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Physical Review A

Fluctuation corrections to the mean-field interaction for a trapped Fermi gas

Jin Woo Jun

Published Apr 17, 2003• 2 citations
Publisher

Abstract

Assuming that the scattering length properly describes the interaction, and taking into account the lowest-order correction effects to the strong mean-field interaction, we find that the density distributions of a Fermi gas in each spin state do not change significantly under the BCS transition, but the critical temperature is reduced significantly in comparison with the case without the fluctuation effect. The inclusion of the fluctuation correction effect for a large value of scattering length affects the expansion of a Fermi gas only in a minor way The expansion of a Fermi gas depends in a crucial way on whether the system is in the collisionless regime or in the hydrodynamic regime.

DOI
10.1103/physreva.67.043608

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Assuming that the scattering length properly describes the interaction, and taking into account the lowest-order correction effects to the strong mean-field interaction, we find that the density distributions of a Fermi gas in each spin state do not change significantly under the BCS transition, but the critical temperature is reduced significantly in comparison with the case without the fluctuation effect. The inclusion of the fluctuation correction effect for a large value of scattering length affects the expansion of a Fermi gas only in a minor way The expansion of a Fermi gas depends in a crucial way on whether the system is in the collisionless regime or in the hydrodynamic regime.
We calculate the effects of two-body interactions on the low-frequency oscillations of a normal Fermi gas confined in a harmonic trap. The mean-field contribution to the collective frequencies is evaluated in the collisionless regime using a sum-rule approach. We also discuss the transition between the collisionless and hydrodynamic regimes with special emphasis on the spin dipole mode in which two atomic clouds occupying different spin states oscillate in the opposite phase. The spin dipole mode is predicted to be overdamped in the hydrodynamic regime. The relaxation time is calculated as a function of temperature and the effects of Fermi statistics are explicitly pointed out.
We investigate the 1/N1/N1/N expansion proposed recently as a strategy to include quantum fluctuation effects in the nonrelativistic, attractive Fermi gas at and near unitarity. We extend the previous results by calculating the next-to-leading order corrections to the critical temperature along the whole crossover from Bardeen-Cooper-Schrieffer (BCS) superconductivity to Bose-Einstein condensation. We demonstrate explicitly that the extrapolation from the mean-field approximation, based on the 1/N1/N1/N expansion, provides a useful approximation scheme only on the BCS side of the crossover. We then apply the technique to the study of strongly interacting relativistic many-fermion systems. Having in mind the application to color superconductivity in cold dense quark matter, we develop, within a simple model, a formalism suitable to compare the effects of order parameter fluctuations in phases with different pairing patterns. Our main conclusion is that the relative correction to the critical temperature is to a good accuracy proportional to the mean-field ratio of the critical temperature and the chemical potential. As a consequence, it is significant even rather deep in the BCS regime, where phenomenologically interesting values of the quark--quark coupling are expected. Possible impact on the phase diagram of color-superconducting quark matter is discussed.
Using a scattering length parametrization of the crossover from a BCS state to a Bose-Einstein condensate as well as the local density approximation for the density profile, we calculate the momentum distribution of a harmonically trapped atomic Fermi gas at zero temperature. Various interaction regimes are considered, including the BCS phase, the unitarity limit, and the molecular regime. We show that the relevant parameter which characterizes the crossover is given by the dimensionless combination N1/6a/aho,{N}^{1/6}{a/a}_{\mathrm{ho}},N1/6a/aho​, where N is the number of atoms, a is the scattering length, and aho{a}_{\mathrm{ho}}aho​ is the oscillator length. The width of the momentum distribution is shown to depend in a crucial way on the value and sign of this parameter. Our predictions can be relevant for experiments on ultracold atomic Fermi gases near a Feshbach resonance.
We study the effect of the induced interaction on the superfluid transition temperature of a spin-polarized Fermi gas. In the BCS limit, the polarization is very small in the superfluid state, and the effect of the induced interaction is almost the same as in the spin-balanced case. The temperature Tt{T}_{t}Tt​ and the polarization Pt{P}_{t}Pt​ of the tricritical point are both reduced from mean-field results by a factor about 2.222.222.22. This reduction is also significant beyond the BCS limit. In the unitary limit, we find (Pt,Tt/TF)=(0.42,0.16)({P}_{t},{T}_{t}/{T}_{F})=(0.42,0.16)(Pt​,Tt​/TF​)=(0.42,0.16), in comparison with mean-field and experimental results.
The physics of quantum degenerate atomic Fermi gases in uniform as well as in harmonically trapped configurations is reviewed from a theoretical perspective. Emphasis is given to the effect of interactions that play a crucial role, bringing the gas into a superfluid phase at low temperature. In these dilute systems, interactions are characterized by a single parameter, the sss-wave scattering length, whose value can be tuned using an external magnetic field near a broad Feshbach resonance. The BCS limit of ordinary Fermi superfluidity, the Bose-Einstein condensation (BEC) of dimers, and the unitary limit of large scattering length are important regimes exhibited by interacting Fermi gases. In particular, the BEC and the unitary regimes are characterized by a high value of the superfluid critical temperature, on the order of the Fermi temperature. Different physical properties are discussed, including the density profiles and the energy of the ground-state configurations, the momentum distribution, the fraction of condensed pairs, collective oscillations and pair-breaking effects, the expansion of the gas, the main thermodynamic properties, the behavior in the presence of optical lattices, and the signatures of superfluidity, such as the existence of quantized vortices, the quenching of the moment of inertia, and the consequences of spin polarization. Various theoretical approaches are considered, ranging from the mean-field description of the BCS-BEC crossover to nonperturbative methods based on quantum Monte Carlo techniques. A major goal of the review is to compare theoretical predictions with available experimental results.
We examine the contribution of pairing fluctuations to the superfluid order parameter for harmonically trapped atomic Fermi gases in the BCS regime. In the limit of small systems we consider, both analytically and numerically, their space and temperature dependence. We predict a parity effect, i.e., that pairing fluctuations show a maximum or a minimum at the center of the trap, depending on the value of the last occupied shell being even or odd. We propose to detect pairing fluctuations by measuring the density-density correlation function after a ballistic expansion of the gas.