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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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    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.
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    Squeezing atoms using a confinement potential : a thesis presented in fulfillment of the requirements for the degree of Master of Science in Mathematical Physics, Massey University, Albany, New Zealand
    (Massey University, 2010) Coxe, Julianne Neilson
    Understanding the complexities of the interior of planets and stars requires the help of analyzing the effects of high pressures on certain elements believed to be found within. The Hartree-Fock method uses the Schr¨odinger equation, Kummer’s differential equations and a confinement potential to simulate an atom being squeezed to high pressures. The Hartree-Fock method was used to calculate the total energies of atoms. After being compared to Gaussian03 and VASP, the results were deemed accurate. It was also observed that the pressure versus density data closely approximated those pairs found in outer space in the interiors of, for example, Jupiter.
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    Multisymplectic integration : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematical Physics at Massey University, Palmerston North, New Zealand
    (Massey University, 2007) Ryland, Brett Nicholas
    Multisymplectic integration is a relatively new addition to the field of geometric integration, which is a modern approach to the numerical integration of systems of differential equations. Multisymplectic integration is carried out by numerical integrators known as multisymplectic integrators, which preserve a discrete analogue of a multisymplectic conservation law. In recent years, it has been shown that various discretisations of a multi-Hamiltonian PDE satisfy a discrete analogue of a multisymplectic conservation law. In particular, discretisation in time and space by the popular symplectic Runge–Kutta methods has been shown to be multisymplectic. However, a multisymplectic integrator not only needs to satisfy a discrete multisymplectic conservation law, but it must also form a well-defined numerical method. One of the main questions considered in this thesis is that of when a multi-Hamiltonian PDE discretised by Runge–Kutta or partitioned Runge–Kutta methods gives rise to a well-defined multisymplectic integrator. In particular, multisymplectic integrators that are explicit are sought, since an integrator that is explicit will, in general, be well defined. The first class of discretisation methods that I consider are the popular symplectic Runge–Kutta methods. These have previously been shown to satisfy a discrete analogue of the multisymplectic conservation law. However, these previous studies typically fail to consider whether or not the system of equations resulting from such a discretisation is well defined. By considering the semi-discretisation and the full discretisation of a multi-Hamiltonian PDE by such methods, I show the following: • For Runge–Kutta (and for partitioned Runge–Kutta methods), the active variables in the spatial discretisation are the stage variables of the method, not the node variables (as is typical in the time integration of ODEs). • The equations resulting from a semi-discretisation with periodic boundary conditions are only well defined when both the number of stages in the Runge–Kutta method and the number of cells in the spatial discretisation are odd. For other types of boundary conditions, these equations are not well defined in general. • For a full discretisation, the numerical method appears to be well defined at first, but for some boundary conditions, the numerical method fails to accurately represent the PDE, while for other boundary conditions, the numerical method is highly implicit, ill-conditioned and impractical for all but the simplest of applications. An exception to this is the Preissman box scheme, whose simplicity avoids the difficulties of higher order methods. • For a multisymplectic integrator, boundary conditions are treated differently in time and in space. This breaks the symmetry between time and space that is inherent in multisymplectic geometry. The second class of discretisation methods that I consider are partitioned Runge– Kutta methods. Discretisation of a multi-Hamiltonian PDE by such methods has lead to the following two major results: 1. There is a simple set of conditions on the coefficients of a general partitioned Runge– Kutta method (which includes Runge–Kutta methods) such that a general multi- Hamiltonian PDE, discretised (either fully or partially) by such methods, satisfies a natural discrete analogue of the multisymplectic conservation law associated with that multi-Hamiltonian PDE. 2. I have defined a class of multi-Hamiltonian PDEs that, when discretised in space by a member of the Lobatto IIIA–IIIB class of partitioned Runge–Kutta methods, give rise to a system of explicit ODEs in time by means of a construction algorithm. These ODEs are well defined (since they are explicit), local, high order, multisymplectic and handle boundary conditions in a simple manner without the need for any extra requirements. Furthermore, by analysing the dispersion relation for these explicit ODEs, it is found that such spatial discretisations are stable. From these explicit ODEs in time, well-defined multisymplectic integrators can be constructed by applying an explicit discretisation in time that satisfies a fully discrete analogue of the semi-discrete multisymplectic conservation law satisfied by the ODEs. Three examples of explicit multisymplectic integrators are given for the nonlinear Schr¨odinger equation, whereby the explicit ODEs in time are discretised by the 2-stage Lobatto IIIA– IIIB, linear–nonlinear splitting and real–imaginary–nonlinear splitting methods. These are all shown to satisfy discrete analogues of the multisymplectic conservation law, however, only the discrete multisymplectic conservation laws satisfied by the first and third multisymplectic integrators are local. Since it is the stage variables that are active in a Runge–Kutta or partitioned Runge– Kutta discretisation in space of a multi-Hamiltonian PDE, the order of such a spatial discretisation is limited by the order of the stage variables. Moreover, the spatial discretisation contains an approximation of the spatial derivatives, and thus, the order of the spatial discretisation may be further limited by the order of this approximation. For the explicit ODEs resulting from an r-stage Lobatto IIIA–IIIB discretisation in space of an appropriate multi-Hamiltonian PDE, the order of this spatial discretisation is r - 1 for r = 10; this is conjectured to hold for higher values of r. For r = 3, I show that a modification to the initial conditions improves the order of this spatial discretisation. It is expected that a similar modification to the initial conditions will improve the order of such spatial discretisations for higher values of r.