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    Towards exact numerical solutions of quantum many-body problems in ultracold Bose gases : a dissertation presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics at Massey University, Albany, New Zealand
    (Massey University, 2022) Yang, Mingrui
    The main objective of this thesis is to examine how the full configuration interaction quantum Monte Carlo (FCIQMC) method can be best utilized for studying ultracold Bose gases. FCIQMC is a stochastic approach for finding the ground state of a quantum many-body Hamiltonian. It is based on the dynamical evolution of a walker population in Hilbert space, which samples the ground state configuration vector over many iterations. The method has been previously applied to studies of the electronic structure of molecules, solids and certain spin models, as well as recently to ultracold Fermi gases. Whereas in this work we are interested in using the method to examine ultracold bosonic atoms. In this thesis, we cover methodological developments and applications of the FCIQMC method. Two main themes are covered in this thesis: methodological developments and applications of the full configuration interaction quantum Monte Carlo method. Firstly, we present a modification of the original protocol in FCIQMC for walker population control of Booth et al. [J. Chem. Phys. 131, 054106 (2009)] in order to achieve equilibration at a pre-defined average walker number and to avoid walker number overshoots. Next, we investigate a systematic statistical bias found in FCIQMC, known as the population control bias, that originates from controlling a walker population with a fluctuating shift parameter and can become large in bosonic systems. We use an exactly solvable stochastic differential equation to model the bias. Lastly, we showcase an application of FCIQMC in studying the properties of the lowest-energy momentum eigenstates, known as yrast states, of Bose gases coupled with a mobile impurity in one spatial dimension. Based on the results of our computations, we identify different dynamical regimes: the polaron and depleton regimes and transitions between them.
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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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    Ultra-cold bosons in one-dimensional single- and double-well potentials : a thesis submitted in partial fulfilment of the requirements for the degree of Masterate of Science at Massey University
    (Massey University, 2010) Gulliksen, Jake Steven
    A variationally optimised basis allows an accurate description of the quantum behaviour of ultra-cold atoms, even in the strongly correlated regime. A rescaling scheme corrects discrepancies caused by using a reduced Hilbert space. This approach also allows the modelling of experimentally realizable double-well potentials, which still reveals the maximally-entangled states seen in fixed basis models. Time dynamics of these double-well systems show macroscopic tunnelling between wells for bosons with a sufficient interaction strength. The many-body problem of interacting bosons in the highly-correlated regime is difficult. The number of basis states needed to describe this quantum system accurately quickly grows beyond computational reach. Rescaling the interaction strength proves a simple and effective method of calculating exact eigenvalues in a reduced Hilbert space. Bosonic systems in the double-well potential are investigated next. First, how different eigen-states depend on the interaction strength is examined. The variationally optimised method has advantages over a standard fixed basis method with the ability to model experimentally viable systems and explore more stronglycorrelated regimes. Secondly, tunnelling dynamics in the double well are studied, specifically for a system where all particles initially occupy a single well. Oscillations corresponding to collective tunnelling between wells are found in regimes where there are zero interactions or bosons lie in a maximally-entangled state. What governs the dynamics outside these two regimes is also considered.