 2m[ɛ cut V HO (x)], (4.13) for the infrared cutoff. The minimum resolvable length in the coherent region, λ min, is found at V HO (x) = 0 as λ min = 2π h 2mɛcut. (4.14) Similarly we find the largest spatial extent due to our cutoff by taking K c (x) = 0, so that in the plane z = 0, m 2 ω2 rr 2 c ɛ cut, (4.15) 39

50 Chapter 4. Persistent Current Formation 4.3. Theoretical Description of the System where R c defines the transverse position cutoff. Hence we find R c = 2ɛcut mω 2 r, (4.16) where R c is the largest spatial length scale represented by the basis. So, our potential can be described by the basis {ɛ n < ɛ cut } provided the spatial width of the Gaussian potential (σ 0 ) satisfies λ min σ 0 R c. (4.17) This criterion is satisfied by the experimental system (see figure 4.4), where we use a value of ɛ cut described later in (4.32). Note also that as the stirring procedure only causes a displacement of the Gaussian of 5µm σ 0, it is not difficult to ensure that (4.17) holds for the entire stir dynamics. The primary effect of the extra Gaussian potential on the incoherent region is to shift the chemical potential. We include this in what follows, and neglect any further changes to the incoherent region states, adopting the semiclassical treatment previously used for a harmonic external potential. Here we give a brief description of our procedure [12, 18, 85], which we also use in chapters 5 and 6 for systems confined by a harmonic external potential. We describe the incoherent region by a single particle Wigner function for a Bose-Einstein distribution in the local-density approximation where F I (x, k) = 1 exp [( hω(x, k) µ)/k B T ] 1, (4.18) hω(x, k) = h2 k 2 2m + V HO(x). (4.19) This gives a good description when the incoherent region (which thermalizes rapidly compared to the coherent region) is in equilibrium with a well-defined temperature T and chemical potential µ. For the simulations we consider in this chapter, and throughout this thesis, this is a good assumption. Although the stirring of the Gaussian potential creates highly non-equilibrium vortex dynamics in the low energy modes, the incoherent region will be only weakly affected by this because the Gaussian potential is well contained in the coherent region (see (4.17)). Hence the incoherent region will be in near equilibrium, so our description of the incoherent region in terms of (4.19) is valid. Note that our description of the incoherent region is in the laboratory frame, so the thermal cloud is non-rotating. 40

51 Chapter 4. Persistent Current Formation 4.3. Theoretical Description of the System ǫ cut 75 E (units of h ω) V toroidal trap V m 0 harmonic trap Gaussian potential λ min r (µm) R c Figure 4.4: Plot of the Gaussian potential (red), harmonic potential (blue), and the combined harmonic Gaussian potential (black) as a function of radial position. We also show the value of the energy cutoff (ɛ cut ), found from (4.32), from which the values λ min and R c are found describing the smallest and largest length scales of the coherent region defined by this cutoff. As the Gaussian width, σ 0 = 23 µm, lies well inside these spatial limits, the extra Gaussian potential can be described by the single-particle basis. Atom Number Atoms above the cutoff primarily reside at large radii and hence are largely insensitive to interactions with coherent region atoms. The dominant effect of interactions is included in the Thomas-Fermi chemical potential, thus we describe atoms above the energy cutoff using an ideal Bose-Einstein distribution. The properties of the ideal gas description of (4.18) and (4.19), including the effect of the cutoff, can be expressed in terms of the incomplete 41

52 Chapter 4. Persistent Current Formation 4.3. Theoretical Description of the System Bose-Einstein function defined as g ν (z, y) = 1 dxx ν 1 (ze x ) l Γ(ν) y l=1 z l Γ(ν, yl) l ν Γ(ν), (4.20) l=1 where Γ(ν, x) x dyyν 1 e y is the incomplete gamma function. For example, we can write N I as [18] N I = g 3 (e βµ, βɛ cut )/(β h ω) 3, (4.21) where β = 1/(k B T ) is the inverse temperature, and where we use µ = µ V m since the integration is taken over the incoherent region in a semiclassical picture. Thus (4.21) gives the number of atoms of a non-interacting gas obeying the Bose-Einstein distribution at energies above the energy cutoff ɛ cut. Growth Rate The full growth rate in the SPGPE is given by [37, 18] G(x) u 2 (2π) 5 h 2 d 3 K 1 d 3 K 2 d 3 K 3 F (x, K 1 )F (x, K 2 ) Ω 1 [1 + F (x, K 3 )] 123 (0, 0), (4.22) where 123 (k, ɛ) δ(k 1 + K 2 K 3 k)δ(ω 1 + ω 2 ω 3 ɛ/ h) conserves energy and momentum during the collision. This growth rate G(x) can be calculated for near equilibrium situations where the incoherent region is well described by an ideal semiclassical Bose-Einstein distribution (4.18) at chemical potential µ and temperature T [18]. In this case we adopt the notation G(x) = γ, where γ = γ 0 k=1 e βµ(k+1) [ ] e βµ 2 e Φ, 1, k, (4.23) 2βɛcutk βɛcut e and where the rate constant is γ 0 = 4m(a s k B T ) 2 /π h 3, and Φ[x, y, z] is the Lerch transcendent defined by Φ[z, s, a] = k=0 42 z k (a + k) s. (4.24)

53 Chapter 4. Persistent Current Formation 4.4. Numerical Procedure This spatially constant growth rate is exact for the inner spatial region of a harmonic trap (where V (x) 2ɛ cut /3), and is a good approximation elsewhere where there is only a weak spatial dependence [18]. 4.4 Numerical Procedure Overview The experiment detailed in section 4.2 is ideally suited to model using the SPGPE. The experiment is in the high temperature regime with T/T c 0.9, and is extremely well characterized. Previous SPGPE usage has involved a harmonic external trapping potential [98, 18, 85]. Here we detail how we apply the SPGPE to the harmonic Gaussian external potential, describing how we determine appropriate SPGPE parameters in order to model the experimental system. We also detail how we use the SPGPE and DPGPE to model the dynamics of the persistent current experiment Equations of Motion SPGPE We are primarily concerned with modeling the experiment with the SPGPE to give a quantitative comparison between theory and experiment. The SPGPE as shown in section 3.7 is given by (S)dψ C (x, t) = P C { ī h L Cψ C (x, t)dt } γ + k B T (µ L C) ψ C (x, t)dt + dw γ (x, t), (4.25) where dw γ (x, t)dw γ (x, t) = 2γδ C (x, x )dt, (4.26) and where we have used γ as the growth rate, G(x), as shown in section Note that L C is given by (3.50), which is in terms of the full potential V (x, t) in (4.1). The SPGPE is a grand-canonical theory where the system is characterized by the temperature (T ) and chemical potential (µ) of the reservoir, the cutoff energy ɛ cut, and the growth rate γ. We describe how we determine these SPGPE parameters in section

54 Chapter 4. Persistent Current Formation 4.4. Numerical Procedure DPGPE In this chapter we also simulate the dynamics using the DPGPE. The DPGPE is of the form (see section 3.8) i h ψ C(x, t) t { = P C L C ψ C (x, t) i hγ } k B T (L C µ) ψ C (x, t). (4.27) As the DPGPE has no noise term, we use the DPGPE to help study the influence that thermal fluctuations have on the dynamics of the persistent current experiment. We use the same parameters for the DPGPE simulations as used in the SPGPE simulations, shown in section 4.4.3, with the only difference being that the noise is set to zero Simulation Parameters As mentioned in section 4.4.2, the SPGPE is a grand-canonical theory with control parameters of the reservoir temperature T and chemical potential µ, cutoff energy ɛ cut, and growth rate γ. However experiments usually only measure T, and the total atom number N. For a given temperature a choice of µ(t, N) can be made to give a consistent total atom number, while ɛ cut (T, N) is important to ensure the SPGPE cutoff occupation criteria is satisfied. The problem of finding µ(t, N) and ɛ cut (T, N) for the harmonically trapped Bose gas has been previously addressed, using a Hatree-Fock density of states to approximate condensate and thermal cloud atomic interactions [85]. For the case of a toroidally trapped Bose gas, we use a less general approach which is suitable for estimating µ(t, N) for the regime of the experiment. The results presented in section 4.3.3, describing the properties of a toroidally trapped Bose gas in the harmonic toroid approximation, form the basis of our parameter estimation procedure. Chemical Potential The first task in modeling the experiment is to produce an SPGPE equilibrium state, with temperature and atom number matching the experiment immediately prior to the stir sequence (immediately prior to the spin in figure 4.1). This requires finding an appropriate chemical potential to give the desired total atom number. The total atom number N is given by N = N C + N I, (4.28) 44

