ENGLISH ABSTRACT : Motivated by recent experiments, we study the time evolution of a two-species Bose-Einstein condensate which is coupled to a bosonic bath. For the particular condensate, unconventional thermodynamics have recently been predicted. To study these thermal properties we find the conditions under which this open quantum system thermalises—equilibrates to the Gibbs state describing the canonical ensemble. The condensate is mapped from its bosonic representation, describing N interacting bosons, onto a Schwinger spin representation, with spin angular momentum S = 2N. The corresponding Hamiltonian takes the form of a Lipkin-Meshkov-Glick (LMG) model. We find that the total system-bath Hamiltonian is too difficult to solve. Fortunately, in the case where the LMG model is only weakly coupled to a near-memoryless bath, we may derive an approximate differential equation describing the LMG model’s evolution. Upon further approximations this equation describes a quantum dynamical semigroup—preserving all the rules of quantum mechanics. The generator of this semigroup, known as the Lindbladian, is our main object of interest. Its nullspace is populated by the stationary state(s), which we wish to compare to the canonical ensemble. If the only state in this nullspace is the Gibbs state, then the system will thermalise to that state. For extensive bath temperatures we show that the Gibbs state is stationary in the thermodynamic limit N = S/2 → ∞. Numerically we find that this is the only stationary state, meaning the system thermalises. To prove thermalisation for the intensive temperature case we diagonalise the Lindbladian. To do this we first perform a Holstein-Primakoff (HP) mapping from spins onto bosons. Such a mapping may be approximated in the region of interest—close to the stationary state(s)—given one has knowledge of that regions expectation values, such as the magnetisation hSz ieq. The new approximated Lindbladian may then be diagonalised via a Bogoliubov transform. To find these expectation values we study the observables’ time evolution in the Heisenberg picture. We find that energy dissipation happens on a time scale purely characterised by the weak coupling. This allows us to set up an approximate differential equation for the energy which we solve, hence finding the equilibrium energy. Using this energy together with spin coherent states we are then able to find hSz ieq and hS 2 xieq. Performing the HP mapping in the region of interest then leaves a quadratic, in bosonic ladder operators, Lindbladian which we subsequently diagonalise with a Bogoliubov transform. The stationary states are found to be approximate Gibbs states. In the non-degenerate phase of the LMG model we find a single stationary state, thus proving thermalisation. In contrast, we find two (orthogonal) near stationary states in degenerate phase. Numerically we find, for finite system size, that these two states form an equal fraction of the true stationary state which is a Gibbs state. However the numerics further indicates that the spectral gap between this Gibbs state and its closest state tends to zero in the thermodynamic limit. As such we expect two (orthogonal) stationary states in the degenerate phase as S → ∞.
ENGLISH ABSTRACT : Motivated by recent experiments, we study the time evolution of a two-species Bose-Einstein condensate which is coupled to a bosonic bath. For the particular condensate, unconventional thermodynamics have recently been predicted. To study these thermal properties we find the conditions under which this open quantum system thermalises—equilibrates to the Gibbs state describing the canonical ensemble. The condensate is mapped from its bosonic representation, describing N interacting bosons, onto a Schwinger spin representation, with spin angular momentum S = 2N. The corresponding Hamiltonian takes the form of a Lipkin-Meshkov-Glick (LMG) model. We find that the total system-bath Hamiltonian is too difficult to solve. Fortunately, in the case where the LMG model is only weakly coupled to a near-memoryless bath, we may derive an approximate differential equation describing the LMG model’s evolution. Upon further approximations this equation describes a quantum dynamical semigroup—preserving all the rules of quantum mechanics. The generator of this semigroup, known as the Lindbladian, is our main object of interest. Its nullspace is populated by the stationary state(s), which we wish to compare to the canonical ensemble. If the only state in this nullspace is the Gibbs state, then the system will thermalise to that state. For extensive bath temperatures we show that the Gibbs state is stationary in the thermodynamic limit N = S/2 → ∞. Numerically we find that this is the only stationary state, meaning the system thermalises. To prove thermalisation for the intensive temperature case we diagonalise the Lindbladian. To do this we first perform a Holstein-Primakoff (HP) mapping from spins onto bosons. Such a mapping may be approximated in the region of interest—close to the stationary state(s)—given one has knowledge of that regions expectation values, such as the magnetisation hSz ieq. The new approximated Lindbladian may then be diagonalised via a Bogoliubov transform. To find these expectation values we study the observables’ time evolution in the Heisenberg picture. We find that energy dissipation happens on a time scale purely characterised by the weak coupling. This allows us to set up an approximate differential equation for the energy which we solve, hence finding the equilibrium energy. Using this energy together with spin coherent states we are then able to find hSz ieq and hS 2 xieq. Performing the HP mapping in the region of interest then leaves a quadratic, in bosonic ladder operators, Lindbladian which we subsequently diagonalise with a Bogoliubov transform. The stationary states are found to be approximate Gibbs states. In the non-degenerate phase of the LMG model we find a single stationary state, thus proving thermalisation. In contrast, we find two (orthogonal) near stationary states in degenerate phase. Numerically we find, for finite system size, that these two states form an equal fraction of the true stationary state which is a Gibbs state. However the numerics further indicates that the spectral gap between this Gibbs state and its closest state tends to zero in the thermodynamic limit. As such we expect two (orthogonal) stationary states in the degenerate phase as S → ∞.
