We study Bose-Einstein condensates at the low-energy limit and show that their collective dynamics exhibit interesting quantum dynamical behavior. The system undergoes a dynamical quantum phase transition after a sudden quench into a properly distributed static force, provided its dispersion relation is linear. We corroborate the occurrence of a dynamical phase transition by calculating the Fisher zeros of the Loschmidt amplitude and showing that they cross the real time axis in the thermodynamic limit. A connection is established between the order of nonanalycity in the return rate function and the spectral density of the force. Furthermore, it is shown that a logarithmic or power-law scaling holds at the critical times depending on the displacement spectrum. The scaling behaviors are studied for three different cases. Eventually, a scheme for the quantum simulation of such a dynamical phase transition and its verification is proposed. We show that by taking into account the environmental effects the behavior remains observable.
We study Bose-Einstein condensates at the low-energy limit and show that their collective dynamics exhibit interesting quantum dynamical behavior. The system undergoes a dynamical quantum phase transition after a sudden quench into a properly distributed static force, provided its dispersion relation is linear. We corroborate the occurrence of a dynamical phase transition by calculating the Fisher zeros of the Loschmidt amplitude and showing that they cross the real time axis in the thermodynamic limit. A connection is established between the order of nonanalycity in the return rate function and the spectral density of the force. Furthermore, it is shown that a logarithmic or power-law scaling holds at the critical times depending on the displacement spectrum. The scaling behaviors are studied for three different cases. Eventually, a scheme for the quantum simulation of such a dynamical phase transition and its verification is proposed. We show that by taking into account the environmental effects the behavior remains observable.
We study the dynamics of two-component atomic Bose gases initially in a mixture encountering a sudden quench of the interspecies interactions. The dynamics above the critical temperature Tc is studied using a leading-order large-N approximation that predicted a phase transition from mixing to phase separation as a function of the interspecies coupling. Here, we explore the dynamics of this phase transition following a quench and compare our results to those found at zero temperature using the time-dependent Gross-Pitaevskii equations which ignore quantum and thermal fluctuations. In the regime above Tc where no condensate is present, however, the time evolution of the densities following the quench exhibits features similar to those found at zero temperature where only the condensates contribute to the densities. When the interspecies interaction jumps above the critical value, we observe dynamical transitions from a homogeneous mixture to a phase-separated structure for both cases. Our simulations suggest that at temperatures above Tc where no condensate is present this dynamical transition should still be observable in experiments.
Dynamical quantum phase transition is a nonstationary analog of the equilibrium quantum phase transitions that may occur after global quenches across a quantum phase transition in systems. In this work, we investigate the quench dynamics of single-particle tight-binding quasiperiodic lattices, where the quenching process originates and terminates within the same phases. Intriguingly, the system under quench dynamics undergoes an energy-dependent dynamical quantum phase transition, characterized by the singularities of the Loschmidt echo in time scale. Specifically, we find disentangling behavior between static and dynamical quantum phase transitions for eigenenergies excluding band edges. Remarkably, the connection between static and dynamical quantum phase transitions does not hold generally for all eigenenergies. Our findings suggest the existence of dynamical phase transitions inside the band energies without crossing the critical point under certain conditions. Our results pave the way to a deeper understanding of the energy-dependent nonequilibrium quench dynamics.
Abstract Phase transitions in nonequilibrium dynamics of quantum many-body system, known as dynamical phase transitions (DPTs), play an important role for understanding various dynamical phenomena observed in different branches of physics. In general, there are two types of DPTs, the first one is characterized by distinct evolutionary behaviors of a physical observable, while the second one is marked by the vanishing overlap between the time-evolved and initial states. Here, we focus on exploring such DPTs from both quantum and semiclassical perspectives in a spinor Bose–Einstein condensate (BEC), an ideal platform for investigating nonequilibrium dynamics. Utilizing the sudden quench process, we demonstrate that the system exhibits both types of DPTs as the control parameter is quenched through the critical value, referring to as the critical quenching. We show analytically how to determine the critical quenching via the semiclassical approach and carry out a detailed examination of both semiclassical and quantum signatures of DPTs. In particular, we reveal that the occurrence of DPTs is triggered by the separatrix in the underlying semiclassical system. Our findings offer deeper insights into the properties of DPTs and verify the usefulness of semiclassical analysis for studying DPTs in quantum systems with well-defined semiclassical limit.
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 demonstrate the existence of a dynamical quantum phase transition (DQPT) in a dissipative collective-spin model that exhibits the boundary time crystal (BTC) phase. We initialize the system in the ground state of the Hamiltonian in either the BTC or the non-BTC phase, and drive it across the BTC transition. The driving is done by an abrupt quench or by a finite-time linear ramp of a Hamiltonian control parameter under Markovian Lindblad dynamics. We diagnose DQPTs through zeros of the fidelity-based Loschmidt echo between the initial state and the evolving mixed state, which induce nonanalytic cusp-like features in the associated rate function. For quenches into the BTC phase, the Loschmidt echo exhibits repeated zeros due to the emergent time-periodic steady state, whereas for quenches into the non-BTC phase, the overlap vanishes and remains zero once the dynamics relaxes to a stationary state. We further show that the DQPT persists under the ramp protocol followed by unitary evolution with the final Hamiltonian. Finally, we analyze the finite-size scaling of the first critical time and find convergence to a constant in the thermodynamic limit, with distinct power-law approaches for the quench and the ramp protocols.
Critical properties of the dynamical phase transition in the quenched generalized Bose-Anderson impurity model are studied in the mean-field limit of an infinite number of channels. The transition separates the evolution toward ground state and toward the branch of stable excited states. We perform numerically exact simulations of a close vicinity of the critical quench amplitude. The relaxation constant describing the asymptotic evolution toward ground state, as well as asymptotic frequency of persistent phase rotation and number of cloud particles at stable excited state are power functions of the detuning from the critical quench amplitude. The critical evolution (separatrix between the two regimes) shows a non-Lyapunov power-law instability arising after a certain critical time. The observed critical behavior is attributed to the irreversibility of the dynamics of particles leaving the cloud and to memory effects related to the low-energy behavior of the lattice density of states.