Kohn's theorem in a superfluid Fermi gas with a Feshbach resonance

TL;DR

This paper proves that the Kohn theorem holds in superfluid Fermi gases with Feshbach resonances within certain theoretical frameworks, and explores how the dipole mode frequency depends on Feshbach resonance effects.

Contribution

It demonstrates the validity of Kohn's theorem in superfluid Fermi gases using HFB-GRPA and NSR-GRPA formalisms, including Feshbach resonance effects, and analyzes the eigenfunctions of the Kohn mode.

Findings

Kohn's theorem is valid in HFB-GRPA and NSR-GRPA formalisms.

Feshbach resonance influences the dipole mode frequency when trap frequencies differ.

Eigenfunctions of the Kohn mode exhibit unusual features in a Fermi superfluid.

Abstract

We investigate the dipole mode in a superfluid gas of Fermi atoms trapped in a harmonic potential. According to Kohn's theorem, the frequency of this collective mode is not affected by an interaction between the atoms and is always equal to the trap frequency. This remarkable property, however, does not necessarily hold in an approximate theory. We explicitly prove that the Hartree-Fock-Bogoliubov generalized random phase approximation (HFB-GRPA), including a coupling between fluctuations in the density and Cooper channels, is consistent with both Kohn's theorem as well as Goldstone's theorem. This proof can be immediately extended to the strong-coupling superfluid theory developed by Nozi\'eres and Schmitt-Rink (NSR), where the effect of superfluid fluctuations is included within the Gaussian level. As a result, the NSR-GRPA formalism can be used to study collective modes in the BCS-BEC crossover region in a manner which is consistent with Kohn's theorem. We also include the effect of a Feshbach resonance and a condensate of the associated molecular bound states. A detailed discussion is given of the unusual nature of the Kohn mode eigenfunctions in a Fermi superfluid, in the presence and absence of a Feshbach resonance. When the molecular bosons feel a different trap frequency from the Fermi atoms, the dipole frequency is shown to {\it depend} on the strength of effective interaction associated with the Feshbach resonance.