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Physical Review E

Ground-state computation of Bose-Einstein condensates by an imaginary-time quantum lattice Boltzmann scheme

Silvia Palpacelli, Sauro Succi A, Renato Spigler

Published Sep 28, 2007• 42 citations
Publisher

Abstract

The multidimensional formulation of the quantum lattice Boltzmann (qLB) scheme is extended to the case of nonlinear quantum wave equations. More specifically, imaginary-time formulations of the qLB scheme are developed and applied to the numerical computation of the ground state of the Gross-Pitaevskii equation in one and two spatial dimensions. The calculation is validated through detailed comparison with other numerical methods, as well as with analytical results based on the Thomas-Fermi approximation. The linear scaling of the time-step size with the spatial mesh spacing, a distinctive feature of the present quantum kinetic approach, is also numerically demonstrated.

DOI
10.1103/physreve.76.036712

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The multidimensional formulation of the quantum lattice Boltzmann (qLB) scheme is extended to the case of nonlinear quantum wave equations. More specifically, imaginary-time formulations of the qLB scheme are developed and applied to the numerical computation of the ground state of the Gross-Pitaevskii equation in one and two spatial dimensions. The calculation is validated through detailed comparison with other numerical methods, as well as with analytical results based on the Thomas-Fermi approximation. The linear scaling of the time-step size with the spatial mesh spacing, a distinctive feature of the present quantum kinetic approach, is also numerically demonstrated.
The multidimensional formulation of the quantum lattice Boltzmann (QLB) scheme is validated against analytical solutions of the time-dependent nonrelativistic Schrödinger equation in two and three spatial dimensions. As a result, we demonstrate the viability of the quantum lattice Boltzmann scheme for the numerical solution of the time-dependent Schrödinger equation in multiple spatial dimensions. In addition, the links between the QLB scheme and sequential splitting methods for partial differential equations are also clarified.
The derivation of the quantum lattice Boltzmann model is reviewed with special emphasis on recent developments of the model, namely, the extension to a multi-dimensional formulation and the application to the computation of the ground state of the Gross-Pitaevskii equation (GPE). Numerical results for the linear and nonlinear Schrödinger equation and for the ground state solution of the GPE are also presented and validated against analytical results or other classical schemes such as Crank-Nicholson.
The phenomenon of Anderson localization in expanding one-dimensional Bose-Einstein condensates is investigated by numerically solving the Gross-Pitaevskii equation with a random speckle potential. To this purpose, a quantum lattice Boltzmann (QLB) method is used, and compared with a standard Crank-Nicolson scheme. The QLB simulations show evidence of Anderson localization even for relatively low-energy condensates, with a healing length as large as one-tenth of the Thomas-Fermi length. Moreover, very long-time simulations, lasting up to 15 000 optical confinement periods, indicate that the Anderson localization degrades in time, although at a very slow pace. In particular, the inverse localization length is found to decay according to a t;{-1/3} law. This lends support to the idea that localized wave functions, although not strictly ground states, represent extremely long-lived metastable states of the expanding condensate.
In this paper, we mainly review recent results on mathematical theory andnumerical methods for Bose-Einstein condensation (BEC),based on the Gross-Pitaevskii equation (GPE). Starting from the simplest case with one-component BEC of the weakly interacting bosons, we study the reduction of GPE to lower dimensions, the ground states of BEC including the existence and uniqueness as well as nonexistence results, and the dynamics of GPE including dynamical laws, well-posedness of the Cauchy problem as well as the finite time blow-up. To compute the ground state, the gradient flow with discrete normalization (or imaginary time) method is reviewed and various full discretization methods are presented and compared. To simulate the dynamics, both finite difference methods and time splitting spectral methods are reviewed, and their error estimates are briefly outlined. When the GPE has symmetric properties, we show how to simplify the numerical methods. Then we compare two widely used scalings, i.e. physical scaling (commonly used) and semiclassical scaling, for BEC in strong repulsive interaction regime (Thomas-Fermi regime), and discuss semiclassical limits of the GPE. Extensions of these results for one-component BEC are thencarried out for rotating BEC by GPE with an angular momentum rotation, dipolar BEC by GPE with long range dipole-dipole interaction, and two-component BEC by coupled GPEs. Finally, as a perspective, we show briefly the mathematical models forspin-1 BEC, Bogoliubov excitation and BEC at finite temperature.
We review some modelling and numerical aspects in quantum kinetic theory for a gas of interacting bosons and we try to explain what makes Bose-Einstein condensation in a dilute gas mathematically interesting and numerically challenging. Particular care is devoted to the development of efficient numerical schemes for the quantum Boltzmann equation that preserve the main physical features of the continuous problem, namely conservation of mass and energy, the entropy inequality and generalized Bose-Einstein distributions as steady states. These properties are essential in order to develop numerical methods that are able to capture the challenging phenomenon of bosons condensation. We also show that the resulting schemes can be evaluated with the use of fast algorithms. In order to study the evolution of the condensate wave function the Gross-Pitaevskii equation is presented together with some schemes for its efficient numerical solution. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Numerical simulations with previous formulations of the quantum lattice Boltzmann (QLB) scheme in three spatial dimensions showed significant lack of isotropy. In two or more spatial dimensions the QLB approach relies upon operator splitting to decompose the time evolution into a sequence of applications of the one-dimensional QLB scheme along coordinate axes. Each application must be accompanied by a rotation of the wave function into a basis of chiral eigenstates aligned along the relevant axis. The previously observed lack of isotropy was due to an inconsistency in the application of these rotations. Once this inconsistency is removed, the QLB scheme is shown to exhibit isotropic behavior to within a numerical error that scales approximately linearly with the lattice spacing. This establishes the viability of the QLB approach in two and three spatial dimensions.