Abstract:
The Ginzburg-Landau (GL) equations for a d-wave superconductor are derived within the context of two microscopic lattice models used to describe the cuprates: the extended Hubbard model and the Antiferromagnetic-van Hove model. Both models have pairing on nearest-neighbour links, consistent with theories for d-wave superconductivity mediated by spin fluctuations. Analytical results obtained for the extended Hubbard model at low electron densities and weak-coupling are compared to results reported previously for a d-wave superconductor in the continuum. The variation of the coefficients in the GL equations with carrier density, temperature, and coupling constants are calculated numerically for both models. The relative importance of anisotropic higher-order terms in the GL free energy is investigated, and the implications for experimental observations of the vortex lattice are considered.

Abstract:
The properties of a rotating Bose-Einstein condensate confined in a prolate cylindrically symmetric trap are explored both analytically and numerically. As the rotation frequency increases, an ever greater number of vortices are energetically favored. Though the cloud anisotropy and moment of inertia approach those of a classical fluid at high frequencies, the observed vortex density is consistently lower than the solid-body estimate. Furthermore, the vortices are found to arrange themselves in highly regular triangular arrays, with little distortion even near the condensate surface. These results are shown to be a direct consequence of the inhomogeneous confining potential.

Abstract:
The ground and excited states of a weakly interacting and dilute Bose-Einstein condensed gas, confined in a completely anisotropic harmonic oscillator potential, are determined at zero temperature within the Bogoliubov approximation. The numerical calculations employ a computationally efficient procedure based on a discrete variable representation (DVR) of the Hamiltonian. The DVR is efficient for problems where the interaction potential may be expressed as a local function of interparticle coordinates. In order to address condensates that are both very large (millions of atoms) and fully anisotropic, the ground state is found using a self-consistent field approach. Experience has demonstrated, however, that standard iterative techniques applied to the solution of the non-linear partial differential equation for the condensate are non-convergent. This limitation is overcome using the method of direct inversion in the iterated subspace (DIIS). In addition, the sparse structure of the DVR enables the efficient application of iterative techniques such as the Davidson and/or Lanczos methods, to extract the eigenvalues of physical interest. The results are compared with recent experimental data obtained for Bose-Einstein condensed alkali metal vapors confined in magnetic traps.

Abstract:
The increase in fatigue life under a constant applied load, due to the presence of crack shielding, was studied. Time to failure was estimated by using a mathematical function that describes the crack shielding developed during subcritical crack growth. Such results were compared with experimental fatigue lives obtained in Y-TZP/PSZ materials. From this it may be concluded that the initial slope of crack shielding curves greatly influences the time to failure. Se ha estudiado el aumento en el tiempo a rotura bajo la aplicación de una carga constante debido a la presencia de apantallamiento. Este tiempo se evaluó considerando una función matemática que describe el apantallamiento desarrollado en propagación subcrítica y los valores obtenidos se comparan con resultados experimentales correspondientes a materiales Y-TZP/PSZ. Se concluye que la pendiente inicial de la curva de apantallamiento ejerce gran influencia en la vida a fatiga del material.

Abstract:
A set of necessary and sufficient conditions are derived for the equivalence of an arbitrary pure state and a graph state on n qubits under stochastic local operations and classical communication (SLOCC), using the stabilizer formalism. Because all stabilizer states are equivalent to a graph state by local unitary transformations, these conditions constitute a classical algorithm for the determination of SLOCC-equivalence of pure states and stabilizer states. This algorithm provides a distinct advantage over the direct solution of the SLOCC-equivalence condition for an unknown invertible local operator S, as it usually allows for easy detection of states that are not SLOCC-equivalent to graph states.

Abstract:
Twin boundaries in orthorhombic d-wave superconductors are investigated numerically using the Bogoliubov-deGennes formalism within the context of an extended Hubbard model. The twin boundaries are represented by tetragonal regions of variable width, with a reduced chemical potential. For sufficiently large twin boundary width and change in chemical potential, an induced s-wave component may break time-reversal symmetry at a low temperature. This temperature, and the magnitude of the complex component, are found to depend strongly on electron density. The results are compared with recent tunneling measurements.

Abstract:
The Density Matrix Renormalization Group algorithm is used to characterize the ground states of one-dimensional coupled cavities in the regime of low photon densities. Numerical results for photon and spin excitation densities, one- and two-body correlation functions, superfluid and condensate fractions, as well as the entanglement entropy and localizable entanglement are obtained for the Jaynes-Cummings-Hubbard (JCH) model, and are compared with those for the Bose-Hubbard (BH) model where applicable. The results indicate that a Tonks-Girardeau phase, in which the photons are strongly fermionized, appears between the Mott-insulating and superfluid phases as a function of the inter-cavity coupling. In fact, the superfluid density is found to be zero in a wide region outside the Mott-insulator phase boundary. The presence of two different species of excitation (spin and photon) in the JCH model gives rise to properties with no analog in the BH model, such as the (quasi)condensation of spin excitations and the spontaneous generation of entanglement between the atoms confined to each cavity.

Abstract:
This paper presents a mean-field numerical analysis, using the full three-dimensional time-dependent Gross-Pitaevskii equation (GPE), of an experiment carried out by Orzel et al. [Science 291, 2386 (2001)] intended to show number squeezing in a gaseous Bose-Einstein condensate in an optical lattice. The motivation for the present work is to elucidate the role of mean-field effects in understanding the experimental results of this work and those of related experiments. We show that the non-adiabatic loading of atoms into optical lattices reproduces many of the main results of the Orzel et al. experiment, including both loss of interference patterns as laser intensity is increased and their regeneration when intensities are lowered. The non-adiabaticity found in the GPE simulations manifests itself primarily in a coupling between the transverse and longitudinal dynamics, indicating that one-dimensional approximations are inadequate to model the experiment.

Abstract:
The mean-field properties of finite-temperature Bose-Einstein gases confined in spherically symmetric harmonic traps are surveyed numerically. The solutions of the Gross-Pitaevskii (GP) and Hartree-Fock-Bogoliubov (HFB) equations for the condensate and low-lying quasiparticle excitations are calculated self-consistently using the discrete variable representation, while the most high-lying states are obtained with a local density approximation. Consistency of the theory for temperatures through the Bose condensation point requires that the thermodynamic chemical potential differ from the eigenvalue of the GP equation; the appropriate modifications lead to results that are continuous as a function of the particle interactions. The HFB equations are made gapless either by invoking the Popov approximation or by renormalizing the particle interactions. The latter approach effectively reduces the strength of the effective scattering length, increases the number of condensate atoms at each temperature, and raises the value of the transition temperature relative to the Popov approximation. The renormalization effect increases approximately with the log of the atom number, and is most pronounced at temperatures near the transition. Comparisons with the results of quantum Monte Carlo calculations and various local density approximations are presented, and experimental consequences are discussed.

Abstract:
Dark soliton states of Bose-Einstein condensates in harmonic traps are studied both analytically and computationally by the direct solution of the Gross-Pitaevskii equation in three dimensions. The ground and self-consistent excited states are found numerically by relaxation in imaginary time. The energy of a stationary soliton in a harmonic trap is shown to be independent of density and geometry for large numbers of atoms. Large amplitude field modulation at a frequency resonant with the energy of a dark soliton is found to give rise to a state with multiple vortices. The Bogoliubov excitation spectrum of the soliton state contains complex frequencies, which disappear for sufficiently small numbers of atoms or large transverse confinement. The relationship between these complex modes and the snake instability is investigated numerically by propagation in real time.