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Journal of Physics B Atomic Molecular and Optical Physics

Ground state and quadrupole excitations of anisotropically trapped Fermi superfluids with arbitrary atom numbers in the BCS–BEC crossover

Zhou Li, Changru Ma, Yong-li Ma

Published Nov 19, 2007• 9 citations
Publisher

Abstract

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.

DOI
10.1088/0953-4075/40/24/004

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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 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.
We present a quasi-one-dimensional (Q1D) Bose–Fermi mixture model of strongly interacting Fermi gases through the confinement-induced resonance (CIR) to describe 3D fermions trapped in a Q1D geometry along the z axis. Based on the 1D Bethe ansatz method, we analyse the ground and excited states of the Q1D system and derive their integral equations in terms of the momentum distributions in the thermodynamic limit. By applying a Legendre expansion method, the crossover from the BCS superfluid of singlet pairs to the BEC of molecules is calculated in the ground and excited states. We then obtain the dependence of several physical quantities on the inverse of a dimensionless coupling parameter a ⊥ / a with a dimensionless density parameter na ⊥ of Q1D where is the transverse size and ω ⊥ is the 2D harmonic trap frequency in the xy plane. Our results can go back to the smooth 1D crossover when na ⊥ → 0. The theoretical results quantitatively show that the strongly coupling Fermi gas in the Q1D behaves as a Bose–Fermi mixture near the crossover.
Abstract In this work, we present a theoretical study for the collective oscillation modes, i.e. quadrupole, radial and axial mode, of a mixture of Bose and Fermi superfluids in the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a molecular Bose–Einstein condensate (BEC) in harmonic trapping potentials with cylindrical symmetry of experimental interest. To this end, we start from the coupled superfluid hydrodynamic equations for the dynamics of Bose–Fermi superfluid mixtures and use the scaling theory that has been developed for a coupled system. The collective oscillation modes of Bose–Fermi superfluid mixtures are found to crucially depend on the overlap integrals of the spatial derivations of density profiles of the Bose and Fermi superfluids at equilibrium. We not only present the explicit expressions for the overlap density integrals, as well as the frequencies of the collective modes provided that the effective Bose–Fermi coupling is weak, but also test the valid regimes of the analytical approximations by numerical calculations in realistic experimental conditions. In the presence of a repulsive Bose–Fermi interaction, we find that the frequencies of the three collective modes of the Bose and Fermi superfluids are all upshifted, and the change speeds of the frequency shifts in the BCS–BEC crossover can characterize the different groundstate phases of the Bose–Fermi superfluid mixtures for different trap geometries.
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.
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.