Structures and proximity effects of inhomogeneous population-imbalanced Fermi gases with pairing interactions

TL;DR

The paper addresses how population‑imbalanced Fermi gases in a quasi‑1D box host coexisting BCS, FFLO, and normal phases when pairing strength or spin polarization varies in space. It employs the Bogoliubov–de Gennes framework to compute self‑consistent order parameters, pair correlations, and their momentum spectra, identifying characteristic scales such as the BCS coherence length \\(\xi_{BCS}\\) and the FFLO momentum \\(q = k_{F\uparrow} - k_{F\downarrow}\\. The study reveals proximity effects where FFLO correlations penetrate normal and BCS regions, produces a buffer FFLO phase at certain interfaces, and demonstrates that momentum‑space signatures (peaks at \\(\tilde{q}\\)) robustly indicate finite‑momentum pairing across interfaces. The results have experimental relevance for box traps and spatially controlled interactions in ultracold atoms, with implications for observing multi‑phase coexistence and interfacial physics in low‑dimensional fermionic systems.

Abstract

By introducing spatially varying profiles of pairing interaction or spin polarization to quasi one-dimensional two-component atomic Fermi gases confined in box potentials, we analyze the ground state structures and properties when multiple phases coexist in real space by implementing the Bogoliubov--de~Gennes equation suitable for describing inhomogeneous fermion systems. While the BCS, Fulde--Ferrell--Larkin--Ovchinnikov (FFLO), and normal phases occupy different regions on the phase diagram when the parameters are uniform, a spatial change of pairing strength or spin polarization can drive the system from the FFLO phase to a normal gas or from a BCS superfluid to the FFLO phase in real space. The FFLO phase exhibits its signature modulating order parameter at the FFLO momentum due to population imbalance, and the pair correlation penetrates the polarized normal phase and exhibits proximity effects. Meanwhile, the BCS phase tends to repel population imbalance and maintain a plateau of pairing. Interestingly, a buffer FFLO phase emerges when the spatial change attempts to join the BCS and normal phase in the presence of spin polarization. By analyzing the pairing correlations, interfacial properties, and momentum-space spectra of the inhomogeneous structures, relevant length- and momentum- scales and their interplay are characterized. We also briefly discuss implications of inhomogeneous multi-phase atomic Fermi gases with population imbalance.

Figures (7)

• Figure 1: Phase diagram of two-component Fermi gases in a 1D box with uniform attractive coupling constant $\tilde{g}$ and spin-polarization field $\tilde{h}$. The BCS and FFLO superfluid phases as well as the normal phase are labeled. The arrows (a)–(d) indicate later constructions of inhomogeneous systems by joining two phases together using an inhomogeneous profile of $\tilde{g}$ or $\tilde{h}$. The dashed line in the lower-left corner represents our best estimations due to the blurring features when both pairing and spin-polarization are small.
• Figure 2: Joint FFLO–normal configuration generated by an inhomogeneous pairing interaction profile with $\tilde{g}_L = 2$ and $\tilde{g}_R = 0$, as indicated by arrow (a) in Fig. \ref{['fig:phase_diagram']}. The vertical dashed lines mark where $\tilde{g}$ changes. Here $\tilde{\mu}=1$ and $\tilde{h}=0.35$ are uniform. (a) The order parameter $\Delta(x)$ displays the FFLO oscillations on the left half and vanishes in the normal region. (b) The pair correlation function $F(x)$ exhibits oscillations on the left side and penetrates into the noninteracting region.(c) Spin-resolved densities $\rho_\sigma(x)$ reflect the population imbalance. (d) Dimensionless Fourier amplitudes $\tilde{F}(\tilde{k})$ of the left (blue solid line) and right (red dash line) halves of $F(x)$. The FFLO region shows a sharp peak at the FFLO momentum $\tilde{q} = (k_{F\uparrow} - k_{F\downarrow})/k_F^0$. The normal region also has a weaker peak at $\tilde{q}$ (magnified in the inset), demonstrating proximity-induced FFLO correlations.
• Figure 3: Joint FFLO–BCS configuration generated by an inhomogeneous profile of $\tilde{h}_L=0.35, \tilde{h}_R=0$ for the left and right halves of the system, corresponding to arrow $b$ in Fig. \ref{['fig:phase_diagram']}. Here $\tilde{\mu} = 1$ and $\tilde{g} = 2$ are uniform. The vertical dashed lines mark where $\tilde{h}$ changes. (a) The order parameter $\Delta(x)$ exhibits oscillations (a plateau) in the FFLO (BCS) region. (b) The pair correlation function $F(x)$ shows similar behavior. (c) The spin-resolved densities $\rho_{\sigma}(x)$ are imbalanced (balanced) on the FFLO (BCS) side. (d) Fourier spectra of the bulk $F(x)$ on the left (solid) and right (dashed) sides. The left FFLO region shows a sharp peak at the FFLO momentum $\tilde{q}$ while the right BCS region exhibits no observable feature.
• Figure 4: Joint FFLO–BCS configuration by an inhomogeneous profile of $\tilde{g}_L=2, \tilde{g}_R=4$ for the left and right halves of the system, corresponding to arrow $c$ in Fig. \ref{['fig:phase_diagram']}. The vertical dashed lines mark where $\tilde{g}$ changes. Here $\tilde{\mu} = 1$ and $\tilde{h} = 0.35$ are uniform. (a) The order parameter $\Delta(x)$ exhibits oscillations (a plateau) in the FFLO region (BCS) region. (b) The pair correlation function $F(x)$ follows a similar structure. (c) The densities $\rho_{\sigma}(x)$ are imbalanced (balanced) in the FFLO (BCS) region with fluctuations near the hard-wall boundaries. (d) Fourier spectra of the bulk $F(x)$ on the left (solid line) and right (dashed line) sides, showing the FFLO momentum $\tilde{q}$ on the left and a flat line on the right.
• Figure 5: Joint BCS-normal configuration generated by an inhomogeneous pairing-interaction profile with $\tilde{g}_{L} = 2$ and $\tilde{g}_{R} = 0$ for the left and right halves, corresponding to arrow $d$ in Fig. \ref{['fig:phase_diagram']}. The vertical dashed lines mark where $\tilde{g}$ changes. Here $\tilde{\mu} = 1$ and $\tilde{h} = 0.15$ are uniform. (a) The order parameter $\Delta(x)$ is nearly uniform in the BCS region and vanishes in the normal region. (b) The pair correlation function $F(x)$ shows a plateau on the BCS side but exhibits oscillations characteristic of the FFLO pair correlation. (c) The spin-resolved densities $\rho_{\sigma}(x)$ are balanced in the BCS region and become imbalanced in the normal region. (d) Bulk Fourier spectra of $F(x)$ from the left (solid line) and right (dashed line) halves. The BCS side is featureless but the normal side displays a peak at the FFLO momentum $\tilde{q}$, indicating a buffer of the FFLO phase as the system changes from the BCS to polarized normal regions.