This thesis investigates thermalisation, correlations and entanglement in Bose-Einstein condensates. Bose-Einstein condensates are ultra-cold collections of identical bosonic atoms which accumulate in a single quantum state, forming a mesoscopic quantum object. They are clean and controllable quantum many-body systems that permit an unprecedented degree of experimental flexibility compared to other physical systems. Further, a tractable microscopic theory exists which allows a direct and powerful comparison between theory and experiment, propelling the field of quantum atom optics forward at an incredible pace. Here we explore some of the fundamental frontiers of the field, examining how non-classical correlations and entanglement can be created and measured, as well as how non-classical effects can lead to the rapid heating of atom clouds. We first investigate correlations between two weakly coupled condensates, a system analogous to a superconducting Josephson junction. The ground state of this system contains non-classical number correlation arising from the repulsion between the atoms. Such states are of interest because they may lead to more precise measurement devices such as atomic gyroscopes. Unfortunately thermal fluctuations can destroy these correlations, and great care is needed to experimentally observe non-classical effects. We show that adiabatic evolution can drive the isolated quantum system out of thermal equilibrium and decrease thermal noise, in agreement with a recent experiment [Esteve et al. Nature 455, 1216 (2008)]. This technique may be valuable for observing and using quantum correlated states in the future. Next, we analyse the rapid heating that occurs when a condensate is placed in a moving periodic potential. The dynamical instability responsible for the heating was the subject of much uncertainty, which we suggest was due to the inability of the mean-field approximation to account for important spontaneous scattering processes. We show that a model including non-classical spontaneous scattering can describe dynamical instabilities correctly in each of the regimes where they have been observed, and in particular we compare our simulations to an experiment performed at the University of Otago deep inside the spontaneous scattering regime. Finally, we proposed a method to create and detect entangled atomic wave-packets. Entangled atoms are interesting from a fundamental perspective, and may prove useful in future quantum information and precision measurement technologies. Entanglement is generated by interactions, such as atomic collisions in Bose-Einstein condensates. We analyse the type of entanglement generated via atomic collisions and introduce an abstract scheme for detecting entanglement and demonstrating the Einstein-Podolsky-Rosen paradox with ultra-cold atoms. We further this result by proposing an experiment where entangled wave-packets are created and detected. The entanglement is generated by the pairwise scattering that causes the instabilities in moving periodic potentials mentioned above. By careful arrangement, the instability process can be controlled to to produce two well-defined atomic wave-packets. The presence of entanglement can be proven by applying a series of laser pulses to interfere the wave-packets and then measuring the output populations. Realising this experiment is feasible with current technology.
This thesis investigates thermalisation, correlations and entanglement in Bose-Einstein condensates. Bose-Einstein condensates are ultra-cold collections of identical bosonic atoms which accumulate in a single quantum state, forming a mesoscopic quantum object. They are clean and controllable quantum many-body systems that permit an unprecedented degree of experimental flexibility compared to other physical systems. Further, a tractable microscopic theory exists which allows a direct and powerful comparison between theory and experiment, propelling the field of quantum atom optics forward at an incredible pace. Here we explore some of the fundamental frontiers of the field, examining how non-classical correlations and entanglement can be created and measured, as well as how non-classical effects can lead to the rapid heating of atom clouds. We first investigate correlations between two weakly coupled condensates, a system analogous to a superconducting Josephson junction. The ground state of this system contains non-classical number correlation arising from the repulsion between the atoms. Such states are of interest because they may lead to more precise measurement devices such as atomic gyroscopes. Unfortunately thermal fluctuations can destroy these correlations, and great care is needed to experimentally observe non-classical effects. We show that adiabatic evolution can drive the isolated quantum system out of thermal equilibrium and decrease thermal noise, in agreement with a recent experiment [Esteve et al. Nature 455, 1216 (2008)]. This technique may be valuable for observing and using quantum correlated states in the future. Next, we analyse the rapid heating that occurs when a condensate is placed in a moving periodic potential. The dynamical instability responsible for the heating was the subject of much uncertainty, which we suggest was due to the inability of the mean-field approximation to account for important spontaneous scattering processes. We show that a model including non-classical spontaneous scattering can describe dynamical instabilities correctly in each of the regimes where they have been observed, and in particular we compare our simulations to an experiment performed at the University of Otago deep inside the spontaneous scattering regime. Finally, we proposed a method to create and detect entangled atomic wave-packets. Entangled atoms are interesting from a fundamental perspective, and may prove useful in future quantum information and precision measurement technologies. Entanglement is generated by interactions, such as atomic collisions in Bose-Einstein condensates. We analyse the type of entanglement generated via atomic collisions and introduce an abstract scheme for detecting entanglement and demonstrating the Einstein-Podolsky-Rosen paradox with ultra-cold atoms. We further this result by proposing an experiment where entangled wave-packets are created and detected. The entanglement is generated by the pairwise scattering that causes the instabilities in moving periodic potentials mentioned above. By careful arrangement, the instability process can be controlled to to produce two well-defined atomic wave-packets. The presence of entanglement can be proven by applying a series of laser pulses to interfere the wave-packets and then measuring the output populations. Realising this experiment is feasible with current technology.
