Continuously measured interacting quantum systems almost invariably heat, causing loss of quantum coherence. Here, we study Bose-Einstein condensates (BECs) subject to repeated weak measurement of the atomic density and describe several protocols for generating a feedback signal designed to remove excitations created by measurement back-action. We use a stochastic Gross-Pitaevskii equation to model the system dynamics and find that a feedback protocol utilizing momentum-dependent gain and filtering can effectively cool both one-dimensional (1D) and two-dimensional systems. The performance of these protocols is quantified in terms of the steady-state energy, entropy, and condensed fraction. These are the first feedback cooling protocols demonstrated in two dimensions, and in one dimension our optimal protocol reduces the equilibrium energy by more than a factor of 100 as compared with a previous cooling protocol developed using the same methodology. We also use this protocol to quench-cool 1D BECs from noncondensed highly excited states and find that they rapidly condense into a far-from-equilibrium state with energy orders of magnitude higher than the equilibrium ground-state energy for that condensate fraction. We explain this in terms of the near integrability of our 1D system, whereby efficiently cooled low-momentum modes are effectively decoupled from the energetic ``reservoir'' of the higher-momentum modes. We observe that the quench-cooled condensed states can have nonzero integer winding numbers described by quantized supercurrents.
Continuously measured interacting quantum systems almost invariably heat, causing loss of quantum coherence. Here, we study Bose-Einstein condensates (BECs) subject to repeated weak measurement of the atomic density and describe several protocols for generating a feedback signal designed to remove excitations created by measurement back-action. We use a stochastic Gross-Pitaevskii equation to model the system dynamics and find that a feedback protocol utilizing momentum-dependent gain and filtering can effectively cool both one-dimensional (1D) and two-dimensional systems. The performance of these protocols is quantified in terms of the steady-state energy, entropy, and condensed fraction. These are the first feedback cooling protocols demonstrated in two dimensions, and in one dimension our optimal protocol reduces the equilibrium energy by more than a factor of 100 as compared with a previous cooling protocol developed using the same methodology. We also use this protocol to quench-cool 1D BECs from noncondensed highly excited states and find that they rapidly condense into a far-from-equilibrium state with energy orders of magnitude higher than the equilibrium ground-state energy for that condensate fraction. We explain this in terms of the near integrability of our 1D system, whereby efficiently cooled low-momentum modes are effectively decoupled from the energetic ``reservoir'' of the higher-momentum modes. We observe that the quench-cooled condensed states can have nonzero integer winding numbers described by quantized supercurrents.
Abstract We theoretically investigate measurement-based feedback control over the motional degrees of freedom of an oblate quasi-2D atomic Bose-Einstein condensate (BEC) subject to continuous density monitoring. We develop a linear-quadratic-Gaussian (LQG) model that describes the multi-mode dynamics of the condensate's collective excitations under continuous measurement and control. Crucially, the multi-mode cold-damping feedback control we consider uses a realistic state-estimation scheme that does not rely upon a particular model of the atomic dynamics. We present analytical results showing that collective excitations can be cooled to below single-phonon average occupation (ground-state cooling) across a broad parameter regime, and identify the conditions under which the lowest steady-state phonon occupation is asymptotically achieved. Further, we develop multi-objective optimization methods that explore the trade-off between cooling speed and the final energy of the cloud, and provide numerical simulations demonstrating the ground-state cooling of the lowest ten motional modes above the condensate ground state. Our investigation provides concrete guidance on the feedback control design and parameters needed to experimentally realize a feedback-cooled BEC.
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.
We report on the experimental observation of stimulated cooling in the non-equilibrium Bose-Einstein condensate (BEC) of weakly interacting exciton-polaritons from approximately room temperature down to 20K. By resolving the condensate in energy-momentum space and performing interferometric measurements, we distinguish the condensate from thermalized particles yet occupying excited states macroscopically. In contrast to the analytical quantum theories of non-equilibrium BEC [Shishkov et al., Phys. Rev. Lett. 128, 065301 (2022)], we observe segmentation of the particle density along the excited states into two fractions both following Bose-Einstein distribution, albeit with different effective temperatures and chemical potentials. Our results indicate that the temperature of the weakly interacting Bose gas is universally set by the density-dependent chemical potential, revealing a defining property of non-equilibrium BECs. Finally, we demonstrate that the stimulated nature of the cooling process directly governs the emergence of quantum coherence of the condensate and shapes the dissipative properties of the excited states.
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.
Weak measurement in tandem with real-time feedback control is a new route toward engineering novel nonequilibrium quantum matter. Here we develop a theoretical toolbox for quantum feedback control of multicomponent Bose-Einstein condensates (BECs) using backaction-limited weak measurements in conjunction with spatially resolved feedback. Feedback in the form of a single-particle potential can introduce effective interactions that enter into the stochastic equation governing system dynamics. The effective interactions are tunable and can be made analogous to Feshbach resonances-spin independent and spin dependent-but without changing atomic scattering parameters. Feedback cooling prevents runaway heating due to measurement backaction and we present an analytical model to explain its effectiveness. We showcase our toolbox by studying a two-component BEC using a stochastic mean-field theory, where feedback induces a phase transition between easy-axis ferromagnet and spin-disordered paramagnet phases. We present the steady-state phase diagram as a function of intrinsic and effective spin-dependent interaction strengths. Our result demonstrates that closed-loop quantum control of Bose-Einstein condensates is a powerful tool for quantum engineering in cold-atom systems.
Ultra-cold atomic gases are an ideal platform for precision measurement devices and analogue quantum simulators, which will prove instrumental in unlocking the secrets of quantum gravity and high-temperature superconductivity. However, current experimental techniques cannot cool atomic gases to simulatenously both the low entropies and high particle numbers necessary for these applications. A promising alternative is feedback cooling: using continuous-measurement feedback control to damp out energy fluctuations and cool a gas. So far, feedback cooling has been primarily studied at zero temperature, with the only finite temperature simulation achieved via a computationally expensive numerical method restricted to bosons. This thesis develops a perturbative model for the feedback cooling of a finite temperature condensed Bose gas using Bogoliubov theory, with the aim of deriving dynamic equations for the system that are both analytically tractable, and allow for fast numerical prototyping of new feedback control schemes. Using the measurement-feedback model of Szigeti \textit{et al.} \cite{szigeti_continuous_2009,szigeti_feedback_2010}, we derive a low temperature perturbative model for feedback cooling of a Bose gas in an arbitrary trapping and control potential. Using this general model, we then derive a model for the dynamics of a Bose gas in a hard box trap being cooled with an energy damping control. We complete preliminary simulations of this model in the no-backaction conditional measurement limit, damping out density fluctuations in a gas of 90% condensate fraction and cooling it to 93.5±1%. We find that, in this limit, the dynamics of the gas are largely independent of number but significantly depend on the inter-particle interaction. We also find an optimal energy damping control strength in this limit. However, our model is not very efficient for simulation, particularly for a large number of particles and measurement strength. As an alternative, we propose an approximation scheme in which steady-state analytic solutions could be obtained from the model. Finally, we propose two methods to develop a Bogoliubov model for the feedback cooling of fermions, which would be the first finite temperature model for the Fermi gas case.