55 Chapter 4. Persistent Current Formation 4.4. Numerical Procedure where the number of particles in the coherent region is N C = d 3 x ψ C (x) 2, (4.29) and the number in the incoherent region is found from (4.21) as described in section The SPGPE chemical potential is defined relative to the ground state of the harmonic oscillator potential. The harmonic toroid chemical potential (µ HT ) is expressed relative to the trap minimum V m, so a first estimate of the SPGPE chemical potential is µ(t, N) = µ HT (N 0 ) + V m. (4.30) We determine µ HT using equation (4.10) for N 0 = atoms in the Bose-Einstein condensate, and we set T = 98 nk which lies within the uncertainty of the temperature measured in the experiment at the start of the spin procedure (see table 4.1). For the parameters of the experiment we find from (4.5) that V m = 14.9 h ω, and from (4.10) that µ HT = 25.6 h ω, hence we estimate the SPGPE chemical potential as µ = 40.5 h ω. However µ HT is calculated in the harmonic toroid approximation, so needs to be modified to give a suitable estimate of µ as an SPGPE parameter. Hence we use µ = µ HT + V m δ, (4.31) where δ is a small correction which we determine, to account for details of the potential. In table 4.2 we show the dependence that the total atom number has on δ. Table 4.2 shows that if α = 0, that is we use (4.30) to determine µ, then we get a total number of atoms that is larger than required. We see that using a correction of δ = 2 h ω gives a better choice of µ, with N in this case. In numerically modeling this experiment, we need to have a system size within the experimental uncertainty in atom number. The most interesting case for this work is for the smallest system size, since with a smaller atom number more vortices are nucleated during the stirring process. We find that a value of δ = 6.5 h ω gives a total number near the lower value of the experimental uncertainty. Energy Cutoff We also need to determine an appropriate cutoff energy, ɛ cut, so that the cutoff occupation is of order unity. We previously estimated this for the harmonically trapped Bose gas using the Hatree-Fock density of states [85]. In the present case of the toroidal potential, we do 45

56 Chapter 4. Persistent Current Formation 4.4. Numerical Procedure δ ( h ω) Experimental Values N ( 10 6 ) N expt 3.05(0.4) Table 4.2: Table showing how the total number in SPGPE simulations varies with chemical potential. The different values of chemical potential are parameterized by δ, as in (4.31).We see δ = 2 h ω gives a total number in SPGPE simulations consistent with N expt. not have a similar procedure. We choose the cutoff energy as ɛ cut = 2.05µ = 83.1 h ω, (4.32) where we use the chemical potential determined from (4.30). Our choice of the cutoff given by (4.32) is driven by the requirement that it must be sufficiently high so that the single particle basis will accurately describe the interacting modes of the system [12]. In practice this requirement is satisfied if the cutoff energy is of order 2 to 3 times larger than µ, hence we initially estimate ɛ cut = 2.05µ. We check the validity of this choice of cutoff by finding the average cutoff occupation. The mean occupation at the cutoff can be found easily by time averaging the amplitudes of the classical field (3.37), α i. The average occupation of the ith single particle mode φ i (x) in equilibrium is n i = 1 N s N s n=1 α i (t n ) 2, (4.33) where N s is the number of samples used in the time averaging. For a consistently chosen cutoff the interacting energy eigenmodes at the cutoff ɛ cut are, to a good approximation, single particle states. This means the occupation at the cutoff can be found directly from (4.33) by finding the lowest value of the occupation when considering all the modes in the classical field. In Figure 4.5 we show the occupation of the least occupied single particle modes, found by using (4.33). We determine the occupation of a SPGPE equilibrium state for the upper and lower values of chemical potential determined from (4.31), δ = 0 h ω and δ = 6.5 h ω, where in both cases we have the same the cutoff energy (4.32). In both cases the occupation is of order one at the cutoff, so we use the chosen value of ɛ cut for our toroidal simulations. 46

57 Chapter 4. Persistent Current Formation 4.4. Numerical Procedure α = 0 α = 6.5 Occupation Least Occupied Single Particle States Figure 4.5: Occupation number of the 50 least occupied single particle energy modes in a toroidal ground state with ɛ cut = 83.1 h ω, as in (4.32). The two cases correspond to different chemical potential values determined by (4.31), with α = 0 (red squares) and α = 6.5 (black circles). Growth Rate Once the reservoir parameters T and µ, and the cutoff energy ɛ cut are determined, we calculate the growth rate using (4.23). This equation explicitly determines γ γ(t, N, ɛ cut ) in a regime which is valid for our calculations, since in this system the incoherent region is well described by an ideal semiclassical Bose-Einstein distribution. As we determine µ(t, N) prior to simulation to give a consistent total atom number, our method provides a complete treatment of the dissipative process leading to particle exchange between the incoherent and coherent regions, without any fitting parameters Initial States We create equilibrium states of the SPGPE by evolving 4.25) with appropriate parameters of T, µ, and ɛ cut, using an initial state consisting of a harmonically trapped Thomas-Fermi 47

58 Chapter 4. Persistent Current Formation 4.4. Numerical Procedure ground state with density [23] n 0 (x) = 1 U 0 (µ TF V HO (x)) µ TF V HO (x) 0 0 µ TF V HO (x) < 0, (4.34) using a high value for the growth rate (γ 0.7) to increase the rate of evolution to equilibrium. Using the Thomas-Fermi density as the initial state is important as although any random state will eventually evolve to a equilibrium state with temperature T and chemical potential µ, due to the toroidal geometry any vortices which appear from the SPGPE evolution of a random state can become pinned to the center of the toroid. Using a Thomas-Fermi density as the initial state means we consistently create a SPGPE equilibrium state with no circulation. We generate a ground state of the DPGPE in the same manner, by evolving the DPGPE (4.27) from an initial Thomas-Fermi ground state (4.34). Again we use a high growth rate of γ 1 to make the evolution to equilibrium more rapid Dynamics Hold 1 Dynamics We model the stirring process by using a time dependent Gaussian in the Harmonic Gaussian potential (4.1), where the center of the Gaussian is set by ( x(t), ȳ(t)) = r 0 (1 cos(κt), sin(κt)). (4.35) This moves the Gaussian potential barrier of height V 0 in a circle of radius r 0, about the point (x, y) = (r 0, 0), at a frequency of κ. Note that our simulation method moves the obstacle, whereas the experiment moves the trap with the barrier in a fixed position. We set r 0 = µm, and κ = 2π/(333 ms) = 6π s 1 to model the stirring, and then set ( x, ȳ) = (0, 0) at the completion of the stir after t = 333 ms. In table 4.3, we detail all parameters used in the SPGPE simulations. We show the parameters used for modeling the dynamics of hold 1 in the experiment, which corresponds to the first 1.5 s of time evolution including the stir. 48

59 Chapter 4. Persistent Current Formation 4.4. Numerical Procedure Parameter Hold 1 (SPGPE) Hold 2 (SPGPE) T (nk) µ ( h ω) N(µ, T ) (10 6 ) γ h/k B T (10 4 ) ɛ cut ( h ω) V 0 ( h ω) 1 58, σ (µm) 23/ 2 23/ 2 r 0 (µm) κ (s 1 ) 6π 0 1 The second value of V 0 for hold 1 corresponds to a different set of simulations where we use the highest value of V 0 within experimental uncertainty (see section 4.6.4). Table 4.3: Table showing all parameters used in SPGPE calculations to model both hold 1 and hold 2 of the experimental sequence. Note that κ = r 0 = 0 for hold 2 since this occurs after the spin sequence (after t = 1.5 s), so we do not simulate the spin procedure with hold 2 parameters. We use the same parameters as in hold 1 for the DPGPE calculations (section 4.5), only the noise is set to zero. Hold 2 Dynamics In the experiment after hold 1 is completed, the RF is abruptly reduced from 4 MHz to 3.5 MHz, which induces extra evaporative cooling. In terms of modeling this using the SPGPE, we abruptly change the chemical potential. Using the methods above in section 4.4.3, we find a chemical potential which gives the desired final atom number for the required temperature (N at T 50 nk, see table 4.1). The SPGPE parameters used for hold 2 are also shown in table 4.3. To simulate the dynamics of hold 2, we first simulate the hold 1 dynamics for t = 1.5 s. Then after t = 1.5 s when hold 2 begins, we simply switch µ and T to the hold 2 values for each trajectory, while keeping the energy cutoff fixed. Overview We performed simulations investigating the dynamics of the persistent current formation experiment, using a number of different parameters. In section 4.5 we show DPGPE calculations using hold 1 parameters throughout the entire evolution. We do not consider hold 2 within the DPGPE treatment. 49

