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    Quantum description of dark solitions in one-dimensional quantum gases : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics at Massey University, Albany, New Zealand
    (Massey University, 2017) Shamailov, Sophie S
    The main objective of this thesis is to explain, from the quantum-mechanical point of view, the nature of dark solitons in one-dimensional cold-atom systems. Models of bosons and fermions with contact interactions on a ring are exactly solvable via the Bethe ansatz, and support so-called type-II elementary excitations. These have long been associated with dark solitons of the Gross-Pitaevskii equation due to the similarity of the dispersion relation, despite the completely different physical properties of the states. Fully understanding this connection is our primary aim. We begin by reviewing the Gross-Pitaevskii equation and its dark soliton solutions. Next, we solve the mean-field problem of two coupled one-dimensional Bose-Einstein condensates, with special emphasis on Josephson vortices and their dispersion relation. Predictions are given for possible experimental detection. Then we give a derivation that justifies a method for the extraction of the so-called missing particle number from the dispersion relation of solitonic excitations. A derivation of the finite Bethe ansatz equations for the Lieb-Liniger and Yang-Gaudin models follows. These describe a single species of bosons and two component fermions, respectively. We review the elementary excitations of the Lieb-Linger model, and carry out a comprehensive study of the (much richer) excitations of the Yang-Gaudin model. The thermodynamic limit Bethe ansatz equations for all states of interest in both models are derived, and the missing particle number and the closely-related phase-step are extracted from the dispersion relations. Next, we develop a method for approximating the finite-system dispersion relation of solitonic excitations from the thermodynamic limit results. Finally, we show that the single particle density and phase profiles of appropriately formed superpositions of type-II states with different momenta exhibit solitonic features. Through this idea, the missing particle number and phase step extracted from the dispersion relation gain physical meaning. Moreover, we use a convolution model to extract the fundamental quantum dark soliton length scale across the range of interactions and momenta. The insight gained in the bosonic case is used to make inferences about dark solitons in the fermionic case. Furthermore, we study the Hess-Fairbank effect in the repulsive Yang-Gaudin model and the fermionic super Tonks-Girardeau regime.
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    Non-equilibrium dynamics from few- to many-body systems : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics at Massey University, Albany, New Zealand
    (Massey University, 2017) Cosme, Jayson Gutierrez
    We study different nonequilibrium phenomena of isolated quantum systems ranging from few- to many-body interacting bosons. Firstly, we have suggested the dynamics of the center-of-mass motion to sensitively detect unconverged numerical many-body dynamics in potential with separable quantum motion of the center of mass. As an example, we consider the time evolution of attractive bosons in a homogenous background and use it to benchmark a specific numerical method based on variational multimode expansion of the many-body wave function - the Multicon gurational time-dependent Hartree for bosons (MCTDHB). We demonstrate that the simplified convergence criterion based on a threshold value for the least occupied mode function fails to assure qualitatively correct result while our suggested convergence test based on the center-of-mass motion correctly detects the deviation of numerical results from the exact results. Recent technological progress in manipulating low-entropy quantum states has motivated us to study the phenomenon of interaction blockade in bosonic systems. We propose an experimental protocol to observe the expected bosonic enhancement factor in this blockade regime. Specifically, we suggest the use of an asymmetric double-well potential constructed by superposition of multiple optical tweezer laser beams. Numerical simulations using the MCTDHB method predict that the relevant states and the expected enhancement factor can be observed. In the second half of the thesis, we have investigated the onset of quantum thermalization in a two-level generalization of the Bose-Hubbard dimer. To this end, the relaxation dynamics following a quench is studied using two numerical methods: We study different nonequilibrium phenomena of isolated quantum systems ranging from few- to many-body interacting bosons. Firstly, we have suggested the dynamics of the center-of-mass motion to sensitively detect unconverged numerical many-body dynamics in potential with separable quantum