Quantum simulation of competing orders with fermions in quantum optical lattices

Ultracold Fermi atoms confined in optical lattices coupled to quantized modes of an optical cavity are an ideal scenario to engineer quantum simulators in the strongly interacting regime. The system has both short-range and cavity-induced long-range interactions. We propose such a scheme to investigate the coexistence of superfluid pairing, density order, and quantum domains having antiferromagnetic or density order in the Hubbard model in a high-finesse optical cavity at $T=0$. We demonstrate that those phases can be accessed by properly tuning the linear polarizer of an external pump beam via the cavity back-action effect, while modulating the system doping. This allows one to emulate the typical scenarios of analog strongly correlated electronic systems.

Figure 1

Schematic representation of the model. A Fermi-Hubbard chain is placed inside a single mode standing-wave optical cavity. A linearly polarized pump beam with incidence perpendicular to the cavity along the $z$ axis, creates an electromagnetic field that couples light to the atomic modes. The polarization disk (purple) shows the choice of polarization angle $\theta$ measured from the $-x$ polarization axis. The light pumped into the system induces density wave order maximally for $\theta =0$ (bottom), while antiferromagnetic order is maximal for $\theta =\pi /2$ (top).

Figure 2

Competition of quantum phases by tuning the angle $\theta$ for given values of the doping. Order parameters ${\mathcal{O}}_{\mathrm{DW}}$, ${\mathcal{O}}_{\mathrm{M}}$, and ${\mathcal{O}}_{\eta }$ as functions of the filling factor and the angle of polarization $\theta$. In panel (a), at half-filling an AFM phase is produced with $\theta =\pi /2$. Properly changing the angle of the linear polarized beam to $\theta =3\pi /8$, a PDW phase emerges, doping the system by one or two holes/pairs. For values of doping far beyond half-filling and $\theta =\pi /4$, a ${\mathrm{SF}}_{\eta }$ phase dominates. In panel (b), a DW insulator is produced at half-filling with $\theta =3\pi /10$. Doping by one pair/hole and changing the polarization angle to $\theta =2\pi /5$, the system is AFM. A ${\mathrm{SF}}_{\eta }$ phase can be accessed away from half-filling by doping further and increasing the polarization angle to $\theta =\pi /4$. Parameters are $N_s=10$, $t/\left|U\right|=0.1$, $g_{\mathrm{eff}}/\left|U\right|=-0.75$ (a), and $g_{\mathrm{eff}}/\left|U\right|=-1$ (b).

Figure 3

Phase diagram of fermions in QOL with balanced population. We show competition of QP for different doping in the strongly attractive interacting limit $\left|U\right|\gg t$. We plot the QP in the system for several values of doping as a function of the detuning in terms of $g_{\mathrm{eff}}$ and the angle $\theta$. Doping is $\rho =1-\frac{4}{N_s}$ (a), $\rho =1-\frac{2}{N_s}$ (b), and $\rho =1$ (c). As the system is doped with holes [(c) to (a)] the regions in the phase diagram that support a DW insulator and AFM shrink while the intermediate PDWs enlarge. Depending on the value of $g_{\mathrm{eff}}$ one can access trajectories such as the ones portrayed in Fig. 2. Parameters are $N_s=10$, $t=0.1$, $N_{\uparrow }=N_{\downarrow }$, $N_{\sigma }=\langle {\sum }_i{\widehat{n}}_{i,\sigma }\rangle$.