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Journal of Statistical Mechanics Theory and Experiment

Theoretical results of strongly interacting quasi-one-dimensional Fermi gases in the BCS–BEC crossover

Zhou Li, Hang Dong, Yong-li Ma

Published Dec 16, 2014• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1742-5468/2014/12/p12016

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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.
We propose an eigenfunction expansion method to solve the integral equations raised by one-dimensional δ-interaction integrable models. Based on this method, the crossover from BCS superfluid of singlet pairs to BEC of molecules is exactly analyzed in both ground and excited states in terms of the momentum distributions. In the exact one-dimensional model, we derive the expressions and calculate the dependence of several physical quantities on the inverse of the dimensionless coupling parameter γ. These physical quantities include the ground state energy, chemical potential, momentum cutoff, charge velocity, excitation energy, excitation momentum, spin gap and spin velocity. Our exact results agree well with previous numerical results. In the practical quasi-one-dimensional model, we apply the method to the ground state and the results give a clear BCS–BEC crossover picture.
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 consider a two-component gas of fermionic atoms confined to a quasi-two-dimensional (quasi-2D) geometry by a harmonic trapping potential in the transverse direction. We construct a mean-field theory of the BCS-BEC crossover at zero temperature that allows us to extrapolate to an infinite number of transverse harmonic oscillator levels. In the extreme BEC limit, where the confinement length exceeds the dimer size, we recover 3D dimers confined to two dimensions with weak repulsive interactions. However, even when the interactions are weak and the Fermi energy is less than the confinement frequency, we find that the higher transverse levels can substantially modify fermion pairing. We argue that recent experiments on pairing in quasi-2D Fermi gases [Y. Zhang et al., Phys. Rev. Lett. 108, 235302 (2012)] have already observed the effects of higher transverse levels.
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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.
Abstract We study the crossover from the Bardeen–Cooper–Shrieffer regime to the Bose–Einstein-condensation regime in an anisotropic trapped quantum gas of ultracold fermionic atoms. Using an effective two-dimensional Hamiltonian with renormalized interactions between atoms and dressed molecules, we extend the Gaussian pair fluctuation theory to quasi-two-dimensional systems. The equation of state, pair size and sound velocity as well as the convexity parameters of the Goldstone mode are investigated to show how Fermi superfluidity is affected by reduced dimensionality and Gaussian fluctuations at zero temperature in a wide range of crossover. We compare our results with the recent experiment of 6 L i atomic gases and find good agreements.