motion of the center of mass. As an example, we consider the time evolution of attractive bosons in a homogenous background and use it to benchmark a specific numerical method based on variational multimode expansion of the many-body wave function - the Multicon gurational time-dependent Hartree for bosons (MCTDHB). We demonstrate that the simplified convergence criterion based on a threshold value for the least occupied mode function fails to assure qualitatively correct result while our suggested convergence test based on the center-of-mass motion correctly detects the deviation of numerical results from the exact results. Recent technological progress in manipulating low-entropy quantum states has motivated us to study the phenomenon of interaction blockade in bosonic systems. We propose an experimental protocol to observe the expected bosonic enhancement factor in this blockade regime. Specifically, we suggest the use of an asymmetric double-well potential constructed by superposition of multiple optical tweezer laser beams. Numerical simulations using the MCTDHB method predict that the relevant states and the expected enhancement factor can be observed. In the second half of the thesis, we have investigated the onset of quantum thermalization in a two-level generalization of the Bose-Hubbard dimer. To this end, the relaxation dynamics following a quench is studied using two numerical methods: (1) full quantum dynamics and (2) semiclassical phase-space method. We rely on arguments based on the eigenstate thermalization hypothesis (ETH), quantum chaos as seen from the distribution of level spacings, and the concept of chaotic eigenstates in demonstrating equilibration dynamics of local observables in the system after an integrability-breaking quench. The same issue on quantum thermalization can be viewed from a different perspective using semiclassical phase-space methods. In particular, we employ the truncated Wigner approximation (TWA) to simulate the quantum dynamics. In this case, we show that the marginal distributions of the individual trajectories which sample the initial Wigner distribution are in good agreement with the corresponding microcanonical distribution.
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    Advances in classical and quantum wave dynamics on quasiperiodic lattices : a dissertation submitted for the degree of Doctor of Philosophy in Physics, Centre for Theoretical Chemistry and Physics, New Zealand Institute for Advanced Study, Massey University, Albany, New Zealand
    (Massey University, 2016) Danieli, Carlo
    Lattices and discrete networks are cornerstones of a number of scientific subjects. In condensed matter, optical lattices allowed the experimental realization of several theoretically predicted phenomena. Indeed, these structures constitute ideal benchmarks for light and wave propagation experiments involving interacting particles, such as clouds of ultra-cold atoms that Bose-Einstein condensate. Moreover, they allow experimental design of particular lattice topologies, as well as the implementation of several classes of spatial perturbations. For example, Anderson localization being observed for the first time in atomic Bose-Einstein condensate experiments and Aubry-André localization discovered with light propagating through networks of optical waveguide. This thesis considers different types of lattices in the presence of quasiperiodic modulations, mainly the celebrated Aubry-André potential. Particular attention will be given to spectral properties of models, localization features of eigenmodes and the transition from delocalized (metallic) eigenstates to localized (insulating) ones within the energy spectrum. We additionally discuss the relation between the model’s properties and the dynamics of particles hopping along the lattice. After introducing the linear discrete Schrödinger equation, we first discuss the spectral properties of the Aubry-André model. We then study the transition between metallic and insulating regimes of a class of quasiperiodic potentials constructed as an iterative superposition of periodic potentials with increasing spatial period. Next, we discuss the Aubry-André perturbation of flat-band topologies, their energy-dependent transition (mobility edge), which can be expressed in analytical forms in case of specific onsite energy correlations, highlighting existence of zeroes, singularities and divergences. We then discuss two cases of driven one-dimensional lattices, namely an Aubry-André chain with a weak time-space periodic driving and an Anderson chain with a quasiperiodic multi-frequency driving. We show anaytically and numerically how drivings can lift the respective localization and generate delocalization by design. Finally we discuss the problem of the possible generation of correlated metallic states of two interacting particles problem in one dimensional Aubry-André chains, under a coherent drive of the interaction.
