Quantum lattice Boltzmann methods for the linearand nonlinear Schrödinger equation in several dimensions

Abstract

In the last decade the lattice kinetic approach to fluid dynamics, and notably the Lattice Boltzmann (LB) method, has consolidated into a powerful alternative to the discretization of the Navier-Stokes equations for the numerical simulation of a wide range of complex fluid flows. However, to date, the overwhelming majority of LB work has been directed to the investigation of classical (non quantum) fluids. Nonetheless a small group of authors have also investigated lattice kinetic formulations of quantum mechanics which led to the definition of the so-called quantum lattice gas methods for solving linear and nonlinear Schrodinger equations. The earliest LB model for quantum motion was proposed by Succi and Benzi in 1993 and it built upon a formal analogy between the Dirac equations and a Boltzmann equation satisfied by a complex distribution function. This first quantum lattice Boltzmann (qLB) scheme was formulated in multi- dimensions but it was numerically validated only in one space dimension. Indeed, the first result of this thesis is the effective numerical extension and validation of the multi-dimensional qLB scheme. In particular, we present a numerical study of the two- and three- dimensional qLB model, based on an operator splitting approach. Our results show a satisfactory agreement with the analytical solutions, thereby demonstrating the validity of the three-step stream-collide-rotate theoretical structure of the multi-dimensional qLB scheme. Moreover, we extend the qLB model by developing an imaginary-time version of the scheme in order to compute the ground state solution of the Gross-Pitaevskii equation (GPE). The GPE is commonly used to describe the dynamics of zero-temperature Bose-Einstein condensates (BEC) and it is a nonlinear Schrodinger equation with a cubic nonlinearity. The ground state solution of the GPE is the eigenstate which corresponds to the minimum energy level. Typically, this minimizer is found by applying to the GPE a transformation, known as Wick rotation, which consists on "rotating" the time axis on the complex plane so that time becomes purely imaginary. With this rotation of the time axis, the GPE becomes a diffusion equation with an absorption/emission term given by the nonlinear potential. Thus, the basic idea behind the imaginary-time qLB model is to apply the Wick rotation to the real-time qLB scheme. The imaginary-time qLB scheme is also extended to multi-dimensions by using the same splitting operator approach already applied to the real-time qLB model. In addition, we apply the qLB scheme to the study of the dynamics of a BEC in a random potential, which is a very active topic in present time research on condensed matter and atomic physics research. In particular, we investigate the conditions under which an expanding BEC in a random speckle potential can exhibit Anderson localization. Indeed, it is well known that disorder can profoundly affect the behavior of quantum systems, Anderson localization being one of the most fascinating phenomena in point. Here, we explore the use of qLB for the case of nonlinear interactions with random potentials and, in particular, we investigate the mechanism by which the localized state of the BEC is modified by the residual self-interaction in the (very) long-time term evolution of the condensate. These studies have demonstrated the viability of the qLB model as numerical algorithm for solving linear and nonlinear Schrodinger equations for both the time-dependent and ground state solutions, even in external random potentials. Such lattice kinetic methods for quantum mechanics represent interesting numerical schemes, which can be easily implemented and retain the usual attractive features of LB methods: simplicity, computational speed, straight- forward parallel implementation.