Energy and Chemical Potential Asymptotics for the Ground State of Bose-Einstein Condensates in the Semiclassical Regime

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 trapping 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 parameter βd for energy and chemical potential of the ground and all excited states in both weakly interacting regime, i.e. βd→0 and strongly repulsive interacting regime, i.e. βd→∞, respectively. 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 βd for the energy and chemical potential of the ground state in semiclassical regime, i.e. βd→∞. Our extensive numerical results confirm all our asymptotic approximations, provide convergence rate and suggest several very interesting conclusions.

Department(s)

Mathematics and Statistics

Keywords and Phrases

Bose-Einstein condensation; Gross-Pitaevskii equation; box potential; energy; chemical potential; harmonic oscillator potential; double well potential; optical lattice potential; ground state; excited state; semiclassical regime

Document Type

Newsletter

Document Version

Citation

File Type

text

Language(s)

English

Rights

© 2007 Institute of Mathematics Academia Sinica, All rights reserved.

Publication Date

01 Jan 2007

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