60 Chapter 4. Persistent Current Formation 4.5. DPGPE Treatment We show the results of the SPGPE simulations in section 4.6 where we consider a range of different parameters. We perform SPGPE simulations with the parameters of hold 1 for the entire evolution in section We then consider the effect that hold 2 has on the vortex dynamics in section 4.6.3, where we change the parameters to those for hold 2 after t = 1.5 s. Then we investigate the effect of varying the barrier height V 0 in section 4.6.4, where we perform simulations with V 0 = 67 h ω which is the highest barrier height within experimental uncertainty used in experiment. 4.5 DPGPE Treatment In this section we present results from simulations using the DPGPE. We show the DPGPE results first as although it does not give a complete description of the finite temperature physics since it neglects thermal fluctuations, it gives the visually clearest view of the type of vortex physics occurring in the experiment within a phenomenological description. It also gives a clear comparison for the SPGPE results in section Numerical Overview We evolve the DPGPE (4.27) with an initial state as prescribed in section We use the same parameters as shown in table 4.3 for hold 1, only we set T = 0 and there is no noise since we are using the DPGPE. We only use the parameters of hold 1, so we do not simulate the dynamics in hold 2 where further evaporative cooling is done in experiment. The hold 2 dynamics are considered in the SPGPE treatment in section We only need to find a single DPGPE trajectory since the evolution equation is not stochastic. However the large system size requires many modes to numerically simulate, and is thus computationally intensive. The computing time required to simulate the dynamics until all vortices have decayed from the system with the DPGPE was more than 3 months of wall clock time on a single processor for a single trajectory Results Figures 4.6 and 4.7 show the dynamics from DPGPE evolution. Figure 4.6 shows the spin sequence, where the circular arrow at t = 0 s depicts the circular motion of the initially central repulsive potential. Note that the t we use is relative to the start of the stir, while the experimental t h is relative to the end of the s stir. 50

61 Chapter 4. Persistent Current Formation 4.6. SPGPE Treatment We see nucleation of a number of vortices at t = 0.17 s, and after t = 0.24 s many vortex dipoles can be clearly seen in the phase profile. The majority of vortex dipoles are formed when the repulsive barrier is dragged closest to the exterior boundary of the condensate. This corresponds to the barrier being dragged through a region of lower density, and hence a region of lower critical velocity, so more vortices can be nucleated. The asymmetric nature of the stirring process leads to a non-uniform distribution of vortices (t = 0.33 s). The stirring process creates a Bose-Einstein condensate with non-zero angular momentum, since the more centrally located vortex in the dipole pair is of the same charge for all dipoles formed. From this turbulent state of the Bose-Einstein condensate at the end of the stir, we see the subsequent decay of vortices leading to a persistent current (figure 4.7). The vortex dynamics in this system are very complex, and a number of different decay processes are observed. The repulsive barrier acts as a vortex pinning source, and can split a vortex dipole by attracting the more central vortex thus leaving the other vortex free. Depending on the location of the vortices, they can decay toward the center of the condensate to the pinning potential, or decay to the condensate boundary. Figure 4.7 shows the effect of the types of decay where as time evolves we observe that a number of vortices become pinned, and the remaining free vortices decay to the boundary, or mutually annihilate either in the bulk fluid or on the pinning potential. The DPGPE calculations are useful as the dynamics of the vortices are clear to see without the extra complexity of thermal fluctuations. However we see that the absence of thermal fluctuations leads to very long decay times for the vortices. After t = 7 s there are still 3 free vortices as well as a number pinned to the center of the toroid, while experimentally for t 5 s there is little sign of any free vortices. We see one free vortex still in the system after 34 s which decays to the condensate boundary, leaving a charge 3 persistent current. Further discussion of the quantitative results from all calculations is given in section SPGPE Treatment In this section we present results from SPGPE simulations used to model the experiment. The SPGPE gives results that can be interpreted quantitatively in contrast to the DPGPE simulations. We show the accuracy of the SPGPE as a quantitative tool by comparing SPGPE simulations with experimental results in this section. 51

62 Chapter 4. Persistent Current Formation 4.6. SPGPE Treatment 40 Column Density t =0s Phase (z =0) y (µm) t =0.17s y (µm) t =0.24s y (µm) t =0.33s y (µm) x (µm) x (µm) Figure 4.6: Images of sequence of the stir for a DPGPE simulation, showing the column density and phase for different times. The sequence is described in detail in the text. 52

63 Chapter 4. Persistent Current Formation 4.6. SPGPE Treatment 40 Column Density t = 0.67s Phase (z = 0) y (µm) t = 1.46s y (µm) t = 7s y (µm) t = 34s y (µm) x (µm) x (µm) Figure 4.7: Image sequence of a DPGPE simulation, showing the column density and phase for different times after the spin procedure. 53

64 Chapter 4. Persistent Current Formation 4.6. SPGPE Treatment Numerical Overview We use the SPGPE (4.25) to evolve a SPGPE ground state found using the procedure in section 4.4.4, using a range of parameters as specified in section 4.4.5, and shown in table 4.3. We perform two sets of simulations using the parameters of hold 1 for the entire simulation, with the barrier height V 0 different in each set. We also calculate a third set of trajectories, where we model the hold 2 dynamics by changing the parameters to those of hold 2 after t = 1.5 s. As with the DPGPE simulations in section 4.5, the SPGPE calculations are also numerically challenging due to the large system size. Individual SPGPE trajectories are also more numerically challenging than the DPGPE due to the presence of fluctuations. This is coupled with the fact that the SPGPE is a stochastic equation of motion, so for each parameter set we must average over a sufficiently large ensemble of individual trajectories. For each set of parameters we calculate 16 SPGPE trajectories. The computational time required for each trajectory was approximately 6 weeks of wall clock time on a single processor, which is less than that for the DPGPE simulation since the vortex lifetimes are much smaller in the SPGPE treatment so less simulation time is required (see section 4.6.5). We estimate the total computing time used to determine the entire ensemble of data at approximately 4 years on a single processor. We emphasize that in these SPGPE calculations, all parameters including the damping rate are determined in a consistent manner prior to simulation. Thus we have a description of thermal damping from first principles, enabling a true quantitative test of the SPGPE with experiment Hold 1 Parameters In figure 4.8 and figure 4.9, we show the results of a single SPGPE trajectory for the parameters shown in table 4.3 for hold 1, with a barrier height of V 0 = 58 h ω. The entire time evolution is based on the hold 1 parameters only, we look at the effects of hold 2 in the next section Figure 4.8 shows the spin sequence, where the circular arrow at t = 0 s shows the circular motion of the initially central repulsive potential. As with the DPGPE calculations, we see during the initial 0.33 s stir that many vortex dipoles are nucleated. Due to the size of the Bose-Einstein condensate the vortices are hard to see in the column density, however they are more easily identified in the phase slice. The qualitative picture of the dynamics is essentially the same here as in the DPGPE calculation. 54

65 Chapter 4. Persistent Current Formation 4.6. SPGPE Treatment The formation of multiple vortex dipoles is clear after t = 0.27 s. The subsequent vortex dynamics are complicated, where the vortices decay and become pinned to the repulsive barrier, or annihilate either at the condensate boundary or internally via vortex/antivortex annihilation. The major difference in the SPGPE calculations compared to the DPGPE is the timescale of the vortex decay. Figure 4.9 shows the decay of the vortices after the stir. After t = 1.57 s there are only 2 free vortices that can be clearly observed, with an additional 4 quanta pinned to the center of the toroid, essentially equivalent to the state of the DPGPE simulation after t = 7 s. Finally after nearly 3 s of SPGPE evolution the Bose-Einstein condensate forms a charge 4 persistent current, while the DPGPE calculation still shows free vortices after t = 30 s and has only 3 quanta pinned to the Gaussian potential. We infer from these results that thermal fluctuations play a key role in more rapidly driving the decay of the system to a stable persistent current SPGPE Treatment of Hold 2 We have shown both DPGPE and SPGPE calculations where we have used the parameters for hold 1 throughout the entire simulation. Hold 2 occurs after t = 1.5 s of time evolution, and at this point the main dynamics have been observed and all that remains is the decay of a few free vortices (see figure 4.9 at t = 1.57 s). Hence we assumed implementing the change of parameters for hold 2 at t = 1.5s will have little effect on the remaining simulation, meaning we initially did all calculations with hold 1 parameters only. To test this we adjusted the parameters of our ensemble of SPGPE trajectories to that of hold 2 (see table 4.3) after t = 1.5 s, and continued SPGPE evolution with the new parameters until all free vortices had decayed. From inspection of individual trajectories, we find no significant change in the persistent current dynamics except that rapid rotation is observed in the phase as the thermal reservoir becomes out of equilibrium with the coherent region due to the sudden drop of the chemical potential and temperature. In table 4.4 we show the average number of quanta that become pinned to the Gaussian barrier once all free vortices have decayed. We see that there is no significant difference in the average number pinned between using only hold 1 parameters and using hold 2 parameters after t = 1.5 s. The number of vortices pinned is in the range of 3-5 as seen experimentally. 55

