Fermionic Superradiance in a Transversely Pumped Optical Cavity

Following the experimental realization of Dicke superradiance in Bose gases coupled to cavity light fields, we investigate the behavior of ultracold fermions in a transversely pumped cavity. We focus on the equilibrium phase diagram of spinless fermions coupled to a single cavity mode and establish a zero temperature transition to a superradiant state. In contrast to the bosonic case, Pauli blocking leads to lattice commensuration effects that influence self-organization in the cavity light field. This includes a sequence of discontinuous transitions with increasing atomic density and tricritical superradiance. We discuss the implications for experiment.

Figure 1

Equilibrium phase diagram of fermions in a transversely pumped cavity; see inset. As the pumping is increased, there is a transition to a superradiant (SR) state, where the fermions spontaneously “self-organize” in the self-consistent light field. The panels correspond to partial filling of (a) the first and (b) second bands of the emergent lattice. At large detuning the cavity field (gray scale) grows continuously above a critical pump field (solid blue line), while at smaller detuning the transition is discontinuous (double red line). These first- and second-order boundaries join differently at different fillings; for $n_F\lesssim 1$ they meet at a tricritical point (circle), while at higher fillings there is a critical end point (diamond). The first-order boundary in (b) corresponds to a liquid-gas-type transition within the SR phase. The unstable region $\tilde{\omega }<2n_F$ will be stabilized by cavity loss; see text. The dashed line is the spinodal extension of the continuous line.

Figure 2

(a) Atomic bands versus $\varphi$ for $\eta =0.8$. When filling the second or higher bands, $f(\varphi )$ is nonmonotonic, thus allowing first-order liquid-gas-type transitions in the SR phase. The bands are labeled by the symmetry of the Wannier orbitals at large $\varphi$. Panels (b),(c) illustrate a distortion of the FS on crossing the liquid-gas boundary; the values of $\tilde{\omega }$ are indicated by crosses in Fig. 1. The FS delimits a partially filled second band and is shown in a reduced zone scheme. Occupied states are shaded.

Figure 3

(a) Equilibrium phase diagram as a function of $n_F$ at $\eta =0.8$. The lines and the gray scale are as in Fig. 1. A critical point (empty circle) terminates the liquid-gas-like transition at $n_F\gtrsim 1$. The second-order normal-SR transition shows a peak near $n_F\simeq 1$ reflecting nesting; see text. (b) Evolution with temperature. (c) Effect of a confining potential, $V(r)=E_R(r/r_0{)}^{\alpha }$, as shown in the inset; the solid line corresponds to a hard-wall potential. (d) Impact of cavity losses. In (b)–(d) we focus on the change to the (second-order) normal-SR boundary, and other features are suppressed.