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arXiv:2503.09671 · Physical Review A

Two-component atomic Fermi superfluid with spin-orbital coupling in thin-spherical-shell geometry

Yan He, Chih-Chun Chien

Published May 14, 2025• 0 citations
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Abstract

We present a theory of two-component atomic Fermi superfluid with tunable pairing interaction in a thin spherical shell subject to spin-orbit coupling (SOC). By incorporating SOC into the Fermi superfluid in the BCS-Bose Einstein condensation (BEC) crossover, we obtain the energy spectrum and equations of state. While the order parameter and chemical potential are suppressed by SOC on the BCS side, the former remains positive but the latter may be pushed to negative values by SOC. Meanwhile, the compressibility exhibits kinks as the pairing interaction or SOC varies, indicating singular behavior of higher derivatives of the free energy despite the presence of the superfluid order parameter. The minimum of the energy dispersion indicates a decoupling of the energy gap from the order parameter, and the compressibility kinks occur when the energy gap approaches 0. We found the particle-hole mixing in the Fermi superfluid essential for the compressibility kinks since a Fermi gas with the same SOC but no pairing interaction only exhibits complicated dispersions but no singular behavior. Therefore, our results show that a combination of superfluid, SOC, and geometry can lead to interesting phenomena. We also discuss possible experimental realizations and implications.

