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Physica Scripta

Mean-field Wigner function of Bose–Einstein condensates in the Thomas–Fermi limit

Julian D. Teske, M R Besbes, B Okhrimenko, R. Walser

Published Oct 26, 2018• 5 citations
Publisher

Abstract

Abstract We study the mean-field phase-space distribution of the ground state of an anisotropic, harmonically trapped Bose–Einstein condensate at zero temperature. First, we derive the Fourier transform of the ground-state wave function in the Thomas–Fermi limit in d dimensions. Second, we obtain analytical approximations for the Thomas–Fermi Wigner function in one and three dimensions with a factorization ansatz of the coherence function. These approximate expressions are compared with exact numerical simulations of the Gross–Pitaevskii equation.

DOI
10.1088/1402-4896/aae516

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Abstract We study the mean-field phase-space distribution of the ground state of an anisotropic, harmonically trapped Bose–Einstein condensate at zero temperature. First, we derive the Fourier transform of the ground-state wave function in the Thomas–Fermi limit in d dimensions. Second, we obtain analytical approximations for the Thomas–Fermi Wigner function in one and three dimensions with a factorization ansatz of the coherence function. These approximate expressions are compared with exact numerical simulations of the Gross–Pitaevskii equation.
In this thesis we study a two-component Bose-Einstein Condensate system in 2D, using a combination of approximate analytical and exact numerical methods of solving the Gross-Pitaevskii equation (GPE). We discuss some of the ways of finding approximate ground state solutions to the GPE, one of them being the Thomas-Fermi approximation. Using the Thomas-Fermi approximation, one can separate the system into two regimes; one where both components are disks and one where one of the components is a disk and the other component is an annulus. However, the Thomas-Fermi approximation does not hold true beyond a critical value. We demonstrate a method using which one can get more precise solutions for the GPE. This involves using a method of approximation of the GPE near the critical boundaries, which has been previously used for a single component Bose gas. This can be adapted for a two-component system in order obtain explicit solutions for a BEC in a 2D harmonic trap with repulsive interactions. We show in detail how this approximation works for the different regimes suggested by the Thomas-Fermi analysis and also for more generalized cases where one has different masses, particle numbers and trapping frequencies for the two components. This thesis also talks about how one could analyse the system using this approximation in 1D and 3D, shedding more light on what might be the parameters affecting the precision of the solution.
We derive a unified set of scaling equations for Bose-Einstein condensates in time-dependent harmonic traps, connecting the weakly interacting Gaussian regime to the strongly interacting Thomas-Fermi regime. The exact Gross-Pitaevskii ground-state density is taken as a fixed profile in rescaled coordinates, with its dynamics described by three scaling factors corresponding to compression or expansion along the three spatial directions. The resulting equations contain no adjustable parameter, since their coefficients are determined once from the initial state. They recover analytically the Gaussian variational and Thomas-Fermi scaling equations in their respective limits and satisfy an axis-resolved virial theorem. The approach also yields the global phase and the low-lying collective-mode frequencies of the condensate. We benchmark the model against three-dimensional Gross-Pitaevskii simulations and against measured expansion energies. The model remains accurate across all interaction regimes, in strongly anisotropic traps, and along time-dependent sequences including a relatively fast quench, up to the point where the underlying frozen-profile hypothesis, shared by all three scaling approaches, breaks down.
In this work, we study phase transitions in dilute Bose-Einstein Condensates theoretically. The Gross-Pitaevskii equation (GPE) is applied to describe the properties of dilute Bose gases near zero temperature for various confining geometries. Then, using the harmonic trap, the Thomas Fermi equation has been investigated. The Bose-Hubbard Model has been also investigated using the mean field approach. It is indicated that Bose-Einstein condensation is a second order phase transition. We also presented an exactly solvable phase transition model in which the phase transition is purely statistically derived. It is found out that the mean field theory can be applied to a number of physical systems so as to study the phenomena of Berezinsky-Kosterlitz- Thouless (BKT) phase transitions. Keywords : Phase Transition, BEC, GPE, Bose-Hubbard Model, Berezinsky-Kosterlitz-Thouless (BKT), Mean Field equation
Abstract This chapter uses Gross–Pitaevskii theory in order to calculate the order parameter relative to the ground state of a weakly interacting Bose gas confined by an external harmonic potential. Both the density and the momentum distribution are calculated and an explicit derivation of the virial theorem is presented. The chapter discusses the Thomas–Fermi limit, where the effects of quantum pressure are neglected. The importance of both finite-size and beyond-mean-field effects are discussed. The chapter finally derives solutions for the Gross–Pitaevskii equation in the case of attractive interactions and show that the quantum pressure plays a crucial role in preventing collapse when the number of atoms is small.
The state of a rotating Bose-Einstein condensate in a harmonic trap is modeled by a wave function that minimizes the Gross-Pitaevskii functional. The resulting minimization problem has two new features compared to other similar functionals arising in condensed matter physics, such as the Ginzburg-Landau functional. Namely, the wave function is defined in all the plane and is normalized relative to the L2L^2L2-norm. This paper deals with the situation when the coupling constant tends to 000 (Thomas-Fermi regime) and the rotation speed is large compared with the first critical speed. It is given the leading order estimate of the ground state energy together with the location of the vortices of the minimizing wave function in the bulk of the condensate. When the rotation speed is inversely proportional to the coupling constant, the condensate is confined in an elliptical region whose conjugate diameter shrinks and whose transverse diameter expands as the rotation speed increases.