Beyond-mean-field stability and dissipative regime of cavity-induced finite-momentum (FFLO-like) pairing in ultracold lattice Fermi gases

Zheng and Wang (Phys. Rev. A 101, 023612 (2020)) showed at zero-temperature Bogoliubov-de Gennes mean-field level that a transversely pumped optical cavity coupled to a two-component Fermi gas in an optical lattice mediates an effectively infinite-range interaction that scatters atom pairs by $\pm 2k_c$ , producing a Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like superfluid in which a finite center-of-mass (COM) momentum order parameter $\Delta_{2k_c}$ coexists with the conventional $\Delta_0$ and remains nonzero whenever the cavity-induced interaction is present. The open problem is to establish the fate of this phase beyond the approximations of that analysis: does finite-COM pairing survive pair and cavity-field fluctuations across the BCS-BEC crossover, how do finite cavity decay $\kappa$ and the driven steady state modify it, and what is the quantitative phase structure for generic momentum-transfer ratios $k_c/k_L$ and in one dimension? Adjacent fluctuation-inclusive and dissipative steady-state methods now exist but have not been applied to this pairing channel, so the problem is open and actively approachable.

Background

In a conventional superconductor or superfluid, Cooper pairs have zero center-of-mass (COM) momentum. The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, proposed in 1964-1965 for spin-imbalanced systems, instead pair fermions between mismatched Fermi surfaces so that pairs carry a finite COM momentum and the order parameter is spatially modulated. Despite decades of work in solids and in ultracold atomic gases with spin imbalance, an unambiguous FFLO state has never been observed: it occupies a thin sliver of the phase diagram and competes with phase separation and density order. Cavity quantum electrodynamics with quantum gases offers a different route that needs no spin imbalance at all. A two-component Fermi gas trapped in an optical lattice is placed in a high-finesse optical cavity and driven by a transverse pump laser; photons scattered between the pump and the cavity mode mediate an effectively infinite-range density-density interaction between the atoms. When the cavity decay rate $\kappa$ is small compared to the cavity-pump detuning $\Delta_{\rm ca}$ , the cavity field can be adiabatically eliminated (replaced by its instantaneous steady value, valid when the atoms evolve slowly compared to the cavity response), and the resulting interaction is modulated in real space at the cavity wave vector. In momentum space it transfers a single, discrete momentum $2k_c$ (twice the transverse component of the cavity photon wave vector) between atoms, so two-body scattering can move a pair's COM momentum by $\pm 2k_c$ even in a balanced gas. Zheng and Wang (2020) analyzed this setup within Bogoliubov-de Gennes (BdG) theory - a mean-field framework that decouples the interacting Hamiltonian into a quadratic quasiparticle problem and determines the pairing order parameters $\Delta_0$ and $\Delta_{2k_c}$ by minimizing the thermodynamic potential - for a lattice Fermi gas with a contact attraction $U_{fr}$ tuned through a Feshbach resonance (a magnetically tunable two-body scattering resonance) and a cavity-induced attraction $U_{ac}$ . They found at mean-field level that the FFLO-like phase, with both $\Delta_0\neq 0$ and $\Delta_{2k_c}\neq 0$ , is the ground state, and that $\Delta_{2k_c}$ is nonzero for arbitrarily weak $U_{ac}$ . The authors themselves flag three caveats that define the open problem. First, BdG theory is not quantitatively accurate along the BCS-BEC crossover (the continuous tuning from weakly paired Bardeen-Cooper-Schrieffer Cooper pairs to tightly bound bosonic molecules by varying the contact interaction, typically described by the scattering parameter $1/k_Fa$ , with $a$ the s-wave scattering length and $k_F$ the Fermi wave vector), and it neglects pair fluctuations entirely; near unitarity (the resonant limit $1/k_Fa=0$ of divergent scattering length) and on the BEC side such fluctuations are large. Second, the treatment neglects cavity photon loss, keeping only the $\kappa\ll\Delta_{\rm ca}$ limit, whereas the pumped cavity is intrinsically a driven-dissipative system whose steady state must in general be found as a non-equilibrium problem. Third, the many-body behavior for generic momentum-transfer ratios $k_c/k_L$ - including rational values $k_c=k_Lp/q$ , where the lattice band splits into $q$ subbands and pairs can carry COM momenta $\pm 2q'k_c$ with $q'=0,1,\dots,q$ - and the phase structure in one dimension, where Luttinger-liquid physics (the standard low-energy description of one-dimensional interacting fermions in terms of collective density waves rather than particle-like quasiparticles) and competition between superconducting and charge-density-wave (CDW) tendencies dominate, were discussed only qualitatively. Subsequent work has sharpened the landscape around, but not into, this problem: Frank, Pini, Lang, and Piazza (2025) fully solved the competing Fermi-surface instabilities of a continuum polarized Fermi gas with a single-wave-vector cavity-mediated interaction at mean-field level, finding density-wave order dominant on the attractive side and non-superradiant (occurring without macroscopic buildup of the intracavity photon field) superfluid phases with degenerate zero- and finite-COM pairing channels on the repulsive side at low temperature; Tolle, Sheikhan, Giamarchi, Kollath, and Halati (2025) developed a fluctuation-inclusive steady-state treatment of fermionic atoms in a dissipative cavity at finite $\kappa$ , but for the self-organization (density-wave) channel; Chakraborty and Piazza (2020) showed that photon fluctuations strongly modify photon-mediated pairing in two-dimensional electron systems; and the experiments of Helson et al. (2023) and Zwettler et al. (2025) realized photon-mediated density-wave order and charge-density-wave/pair-density-wave interference (spatial modulation of the short-range pair correlation function) in unitary Fermi gases (gases at infinite scattering length). None of these addresses the stability, magnitude, or observability of the cavity-induced finite-COM pairing itself.

Problem Statement

For a two-component Fermi gas in an optical lattice coupled to a transversely pumped high-finesse optical cavity - where adiabatic elimination of the cavity field yields an effectively infinite-range interaction transferring momentum $\pm 2k_c$ between atoms, and where zero-temperature BdG mean-field theory predicts an FFLO-like ground state with order parameters $\Delta_0\neq 0$ and $\Delta_{2k_c}\neq 0$ coexisting for any nonzero cavity-induced coupling - determine the quantitative fate of the cavity-induced finite-COM pairing beyond the mean-field, $\kappa\to 0$ approximation. The single overarching objective is to establish whether this phase is stable and observable under realistic conditions, via three scientifically linked strands: (i) include quantum and thermal pair fluctuations and determine the stability region, magnitude, and experimental signatures of the finite-COM pairing across the BCS-BEC crossover, taking into account that cavity self-consistency feeds fluctuation corrections back into the mediated interaction itself; (ii) treat the driven-dissipative atom-cavity system at finite cavity decay rate $\kappa$ , computing the non-equilibrium steady state and establishing how finite photon loss and pumping alter the pairing; and (iii) map the many-body phase diagram quantitatively for generic momentum-transfer ratios $k_c/k_L$ , including the rational cases $k_c=k_Lp/q$ where the band splits into $q$ subbands and pairs can carry COM momenta $\pm 2q'k_c$ ( $q'=0,1,\dots,q$ ), and in one-dimensional regimes where superconducting, charge-density-wave, and Luttinger-liquid tendencies compete. The quantitative question is whether cavity-induced finite-COM pairing survives these corrections - in which parameter regimes, with what magnitude, and with what observable signatures such as momentum-resolved pair correlations or quasiparticle spectra - or is destabilized, and what replaces it in that case.

References

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