66 Chapter 4. Persistent Current Formation 4.6. SPGPE Treatment Column Density 40 Phase (z = 0) t = 0s y (µm) 20 0 ï20 ï40 40 t = 0.13s y (µm) 20 0 ï20 ï40 40 t = 0.27s y (µm) 20 0 ï20 ï40 40 t = 0.33s y (µm) 20 0 ï20 ï40 ï40 ï ï40 ï x (µm) x (µm) Figure 4.8: Images of sequence of the stir for a single SPGPE trajectory, showing the column density and phase for different times. The sequence is described in detail in the text. 56

67 Chapter 4. Persistent Current Formation 4.6. SPGPE Treatment Column Density Phase (z = 0) 40 t = 0.51s y (µm) t = 0.83s y (µm) t = 1.57s y (µm) t = 2.80s y (µm) x (µm) x (µm) Figure 4.9: Image sequence of a single SPGPE trajectory, showing the column density and phase for different times after the spin procedure. 57

68 Chapter 4. Persistent Current Formation 4.6. SPGPE Treatment Simulation Parameters Average Number of Quanta Pinned Hold Hold 1 and Hold Hold 1 with higher potential barrier 4.50 DPGPE Treatment 3 Table 4.4: Average number quanta pinned to the repulsive potential once all free vortices have decayed from the system Effect of Barrier Height The results shown so far have all been based on using a Gaussian barrier with V 0 = 131 hω r = 58 h ω. We calculated another set of SPGPE trajectories using the parameters of hold 1 for all t, except we changed the barrier height to V 0 = 151 hω r = 67 h ω, corresponding to the upper value within experimental uncertainty. We observe the same dynamics as with the lower barrier height, but there is a slight increase in the number of vortices pinned (see table 4.4), as a pinning potential with a higher barrier height can support a larger number of vortices. We see an average number pinned of 4.5, which is still close to what is seen experimentally Average Number of Vortices In figure 4.10, we plot the average number of vortices found from the numerical simulations as a function of time. We plot this data with the experimental data, so we use the hold time t h where time is measured from the end of the s stir. The experimental data is the same as in figure 4.3, where the black boxes correspond to the number of vortices pinned to the blue-detuned laser. The SPGPE result for the parameters of hold 1 only with the high barrier height is shown by the red cross, while the SPGPE result when using the lower barrier height is shown by the blue diagonal cross. The blue vertical cross corresponds to the SPGPE simulations where the lower barrier height is used, but the parameters are changed to that of hold 2 at the appropriate time. We plot the results for each SPGPE parameter set at the time at which a stable persistent current has formed, that is when there are no free vortices left in the system. We see the number of vortices averaged over 16 trajectories agrees with experiment at t h 6 s. Thus the SPGPE predicts the right number of pinned vortices, with all free vortices decaying on the correct timescale. The results for the lower barrier height lie just inside the 58

69 Chapter 4. Persistent Current Formation 4.6. SPGPE Treatment Figure 4.10: A modified version of figure 4.3 to highlight how the numerical results compare with experiment. As before the black boxes represent the average number of vortices found experimentally, after 3 s of hold time after ramp-down of the toroidal beam to resolve individual vortices. The red cross shows the average number of vortices from SPGPE simulations with the parameters of hold 1 where the higher potential barrier of V 0 = 151 hω r = 67 h ω is used. Both the blue crosses correspond to SPGPE simulations with the lower barrier height. The diagonal blue cross is the result for using the parameters of hold 1 throughout the entire SPGPE evolution, while the vertical blue cross shows the result where the parameters are changed to that of hold 2 after t = 1.5 s (or t h = s). The black crosses show the vortex number from the DPGPE. lower error experimental bar. Similarly the results for the higher barrier lie just under the higher experimental error bar. For the simulations where the hold 2 parameters are used, we see a slightly longer time is required for the decay of all the free vortices. This is since the system is at a much cooler temperature during hold 2 (T/T c 0.6), however the extra time required is relatively small and the same number of pinned vortices are seen as with 59

70 Chapter 4. Persistent Current Formation 4.7. Summary simply using the hold 1 parameters. We don t show any other data points for the SPGPE simulations since at this point all vortices are pinned, and there will be no loss of vortices as seen experimentally due to magnetic field drift and beam misalignment. We also show results for the DPGPE calculation (black cross) which highlights the quantitative differences between the methods. We see after t h 6 s the DPGPE shows more than twice the amount of vortices in the system compared to the SPGPE, and the experiment. Despite using a physically consistent damping rate, dissipation on its own leads to a completely wrong timescale of vortex decay. After t h 30 s the DPGPE still predicts too many vortices, with free vortices in the system still undergoing decay. From table 4.4 we see the DPGPE predicts that 3 quanta become pinned, which is less than the SPGPE predicts and does not correspond to how many become pinned experimentally (assuming an ideal experiment with no drop in vortex number as t h increases). Hence properly accounting for thermal fluctuations is vital in making quantitatively correct calculations in the high temperature regime. 4.7 Summary We have modeled the persistent current experiment performed at the University of Arizona [70] using the SPGPE and DPGPE. In doing so we have extended the SPGPE from the harmonically trapped Bose gas to the toroidal system, systematically finding suitable simulation parameters which match the experimental system. Hence we are able to perform the first calculations quantitatively modeling an experiment using the SPGPE without fitting any parameters, giving a true test of the theory. We find the SPGPE quantitatively agrees with experiment, accurately predicting the number of stable vortices in the toroid, as well as the timescale of the decay of the vortices. To the best of our knowledge, this is the first quantitative agreement with experiment seen with a theory of damping using no fitting parameters. DPGPE simulations show clearly the complex vortex dynamics that occur through the stirring process which nucleates vortices. However dissipation on its own leads to a much slower decay rate of the vortices than is seen experimentally, and less quanta are pinned compared to the SPGPE. This clearly shows the importance that thermal fluctuations have on correctly describing vortex dynamics in the high temperature regime [85]. The main limitation of these calculations is that the incoherent region in our formalism is non-rotating. The nucleation of vortices adds angular momentum to the system, which 60

71 Chapter 4. Persistent Current Formation 4.7. Summary in reality induces rotation of the thermal cloud. This rotation stabilizes vortices [33], thus the size of the persistent currents we observe may be altered with a full treatment of the incoherent region and its coupling to the coherent region. However this experiment is in the high temperature regime, so the angular momentum gained by the condensate will be insignificant in comparison to the size of the thermal cloud. Hence using a non-rotating thermal cloud is likely to be a good approximation of the actual system, which is justified by the quantitative agreement we see between the SPGPE and experiment. 61

72 62

73 Chapter 5 Kelvon-Induced Decay of Quantized Vortices 5.1 Introduction In this chapter we systematically investigate the effect that condensate geometry has on the decay of a quantized vortex in a finite temperature system using the stochastic projected Gross-Pitaevskii equation. We examine an initially centrally located vortex which is thermodynamically unstable in a non-rotating system, hence dissipation leads to vortex decay [82, 33]. Such decay has been observed experimentally by Anderson et al. [4], where they observed that the vortex spirals out of the condensate. The decay of a singly quantized vortex has proven to be a useful process to test finite temperature theory. A range of different work on this problem using different methods has been previously reported. Duine et al. [29] studied the dynamics of a straight line vortex using a variational approach to the stochastic Gross-Pitaevskii equation derived by Stoof [92, 93]. Single vortex decay has been simulated using the Zaremba-Nikuni-Griffin formalism [51], while Schmidt et al. [87] used a classical field formalism to study the dissipative dynamics of a vortex in a trapped Bose-Einstein condensate. Madarassy et al. studied vortex decay using a damped GPE with purely phenomenological damping [63]. C-field methods have been previously used to look at vortex dynamics using Hamiltonian classical field methods [101, 102, 103], and the SPGPE [85]. Theoretical studies of the effect of a thermal cloud on vortex dynamics are numerous, however there is a current lack of investigation into the effect condensate geometry has on vortex decay. It is known that like their classical counterparts, quantum vortices can 63