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.
The field of atom optics has progressed rapidly over the past 20 years since the realisation of Bose-Einstein condensation, such that the wave behaviour of atomic gases is now routinely demonstrated. Furthermore, the study of quantum atom optics, which goes beyond a ‘mean-field’ description of quantum systems to consider the behaviour of single particles, has demonstrated both the similarities between photons and massive species, and their differences as a result of the internal structure and external interactions of atoms. An important class of observable quantities which allow such effects to be measured are nth order correlation functions, which can be interpreted as a result of either particle or wave behaviour. These functions provide a statistical description of fluctuations in n-tuples of particles in a source, which rigorously defines concepts such as coherence. The quantum statistics of a Bose-Einstein condensate should be the same as that for an optical laser, while an ideal thermal Bose gas matches the behaviour of incoherent light. However, correlation measurements can also be used to quantify the influence of interactions, dimensionality, confining potentials and waveguides, and the difference in quantum statistics between fermions and bosons, which illustrates the rich range of behaviour exhibited by atomic gases. In this thesis, several aspects of quantum atom optics are explored with experiments using ultracold metastable helium, a species with the unique advantage of facilitating simple single-atom detection with high resolution, while still allowing Bose-Einstein condensates to be formed. The coherence of atomic systems is shown to be maintained when outcoupled as pulsed atom lasers, and the long-range order characteristic of Bose-Einstein condensates is demonstrated to third order for the first time. Conversely, thermal bunching is observed for a variety of atomic systems, including the measurement of correlation functions up to sixth order with near-ideal interference contrast. These results clearly demonstrate the correspondence between the quantum statistics of photons and atoms as was formalised by Glauber, as well as confirming the validity of applying Wick’s theorem to simplify the statistics of atomic gases. Correlation functions are also shown to be an ideal tool to probe the quantum state of an ultracold gas, and were used to observe the phenomenon of transverse condensation in an elongated Bose gas, as well as characterise the mode occupancy of matter waves guided by an optical potential. Ultracold metastable helium is also suitable for exploring other fundamental topics in quantum optics such particle/wave duality. The notion of complementarity stimulated long running philosophical discussions about how apparently mutually exclusive behaviours can coexist, which culminated in Wheeler devising his famous ‘delayed choice’ gedankenexperiment. A proposed experimental method to realise Wheeler’s experiment with ultracold atoms is discussed, and preliminary measurements presented which indicate that the completion of this experiment could be achieved in the near future. Not only is this of interest in its own right, but the implementation of this experiment has also developed techniques which may enable further studies in quantum atom optics such as investigations of the Hong-Ou-Mandel effect and quantum entanglement with massive particles.
The three experiments described in this thesis investigate fundamental properties of ultracold atoms. Using laser cooling and evaporative cooling, a dilute gas of sodium atoms is cooled to ~100 nK. Under these circumstances a Bose-Einstein condensate (BEC) forms, where millions of atoms collapse into the lowest energy state of the system and share a macroscopic wavefunction. The experiments are done in an ultrahigh vacuum and the atoms are manipulated remotely using laser beams, magnetic fields, and RF-fields. Using ultracold atoms allows unprecedented control over the internal and external degrees of freedom of the system, making it very suitable for precision measurements. Ultracold atoms also serve as a model system for more complicated or remote fields such as solid-state physics and cosmology. In the first experiment the refractive index of an ultracold bosonic gas is studied. The atomic clouds are analyzed for temperature and number of particles using nearly non-destructive phase-contrast imaging. After each pulse of probe light a small fraction of the atoms is lost, while the cloud is simultaneously slightly heated. This makes it possible to study the scattering rate as a function of temperature using only a single sample. It is observed that the scattering rate increases below the critical temperature for BEC by more than a factor of 3 compared to the classical value. This enhancement is in good agreement with the predictions by theoretical work that takes two-body correlation and the Lorentz-Lorenz correction into account. The method is also used to perform calorimetry on the Bose gas and directly measure the heat capacity of the BEC. The second chapter deals with magnetism in ultracold gases. When cooling an ultracold gas containing different spin components below the critical temperature for BEC, stable magnetic domains form. The appearance of these domains and the dynamics of the domain walls are studied. The spin mixture is prepared by performing an RF-sweep on ground state atoms in the presence of a magnetic field. Spin dynamics are induced by applying magnetic gradients. Spin drag, which is the transport coefficient responsible for damping spin currents, was measured in a two-component system. Furthermore a new method for spin resolved phase contrast imaging is presented, which is used to study time evolution in situ non-destructively. The last experiment focuses on spontaneous symmetry breaking. When passing a continuous phase transition, spontaneous symmetry breaking causes the development of domains with an independently chosen order parameter. The domains are separated by boundary defects. In this experiment the spontaneous creation of solitons was observed, which are boundary defects in the phase of the condensate, as well as magnetic domain walls, which are the boundary defects in the magnetization. Through the Kibble-Zurek mechanism it is possible to determine the critical exponents that characterize the phase transition. The critical exponents are predicted to be universal, i.e. do not depend on the microphysics of the system, so the results from our ultracold atoms lab can help answer universal questions for example from cosmology and from the early universe.
