Theoretical studies of Bose-Einstein condensates.

摘要

This thesis is a theoretical study of Bose-Einstein Condensation (BEC) in harmonically-trapped, weakly-interacting dilute gases. The motivation for this study is the experimental realization of BEC in trapped alkali gases since 1995. The weak inter-particle interactions and diluteness of the gases allow for a fairly accurate mean-field treatment and justifies a discrete quasi-particle description as we use in this work.; This thesis considers three specific topics in the rapidly growing field of BEC theory: (i) interference effects in BEC, (ii) interaction-induced energy shifts and damping of excitations of condensates and (iii) the properties of highly anisotropic condensates. The results are summarized here:; i. We show that if two multiply occupied boson modes are in eigenstates of the Hermitian relative phase operator, then the visibility of fringes formed by the interference between the modes is necessarily less than unity. For large total occupation numbers the visibility V ≤ π/4. States with definite relative phase and unit visibility do exist. They are related to coherent states and are not orthogonal (not eigenstates of a Hermitian phase operator). This visibility limitation may make it possible to investigate experimentally the physical role of the relative phase eigenstates in interference measurements on BEC.; ii. We evaluate analytically the asymptotic energy shifts of the high energy Bogoliubov quasi-particle modes. In spherical geometry, those modes display a 1/$n$ dependence on their number of radial nodes n but only a weak dependence on their angular momenta l. We obtain similar results for cylindrical geometry. We derive an implicit equation for the widths or decay rates γ of the modes from an assumption of exponential decay. We use the equation to do a detailed numerical study of the trends in the behavior of the widths as a function of temperature, energy, particle number and scattering lengths. In particular, we find that widths due to Landau decay rise rapidly at low n and then declines, while the Beliaev widths rise slowly with n. As temperature → 0, the Beliaev widths reach a constant (>0 for n > 0), while the Landau widths go to zero. The decay rate has a roughly linear dependence on the s-wave scattering length.; iii. For condensates with extreme anisotropy the kinetic energy in the squeezed direction cannot be neglected unless the atom number is extremely large. Therefore, the usual Thomas-Fermi approximation (TFA) breaks down. Alternative analytic forms valid for such condensates are presented for the condensate wavefunction and various physical parameters, and they are compared with results from accurate numerical methods and the TFA. These analytic expressions are used to estimate the eigenvalues and eigenfunctions of excitations of highly anisotropic condensates and to obtain semi-analytic expressions for their widths.