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Physical Review A

Collective modes of low-dimensional superfluid Fermi gases in the BCS-BEC crossover: Time-dependent variational analysis

Yu Long Zhou, Guoxiang Huang

Published Feb 12, 2007• 22 citations
Publisher

Abstract

We investigate collective modes and free expansions of quasi-one- and quasi-two-dimensional (Q1D and Q2D) ultracold Fermi gases in the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC). We solve a superfluid order parameter equation valid for the BCS-BEC crossover by employing a time-dependent variational method. We take a trial wave function of hybrid Gaussian-parabolic type, which not only reflects the low-dimensional character of the system but also allows an essentially analytical approach for the problem. We present Q1D and Q2D criteria that are valid in various superfluid regimes and show clearly the relation between the maximum condensed particle number and the parameters of the trapping potential as well as the atom-atom interaction. We demonstrate that, due to the small particle number in Q1D and Q2D condensates, the contribution to oscillating frequencies of collective modes by the quantum pressure in the strong-confinement direction is significant and hence the Thomas-Fermi approximation cannot be used. We also show that the free expansion of Q1D and Q2D superfluid Fermi gases in the strong-confinement direction is much faster than that in the weak-confinement direction.

DOI
10.1103/physreva.75.023611

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Physical Review B•Mar 25, 2008•17 citations
We investigate collective modes and free expansions of quasi-one- and quasi-two-dimensional (Q1D and Q2D) ultracold Fermi gases in the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC). We solve a superfluid order parameter equation valid for the BCS-BEC crossover by employing a time-dependent variational method. We take a trial wave function of hybrid Gaussian-parabolic type, which not only reflects the low-dimensional character of the system but also allows an essentially analytical approach for the problem. We present Q1D and Q2D criteria that are valid in various superfluid regimes and show clearly the relation between the maximum condensed particle number and the parameters of the trapping potential as well as the atom-atom interaction. We demonstrate that, due to the small particle number in Q1D and Q2D condensates, the contribution to oscillating frequencies of collective modes by the quantum pressure in the strong-confinement direction is significant and hence the Thomas-Fermi approximation cannot be used. We also show that the free expansion of Q1D and Q2D superfluid Fermi gases in the strong-confinement direction is much faster than that in the weak-confinement direction.
We investigate the collective modes of a quasi-two-dimensional (Q2D) superfluid Fermi gas in Bardeen–Cooper–Schrieffer Bose–Einstein condensation (BCS-BEC) crossover. For solving a generalized Gross–Pitaevskii equation by using a time-dependent variational method, we take a trial wavefunction with the form of hybrid Gaussian-parabolic type, which not only reflects the Q2D character of the system and also allows an essentially analytical approach of the problem. We present a Q2D criterion that is valid for various superfluid regimes and displays clearly the relation between the maximum condensed particle number and the parameters of trapping potential as well as atom–atom interaction. We show that due to the small particle number in the Q2D condensate, the contribution to oscillating frequencies by the quantum pressure in the strong confinement direction is significant and hence a Thomas–Fermi approximation can not be used.
A perturbation method is developed for calculating the oscillating frequencies of collective modes of trapped superfluid Fermi gases from Bardeen-Cooper-Schrieffer (BCS) superfluid to Bose-Einstein condensation (BEC). Based on the complete and orthogonal eigenfunction set obtained recently and a series expansion of the equation of state beyond the mean-field approximation, the corrections of eigenfrequencies and eigenfunctions of all collective modes are given systematically, which are valid for various superfluid regimes in the BCS-BEC crossover beyond the mean-field approximation. Expressions of frequency correction of the collective modes for both spherical and axial symmetric traps excited near the BCS, BEC, and unitarity limits are provided explicitly, and their dependence on interatomic interaction strength and anisotropic trapping parameter is also discussed in detail.
We study the Bogoliubov collective excitations of harmonically trapped superfluid Fermi gases in the crossover from Bardeen-Cooper-Schrieffer (BCS) superfluid to Bose-Einstein condensate (BEC) beyond Thomas-Fermi (TF) limit. Starting from a generalized Gross-Pitaevskii equation and an equation of state valid for the whole crossover, we derive Bogoliubov--de Gennes (BdG) equations for low-lying collective modes at zero temperature. We use a Fetter-like variational ground state wave function to remove the noncontinuity of slope at the boundary of condensate, which appears in the TF limit. We solve the BdG equations analytically and obtain explicit expressions for all eigenvalues and eigenfunctions, valid for various crossover regimes and for traps with spherical and axial symmetries. We discuss the feature of these collective excitations in the BCS-BEC crossover and show that the theoretical result obtained agrees with available experimental data near and beyond the TF limit.
我们调查集体模式一伪二维(Q2D ) 在 Bardeen-Cooper-Schrieffer Bose 爱因斯坦的超流体费密气体冷凝作用(BCS-BEC ) 转线路。为由使用一个时间家属解决一个概括 Gross-Pitaevskii 方程变化方法，我们与混合 Gaussian 抛物线的类型的形式拿一个试用波浪函数，它不仅反映系统的 Q2D 字符并且也允许这个问题的一条实质上分析的途径。我们在场为各种各样的超流体政体是有效的并且清楚地显示在最大值之间的关系的一个 Q2D 标准压缩了粒子数字和套住潜力以及原子原子相互作用的参数。我们由于在 Q2D 冷凝物的小粒子数字，由量的震荡的频率在强壮的监禁方向迫使的 contribute to 是重要的表演和因此 Thomas 费密近似不能被使用。
We investigate the ground state and quadrupole excitations of an anisotropically trapped Fermi superfluid gas with arbitrary number of atoms in the BCS–BEC crossover based on a time-dependent variational model. A variational trial wavefunction is used to enable an analytical treatment that covers the whole crossover regime. A correct prefactor is validly considered in the power-law equation of state. Then we focus on three quadrupole modes in different cylindrical symmetric traps which correspond to quasi-one-, quasi-two- and three-dimensional (Q1D, Q2D and 3D) superfluids. We note that the contribution to oscillation frequencies by the quantum pressure cannot be neglected and the excitation frequencies in strong confinement direction are much larger than that in the weak confinement direction both in the Q1D and in Q2D cases. The results are compared with the recent experimental and numerical data. Finally, we discuss the free expansion of the condensate.
We investigate the dynamical behavior of large-amplitude collective modes in a superfluid Fermi gas in the crossover from Bardeen--Cooper--Schrieffer (BCS) superfluid to Bose--Einstein condensate (BEC) based on a hydrodynamic approach. We first solve the superfluid hydrodynamic equations that describe the time evolution of fermionic condensates in the BCS-BEC crossover and calculate explicitly the frequency shifts of the collective modes induced by nonlinear effects using the Lindstedt--Poincar\'e method. The result shows that the frequency shifts display different features in different superfluid regimes. We then study the second-harmonic generation of the collective modes under a phase-matching condition, which can be fulfilled by choosing appropriate parameters of the system. The analytical results obtained are checked by numerical simulations and good agreement is found.