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ENERGY AND CHEMICAL POTENTIAL ASYMPTOTICS FOR THE GROUND STATE OF BOSE-EINSTEIN CONDENSATES IN THE SEMICLASSICAL REGIME

Weizhu Bao, Fong Yin Lim, Yanzhi Zhang

Published Jan 1, 2007• 17 citations
OpenAlex

Abstract

Asymptotic approximations for the energy and chemical potential of the ground state in Bose-Einstein condensates are presented in the semiclassical regime with several typical trap-ping potentials. As preparatory steps, we begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE), review several typical external trapping potentials, scale the 3D GPE and show how to reduce it to 1D and 2D GPEs in certain limiting trapping frequency regime. For the 1D box potential, we derive asymptotic approximations up to o(1) in term of the scaled interacting pa-rameter d for energy and chemical potential of the ground and all excited states in both weakly interacting regime, i.e. d! 0 and strongly repulsive interacting regime, i.e. d! 1, respec-tively. For the 1D harmonic oscillator, double well and optical lattice potentials, as well as a more general external potential in high dimensions, we get asymptotic approximations up to o(1) in term of the scaled interacting parameter d for the energy and chemical potential of the ground state in semiclassical regime, i.e. d!1. Our extensive numerical results conrm all our asymp-totic approximations, provide convergence rate and suggest several very interesting conclusions.

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Asymptotic approximations for the energy and chemical potential of the ground state in Bose-Einstein condensates are presented in the semiclassical regime with several typical trap-ping potentials. As preparatory steps, we begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE), review several typical external trapping potentials, scale the 3D GPE and show how to reduce it to 1D and 2D GPEs in certain limiting trapping frequency regime. For the 1D box potential, we derive asymptotic approximations up to o(1) in term of the scaled interacting pa-rameter d for energy and chemical potential of the ground and all excited states in both weakly interacting regime, i.e. d! 0 and strongly repulsive interacting regime, i.e. d! 1, respec-tively. For the 1D harmonic oscillator, double well and optical lattice potentials, as well as a more general external potential in high dimensions, we get asymptotic approximations up to o(1) in term of the scaled interacting parameter d for the energy and chemical potential of the ground state in semiclassical regime, i.e. d!1. Our extensive numerical results conrm all our asymp-totic approximations, provide convergence rate and suggest several very interesting conclusions.
The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.
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.
In this paper, we present a theoretical study of a Bose–Einstein condensate of interacting bosons in a quartic trap in one-, two- and three-dimensions. Using Thomas–Fermi approximation, suitably complemented by numerical solutions of the Gross–Pitaevskii equation, we study the ground-state condensate density profiles, the chemical potential, the effects of cross-terms in the quartic potential, temporal evolution of various energy components of the condensate and width oscillations of the condensate. Results obtained are compared with corresponding results for a bose condensate in a harmonic confinement.
We study ground, symmetric and central vortex states, as well as their energy and chemical potential diagrams, in rotating Bose-Einstein condensates (BEC) analytically and numerically.We start from the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with an angular momentum rotation term, scale it to obtain a four-parameter model, reduce it to a 2D GPE in the limiting regime of strong anisotropic confinement and present its semiclassical scaling and geometrical optics.We discuss the existence/nonexistence problem for ground states (depending on the angular velocity) and find that symmetric and central vortex states are independent of the angular rotational momentum.We perform numerical experiments computing these states using a continuous normalized gradient flow (CNGF) method with a backward Euler finite difference (BEFD) discretization.Ground, symmetric and central vortex states, as well as their energy configurations, are reported in 2D and 3D for a rotating BEC.Through our numerical study, we find various configurations with several vortices in both 2D and 3D structures, energy asymptotics in some limiting regimes and ratios between energies of different states in a strong replusive interaction regime.Finally we report the critical angular velocity at which the ground state loses symmetry, numerical verification of dimension reduction from 3D to 2D, errors for the Thomas-Fermi approximation, and spourous numerical ground states when the rotation speed is larger than the minimal trapping frequency in the xy plane.
In this paper, we present a theoretical study of a Bose-Einstein condensate of interacting bosons in a quartic trap in one, two, and three dimensions. Using Thomas-Fermi approximation, suitably complemented by numerical solutions of the Gross-Pitaevskii equation, we study the ground sate condensate density profiles, the chemical potential, the effects of cross-terms in the quartic potential, temporal evolution of various energy components of the condensate, and width oscillations of the condensate. Results obtained are compared with corresponding results for a bose condensate in a harmonic confinement.
Abstract Employing a general variational method and perturbation theory, we derived explicit solutions for the description of one-dimensional two species Bose–Einstein condensates confined by a harmonic trap potential in an optical lattice. We consider the system of two coupled Gross–Pitaevskii equations (GPE) and derive explicit expressions for the chemical potentials and wavefunctions in terms of the atom–atom interaction parameters and laser intensity. We have compared our results with the numerical solutions of the GPE and performed a quantitative analysis for the both considered methods. We underline the importance of the obtained explicit solutions to characterize the density profile or degree of miscibility of the two components.