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arXiv:0911.0815 · Physical Review A

Self-trapping of a Fermi superfluid in a double-well potential in the Bose-Einstein-condensate–unitarity crossover

Sadhan K. Adhikari, Hong Bing Lu, Han Pu

Published Dec 3, 2009• 55 citations
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Abstract

We derive a generalized Gross-Pitaevskii density-functional equation appropriate to study the Bose-Einstein condensate (BEC) of dimers formed of singlet spin-half Fermi pairs in the BEC-unitarity crossover while the dimer-dimer scattering length $a$ changes from 0 to $\ensuremath{\infty}$. Using an effective one-dimensional form of this equation, we study the phenomenon of dynamical self-trapping of a cigar-shaped Fermi superfluid in the entire BEC-unitarity crossover in a double-well potential. A simple two-mode model is constructed to provide analytical insights. We also discuss the consequence of our study on the self-trapping of an atomic BEC in a double-well potential.

DOI
10.1103/physreva.80.063607

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We derive a generalized Gross-Pitaevskii density-functional equation appropriate to study the Bose-Einstein condensate (BEC) of dimers formed of singlet spin-half Fermi pairs in the BEC-unitarity crossover while the dimer-dimer scattering length aaa changes from 0 to \ensuremath∞\ensuremath{\infty}\ensuremath∞. Using an effective one-dimensional form of this equation, we study the phenomenon of dynamical self-trapping of a cigar-shaped Fermi superfluid in the entire BEC-unitarity crossover in a double-well potential. A simple two-mode model is constructed to provide analytical insights. We also discuss the consequence of our study on the self-trapping of an atomic BEC in a double-well potential.
We investigate tunneling and self-trapping of superfluid Fermi gases under a two-mode ansatz in different regimes of the crossover from Bardeen—Cooper—Schrieffer (BCS) superfluid to Bose—Einstein condensates (BEC). Starting from a generalized equation of state, we derive the coupled equations of relative atom-pair number and relative phase about superfluid Fermi gases in a double-well system and then classify the different oscillation behaviors by the tunneling strength and interactions between atoms. Tunneling and self-trapping behaviors are considered in the whole BCS-BEC crossover in the case of a symmetric double-well potential. We show that the nonlinear interaction between atoms makes the self-trapping more easily realized in BCS regime than in the BEC regime and stability analysis is also given.
Starting from the three-dimensional (3D) time-dependent nonlinear Gross-Pitaevskii equation for a Bose-Einstein condensate (BEC) and density functional (DF) equation for a Fermi superfluid at the unitarity and Bardeen-Cooper-Schrieffer (BCS) limits, we derive effective one- (1D) and two-dimensional (2D) mean-field equations, respectively, for the dynamics of a trapped cigar- and disk-shaped BEC and Fermi superfluid by using the adiabatic approximation. The reduced 1D and 2D equations for a cigar- and disk-shaped Fermi superfluid have simple analytic nonlinear terms and at unitarity produce results for stationary properties and non-stationary breathing oscillation and free expansion in excellent agreement with the solution of the full 3D equation.
Starting from the three-dimensional (3D) time-dependent nonlinear Gross–Pitaevskii equation for a Bose–Einstein condensate (BEC) and the density-functional (DF) equation for a Fermi superfluid at the unitarity and Bardeen–Cooper–Schrieffer (BCS) limits, we derive effective one- (1D) and two-dimensional (2D) mean-field equations, respectively, for the dynamics of a trapped cigar- and disc-shaped BEC and Fermi superfluid by using the adiabatic approximation. The reduced 1D and 2D equations for a cigar- and disc-shaped Fermi superfluid have simple analytic non-linear terms and at unitarity produce results for stationary properties and non-stationary breathing oscillation and free expansion in excellent agreement with the solution of the full 3D equation.
We derive a generalized time-dependent Galilean-invariant density-functional (DF) equation appropriate to study the stationary and non-stationary properties of a trapped Fermi super-fluid in the Bardeen–Cooper–Schrieffer (BCS) to Bose–Einstein condensation (BEC) crossover. This equation is equivalent to a quantum hydrodynamical equation for a Fermi super-fluid. The bulk chemical potential of this equation has the proper (model-independent) dependence on the Fermi–Fermi scattering length in the BCS and BEC limits. We apply this DF equation to the study of the stationary density profile and size of a cigar-shaped Fermi super-fluid of 6 Li atoms, and the results are in good agreement with the experiment of Bartenstein et al in the BCS–BEC crossover. We also apply the DF equation to the study of axial and radial breathing oscillation and our results for these frequencies are in good agreement with experiments in the BCS–BEC crossover.
On the basis of quantum hydrodynamical equations we derive a unitarity Schrödinger equation of a finite trapped superfluid Fermi gas valid in the whole interaction regime from BCS superfluid to BEC. This equation is just the Ginzburg–Laudau-type equation for the fermionic Cooper pairs in the BCS side, the Gross–Pitaevskii-type equation for the bosonic dimers in the BEC side, and a unitarity equation for a strongly interacting Fermi superfluid in the unitarity limit. By taking a modified Gauss-like trial wave function, we solve the unitarity Schrödinger equation, calculate the energy, chemical potential, sizes and profiles of the ground-state condensate, and discuss the properties of the ground state in the entire BCS-BEC crossover regimes.
By considering the contribution of the higher-order term representing the lowest approximation of beyond mean field corrections, we investigate a superfluid Fermi gas confined in a double-well potential in Bose–Einstein Condensation (BEC) side of the Bardeen–Cooper–Schrieffer (BCS) to BEC crossover. Two limited cases of deep BEC regime and BEC regime of BCS–BEC crossover, corresponding to the two-body scattering length a sc is small enough and large enough, respectively. We derive a simple two-mode model that could depict the dynamics effectively. With making thorough analysis on the two-mode model and its corresponding classical Hamiltonian, we find that the Josephson oscillation or self-trapping phenomenon could emerge at certain parameters. We find three kinds of the phase states: Josephson oscillation (JO), oscillating-phase-type self-trapping (OPTST) and running-phase-type self-trapping (RPTST). The dependence of these three phase states on the dimensionless interaction parameter y = 1/(kFa sc ) and the initial system energy are given in this paper.