74 Chapter 5. Kelvon-Induced Decay of Quantized Vortices 5.1. Introduction support long-wavelength helical traveling waves known as Kelvin waves [78, 32]. If enough Kelvin modes are active, the combination of these helical waves cause the vortex to bend. For quantum vortices these are referred to as kelvons and have been predicted and experimentally investigated in Bose-Einstein condensation in the low temperature regime [32, 20]. Their kelvon excitation mechanism has been theoretically investigated [68], along with the microscopic dynamics of Kelvin waves in Bose-Einstein condensates [89], with both studies verifying the experimental observations of Bretin et al. [20]. In this chapter we use the SPGPE to investigate vortex decay in a variety of condensate geometries, ranging from spherical to the oblate 20:1 geometry, to study the effect vortex bending has on its decay at a variety of temperatures. The work of this chapter is in the same spirit as chapter 4, where we used the SPGPE to quantitatively model the persistent current experiment at the University of Arizona [70]. In this investigation of single vortex decay we also simulate realistic systems without any fitting parameters. We choose tractable parameters which make performing a full systematic study of the effect of temperature and condensate geometry feasible; in particular, we consider a smaller atom number (N 10 5 ). We show that the thermal activation of Kelvin waves is a dominant factor in the decay of a vortex for a finite temperature three-dimensional Bose-Einstein condensate. The underlying reason is that Kelvin waves cause the vortex to wobble and emit acoustic radiation, thus (a) (b) z x y Λ 1 Λ 1 Figure 5.1: Schematic of vortex (solid line) bending from the z-axis (dashed line) in a trapped BEC for (a) spherical trapping geometry (Λ = ω z /ω r 1), and (b) an oblate trap (Λ 1). The trap is assumed to have cylindrical symmetry about the z-axis (ω x = ω y ω r ). 64

75 Chapter 5. Kelvon-Induced Decay of Quantized Vortices 5.2. Procedure allowing the vortex to more effectively dissipate energy, and hence decay by moving out to the boundary of the Bose-Einstein condensate. As Kelvin waves are fundamentally threedimensional excitations of the vortex line, they may be suppressed by flattening the system (see figure 5.1). Beyond a certain oblateness we find that the kelvon mode contribution is frozen out and the vortex decay rate reduces to a geometrically invariant value, characterizing a regime of two-dimensional vortex dynamics. We also find that for tighter traps and higher temperatures the matter wave itself crosses over to being quasi-two-dimensional and we find a dynamical signature of the phase-fluctuating condensate: an anomalous increase in the rate of vortex decay. 5.2 Procedure Physical System In this chapter we look at the decay of a single vortex in a harmonically trapped condensate of 87 Rb atoms, with the external trapping potential given by V (x) = m 2 [ ω 2 r (x 2 + y 2 ) + ω 2 zz 2]. (5.1) We define Λ as the degree of trap oblateness ω z Λω r, (5.2) with ω r = Λ 1/3 ω, enforcing constant geometric mean frequency ω 3 ω 2 rω z for arbitrary oblateness. This choice defines a class of geometric transformations of the system that preserve key thermodynamic properties. In particular, for fixed total atom number N, the ideal gas transition temperature T 0 c (N, ω) = h ω k B [ζ(3)] 1/3 N 1/3, (5.3) is invariant under scaling of Λ. This constraint allows the comparison of a class of systems with the same reduced temperature and different geometries. 65

76 Chapter 5. Kelvon-Induced Decay of Quantized Vortices 5.2. Procedure Numerical Procedure We use the SPGPE (3.63) to simulate the decay of a single vortex. The SPGPE has been detailed in section 3.7. Here we outline how we determine appropriate simulation parameters, detail our procedure of creating initial vortex states, and summarize how we numerically detect the vortex. SPGPE Parameters We generate finite temperature equilibrium states using the SPGPE by evolving (3.63) with an appropriate choice of T, µ, and ɛ cut. The choice of µ is essential in keeping the total atom number, N (and hence critical temperature Tc 0 ), constant for different temperatures. Also ɛ cut must be chosen so the occupation of the highest mode in the C region, n cut, is significantly occupied. The validity of the SPGPE as a classical field theory requires this occupation criteria (as discussed in chapter 3). We find suitable SPGPE parameters, µ and ɛ cut which will give the desired N and n cut at temperature T, by using an efficient and simple mean-field approach described in detail in [85]. We treat the incoherent region in the same manner as in section 4.3.4, as this was based on a harmonic external trapping potential. Hence the determined T, µ, and ɛ cut are used to find the growth rate using equation (4.23), so all SPGPE parameters are found in a physically consistent manner prior to simulation. In our simulations we use initial vortex states prepared for a system containing Rb atoms in a trap of (fixed) geometric mean trap frequency ω = 2π 19.7s 1, (5.4) for which Tc 0 = 69.7nK. We look at a range of geometries from spherical to the highly oblate, and for various temperatures in the range 0.78Tc 0 T 0.93Tc 0. The values of chemical potential used to give the desired atom number, and energy cutoff giving a cutoff occupation of approximately 2, are shown in table 5.1. For each parameter set we calculate vortex decay properties by evolving trajectories of the SPGPE, with the number of trajectories required for convergence increasing with temperature [83]. 66

77 Chapter 5. Kelvon-Induced Decay of Quantized Vortices 5.2. Procedure T/Tc µ[ h ω] ɛ cut [ h ω] γ h/k B T [10 4 ] No. of trajectories Table 5.1: Simulation parameters used in SGPE simulations and the number of trajectories found for each temperature regime for a single dimensionality. The chemical potential parameters are chosen so that N T = , the the cutoff energies are chosen so that the cutoff occupation is approximately 2 for each temperature. We look at seven geometries with aspect ratios ranging from 1:1 to 20:1, with ω set by (5.4). As we look at multiple geometries, the total number of trajectories required becomes very large. Vortex State Preparation Firstly we follow the procedure in section to create a SPGPE equilibrium state. We evolve an initial Thomas-Fermi ground state (4.34) with the SPGPE (3.63) with the appropriate simulation parameters. Since equilibrium states are independent of γ we use a damping rate of γ = 0.3, which is larger than a typical value of 10 4 using (4.23), to decrease the time needed to reach equilibrium. This creates an equilibrium state of the SPGPE at temperature T, and chemical potential µ. We create a vortex state by imprinting a vortex phase on an equilibrium SPGPE state, ψ C (x) ψ C (x)e iθ(x), where ( y Θ(x) = arctan. (5.5) x) This creates a singly charged centrally located vortex line, which lies on the cylindrical symmetry axis of the trap (z-axis) at (x, y) = (0, 0). We do not find it necessary to imprint a vortex density profile because at the temperatures under consideration the extra energy of the core with phase-only imprinting is negligible compared with that from thermal fluctuations. As our description of the incoherent region is that of a non-rotating thermal reservoir, this phase imprinting method generates a non-equilibrium state. Vortex Detection To quantify the amount of vortex bending, we locate the planar vortex position (x v, y v ) as a function of z and t. To detect the vortex in a particular x y plane, we locate the radial position of the phase singularity by finding points of non-zero curl of the condensate velocity 67

78 Chapter 5. Kelvon-Induced Decay of Quantized Vortices 5.2. Procedure field in two dimensions. We define a complex vortex coordinate u(z, t) by u(z, t) = (x v (z, t) x(t)) + i(y v (z, t) ȳ(t)), (5.6) where the barred quantities correspond to the mean planar vortex position over the z range at a particular time t (see Fig. 5.2). u(z, t) measures the vortex line displacement from its instantaneous mean. Fluctuations in the phase at the condensate boundary reduce the region where vortex detection may be performed. We also have to account for the decay of the vortex. As time evolves and the vortex precesses toward the condensate boundary, the range of z-values containing the vortex reduces further. The magnitude of the fluctuations increases with temperature, so we must ensure our detection range is valid for all temperatures we consider. We thus limit the vortex detection to the range z = ±W, where W = 1R 2 TFz, which we have found to be the largest window which gives a well defined and unique curl signal over the vortex lifetime for the temperatures considered. α(z,t) u(z,t) 1 2 R TFz = W ᾱ(t) ū(t) Figure 5.2: Schematic showing the definition of the vortex coordinate u(z, t). The solid line represents the vortex line (aligned on the z-axis) at a certain time t. The dotted line represents ū(t) the mean vortex position over the z coordinate. We define u(z, t) by the vector from ū(t) to the vortex line (arrow). Due to boundary fluctuations, our detection region only extends to z = ± 1R 2 TFz = ±W, hence the total region of detection in the z coordinate is 2W. 68

