Research in the field of ultra-cold quantum gases provides a testbed to examine Hamiltonians from other areas of physics such as the Bose-Hubbard Hamiltonian, as well as interesting phenomena such as superconductivity and superfluidity. Of emerging interest in the field is the analysis of non-equilibrium behaviour. This thesis focuses on the non-equilibrium dynamics of Bose gases in two different elongated systems: a continuously-pumped atom laser, and the formation of solitons as a result of a rapid quench through the Bose-Einstein condensation phase transition. We begin by introducing the theoretical tools required to model these systems, including the Stochastic Projected Gross-Pitaevskii formalism that describes the dynamics of a condensate in contact with a thermal cloud. We apply the Stochastic Projected Gross-Pitaevskii formalism to model a Raman-outcoupled atom laser continuously outcoupled from a replenished condensate reservoir at finite temperature. We investigate the first and second order coherence of the output as a function of the temperature of the condensate reservoir. For large momentum kicks given to outcoupled atoms, we show for the first time that the coherence of the atom laser beam is linked to that of the condensate reservoir. We observe a decrease in first order coherence of the condensate reservoir as its temperature is increased, and show that this decrease is caused by the dephasing of thermal fluctuations. We also investigate the effects of varying the outcoupling rate and the momentum kick on the first order coherence of the beam. We demonstrate that the first order coherence of the atom laser beam deviates from that of the condensate reservoir for small momentum kicks which we identify as due to interactions between the condensate and atom laser beam. Thermal density fluctuations in the condensate cause the atom laser beam to experience a stochastic force and the temporal coherence of the atom laser beam is reduced. We find that the condensate reservoir is significantly depleted for large outcoupling rates, resulting in a reduction in the steady state condensate fraction and in the first order coherence of the atom laser beam. In a second study, the momentum linewidth of a guided atom laser outcoupled using a radio-frequency transition is investigated at zero and finite temperature. We examine the momentum distribution of the beam by measuring the transmission of atoms over a potential barrier following the experimental scheme of the group of Professor Alain Aspect at Institut d'Optique. We observe a noisy momentum distribution for outcoupling rates in the weak-outcoupling limit at zero temperature. We show that the noise is caused by bulk excitations within the condensate reservoir as a result of a density depression caused by the non-adiabatic switch-on of outcoupling. We show that these excitations can be prevented by increasing the outcoupling rate to its final value linearly over time. In addition, we demonstrate that a thermally-broadened linewidth could potentially be measured using this experimental technique, and discuss challenges associated with the measurement scheme. Finally, we focus on the non-equilibrium dynamics of condensate growth in elongated geometries. Following a rapid quench through the Bose-Einstein condensation phase transition, we observe the spontaneous appearance of density dips which we identify as solitons. We develop an algorithm for counting these solitons in order to observe the temporal dependence of the soliton number as a function of the quench rate. To compare with the predictions of the Kibble-Zurek mechanism describing topological defect formation as a result of the dynamics of phase transitions in homogeneous systems, we calculated the scaling exponent of the soliton number with respect to the quench rate. Our results were qualitatively consistent with the Kibble-Zurek predictions, but quantitative comparisons were inconclusive. In trapped systems the phase transition proceeds inhomogeneously, with a phase front that propagates in space with a velocity. If this is lower than a critical value it has been predicted that this can inhibit defect formation. However, our results do not show conclusive evidence of this in the parameter regime that we study. We find that the number of solitons reduces as the condensate reaches equilibrium as a result of the damping included in the Stochastic Projected Gross-Pitaevskii formalism.