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.
The observation of non-classical correlations arising in two weakly coupled Bose–Einstein condensates was recently reported by Estève et al (2008 Nature 455 1216). In order to observe relative number fluctuations between the two condensates below the standard quantum limit, they utilized the process of 'adiabatic passage' to drive the system out of thermal equilibrium. They found that this reduced the relative number fluctuations below that expected in thermal equilibrium at the minimum experimentally realizable temperature. We present a theoretical analysis that takes into account the spatial degrees of freedom of the system, allowing us to calculate the expected correlations at finite temperature in the system, and to verify the hypothesis of reduced number fluctuations via adiabatic passage by comparing the dynamics to the idealized model.
Cold atoms trapped in optical lattices (crystals of light) provide a pristine platform for exploring quantum many body physics. Motivated by several recent experiments, this thesis examines the equilibrium and non-equilibrium dynamics of a Bose-Einstein condensate (BEC) loaded in a low dimensional optical lattice in order to realize novel quantum phases. There are two main research directions in this thesis. The first one involves the possibility that exotic order spontaneously forms when two-component bosons are trapped in a honeycomb lattice. My studies on this theme is motivated by the observation of a “twisted superfluid” state in Prof. Klaus Sengstock’s group at Hamburg (Soltan-Panahi et al., Nat. Phys. 8, 71 (2012)). A twisted superfluid involves Bose-Einstein condensation into a state whose order parameter has a spatially varying phase. In chapter 3, I study the stability of a Bose-Einstein condensate towards forming a twisted superfluid within the framework of mean field theory. Despite a exhaustive numerical search I do not find a parameter regime with a twisted superfluid. This search involved all experimentally relevant parameter regimes and therefore mean field theory predicted that the experimentalists should not observe a twisted superfluid. I conclude that the experimental observations were either a manifestation of counter superfluidity or due to interactions during time-of-flight. Subsequent experiments showed that the observations were an artifact of the measurement process. The second research direction in this thesis is an exploration of the stability of periodically driven quantum systems (also known as Floquet systems). Floquet systems can be used to realize exotic non-equilibrium quantum phases which do not have a counterpart in static systems. However, the driving can cause these systems to heat up which presents a major obstacle to creating exotic states. To explore this issue in a concrete example, I model an experiment (Parker, Ha, and Chin, Nat. Phys. 9, 769 (2013)) where a Bose-Einstein condensate loaded in an optical lattice is subjected to periodic shaking. I investigate the stability of this Floquet BEC to interactions. This research direction consists of 3 studies. In chapter 4, I first do this analysis for a purely one-dimensional system and identify a large parameter regime where the BEC is stable. In the next two chapters, I go beyond 1D and consider the role of transverse degrees of freedom. This is because the shaken lattice experiments that I model involves a 1D array of pancakes. I find that this geometry leads to much more dissipation than a purely 1D system. This extra dissipation arises because interactions can transfer energy between different directions. In chapter 5, I consider the extreme case where there is no transverse confinement. I find that in the absence of transverse confinement, a one-dimensional Floquet BEC is generically unstable. Finally, in chapter 6, I consider harmonic transverse confinement modeling the crossover between chapters 4 and 5. I find that as the transverse confinement is made stronger, the atom loss rate initially increases, but beyond a critical transverse confinement, the atom loss disappears due to unavailability of phase space for scattering. I also predict that if the transverse confinement is tuned to the vicinity of certain magic values, the heating rate exhibits a sharp drop. I perform similar analyses for a shaken square lattice and find that generically a low-dimensional Floquet BEC can be stabilized by suitably designing the transverse confinement.