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arXiv:2302.04617 · Physical Review A

BCS-BCS crossover between atomic and molecular superfluids in a Bose-Fermi mixture

Yixin Guo, Hiroyuki Tajima, Tetsuo Hatsuda, Haozhao Liang

Published Aug 2, 2023• 3 citations
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Abstract

We examine theoretically a continuity between atomic and molecular Fermi superfluids in a Bose-Fermi mixture near the Feshbach resonance. Considering a two-channel model with Fermi atoms $f$, Bose atoms $b$, and the closed-channel molecules $F$, we construct a mean-field framework based on the perturbative expansion of the $b\text{\ensuremath{-}}f\text{\ensuremath{-}}F$ Feshbach coupling. The resulting effective Hamiltonian not only exhibits the continuity between atom-atom to molecule-molecule Cooper pairings but also becomes equivalent to the two-band-superconductor model with Suhl-Matthias-Walker--type pair-exchange coupling. We demonstrate how these atomic and molecular Fermi superfluids coexist within the two-band-like superfluid theory. The pair-exchange coupling $ff\ensuremath{\leftrightarrow}FF$ and resulting superfluid gaps $\ensuremath{\langle}ff\ensuremath{\rangle}$ and $\ensuremath{\langle}FF\ensuremath{\rangle}$ are found to be strongly enhanced near the Feshbach resonance due to the interplay between the infrared singularity of Bogoliubov phonons and their Landau damping arising from the coupling with fermions. The pair-exchange coupling can be probed via the observation of the intrinsic Josephson effect between atomic and molecular superfluids.