79 Chapter 5. Kelvon-Induced Decay of Quantized Vortices 5.3. Results 5.3 Results Figure 5.3 shows isosurfaces of the c-field in a variety of condensate geometries at a temperature of T = 0.78Tc 0, after t = 0.61 s of SPGPE evolution. The phase imprinting technique leads to a straight vortex line along the z-axis, but SPGPE evolution causes vortex bending from thermal fluctuations. We look at the system after a sufficient amount of time evolution to ensure the vortex is fully thermalized, which occurs on a short timescale (typically within 0.1 s). We observe a significant amount of vortex bending in the spherical geometry. For flatter geometries, the magnitude of bending lessens and the vortex behaves more like a straight line. Λ =1 Λ =4 z (µm) z (µm) y (µm) x (µm) y (µm) x (µm) Λ =8 Λ = 20 z (µm) z (µm) y (µm) x (µm) y (µm) x (µm) Figure 5.3: Density isosurfaces of the c-field after t = 0.61 s of SPGPE evolution for a range of trap geometries. Red: the full density isosurface. Blue: isosurface restricted to the region of the vortex core. The system temperature is T = 0.78T 0 c. We see that the amount of vortex bending reduces so that the vortex becomes essentially a straight line as the condensate becomes flattened. 69

80 Chapter 5. Kelvon-Induced Decay of Quantized Vortices 5.3. Results Power Spectra To quantify the extent of vortex bending at a given time, we take the Fourier transform with respect to z (denoted by F z ) of u(z, t), and calculate the power spectrum P u (k, t) = F z [u(z, t)] 2, (5.7) where u(z, t) is found from (5.6) for an individual trajectory. Note that this measure of the power spectrum is insensitive to varicose waves, which arise from an axisymmetric perturbation propagating along the vortex line causing modulation of the vortex core diameter [90]. Thus this measure of the power spectrum is entirely due to vortex bending. Figure 5.4 shows the power spectrum for a range of geometries at t = 0.70 s (averaged over trajectories). We rescale the k axis by k, given by k = 2π/(2 W) = π/w, (5.8) which is the wave number corresponding to the largest visible wavelength due to our choice of window length. In all geometries we see the power spectra take a thermalized form, with maximum occupation of the long wavelength bending modes (low values of k) Effect of Geometry: Kelvon Power and Vortex Decay The effect of geometry on the importance of the kelvon modes is clearly visible in figure 5.4, where the power in the spectrum decreases significantly as the system becomes more oblate. It is useful to calculate the total power in the bending modes, P tot = P u (k, t)dk, (5.9) which by Parseval s theorem is related to the line-integrated deviation of the vortex from its mean position. The thermalization of the vortex line from the straight initial conditions is most directly monitored using the kelvon power spectrum. In figure 5.5 we plot the total power P tot, as a function of time for all geometries considered at a temperature of T = 0.81Tc 0. We find that the total power of the kelvon spectrum stabilizes, typically within 0.1 s which is much shorter than the vortex decay time (see figure 5.6). Thus the early transient dynamics have a negligible impact on our calculations of the vortex decay process. However for Λ = 1, we 70

81 Chapter 5. Kelvon-Induced Decay of Quantized Vortices 5.3. Results 2 Ptot /(2W ) Λ = 1 Ptot /(2W ) Λ = k/ k k/ k 0.04 Ptot /(2W ) Λ = k/ k Figure 5.4: Power spectra as a function of k z / k, for a range of geometries at a temperature of T = 0.78T 0 c and with N T = atoms. k = π/w corresponds to largest possible wavelength visible due to our window length. The spectra are calculated by averaging over trajectories after t = 0.70 s. 71

82 Chapter 5. Kelvon-Induced Decay of Quantized Vortices 5.3. Results Ptot /(2W ) Λ = 1 Λ = 2 Λ = 4 Λ = 8 Λ = 12 Λ = t (s) Figure 5.5: The total power per vortex length in vortex bending modes (5.9) as a function of time, at a temperature of T = 0.81Tc 0 for all condensate geometries considered. The vortex length is given by the window size of 2 W. The vortex line rapidly thermalizes in all geometries except for Λ = 1, for which the total power never reaches a stable value. find that the total power increases with time, never reaching a stable value. As our complex vortex coordinate (5.6) measures the vortex displacement relative to the z-axis, any tilting of the vortex off its original axis will make our total power measure invalid. For geometries flatter than Λ = 1, this is not an issue because it is energetically unfavorable for the vortex to increase its length by tilting off the z-axis. However in a spherical system there is no change in vortex length if any tilting occurs, hence we believe such tilting artificially increases the total power as the total power is measured relative to the z-axis in our numerical scheme. We clearly see the effect of geometry on vortex bending in Fig. 5.6 (a) which shows the time averaged total power as a function of trap geometry for a range of temperatures. We average over a time range where the total power is stable, typically in the range of t 0.5 s 72

83 Chapter 5. Kelvon-Induced Decay of Quantized Vortices 5.3. Results Ptot /(2W ) (a) T = 0.78T 0 c T = 0.81T 0 c T = 0.84T 0 c T = 0.86T 0 c (b) 1/ t (s 1 ) Figure 5.6: (a) Average total power per vortex length as a function of trap oblateness Λ. (b) Effective vortex decay rate (1/ t) as a function of Λ. The disconnected points for Λ = 20 break from the trend of vortex decay rate being independent of Λ (see section 5.3.3). Λ to t 3 s (see figure 5.5). Note we do not include the case of Λ = 1 as we do not have a well defined measure of total power in this case. The total power decreases exponentially as the geometry becomes more two-dimensional, beginning to asymptote gradually to zero. In general the total power is larger with increasing temperature, since the thermal fluctuations are increased inducing greater vortex bending. For all temperatures vortex bending is almost entirely absent in highly oblate systems with Λ 8. Figure 5.6(b) shows the effect of condensate geometry on the vortex lifetime, for a range of temperatures. The lifetime is quantified here in terms of the mean first exit time [85, 83], 73

84 Chapter 5. Kelvon-Induced Decay of Quantized Vortices 5.3. Results denoted by t. This is calculated as the ensemble average of the time it takes for the vortex to become indistinguishable from thermal fluctuations at the condensate boundary (in the plane z = 0). The effective vortex decay rate, given by 1/ t, is shown for a range of geometries in figure. 5.6(b). It decreases exponentially with increasing Λ, following the same trend as the total bending mode power (figure 5.6 (a)). However rather than decreasing to zero, the decay rate approaches a constant temperature dependent value, being almost independent of Λ for Λ > 8. This constant is determined by two dimensional vortex decay processes which do not involve vortex bending. Interestingly we see for the highest temperatures the decay rate increases at Λ = 20, but this does not correspond to an increase in bending mode power in figure 5.6 (a). Aside from this anomaly (discussed below in section 5.3.3), we infer a critical oblateness of Λ c 8 for the onset of two-dimensional vortex dynamics in this system Effect of Temperature: Anomalous Decay in Quasi-Two- Dimensional Systems The results of the previous section show that for the system we consider, the vortex enters a two-dimensional dynamics regime at Λ 8, associated with a strong suppression of vortex bending modes and a diminished decay rate. For temperatures in the range [ ]T 0 c in figure 5.7 this holds unchanged, except for the hottest cases which show an anomalous increase in decay rate as Λ increases. We interpret this revival of the decay rate as arising from the system entering a phase fluctuating quasi-two-dimensional regime. To further investigate the high temperature regime it is necessary to more precisely understand our temperature relative to T c. We correct the ideal gas critical temperature for finite size and interaction effects [43], given by T c(corr) = T 0 c + δt fs c + δt int c, (5.10) where Tc 0 is the noninteracting critical temperature for a trapped Bose gas, δtc fs is the first correction to the noninteracting critical temperature due to finite size effects, and δtc int is the correction due to interactions. These are of the form δt fs c = 0.73 ω ω N 1/3 T 0 c, (5.11) 74

85 Chapter 5. Kelvon-Induced Decay of Quantized Vortices 5.3. Results 1/ t (s 1 ) Λ = 1 Λ = 2 Λ = 4 Λ = 8 Λ = 12 Λ = 15 Λ = T /T c(corr) Figure 5.7: Vortex decay rate as a function of relative temperature, for a range of geometries. Here the critical temperature is adjusted to account for interactions and finite size effects (5.10). The highest decay rate (for Λ = 20, not shown in the figure) is at T/T c(corr) = 0.98, giving 1/ t = 3.9 s 1. and δt int c = 1.33 a s ā N 1/6 T 0 c, (5.12) where ω = 1 3 (2ω r + ω z ) is the arithmetic mean frequency, a s is the s-wave scattering length, and ā = ( h/m ω) 1/2. The corrections depend on Λ, since ω = 1 3 (2Λ 1/3 ω + Λ 2/3 ω) = ω 3 (2Λ 1/3 + Λ 2/3 ). In figure 5.7 we show the decay rates versus temperature relative to the corrected critical temperature (5.10). For the lowest temperatures of T 0.9T c(corr), we see the decay rate increasing with temperature and, for a given temperature, the decay rate is lower for a more oblate condensate. However for Λ 12, we see an anomalously rapid increase in the vortex decay rate for temperatures exceeding T 0.95T c(corr). For these cases the system 75