DOI
10.1103/physreva.111.053309

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We present a theory of two-component atomic Fermi superfluid with tunable pairing interaction in a thin spherical shell subject to spin-orbit coupling (SOC). By incorporating SOC into the Fermi superfluid in the BCS-Bose Einstein condensation (BEC) crossover, we obtain the energy spectrum and equations of state. While the order parameter and chemical potential are suppressed by SOC on the BCS side, the former remains positive but the latter may be pushed to negative values by SOC. Meanwhile, the compressibility exhibits kinks as the pairing interaction or SOC varies, indicating singular behavior of higher derivatives of the free energy despite the presence of the superfluid order parameter. The minimum of the energy dispersion indicates a decoupling of the energy gap from the order parameter, and the compressibility kinks occur when the energy gap approaches 0. We found the particle-hole mixing in the Fermi superfluid essential for the compressibility kinks since a Fermi gas with the same SOC but no pairing interaction only exhibits complicated dispersions but no singular behavior. Therefore, our results show that a combination of superfluid, SOC, and geometry can lead to interesting phenomena. We also discuss possible experimental realizations and implications.
We present a theory of a two-component atomic Fermi gas with tunable attractive contact interactions on a spherical shell going through the Bardeen-Cooper-Schrieffer (BCS)--Bose-Einstein condensation (BEC) crossover, inspired by the realizations of spherical bubble traps for ultracold atoms in microgravity. The derivation follows the BCS-Leggett theory to obtain the gap and number equations. The BCS-BEC crossover can be induced by tuning the interaction, and the properly normalized gap and chemical potential exhibit universal behavior regardless of the planar or spherical geometry. Nevertheless, the spherical-shell geometry introduces another way of inducing the crossover by the curvature. The curvature-induced BCS-BEC crossover is made possible by fixing the particle number and interaction strength while shrinking the sphere, causing a reduction to the ratio of the pairing and kinetic energies and pushing the system toward the BCS limit. The saturation of the superfluid density further confirms that the ground state is a Fermi superfluid.
We investigate the effects of Rashba-type spin-orbit coupling (SOC) on the condensed density and superfluid density tensor of a two-component Fermi gas in the BCS-BEC crossover at zero temperature. In anisotropic three dimensions (3D), we find that SOC has an opposite effect on condensation (enhanced) and superfluidity (suppressed in the SOC direction), and this effect becomes most pronounced for very weak interactions and the SOC strength being larger than a characteristic value. Furthermore, as functions of SOC strength, the condensed density changes monotonically for all interaction parameters, while the superfluid density has a minimum when the interaction parameter is below a critical value. We also discuss the isotropic two-dimensional case where analytical expressions for the gap and number equations are obtained and the same phenomena are found as that of the 3D case.
The recent experimental realization of spin-orbit coupling for ultracold atoms has generated much interest in the physics of spin-orbit-coupled degenerate Fermi gases. Although recently the BCS-BEC crossover in three-dimensional (3D) spin-orbit-coupled Fermi gases has been intensively studied, the corresponding two-dimensional (2D) crossover physics has remained unexplored. In this paper, we investigate, both numerically and analytically, the BCS-BEC crossover physics in 2D degenerate Fermi gases in the presence of the Rashba type of spin-orbit coupling. We derive the mean-field gap and atom-number equations suitable for 2D spin-orbit-coupled Fermi gases and solve them numerically and self-consistently, from which the dependence of the ground-state properties (chemical potential, superfluid pairing gap, ground-state energy per atom) on the system parameters (e.g., binding energy, spin-orbit-coupling strength) is obtained. Furthermore, we derive analytical expressions for these ground-state quantities, which agree well with our numerical results within a broad parameter region. Such analytical expressions also agree qualitatively with previous numerical results for 3D spin-orbit-coupled Fermi gases, where analytical results are lacking. We show that, with increasing spin-orbit coupling (SOC) strength, the chemical potential is shifted by a constant determined by the SOC strength. The superfluid pairing gap is enhanced significantly in the BCS limit for strong SOC, but increases only slightly in the BEC limit.
We present a systematic theoretical study of the BCS-BEC crossover in two-dimensional Fermi gases with Rashba spin-orbit coupling (SOC). By solving the exact two-body problem in the presence of an attractive short-range interaction we show that the SOC enhances the formation of the bound state: the binding energy E(B) and effective mass m(B) of the bound state grows along with the increase of the SOC. For the many-body problem, even at weak attraction, a dilute Fermi gas can evolve from a BCS superfluid state to a Bose condensation of molecules when the SOC becomes comparable to the Fermi momentum. The ground-state properties and the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature are studied, and analytical results are obtained in various limits. For large SOC, the BKT transition temperature recovers that for a Bose gas with an effective mass m(B). We find that the condensate and superfluid densities have distinct behaviors in the presence of SOC: the condensate density is generally enhanced by the SOC due to the increase of the molecule binding; the superfluid density is suppressed because of the nontrivial molecule effective mass m(B).
We performed a theoretical investigation on the ground state properties of a two dimensional ultra-cold Fermi superfluid with an anisotropic spin-orbit coupling (SOC). In the absence of Zeeman field, the system evolves from weak coupling BCS regime to strongly interacting BEC regime (BCS-BEC crossover) with increasing either the two-particle interaction strength or SOC parameters. We focused on the behaviors of pairing parameter and density of states (DOS) when increasing the anisotropic parameter of the SOC. Surprisingly, we discovered that the gap parameter decreases with increasing the anisotropic parameters, but the DOS at the Fermi surface shows non-monotonic behavior as a function of the anisotropic parameter. In the presence of the Zeeman field, we discussed a particular type of topological phase transition by obtaining the analytical result of the topological invariant and directly related this quantum phase transition with a sudden change of the ground state wave-function. Effects of higher partial wave pairing terms on this topological phase transition were briefly discussed.
We investigate the pairing physics of a three-component spin-orbit coupled Fermi gas in two spatial dimensions. The three atomic hyperfine states of the system are coupled by the recently realized synthetic spin-orbit coupling (SOC), which mixes different hyperfine states into helicity branches in a momentum-dependent manner. As a consequence, the interplay of spin-orbit coupling and the hyperfine-state-dependent interactions leads to the emergence of Fulde-Ferrell (FF) pairing states with finite center-of-mass momenta even in the absence of the Fermi-surface asymmetry that is usually mandatory to stabilize an SOC-induced FF state. We show that, for different combinations of spin-dependent interactions, the ground state of the system can either be the conventional Bardeen-Cooper-Schrieffer pairing state with zero center-of-mass momentum or be the FF pairing states. Of particular interest here is the existence of a three-component FF pairing state in which every two out of the three components form FF pairing. We map out the phase diagram of the system and characterize the properties of the three-component FF state, such as the order parameters, the gapless contours, and the momentum distributions. Based on these results, we discuss possible experimental detection schemes for the interesting pairing states in the system.