Inhomogeneous (FFLO-type) pairing in trapped spin-imbalanced atomic Fermi gases: the surviving open core of Leggett's 2000 open problems in the theory of ultracold dilute alkali gases

The FFLO (Fulde-Ferrell-Larkin-Ovchinnikov) pairing scheme for spin-imbalanced gases is severely suppressed by trap inhomogeneity (Modawi and Leggett 1997), and no inhomogeneous (FFLO-type) pairing scheme has yet been realized and unambiguously identified in a trapped atomic Fermi gas. One-dimensional signatures are suggestive; the 2018 review of Kinnunen et al. and work through 2026 — including the 2026 joint quantum-gas-microscopy and quantum-Monte-Carlo study of the spin-imbalanced attractive Hubbard model by Feng et al., which maps the regimes where FFLO signatures should appear without claiming an observation — confirm that a clear observation of FFLO-type pairing remains outstanding. The objective is to determine whether, and if so identify, an inhomogeneous (finite-momentum, FFLO-type) pairing scheme for the trapped imbalanced gas that circumvents this suppression, together with its unambiguous identification, stated at the generality of the source (Leggett 2000; trap analysis by Modawi and Leggett 1997).

Background

Dilute gases of alkali atoms confined in magnetic or optical traps and cooled to nanokelvin temperatures have, since 1995, allowed the laboratory study of quantum degeneracy: Bose-Einstein condensation (BEC), the macroscopic occupation of a single quantum state by bosonic atoms, and, for fermionic isotopes, Bardeen-Cooper-Schrieffer (BCS) pairing, the binding of atoms of opposite momentum and spin into Cooper pairs, which underlies superfluidity (frictionless flow) in superconductors and in liquid $^3$He. A. J. Leggett's lecture notes 'Topics in the Theory of the Ultracold Dilute Alkali Gases', delivered at the Thirteenth Physics Summer School (Canberra, 2000) and published both in the school volume and as a Modern Physics Letters B supplement, surveyed the conceptual state of the field and closed with an enumeration of open issues. For Bose gases these concerned the operational meaning of superfluidity — the Hess-Fairbank effect (a superfluid's refusal, due to nonclassical rotational inertia, to be dragged into uniform rotation with its container), persistent currents around annular geometries, and critical velocities below which motion is dissipationless — in condensates with hyperfine (internal-state) degrees of freedom; the internal Josephson effect (coherent oscillations between two hyperfine states of one condensate driven by a coupling field) and the external Josephson effect (tunneling between spatially separated condensates); and the notion of a 'phase standard', a reproducible phase reference established from condensate interference. The Bose lectures condensed Leggett's companion review (Rev. Mod. Phys. 73, 307 (2001)). For Fermi gases the notes analyzed the prospects for Cooper pairing and its detection.

For the Fermi gases the central difficulty was weak coupling. At low energy the two-body interaction is characterized by a single parameter, the s-wave scattering length $a_s$, and BCS theory then gives a transition temperature $T_c$ exponentially small in $1/k_F|a_s|$ (with $k_F$ the Fermi wave number), so the densities required for a measurable $T_c$ with the then-available scattering lengths would drive the gas toward the mechanical (spinodal) collapse familiar from attractive Bose gases, while at $k_F|a_s|\sim 1$ the scattering-length parametrization of the interaction itself becomes questionable. For spin-imbalanced mixtures the notes discussed the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states: pairing states with finite center-of-mass momentum proposed in 1964 for superconductors in strong Zeeman fields, in which pairs carrying nonzero total momentum accommodate the Fermi-surface mismatch between majority and minority spin components. The trap analysis of Modawi and Leggett (J. Low Temp. Phys. 109, 625 (1997)) on spin-polarized $^6$Li found the FFLO region severely suppressed by trap inhomogeneity: in the local-density picture a harmonic trap samples a continuum of densities, so only a thin shell of the cloud can satisfy the narrow FFLO stability conditions at any time.

The subsequent development of the field transformed this landscape. Feshbach resonances — magnetically tunable scattering resonances arising from the coupling of two free atoms to a closed-channel molecular bound state — allow $|a_s|$ to be varied at fixed, low density, which bypasses the spinodal obstacle entirely: resonance-superfluidity theory (2001), condensation of fermionic atom pairs (2004), collective-mode and thermodynamic evidence for superfluidity (2004), and direct imaging of vortex lattices (2005) established superfluidity of strongly interacting $^6$Li and $^{40}$K gases with $T_c/T_F\approx 0.15$-$0.2$, and the $k_F|a_s|\sim 1$ question was subsumed into the BEC-BCS-crossover and unitarity ($|a_s|\to\infty$) physics where universality makes $a_s$ drop out. Spin-imbalanced gases, realized in 2005-2006, were found to phase separate rather than form FFLO states (Zwierlein et al., Science 311, 492 (2006); Partridge et al., PRL 97, 190407 (2006)). Stabilizations of inhomogeneous (FFLO-type) pairing pursued in this literature include geometry — one-dimensional tubes, two-dimensional layers, optical lattices, box or toroidal potentials — and mass imbalance (Kinnunen et al., Rep. Prog. Phys. 81, 046401 (2018); Yanase, Phys. Rev. B 80, 220510 (2009); Liao et al., Nature 467, 567 (2010)). For the FFLO item, the current frontier couples optical-lattice realizations to quantum gas microscopy (site-resolved imaging of individual atoms) and to unbiased many-body computation: the 2026 study of Feng et al. (PNAS 123, e2600303123, DOI 10.1073/pnas.2600303123) realizes the doped, spin-imbalanced attractive two-dimensional Hubbard model with an optical lattice, images it site-resolved, and matches it against auxiliary-field quantum Monte Carlo, identifying the density-magnetization window where FFLO precursors should first appear. The open question is whether an inhomogeneous (finite-momentum, FFLO-type) pairing scheme can circumvent the trap-induced suppression and be realized and unambiguously identified in a trapped spin-imbalanced atomic Fermi gas.

Problem Statement

The open problem, stated at the generality of the source (Leggett 2000; trap analysis by Modawi and Leggett 1997): in a trapped two-spin-component atomic Fermi gas with unequal populations of the two spin states — a mismatch between the majority and minority Fermi surfaces — uniform BCS (Bardeen-Cooper-Schrieffer) pairing, in which opposite-spin atoms pair at zero center-of-mass momentum, is destroyed beyond a critical imbalance, and the finite-momentum FFLO (Fulde-Ferrell-Larkin-Ovchinnikov) pairing states are severely suppressed by trap inhomogeneity. Determine whether, and if so propose, an inhomogeneous (finite-momentum, FFLO-type) pairing scheme for the trapped imbalanced gas that circumvents this suppression, together with its unambiguous identification, or establish quantitatively that no such scheme circumvents the suppression. Admissible schemes are the inhomogeneous (finite-momentum, FFLO-type) pairing variants. Homogeneous breached/gapless pairing, macroscale phase separation into a balanced superfluid plus polarized normal gas, and the polarized normal state are competing phases against which the realized state must be distinguished, not admissible answers. Unambiguous identification means establishing, by quantitative theory together with reproducible experiment, the region of parameter space in which the scheme is stable against all three competing phases (homogeneous breached/gapless pairing, macroscale phase separation, and the polarized normal state), and an observable signature — for example a finite-momentum pairing peak in a structure-factor, noise-correlation, or Bragg probe — that distinguishes the realized state from those competing phases.

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