86 Chapter 5. Kelvon-Induced Decay of Quantized Vortices 5.4. Summary crosses over to being quasi-two-dimensional as the chemical potential becomes the order of the energy to activate degrees of freedom in the tightly confined direction, that is, µ hω z + ɛ 0 3 hω z /2, (5.13) where ɛ 0 is the single particle ground state energy. Note that µ is independent of Λ, as ω is held constant for all simulations. The particular values of µ for the highest temperature considered are shown in table 5.2, where for Λ > 8, we see that µ becomes comparable to 3 hω z /2, hence showing the system becomes quasi-two-dimensional in this regime. In the quasi-two-dimensional regime vortex dynamics are not well understood, but strong phase fluctuations will likely facilitate the rate of vortex decay we observe. We note that the system still has a phase space density sufficiently high that we are not in the Berezinskii-Kosterlitz-Thouless (BKT) regime, so spontaneous vortices are not expected to be significant [45, 88, 10]. This is shown in table 5.2, where we give the phase space density for a range of system geometries at the highest temperature. The critical phase space density required for a two dimensional Bose gas to undergo the BKT transition is given by [79, 10] n BKT λ 2 db = ln ( ) C, (5.14) ḡ where λ db is the thermal de Broglie wavelength, C is a numerical factor given by C 380 [79], and ḡ is a dimensionless interaction strength given by ḡ = 8π a a z, (5.15) where a is the three-dimensional s-wave scattering length, and a z is the harmonic oscillator length (a z = h/mω z ). In table 5.2 we determine the phase space density by calculating the peak density in our system, while excluding any contribution from excited modes in the z direction. This gives a two dimensional density, which is higher in all cases than the factor ln(380/ḡ) which we find to be approximately 9 in all cases. Hence our system does not lie in the BKT regime. 5.4 Summary We have performed a systematic quantitative investigation of the effect condensate geometry has on the decay of a single vortex in the high temperature regime using the SPGPE. We 76

87 Chapter 5. Kelvon-Induced Decay of Quantized Vortices 5.4. Summary Λ µ/ hω z n 2D λ 2 db Table 5.2: Parameters describing the two-dimensionality of the system for a given oblateness at the highest temperature considered of T = 0.93T 0 c. In the quasi-two-dimensional regime, µ/ hω z 3/2, as in (5.13), thus for Λ < 8 the system is in the crossover to the quasi-twodimensional regime. We also show the peak phase space density, which shows we are not in the BKT regime (see text). find that for spherical systems, the vortex is freely deformed from an initial straight-line configuration by the thermal activation of vortex bending modes. This effect diminishes as the condensate is flattened, and bending modes become energetically inaccessible. We give a quantitative measure of the amount of vortex bending, by calculating the power per unit length in the bending modes. We find the total bending mode power decreases exponentially as the condensate becomes increasingly oblate. We calculated the vortex lifetime to find an effective vortex decay rate, which decreases with increasing oblateness similarly to the total bending mode power. The vortex decay rate reaches a constant, temperature dependent value for Λ > Λ c 8. Thus we identify Λ c as the critical oblateness required for the system to enter the regime of two-dimensional vortex dynamics, where vortex bending is inhibited and can not enhance vortex decay. These observations in general hold true for higher temperatures except, in what we refer to as the critical regime, where critical fluctuations associated with the phase transition become significant. In this regime we find that the vortex decay rate is highly sensitive to temperature and geometry, where we see an anomalously large increase in the vortex decay rate for geometries with Λ c Λ, and 0.95T c(corr) T. We associate this anomalous decay with phase-fluctuations due to the system entering the quasi-two-dimensional regime, and thus as a signature of criticality. 77

88 78

89 Chapter 6 Vortex Dipole Decay 6.1 Introduction The final application of the SPGPE we look at is the decay of a vortex dipole. The physics of vortex dipole dynamics in a trapped Bose-Einstein condensate is very rich due to the vortexantivortex interactions. At zero temperature a single vortex will precess about the center of the trap at constant radius [82, 94]. At finite temperature, it will move toward the condensate boundary where there is no atomic density [85]. This motion, caused by dissipation and the inhomogeneous nature of a trapped Bose-Einstein condensate, also drives the dynamics of the vortex dipole. However the situation is complicated for a vortex dipole, as the vortex and antivortex drive each other in a direction perpendicular to the line joining them [31, 53]. There have been a number of previous theoretical investigations of vortex dipoles in trapped Bose-Einstein condensates. Stationary vortex dipole states have been investigated [22, 69]. This work was extended by Pietilä et al. [77] where they considered the stability and dynamics of vortex clusters using a Bogoliubov approach. Individual vortex dipole trajectories have been calculated with a Gross-Pitaevskii treatment [55, 62]. The energetics of a vortex dipole have been considered by Zhou and Zhai [105], where they determine the angular momentum and energy of a vortex dipole in a trapped Bose-Einstein condensate analytically. However there is a current lack of understanding of the dynamics and decay of a vortex dipole in a dissipative system. In this chapter we numerically study such decay using the SPGPE. In a recent experiment by Neely et al. [71], they observe the formation of vortex dipoles and measure their lifetime. Their experiment is at finite temperature, with T 0.6T c, so lies in the high temperature regime applicable to the SPGPE. We model this experiment and 79

90 Chapter 6. Vortex Dipole Decay 6.2. Procedure compare the lifetime of a vortex dipole from SPGPE simulations with experiment, providing another good test of the SPGPE theory. In the experiment by Neely et al., their condensate is oblate (Λ = ω z /ω r 11) and thus appears to be in the two-dimensional vortex dynamics regime found in chapter 5. The vortex dipole annihilation is suppressed since in such a flattened system, vortex bending and vortex interactions are suppressed. We compare our simulation results from the twodimensional vortex regime with simulations calculated in a spherical system, which provides a good comparison with the results of chapter Procedure The general methods used in this chapter are essentially the same as used in chapter 5, except the system parameters are chosen so that the temperature and total atom number in our simulations match the experiment of Neely et al. [71]. We briefly describe our system and numerical procedure here Physical System As in chapter 5, we have a harmonically trapped condensate of 87 Rb atoms, with the external trapping potential given by V (x) = m 2 [ ω 2 r (x 2 + y 2 ) + ω 2 zz 2], (6.1) where again we define Λ as the degree of trap oblateness, ω z Λω r Numerical Procedure In this chapter we use the SPGPE (3.63), as described in section 3.7, to simulate vortex dipole decay. Here we outline how we determine appropriate simulation parameters, and detail our procedure of creating initial vortex dipole states. Simulation Parameters In this chapter we perform SPGPE simulations based on the same system used in the experiment of Neely et al. [71]. In this experiment, they have an oblate system with harmonic 80

91 Chapter 6. Vortex Dipole Decay 6.3. Results trap parameters of (ω r, ω z ) = 2π (8, 90) Hz, thus Λ = The system has a total atom number of N T atoms, a temperature of T 52 nk. We use the same procedure as in chapter 5 to find SPGPE parameters to give states with the correct atom number to match the experimental system. We set ω = 2π(8 2 90) 1/3 Hz = Hz, T = 52 nk, and using the methods of [85] find that a chemical potential and energy cutoff of (µ, ɛ cut ) = (37 h ω, 71 h ω). These parameters give an appropriate occupation of the highest energy modes ( 2 atoms), and the desired atom number of atoms. From these parameters the dimensionless damping rate of hγ/k B T = can be found using equation (4.23). Vortex Dipole State Preparation Again we follow a similar procedure as in chapter 5 to generate a vortex dipole state. We first create a SPGPE equilibrium state as seen in chapters 4 and 5, by evolving a Thomas-Fermi initial state (4.34) with the SPGPE (3.63) until the state has equilibrated. We then create a vortex state by imprinting a vortex dipole phase pattern on an equilibrium SPGPE state, ψ C (x) ψ C (x)e iθ(x), where now ( ) ( ) y y+ y y Θ(x) = arctan arctan. (6.2) x x This creates an axially aligned vortex dipole with positive and negative vortices located at (x, y ± ) = (0, ±s/2), where s is defined as the initial separation between the vortex dipole. 6.3 Results Here we present results of the SPGPE evolution of the decay of a vortex dipole in a finite temperature Bose-Einstein condensate. We look at two particular cases, consisting of vortex dipole decay in a spherical condensate, and in an oblate condensate with Λ Decay in a Spherical Condensate We first consider dipole decay in a spherical trap. We use the SPGPE parameters as given in section 6.2.2, where we set Λ = 1 so ω z = ω r, while keeping ω fixed. As ω is held constant, these parameters give a system with the same atom number and temperature as the oblate configuration considered in the experiment of Neely et al. [71]. Thus these calculations in a spherical system serve as a benchmark for the effect of geometry on vortex dynamics in the 81

