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arXiv:cond-mat/0111548 · Physical Review A

Manipulating the critical temperature for the superfluid phase transition in trapped atomic Fermi gases

Christopher P. Search, Han Pu, W. Zhang, Brian P. Anderson, Pierre Meystre

Published Jun 14, 2002• 10 citations
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Abstract

We examine the effect of the trapping potential on the critical temperature ${T}_{C}$ for the BCS transition to a superfluid state in trapped atomic gases of fermions. ${T}_{C}$ for an arbitrary power-law trap is calculated in the Thomas-Fermi approximation. For anharmonic traps, ${T}_{C}$ can be increased by several orders of magnitude in comparison to a harmonic trap. Our theoretical results indicate that, in practice, one could manipulate the critical temperature for the BCS phase transition by shaping the traps confining the atomic Fermi gases.

DOI
10.1103/physreva.65.063616

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We examine the effect of the trapping potential on the critical temperature TC{T}_{C}TC​ for the BCS transition to a superfluid state in trapped atomic gases of fermions. TC{T}_{C}TC​ for an arbitrary power-law trap is calculated in the Thomas-Fermi approximation. For anharmonic traps, TC{T}_{C}TC​ can be increased by several orders of magnitude in comparison to a harmonic trap. Our theoretical results indicate that, in practice, one could manipulate the critical temperature for the BCS phase transition by shaping the traps confining the atomic Fermi gases.
We examine a dilute two-component atomic Fermi gas trapped in a harmonic potential in the superfluid phase. For experimentally realistic parameters, the trapping potential is shown to have crucial influence on various properties of the gas. Using an effective Hamiltonian, analytical results for the critical temperature, the temperature dependence of the superfluid gap, and the energy of the lowest collective modes are derived. These results are shown to agree well with numerical calculations. We also discuss in more detail a previously proposed method to experimentally observe the superfluid transition by looking at the collective-mode spectrum. Our results are aimed at the present experimental effort to observe a superfluid phase transition in a trapped atomic Fermi gas.
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We derive semiclassical transport equations for a trapped atomic Fermi gas in the BCS phase at temperatures between zero and the superfluid transition temperature. These equations interpolate between the two well-known limiting cases of superfluid hydrodynamics at zero temperature and the Vlasov equation at the critical one. The linearized version of these equations, valid for small deviations from equilibrium, is worked out and applied to two simple examples where analytical solutions can be found: a sound wave in a uniform medium and the quadrupole excitation in a spherical harmonic trap. In spite of some simplifying approximations, the main qualitative results of quantum mechanical calculations are reproduced, which are the different frequencies of the quadrupole mode at zero and the critical temperature and strong Landau damping at intermediate temperatures. In addition we suggest a numerical method for solving the semiclassical equations without further approximations.
In this paper, taking into account the effect of the induced interaction, we calculate the superfluid-transition temperature of ultracold Fermi gases within the framework of the non-self-consistent TTT-matrix approximation. The result shows that the transition temperature is reduced by about 1010%10 at unitarity due to the induced interaction. In the Bardeen-Cooper-Schrieffer (BCS) limit, the critical temperature is about 0.45TcBCS0.45{T}_{c}^{\text{BCS}}0.45TcBCS​ and it is the same as the result of the usual Gorkov and Melik-Barkhudarov correction, whereas in the Bose-Einstein condensate (BEC) limit, the influence of the induced interaction can be neglected and the critical temperature turns out to be the transition temperature of ideal Bose gases. Our results show that the induced interaction cannot be neglected in the study of the BCS-BEC crossover.
We present a detailed study of the BCS pairing transition in a trapped polarized dipolar Fermi gas. In the case of a shallow nearly spherical trap, we find a decrease in the transition temperature as a function of the trap aspect ratio and predict the existence of the optimal trap geometry. The latter corresponds to the highest critical temperature of the BCS transition for a given number of particles. We also derive the phase diagram for an ultracold trapped dipolar Fermi gas in the situation where the trap frequencies can be of the order of the critical temperature of the BCS transition in the homogeneous case, and determine the critical value of the dipole–dipole interaction energy below which the BCS transition ceases to exist. The critical dipole strength is obtained as a function of the trap aspect ratio. Alternatively, for a given dipole strength, there is a critical value of the trap anisotropy for the BCS state to appear. The order parameter calculated at criticality exhibits novel non-monotonic behaviour resulting from the combined effect of the confining potential and the anisotropic character of the interparticle dipole–dipole interaction.