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    Time flow and reversibility in a probabilistic universe : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Philosophy at Massey University
    (Massey University, 1990) Holster, Andrew Thomas
    A fundamental problem in understanding the nature of time is to explain its 'directionality'. The commonplace view is that this directionality is provided by the 'flow of time'. Unfortunately this concept of 'time flow', which seems to make perfect sense to us in our everyday lives, has resisted philosophical and scientific analysis so well that today it is widely regarded as having no place in the scientific account of the world. Instead, various alternative physical concepts of the directionality of time have been developed, principally the notions of the time reversibility of physical laws or theories, and of the time asymmetry of physical processes. It is frequently argued by philosophers of physics that the scientific account of the directionality of time must be framed entirely in terms of these physical notions. The thesis of the present work is that this conclusion has been reached far too hastily. It is argued that the concept of time flow is a legitimate physical concept, and furthermore, that time flow plays a real part in quantum theory. A number of conceptual investigations are necessary to support this argument. Firstly, it is necessary to give an analysis of what a physical theory of time flow might be like, and how it might be empirically established. This is given in Chapter One, which at the same time is an overview of the results of later chapters. It is found in Chapter One that the concept of physical time flow has an important connection with the concept of time reversibility, which makes it necessary to give an analysis of this notion. A detailed discussion of reversibility and time symmetry is given in Chapters Two to Five. Here it is demonstrated that the orthodox analysis of the reversibility of probabilistic theories is flawed. This conclusion allows it to be shown, in Chapter Six, that, contrary to current scientific belief, quantum theory is profoundly irreversible. This result, together with the argument of Chapter One, allows a strong prima facie case for an interpretation of quantum probabilities as involving time flow to be given. However, because of the traditional problems with the notion of time flow, for this interpretation to become respectable it needs to be demonstrated that it is possible to construct a formal model of a physical ontology in which time flow can be represented. This is undertaken in Chapter Seven. In Chapter Eight, various points about the role of probabilities in quantum theory are discussed. Finally, in Chapter Nine, the implications of relativity theory for the proposed theory of time flow are considered. It is found that relativity theory poses a serious problem for a physical theory of time flow, but the implications of relativity theory for the proposed interpretation of quantum probabilities is not clear because of deeper foundational problems with quantum theory.
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    Quantum many-body dynamics of bright matter-wave solitons : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy at Massey University, Albany, New Zealand
    (Massey University, 2011) Ernst, Thomas
    The interplay of particle and resonant wave scattering including nonlinear effects creates systems of diverse and interesting quantum many-body physics. A better understanding of the physics in these systems could lead to new and exiting application exploiting their quantum nature. As an example, in this thesis we investigate the scattering of bright matter-wave solitons in ultracold gases on a square well in one spatial dimension. For this, solutions of the mean-field Gross-Pitaevskii approximation and a full quantum manybody method, the so-called multiconfigurational time-dependent Hartree approach (MCTDH), are compared. The MCTDH method is based on a finite basis set expansion, which naturally leads to errors in system properties, such as energies and densities, when compared to exact results. In this thesis, we propose an efficient solution to this problem by rescaling the interaction strength between the particles. Even for very large interactions in the Tonks-Girardeau limit, the rescaling leads to significant improvements. This is validated by successfully applying the rescaling to problems in ring systems as well as external confinements, such as a harmonic well and a double-well. The MCTDH method is then applied to the soliton scattering problem and compared to results from mean-field calculations. The latter verify that solitons, when scattered on a well, show quantum effects, such as reflection. For the first time, we show that a soliton can be additionally permanently trapped by the well due to resonances with bound states. For this thesis, to extend these results to a full many-body approach, we developed QiwiB. It is a program package implementing the MCTDHB method, which is a derivative of the MCTDH method, but optimised for bosonic systems. Limits for the validity of the MCTDHB approach are addressed by convergence studies on the soliton scattering problem. Furthermore, we demonstrate that the scattering on the well enables the creation of macroscopic binary quantum superposition states, i.e. NOON states. Novel NOON states corresponding to a superposition of a reflected soliton and a trapped soliton are observed. These states are shown to exist for a large range of initial conditions, and a possible experimental realisation is discussed.