School of Physics - Theses

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    On the one-dimensional bose gas
    Makin, Melissa I. ( 2007-02)
    The main work of this thesis involves the calculation, using the Bethe ansatz, of two of the signature quantities of the one-dimensional delta-function Bose gas. These are the density matrix and concomitantly its Fourier transform the occupation numbers, and the correlation function and concomitantly its Fourier transform the structure factor. The coefficient of the delta-function is called the coupling constant; these quantities are calculated in the finite-coupling regime, both expansions around zero coupling and infinite coupling are considered. Further to this, the density matrix in the infinite coupling limit, and its first order correction, is recast into Toeplitz determinant form. From this the occupation numbers are calculated up to 36 particles for the ground state and up to 26 particles for the first and second excited states. This data is used to fit the coefficients of an ansatz for the occupation numbers. The correlation function in the infinite coupling limit, and its first order correction, is recast into a form which is easy to calculate for any N, and is determined explicitly in the thermodynamic limit.
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    The Jaynes-Cummings-Hubbard model
    Makin, Melissa I. ( 2011)
    This thesis comprises an intensive investigation of the Jaynes-Cummings-Hubbard (JCH)system. This Hamiltonian describes a system of coupled photonic cavities, each cavitycontains a single two-level system. This system is rich in the physics it contains. Thepresence of the two-level system provides a mechanism by which the photons may interactwith each other, providing an interesting array of non-linear phenomena. Using the meanfieldapproximation, the phase diagram of this system has been shown to display what areeffectively two different phases - a superfluid phase, and a Mott-insulator phase. In thisthesis we show that by using exact diagonalisation for a finite number of cavities, thereis a rich structure to the phases that goes beyond this dual division of phases. We alsogenerate phase diagrams that could be experimentally realised using an ion trap system.Investigating the time dependent properties of the one-dimensional JCH system, we obtainboth localised and delocalised behaviour, despite having only one excitation in the system.In certain limits the 1D JCH system approximates two Heisenberg spin chains. We findit is also possible in the one excitation, one-dimension time dependent case to activelycontrol the location of the excitation, by means of a potential, thus creating all standardcomponents of linear optics: we specifically investigate waveguides and beamsplitters. Finally,we investigate the use of matrix product states (MPS) to study the one-dimensionalJCH system in the time domain. MPS is used to show how two colliding excitations canshow the signature of photon blockade.