Research in the field of ultra-cold quantum gases provides a testbed to examine Hamiltonians from other areas of physics such as the Bose-Hubbard Hamiltonian, as well as interesting phenomena such as superconductivity and superfluidity. Of emerging interest in the field is the analysis of non-equilibrium behaviour. This thesis focuses on the non-equilibrium dynamics of Bose gases in two different elongated systems: a continuously-pumped atom laser, and the formation of solitons as a result of a rapid quench through the Bose-Einstein condensation phase transition. We begin by introducing the theoretical tools required to model these systems, including the Stochastic Projected Gross-Pitaevskii formalism that describes the dynamics of a condensate in contact with a thermal cloud. We apply the Stochastic Projected Gross-Pitaevskii formalism to model a Raman-outcoupled atom laser continuously outcoupled from a replenished condensate reservoir at finite temperature. We investigate the first and second order coherence of the output as a function of the temperature of the condensate reservoir. For large momentum kicks given to outcoupled atoms, we show for the first time that the coherence of the atom laser beam is linked to that of the condensate reservoir. We observe a decrease in first order coherence of the condensate reservoir as its temperature is increased, and show that this decrease is caused by the dephasing of thermal fluctuations. We also investigate the effects of varying the outcoupling rate and the momentum kick on the first order coherence of the beam. We demonstrate that the first order coherence of the atom laser beam deviates from that of the condensate reservoir for small momentum kicks which we identify as due to interactions between the condensate and atom laser beam. Thermal density fluctuations in the condensate cause the atom laser beam to experience a stochastic force and the temporal coherence of the atom laser beam is reduced. We find that the condensate reservoir is significantly depleted for large outcoupling rates, resulting in a reduction in the steady state condensate fraction and in the first order coherence of the atom laser beam. In a second study, the momentum linewidth of a guided atom laser outcoupled using a radio-frequency transition is investigated at zero and finite temperature. We examine the momentum distribution of the beam by measuring the transmission of atoms over a potential barrier following the experimental scheme of the group of Professor Alain Aspect at Institut d'Optique. We observe a noisy momentum distribution for outcoupling rates in the weak-outcoupling limit at zero temperature. We show that the noise is caused by bulk excitations within the condensate reservoir as a result of a density depression caused by the non-adiabatic switch-on of outcoupling. We show that these excitations can be prevented by increasing the outcoupling rate to its final value linearly over time. In addition, we demonstrate that a thermally-broadened linewidth could potentially be measured using this experimental technique, and discuss challenges associated with the measurement scheme. Finally, we focus on the non-equilibrium dynamics of condensate growth in elongated geometries. Following a rapid quench through the Bose-Einstein condensation phase transition, we observe the spontaneous appearance of density dips which we identify as solitons. We develop an algorithm for counting these solitons in order to observe the temporal dependence of the soliton number as a function of the quench rate. To compare with the predictions of the Kibble-Zurek mechanism describing topological defect formation as a result of the dynamics of phase transitions in homogeneous systems, we calculated the scaling exponent of the soliton number with respect to the quench rate. Our results were qualitatively consistent with the Kibble-Zurek predictions, but quantitative comparisons were inconclusive. In trapped systems the phase transition proceeds inhomogeneously, with a phase front that propagates in space with a velocity. If this is lower than a critical value it has been predicted that this can inhibit defect formation. However, our results do not show conclusive evidence of this in the parameter regime that we study. We find that the number of solitons reduces as the condensate reaches equilibrium as a result of the damping included in the Stochastic Projected Gross-Pitaevskii formalism.
