Topological Superradiant States in a Degenerate Fermi Gas

We predict the existence of a topological superradiant state in a two-component degenerate Fermi gas in a cavity. The superradiant light generation in the transversely driven cavity mode induces a cavity-assisted spin-orbit coupling and opens a bulk gap at half filling. This mechanism can simultaneously drive a topological phase transition in the system, yielding a topological superradiant state. We map out the steady-state phase diagram in the presence of an effective Zeeman field, and identify a critical tetracritical point beyond which the topological and the conventional superraidiant phase boundaries separate. The topological phase transition can be detected from its signatures in either the momentum distribution of the atoms or the variation of the cavity photon occupation due to the nontrivial feedback of the atoms on the cavity field.

Figure 1

(a) A quasi-one-dimensional Fermi gas is coupled to a two-mode optical cavity, which is under both transverse (along $-\widehat{z}$) and longitudinal (along $\widehat{x}$) pumping. (b) Atomic level scheme. (c) Superradiance-induced gap opening, at positions marked by circles, for $m_z<m_c$ at half filling. (d) The persistence of the bulk gap across the SR transition at $m_z>m_c$. In (c) and (d), the solid (dashed) curves are the lowest-band dispersion spectra in the first Brillouin zone before (after) the SR transition. The dispersion spectrum of the spin-down species (solid green) is shifted by half the Brillouin zone relative to that of the spin-up species (solid red) under the cavity-assisted SOC.

Figure 2

TSR in a quasi-one-dimensional lattice with open boundary conditions. (a) The cavity field $|\alpha |$ across the TSR phase transition with increasing ${\eta }_A$. (b) $\theta (x)$ (see main text) before (dotted curve, with ${\eta }_A=1E_r$) and after (solid and dash-dotted curves, with ${\eta }_A=3E_r$) the TSR phase transition on the central six sites. The critical point is at ${\eta }_A^c\sim 2.05E_r$. The solid (dash-dotted) curve corresponds to the TSR phase whose cavity field acquires a positive (negative) real part due to the spontaneous symmetry breaking. (c) A pair of edge states emerge in the superradiance-induced bulk gap as the system crosses the phase boundary. (d) Wave functions for the edge states in (c) at ${\eta }_A=3E_r$. In all cases, we consider a half-filled lattice of 80 sites, with the parameters $k_{B}T=E_r/200$, $m_z=0$, $V_0=5E_r$, $\kappa =100E_r$, ${\mathrm{\Delta }}_A=-10E_r$, and ${\xi }_A=5E_r$. For ${}^6\mathrm{Li}$ atoms, these can be satisfied by choosing $\kappa \sim 7.4\mathrm{MHz}$, $g_A\sim 27.1\mathrm{MHz}$, $|{\mathrm{\Delta }}_A|\sim 0.74\mathrm{MHz}$, $\mathrm{\Delta }\sim 2\mathrm{GHz}$, and $T\sim 17.7\mathrm{nK}$.

Figure 3

Typical momentum distribution of $⟨{\sigma }_z⟩$ (solid) and $⟨{\sigma }_y⟩$ (dashed) in the (a) TSR and (b) SR phases at ${\eta }_A=3E_r$. Inset: corresponding spin textures $⟨\overrightarrow{\sigma }⟩$ in the first Brillouin zone (shown in a loop). The effective Zeeman fields are (a) $m_z=0.171E_r$ and (b) $m_z=0.181E_r$, where the topological phase transition occurs at $m_z=0.176E_r$ for ${\eta }_A=3E_r$. Other parameters are the same as those in Fig. 2.

Figure 4

Steady-state phase diagram at a finite temperature $k_{B}T=E_r/200$. The solid curve is the TSR phase boundary, and the dotted curve is the topological phase boundary between the TSR and the trivial SR states. The thin dashed curve at $m_c\sim 0.132E_r$ is the boundary between the $M$ and the $I$ states, and the dash-dotted curve is the conventional SR phase boundary. The various boundaries merge at a tetracritical point (dot) at ${\eta }_A\sim 2.614E_r$, $m_c\sim 0.132E_r$. Other parameters are the same as those used for Fig. 2. Inset: change of bulk gap before (solid), after (dashed), and right at (dotted) the phase boundaries labeled by arrows.

Figure 5

(a),(b) Momentum distribution of the spin-up (a) and the spin-down (b) atoms in the TSR state (dashed) and the trivial SR state (solid), respectively. The parameters of the dashed [solid] curves are the same as those in Fig. 3 [Fig. 3]. (c),(d) Variation of the cavity-photon number across the TSR-SR topological phase boundary on the phase diagram in Fig. 4, with $m_z=0.03E_r$ (dash dotted), $m_z=0.132E_r$ (dotted), $m_z=0.4E_r$ (solid).