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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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    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.