New Coherence and RIP Analysis for Weak Orthogonal Matching Pursuit

In this paper we define a new coherence index, named the global 2-coherence, of a given dictionary and study its relationship with the traditional mutual coherence and the restricted isometry constant. By exploring this relationship, we obtain more general results on sparse signal reconstruction using greedy algorithms in the compressive sensing (CS) framework. In particular, we obtain an improved bound over the best known results on the restricted isometry constant for successful recovery of sparse signals using orthogonal matching pursuit (OMP).Comment: arXiv admin note: substantial text overlap with arXiv:1307.194

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

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