When a gas of bosonic atoms is cooled sufficiently near to absolute zero, it undergoes a startling transition from a collection of randomly-behaved individual atoms to a phase in which thousands, even millions, of atoms act in collective unison. In this new phase, known as a Bose-Einstein condensate (BEC), the atoms all occupy the exact same quantum mechanical ground state and can be collectively described by a single phase-coherent wavefunction of macroscopic size. But how exactly does this harmonious behaviour arise within a collection of incoherent atoms as their temperature decreases? Does the condensate wavefunction grow from a single central nucleus, or does it emerge simultaneously throughout the system? And how does the situation differ when the gas is cooled suddenly, away from thermodynamic equilibrium? Can non-equilibrium behaviour be inherited by the condensate that forms? In this thesis we aim to answer these and other questions by studying the formation dynamics of Bose-Einstein condensates and the development of phase coherence. The research described herein makes use of several theoretical tools, which we lay out on the workbench in Chapter 2: Semi-classical Hartree-Fock theory provides a mean-field description of a condensate and thermal cloud at equilibrium; Quantum kinetic theory is used to calculate the dynamics of condensate growth into an equilibrium state, from a non-equilibrium thermal cloud; The stochastic projected Gross-Pitaevskii equation (SPGPE) employs a classical-field description to encapsulate the non-equilibrium dynamics of the condensate and low-energy region of the atom cloud, coupled to an equilibrium thermal cloud. This thesis is composed of four principal studies, each constituting a separate chapter.In Chapter 3 we study the formation of a Bose-Einstein condensate in a cigar-shaped 3D harmonic trap, induced by the controlled addition of an attractive ``dimple'' potential along the weak axis in a recent experiment [Garrett et al., Phys. Rev. A 83, 013630 (2011)]. In this manner condensation is induced without cooling due to a localized increase in the phase space density. We perform a quantitative analysis of the thermodynamic transformation in both the sudden and adiabatic regimes for a range of dimple widths and depths. We find good agreement with equilibrium calculations based on self-consistent semi-classical Hartree-Fock theory describing the condensate and thermal cloud. We observe there is an optimal dimple depth that results in a maximum in the condensate fraction. We also study the non-equilibrium dynamics of condensate formation in the sudden turn-on regime, finding good agreement for the observed time dependence of the condensate fraction with calculations based on quantum kinetic theory. In Chapter 4 we study the onset of phase coherence during the formation of a BEC following a sudden quench to below the transition temperature. Using the SPGPE, we simulate a recent experiment performed at ANU in which a sudden RF quench is applied to a dilute gas of metastable Helium atoms confined in an elongated 3D harmonic magnetic trap. At regularly-spaced intervals following the quench, broadband RF pulses are applied to homogeneously outcouple small (<5%) pulses of atoms, which expand and fall to a microchannel plate detector. The second-order correlation function at the detector, postulated to provide a measure of phase coherence of the in-situ cloud from which they were outcoupled, is measured versus time and compared with the condensate growth curve. A delay was observed in the relaxation of the second-order correlation function, relative to the peak in the number of condensate atoms. In our simulations we incorporate the effects of periodic outcoupling both the condensate and thermal cloud, and monitor the time-dependence of first- and second-order correlation functions in both position and momentum space. We observe long-range phase coherence in the final equilibrium state, though mild persistent phase fluctuations are present. We also observe the curious phenomenon of ``superbunching'' - an enhancement of density fluctuations - at specific values of the momentum, which we relate to anomalous correlations of fluctuations. Despite obtaining a good quantitative fit to the experimentally-measured condensate growth curve in our SPGPE simulations, we do not observe any evidence of the delay in coherence observed in the experiment, which remains unexplained. In Chapter 5 we use the SPGPE to study the crossover from a true BEC with long-range phase coherence to a quasicondensate dominated by phase fluctuations, for the case of a Bose gas in an highly-elongated 3D harmonic trap. We use parameters corresponding to an experiment in which a sudden quench resulted in the formation of a non-equilibrium condensate, the dynamics of which we study in the chapter that follows. Sweeping through a range of temperatures and chemical potentials within the crossover region, we reveal the behaviour of phase fluctuations through analysis of correlation functions. We compare the ideal-gas approximation for the transition temperature with the more accurate value obtained via the Binder cumulant. By comparing the phase coherence length with the spatial extent across which density fluctuations are suppressed, we identify the BEC/quasicondensate crossover point. We compare directly with predictions of mean-field theory [Petrov et al., Phys. Rev. Lett. 87, 050404 (2001)] and find that phase fluctuations persist to lower temperatures in our SPGPE simulations. We calculate anomalous correlations of the fluctuation field and show that the resulting correction to the mean-field condensate self-interaction energy is significant within the quasicondensate regime. In Chapter 6 we study the formation of a BEC into a non-equilibrium state by simulating an experiment in which a dilute gas of Rubidium-87 atoms in a highly-elongated 3D harmonic magnetic trap is rapidly quenched to below the transition temperature. Following the quench, a condensate is observed to form with an excessively elongated shape, and subsequently undergo damped quadrupole shape oscillations [Shvarchuck et al., Phys. Rev. Lett. 89, 270404 (2002)]. We simulate this experiment by modifying the SPGPE to incorporate a hydrodynamic thermal cloud, parametrized by axially and temporally-dependent temperature and chemical potential. Comparison against simulations without this axial dependence suggests the hydrodynamic behaviour of the thermal cloud plays a crucial role in the formation of a non-equilibrium condensate. Analyses of the first-order correlation function and the condensate wavefunction reveal the spontaneous formation of quantum vortices, as well as strong phase fluctuations at equilibrium indicating quasicondensate behaviour.