DOI
10.1103/physreva.108.023304

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We examine theoretically a continuity between atomic and molecular Fermi superfluids in a Bose-Fermi mixture near the Feshbach resonance. Considering a two-channel model with Fermi atoms fff, Bose atoms bbb, and the closed-channel molecules FFF, we construct a mean-field framework based on the perturbative expansion of the b\ensuremath-f\ensuremath-Fb\text{\ensuremath{-}}f\text{\ensuremath{-}}Fb\ensuremath-f\ensuremath-F Feshbach coupling. The resulting effective Hamiltonian not only exhibits the continuity between atom-atom to molecule-molecule Cooper pairings but also becomes equivalent to the two-band-superconductor model with Suhl-Matthias-Walker--type pair-exchange coupling. We demonstrate how these atomic and molecular Fermi superfluids coexist within the two-band-like superfluid theory. The pair-exchange coupling ff\ensuremath↔FFff\ensuremath{\leftrightarrow}FFff\ensuremath↔FF and resulting superfluid gaps \ensuremath⟨ff\ensuremath⟩\ensuremath{\langle}ff\ensuremath{\rangle}\ensuremath⟨ff\ensuremath⟩ and \ensuremath⟨FF\ensuremath⟩\ensuremath{\langle}FF\ensuremath{\rangle}\ensuremath⟨FF\ensuremath⟩ are found to be strongly enhanced near the Feshbach resonance due to the interplay between the infrared singularity of Bogoliubov phonons and their Landau damping arising from the coupling with fermions. The pair-exchange coupling can be probed via the observation of the intrinsic Josephson effect between atomic and molecular superfluids.
Fermionic superfluidity in atomic Fermi gases across a Feshbach resonance is normally described by the atom-molecule theory, which treats the closed channel as a noninteracting point boson. In this work we present a theoretical description of the resonant superfluidity in analogy to the two-band superconductors. We employ the underlying two-channel scattering model of Feshbach resonance where the closed channel is treated as a composite boson with binding energy ${\ensuremath{\varepsilon}}_{0}$ and the resonance is triggered by the microscopic interchannel coupling U12{U}_{12}U12​. The binding energy ${\ensuremath{\varepsilon}}_{0}$ naturally serves as an energy scale of the system, which has been sent to infinity in the atom-molecule theory. We show that the atom-molecule theory can be viewed as a leading-order low-energy effective theory of the underlying fermionic theory in the limit ${\ensuremath{\varepsilon}}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$ and ${U}_{12}\ensuremath{\rightarrow}0$, while keeping the phenomenological atom-molecule coupling finite. The resulting two-band description of the superfluid state is in analogy to the BCS theory of two-band superconductors. In the dilute limit ${\ensuremath{\varepsilon}}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$, the two-band description recovers precisely the atom-molecule theory. The two-band theory provides a natural approach to study the corrections because of a finite binding energy ${\ensuremath{\varepsilon}}_{0}$ in realistic experimental systems. For broad and moderate resonances, the correction is not important for current experimental densities. However, for extremely narrow resonance, we find that the correction becomes significant. The finite binding energy correction could be important for the stability of homogeneous polarized superfluid against phase separation in imbalanced Fermi gases across a narrow Feshbach resonance.
We theoretically show that a two-band system with very different masses harbors a resonant pair scattering that leads to novel pairing properties, as highlighted by the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BEC) crossover. Most importantly, the interband pair-exchange coupling induces an effective intraband attraction in each band, enhancing the superfluidity/superconductivity. The effect, a kind of Suhl-Kondo mechanism, is specifically enhanced when the second band has a heavy mass and is incipient (lying close to, but just above, the chemical potential $\ensuremath{\mu}$), which we call a resonant pair scattering. By elucidating the dependence of the effective interactions and gap functions on $\ensuremath{\mu}$, we can draw an analogy between the resonant pair scattering and the Feshbach resonance.
We discuss the superfluid phase transition in a gas of Fermi atoms with a Feshbach resonance. A tunable pairing interaction associated with the Feshbach resonance is shown to naturally lead to the BCS-BEC crossover, where the character of superfluidity continuously changes from the weak-coupling Bardeen-Cooper-Schrieffer (BCS) type to the Bose-Einstein condensation (BEC) of tightly bound molecules, as one decreases the threshold energy 2ν of the Feshbach resonance. We also discuss effects of a trap, as well as the p-wave BCS-BEC crossover adjusted by a p-wave Feshbach resonance.
We study static superfluid properties of alkali-earth-like Fermi atomic systems in the presence of orbital Feshbach resonance. Using a two-band description of the ground state and excited state and a mean-field approximation of the intraband atomic pairing, we investigate the phase transitions and crossover between BCS and Bose-Einstein-condensate (BEC) superfluidity. At zero temperature, we find exact BCS analytical solutions for the mean-field gap equations and number equations. Using these exact solutions, we calculate various static properties, such as superfluid order parameters, chemical potentials, density variations, density profiles, correlation and coherence lengths, ground-state energy, and Tan's contact density across the entire BCS-BEC crossover region. We anticipate that our closed-form analytical results can be used as a benchmark for future experimental and theoretical investigations and will have an impact on the current understanding of two-band superconductors such as MgB2{\mathrm{MgB}}_{2}MgB2​.
We theoretically investigate the superfluid properties of a two-band gas of 173Yb^{173}\mathrm{Yb}173Yb Fermi atoms with an orbital Feshbach resonance (OFR). To describe the BCS-BEC crossover region, we include superfluid fluctuations caused by interband and intraband pairing interactions associated with OFR by extending the strong-coupling theory developed by Nozi\`eres and Schmitt-Rink to the two-band case below the superfluid phase transition temperature; however, the effects of an experimentally inaccessible deep bound state are removed to model a real 173Yb^{173}\mathrm{Yb}173Yb Fermi gas near OFR. We show that the condensate fraction in the upper closed channel gradually becomes smaller than that in the lower open channel as one moves from the strong- to the weak-coupling regime, because the OFR-pairing mechanism tunes the interaction strengths by adjusting the energy difference between the two bands. However, even when the closed-channel band is much higher in energy than the open-channel band in the weak-coupling regime, the magnitude of the superfluid order parameter in the closed channel is found to be still comparable to that in the open channel. As the reason for this, we point out a pair-tunneling effect by the OFR-induced interband interaction. In addition to these superfluid quantities, we also examine collective modes, such as the Goldstone mode, the Schmid (Higgs) mode, and Leggett mode, to clarify how they appear in the spectral weights of pair-correlation functions in each band. Since the realization of a multiband superfluid Fermi gas is a crucial issue in cold Fermi gas physics, our results would contribute to the basic understanding of this type of Fermi superfluid in the BCS-BEC crossover region.
We investigate the homogeneous superfluid mixtures of a Bardeen-Cooper-Schrieffer (BCS) superfluid originating from pairing two-species fermionic atoms and superfluidity stemming from condensation of bosonic atoms. By integrating out the freedoms associated with the BCS superfluid, we derive the fermion-mediated interactions between bosons, which is attractive and can be tuned from long range in the BCS region to short range in the region of Bose-Einstein condensation of molecular dimers. By analyzing the Bogoliubov spectrum and the damping rate of bosonic superfluid, we map out the phase diagram spanned by the boson-fermion mass ratio and the boson-fermion coupling strength, which consists of a phase separation region and two phase-mixing regions with and without Landau damping. The three different phases can coexist at a tricritical point, which moves toward a low boson-fermion mass ratio and a high boson-fermion scattering length as the fermion-fermion interaction strength is tuned up on the BCS side.