92 Chapter 6. Vortex Dipole Decay x (µm) y (µm) x (µm) y (µm) x (µm) z (µm) x (µm) y (µm) t = s z (µm) t = s z (µm) z (µm) t = s x (µm) y (µm) t = s x (µm) y (µm) y (µm) t = s z (µm) x (µm) t = s z (µm) t = s z (µm) z (µm) t = s 6.3. Results y (µm) x (µm) y (µm) Figure 6.1: Isosurfaces of dipole decay in a spherical condensate, with an initial dipole separation of 3 µm. Red: the c-field isosurface. Blue: isosurface restricted to the region of the vortex cores. Vortex decay in the three dimensional regime leads to complex dynamics, where we see decay via vortex ring formation. system, where we are clearly in a regime of three-dimensional vortex dynamics (as discussed in chapter 5. Figure 6.1 shows the decay sequence for a vortex dipole in a spherical trap, where the vortex dipole is phase imprinted with an initial separation of s = 3 µm. The phase imprinted vortices are initially axially aligned with the z-axis, and then rapidly curve as the vortices precess such that the vortex cores are normal to the condensate surface. This long-wavelength bending leads to formation of a vortex ring, which then subsequently dissociates into two vortices at t = 0.5s. As the vortices closely approach again at t = 0.62s, both loops and rings form, reducing the total vortex length and leading to total decay in less than 1s. We thus observe rapid vortex dipole decay via intermediate vortex ring and loop formation enabled by the system lying in the three-dimensional vortex dynamics regime Decay in an Oblate Condensate Here we use the SPGPE to model the dynamics of a charge-one vortex dipole in an oblate Bose-Einstein condensate, as in the experiment of Neely et al. [71]. We use the SPGPE pa82

93 Chapter 6. Vortex Dipole Decay 6.3. Results rameters as described in section to give a system with atoms at temperature of 52 nk, where we use the harmonic trap parameters of the experiment (ω r, ω z ) = 2π (8, 90) Hz. Since in this case Λ = exceeds the critical oblateness for two-dimensional vortex dynamics of Λ c 8 as found in chapter 5, we can expect this system and the corresponding experimental regime to lie in the two dimensional vortex dynamics regime. Experimentally, a single vortex dipole is nucleated by moving the Bose-Einstein condensate past an obstacle above the critical velocity of the system [71]. To make a quantitative comparison with this experiment we need to be able to compare the distance s between the two vortices upon the creation of the vortex dipole in both experiment and simulations, and to relate s to the vortex dipole lifetime in each case. However determining s accurately in experiment is not possible. The experimental spatial resolution was approximately 6µm, which is very close to the vortex separation calculated in Gross-Pitaevskii simulations of the dipole formation dynamics [71]. Thus we can only place an upper bound on s since the vortices are not individually resolvable immediately after nucleation experimentally, meaning we can not make a quantitative comparison between the SPGPE and experiment with no fitting parameters. We can make a comparison with theory by systematically investigating the dependence of the vortex dipole lifetime on s, and comparing the SPGPE dynamics for a range of initial vortex dipole separations with the experimental lifetimes. To do this we phase imprint a vortex dipole into the Bose-Einstein condensate with initial separations of s = 1, 2, 2.5, 3, and 3.5 µm. Note that the condensate radius is 40µm, so these separations are small relative to the system size, but larger than the healing length ξ = 0.2µm. Qualitatively, our simulations differ from the experiment in several ways: in our simulations we have two vortices initially, a high degree of cylindrical and mirror symmetry, and a well defined total atom number. In the experiment there is occasionally a vortex present from the BEC formation process [98], an irreducible (albeit very small) cylindrical asymmetry of the harmonic potential, a mirror asymmetry of the stirring potential, and uncertainty in the total atom number. We do not attempt to model all of this complexity here, confining our study to the basic decay process of internal annihilation in the symmetric system, with fixed total atom number. Figure 6.2 shows the mean vortex number as a function of time from the experimental data [71], compared with the results of our simulations. In our simulations we always find that 83

94 Chapter 6. Vortex Dipole Decay 6.3. Results Average number of vortices µm 2µm 3µm 3.5µm t(s) Figure 6.2: Average number of vortices as a function of time during vortex dipole decay for numerical simulations and experimental data. The dashed lines show SPGPE results for a range of initial vortex dipole separations. Circles show experimental data of reference [71]. The blue diamond at t = 0 s shows the average number of vortices observed in the experimental initial state due to spontaneous vortex formation [98]. the vortices mutually annihilate each other in the center of the BEC, rather than damping at the boundary. The vortices move rapidly when closely approaching each other, nearing the speed of sound c = 0.2cm/s. Consequently most of the orbital time is spent at the outer edge of the BEC, and the time interval in which annihilation may occur is a small fraction of the orbital period ( 10%). Thus the vortex number has an abrupt time dependence for a given initial separation, with characteristic timescale given by the orbital period of the vortex dipole, found numerically and experimentally to be 1.2s and independent of s over a wide range [71]. In general, the s value selects a given orbit of annihilation, within which there is very small variation of lifetime. A summary of the lifetimes measured is given in Table 6.1 for all separations considered. The orbit of annihilation corresponds to which orbit number the dipole is on when it is annihilated, while the lifetime is found from an average of SPGPE trajectories. As the temperature of this system is relatively cold (T/T c 0.6), a minimal number of trajectories are required. We see the number of orbits of the dipole is extremely sensitive to the 84

95 Chapter 6. Vortex Dipole Decay 6.4. Summary separation (µm) orbit of annihilation mean lifetime (s) Table 6.1: Comparison of the orbit of annihilation and the mean vortex dipole lifetime with the initial dipole separation. initial separation, even though this change in separation is small relative to the size of the condensate. The separation of s = 1 µm gives a very small lifetime as the vortex dipole annihilates immediately after creation. The separations s = 2.5µm and s = 3µm lead to almost identical lifetimes of about 2s, corresponding to the significant drop in vortex number seen experimentally for 2s t 2.5s, which we interpret as an experimental indication of the vortex dipole lifetime. In figure 6.3 we plot a time series of simulation data for s = 3µm showing internal annihilation at the end of the second orbit. Interestingly, we observe a tilt in the axis of dipole precession which are also observed in experimental absorption images. 6.4 Summary We have quantitatively simulated the dynamics of the decay of a vortex dipole using the SPGPE. We modeled the experiment of Neely et al. [71], finding that the approximate decay time of a vortex dipole in the experiment ( 2.25 s) is consistent with SPGPE calculations with an initial vortex dipole separation of s µm. We also find the approximate orbit time of the vortex dipole ( 1.2 s) agrees well with experiment. Although the SPGPE gives a vortex decay time consistent with experiment this does not show the same strict quantitative agreement as found in chapter 4, as we have used the dipole separation s as a fitting parameter. To avoid this, improved experimental resolution would be needed. The simulations in the experimental regime are in a geometry (Λ = 11.25) which was found to be in the two-dimensional vortex dynamics regime for closely related parameters in chapter 5. We also determined the SPGPE evolution of a vortex dipole in a spherical system which exhibits three-dimensional vortex dynamics, finding a much quicker decay times for s = 3 µm due to the formation of vortex rings and loops. Note that such bent geometries 85

96 Chapter 6. Vortex Dipole Decay 6.4. Summary y (µm) y (µm) t = 0s x (µm) t =1.67s x (µm) t =2.04s x (µm) t =2.08s x (µm) 1 0 π 0 π Figure 6.3: Column densities and phase slices showing the SPGPE evolution of a vortex dipole initially separated by 3 µm. Subsequent snapshots show the evolution of the vortex dipole on its final orbit, where we observe internal annihilation of the dipole towards the completion of its final orbit. would not be readily visible using the experimental methods of Neely et al. [71]. We thus infer from the experiment of Neely et al. [71], and from the lack of vortex ring and loop formation in the simulations of the oblate system, that the experimental system consisting of a Bose-Einstein condensate with oblateness Λ = is in the two-dimensional vortex dynamics regime. This is consistent with our results from chapter 5, validating that a critical oblateness of Λ c 8 corresponds to the onset of two dimensional vortex dynamics. 86