In this Thesis we study steady state solutions and dynamical evolutions of two– component atomic Bose–Einstein Condensates. We initially investigate the equilibrium properties of condensate mixtures in harmonic trapping potentials at zero temperature. Subsequently we simulate the coupled growth of these condensates by inclusion of damping terms. Finally, we investigate the evolution of coupled Bose gases via the so-called classical–field method. A recent experiment [D. J. McCarron et al., Phys. Rev. A, 84, 011603(R) (2011)] achieved Bose–Einstein Condensation of a two–species 87Rb–133Cs phase segregated mixture in harmonic trapping potentials. Depending on relative atom numbers of the two species, three distinct regimes of density distributions were observed. For these experimental parameters, we investigate the corresponding time–independent ground state solutions through numerical simulations of the coupled Gross–Pitaevskii equations. By including experimentally relevant shifts between the traps, we observe a range of structures including ‘ball and shell’ formations and axially/radially separated states. These are found to be very sensitive to the trap shifts. For all three experimental regimes, our numerical simulations reveal good qualitative agreement. The observed experimental profiles cannot be guaranteed to be fully equilibrated. This, coupled with the rapid sympathetic cooling of the experimental system, leads to a situation where growth may play a determining factor in the density structures formed. To investigate this further, we introduce phenomenological damping to describe the associated condensate growth/decay, revealing a range of transient structures. However, such a model always predicts the predominance of one condensate species over longer evolution times. Work undertaken by collaborators with the more elaborate Stochastic Projected Gross–Pitaevskii equations, which can describe condensate formation by coupling to a heat bath, predicts the spontaneous formation of dark–bright solitons. Motivated by this, we show how the presence of solitons can affect the condensate distribution, thus highlighting the overall dynamical role in the emerging patterns. Finally, we use classical field methods to analyse the evolution of non trapped Bose gases from strongly nonequilibrium initial distributions. The contrast between miscible (overlapping) and immiscible (phase segregated) components gives rise to important distinctions for condensate fractions and the formation of domains and vortices. In addition, splitting the particles of a single component thermalised state into two components is investigated. We then study the effects of suddenly quenching the strength of the interspecies interactions. Under suitable conditions, this quench generates isotropic vortex tangles. While this tangle subsequently decays over time, we propose how a repeat sequence of quenches at regular intervals could be employed to drive the tangle, thereby potentially providing a novel route to the generation of quantum turbulence.
A continuous wave (cw) atom laser will most likely be realised by outcoupling from a reservoir containing a Bose-Einstein condensate at finite temperature. This will mean that the correlation properties and linewidth of the cw atom laser beam will be influenced by the thermal nature of the reservoir. Here we report on simulations using the stochastic Gross-Pitaevskii formalism for finite temperature dynamics to find the steady state properties of the outcoupled beam. We determine how the linewidth and correlation length depend on the condensate size, temperature, and outcoupling properties, and determine the optimum parameters for cw atom laser operation.
