Quantum phases of spinful Fermi gases in optical cavities

We explore the quantum phases emerging from the interplay between spin and motional degrees of freedom of a one-dimensional quantum fluid of spinful fermionic atoms, effectively interacting via a photon-mediating mechanism with tunable sign and strength $g$, as it can be realized in present-day experiments with optical cavities. We find the emergence, in the very same system, of spin- and atomic-density wave ordering, accompanied by the occurrence of superfluidity for $g>0$, while cavity photons are seen to drive strong correlations at all $g$ values, with fermionic character for $g>0$, and bosonic character for $g<0$. Due to the long-range nature of interactions, to infer these results we combine mean-field and exact-diagonalization methods supported by bosonization analysis.

Figure 1

(a) System concept. Atoms in a $\mathrm{\Lambda }$ scheme involving three levels ($|s\rangle , |g\rangle , |e\rangle$) are placed into a linear optical cavity with vacuum-mode frequency ${\omega }_c$, and transversely pumped by a classical field with frequency ${\omega }_p$ and strength $\mathrm{\Omega }$. (b) A large detuning ${\mathrm{\Delta }}_e$ allows the adiabatic elimination of $|e\rangle$; the remaining effective levels $|s\rangle \equiv |\uparrow \rangle$ and $|g\rangle \equiv |\downarrow \rangle$, separated by a transition frequency ${\omega }_a$, are coupled to the cavity via $\hslash g_{\mathrm{eff}}$. The shift ${\delta }_{\uparrow }$ and potential $V_0\left(x\right)$ are embodied in the effective system. (c) Cartoon of spin-density-wave (SDW), atomic-density-wave (ADW), and superfluid (SF) orderings expected from the resulting Hamiltonian.

Figure 2

Mean-field phase diagram as a function of $g$. (a) Order parameters ${\mathrm{\Delta }}_{\mathrm{SDW}}$ ($\ne 0$ for $g<0$) and ${\mathrm{\Delta }}_{\mathrm{SF}}$ ($\ne 0$ for $g>0$), showing no coexistence of SDW and SF phases. Insets: momentum distributions $n\left(k\right)$, showing bosonic ($g<0$) and fermionic ($g>0$) characters. Cartoons represent sketches of fluid structures in terms of average interparticle distance $r_0$, as resulting from pair-distribution functions. (b), (c) Chemical potential $\mu$ and corresponding fraction $2N_0/N$ of spin-paired ($g<0$) and SF-paired ($g>0$) particles.

Figure 3

Ground-state energy $E_0$ as a function of the coupling $g$, for $N=4,8,12$, as obtained with ED, showing that the quantum fluid is strongly correlated at all $g$ values. Insets: momentum distributions for different $g$ values, confirming the bosonic ($g<0$) and fermionic ($g>0$) fluid character of Fig. 2.

Figure 4

Correlation functions in $k$ space for different $g$ values, from ED with $N=8$. In density-density correlations $\langle {\widehat{O}}_{\rho }{\widehat{O}}_{\rho }\rangle$, a peak develops at $k=0$ for $g<0$ and not for $g>0$, and peaks at $\pm 2k_F$ emerge only for $g>0$, signaling the setting of ADW processes. Inset: spin-spin correlations $\langle {\widehat{O}}_{\sigma }{\widehat{O}}_{\sigma }\rangle$; the peaks at $\pm 2k_F$ are driven by SDW and ADW processes.