We derive a Lindblad master equation that approximates the dynamics of a Lipkin-Meshkov-Glick (LMG) model weakly coupled to a bosonic bath. By studying the time evolution of operators under the adjoint master equation we prove that, for large system sizes, these operators attain their thermal equilibrium expectation values in the long-time limit, and we calculate the rate at which these values are approached. Integrability of the LMG model prevents thermalization in the absence of a bath, and our work provides an explicit proof that the bath indeed restores thermalization. Imposing thermalization on this otherwise nonthermalizing model outlines an avenue towards probing the unconventional thermodynamic properties predicted to occur in ultracold-atom-based realizations of the LMG model.
This thesis approaches questions concerning the thermalisation of subsystems of closed quantum systems, prepared in pure states of definite energy but far from equilibrium, under exact unitary evolution. Taking motivation from experiments in the field of ultracold atoms, an extensive study of relaxation to a thermal state in the Hubbard model is presented. The study of small local subsystems in Hubbard-model lattice clusters has led to some interesting findings. Explored are the effects of interactions between fermions, the initial-state energy and the energy uncertainty in the initial state and their effects on relaxation dynamics and thermalisation. The most significant finding is that while subsystem thermalisation is seen for a large range of subsystem-bath coupling strengths, the temporal form of the relaxation varies markedly from exponential decay for weak couplings with a crossover to Gaussian behaviour with increased coupling strength. This is found to hold more generally for random couplings between the subsystem and bath and for bosons as well as fermions, thus demonstrating generality. As well as being demonstrated numerically, this behaviour is derived for a generic class of bi-partite quantum systems which may be described with the use of random matrices. A Brownian motion model is employed to show the exponential to Gaussian crossover when the subsystem-bath coupling matrix takes a banded form. This result agrees well with numerical Hubbard-model results, and yields identical results at short times to those from straight-forward perturbative methods. It is demonstrated that the non-Markovian Gaussian behaviour should also be observable in the limit of macroscopic baths.
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.
Understanding the dynamics of dissipative quantum systems, particularly beyond the weak coupling approximation, is central to various quantum applications. While numerically exact methods provide accurate solutions, they often lack the analytical insight provided by theoretical approaches. In this study, we employ the recently developed method dubbed the effective Hamiltonian theory to understand the dynamics of system-bath configurations without resorting to a perturbative description of the system-bath coupling energy. Through a combination of mapping steps and truncation, the effective Hamiltonian theory offers both analytical insights into signatures of strong couplings in open quantum systems and a straightforward path for numerical simulations. To validate the accuracy of the method, we apply it to two canonical models: a single spin immersed in a bosonic bath and two noninteracting spins in a common bath. In both cases, we study the transient regime and the steady state limit at nonzero temperature and spanning system-bath interactions from the weak to the strong regime. By comparing the results of the effective Hamiltonian theory with numerically exact simulations, we show that although the former overlooks non-Markovian features in the transient equilibration dynamics, it correctly captures non-perturbative bath-generated couplings between otherwise non-interacting spins, as observed in their synchronization dynamics and correlations. Altogether, the effective Hamiltonian theory offers a powerful approach for understanding strong coupling dynamics and thermodynamics, capturing the signatures of such interactions in both relaxation dynamics and in the steady state limit.
We study the dynamics of a Lipkin-Meshkov-Glick model in the presence of Markovian dissipation, with a focus on late-time dynamics and the approach to thermal equilibrium. Making use of a vectorized bosonic representation of the corresponding Lindblad master equation, we use degenerate perturbation theory in the weak-dissipation limit to analytically obtain the eigenvalues and eigenvectors of the Liouvillian superoperator, which in turn give access to closed-form analytical expressions for the time evolution of the density operator and observables. Our approach is valid for large systems but takes into account leading-order finite-size corrections to the infinite-system result. As an application, we show that the dissipative Lipkin-Meshkov-Glick model equilibrates by passing through a continuum of thermal states with damped oscillations superimposed, until finally reaching an equilibrium state with a temperature that in general differs from the bath temperature. We discuss limitations of our analytic techniques by comparing to exact numerical results.
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.