The problem of understanding how a coherent, macroscopic Bose- Einstein condensate (BEC) emerges from the cooling of a thermal Bose gas has attracted significant theoretical and experimental interest over several decades. The pioneering achievement of BEC in weakly interacting dilute atomic gases in 1995 was followed by a number of experimental studies examining the growth of the BEC number, as well as the development of its coherence. More recently, there has been interest in connecting such experiments to universal aspects of nonequilibrium phase transitions, in terms of both static and dynamical critical exponents. Here, the spontaneous formation of topological structures such as vortices and solitons in quenched cold-atom experiments has enabled the verification of the Kibble-Zurek mechanism predicting the density of topological defects in continuous phase transitions, first proposed in the context of the evolution of the early universe. This chapter reviews progress in the understanding of BEC formation and discusses open questions and future research directions in the dynamics of phase transitions in quantum gases. Introduction The equilibrium phase diagram of the dilute Bose gas exhibits a continuous phase transition between condensed and noncondensed phases. The order parameter characteristic of the condensed phase vanishes above some critical temperature T c and grows continuously with decreasing temperature below this critical point. However, the dynamical process of condensate formation has proved to be a challenging phenomenon to address both theoretically and experimentally. This formation process is a crucial aspect of Bose systems and of direct relevance to all condensates discussed in this book, despite their evident system-specific properties. Important questions leading to intense discussions in the early literature include the time scale for condensate formation and the role of inhomogeneities and finite-size effects in “closed” systems. These issues are related to the concept of spontaneous symmetry breaking, its causes, and implications for physical systems (see, for example, Chapter 5 by Snoke and Daley). In this chapter, we give an overview of the dynamics of condensate formation and describe the present understanding provided by increasingly well-controlled cold-atom experiments and corresponding theoretical advances over the past twenty years. We focus on the growth of BECs in cooled Bose gases, which, from a theoretical standpoint, requires a suitable nonequilibrium formalism.
This thesis presents experiments performed to investigate the non-equilibrium dynamics of ultracold bosons. In depth understanding of the behaviour of quantum systems when taken out of equilibrium is of particular interest, especially for the production of quantum computers where thermal effects cause undesirable behaviours. We present our experiments on the thermalization of a87Rb Bose-Einstein condensate taken out of equilibrium by a partial Bragg pulse. We measure the rate of thermalization of the system and measure the temperature of the thermalized state, and compare to numerical simulations. We observe the formation of grey solitons in the thermalized state consistent with the Kibble-Zurek mechanism. We also present experiments on artificial gauge potentials with Bose-Einstein condensates. We benchmark the experiments with a uniform synthetic potential and extend to spin-orbit coupled systems. We take the spin-orbit coupled BEC out of equilibrium through the application of a synthetic electric field and measure the thermalization process of the system. We measure the rate at which the system reaches equilibrium as a function of the final Raman coupling strength, and measure the temperature of the thermalized states. We observe the system thermalize into a spin-orbit coupled state with a the final temperature decreasing with increasing Raman coupling.
Phase transitions abound in the physical world, from the subatomic length scales of quark condensation to the decoupling forces in the early universe. In the Bose-Einstein condensation phase transition, a gas of trapped bosonic atoms is cooled to a critical temperature. Below this temperature, a macroscopic number of atoms suddenly starts to occupy a single quantum state; these atoms comprise the Bose-Einstein condensate (BEC). The dynamics of the BEC phase transition are the focus of this dissertation and the experiments described here have provided new information on the details of BEC formation. New theoretical developments are proving to be valuable tools for describing BEC phase transition dynamics and interpreting new experimental results. With their amenability to optical manipulation and probing along with the advent of new microscopic theories, BECs provide an important new avenue for gaining insight into the universal dynamics of phase transitions in general.Spontaneous symmetry breaking in the system's order parameter may be one result of cooling through a phase transition. A potential consequence of this is the spontaneous formation of topological defects, which in a BEC appear as vortices. We experimentally observed and characterized the spontaneous formation of vortices during BEC growth. We attribute vortex creation to coherence length limitations during the initial stages of the phase transition. Parallel to these experimental observations, theory collaborators have used the Stochastic Gross-Pitaevski Equation formalism to simulate the growth of a condensate from a thermal cloud. The experimental and theoretical statistical results of the spontaneous formation of vortex cores during the growth of the condensate are in good quantitative agreement with one another, supporting our understanding of the dynamics of the phase transition. We believe that our results are also qualitatively consistent with the Kibble-Zurek mechanism, a universal model for topological defect formation.Ultimately, our understanding of the dynamics of the BEC phase transition may lead to a broader understanding of phase transitions in general, and provide new insight into the development of coherence in numerous systems.