Umklapp Superradiance with a Collisionless Quantum Degenerate Fermi Gas

The quantum dynamics of the electromagnetic light mode of an optical cavity filled with a coherently driven Fermi gas of ultracold atoms strongly depends on the geometry of the Fermi surface. Superradiant light generation and self-organization of the atoms can be achieved at low pumping threshold due to resonant atom-photon umklapp processes, where the fermions are scattered from one side of the Fermi surface to the other by exchanging photon momenta. The cavity spectrum exhibits sidebands that, despite strong atom-light coupling and cavity decay, retain narrow linewidth, due to absorptionless transparency windows outside the atomic particle-hole continuum and the suppression of broadening and thermal fluctuations in the collisionless Fermi gas.

Figure 1

Left: Fermi sphere (red circle) of the ultracold, atomic Fermi gas inside the mirrors of an optical cavity (curved, blue) in combined position-momentum representation. $|{\mathbf{k}}_F|$ is the Fermi momentum and $\mathbf{Q}$ is the superposition of the cavity momentum and the momentum of coherent drive laser with amplitude $\mathrm{\Omega }$. Right: Level scheme employed for the superradiant self-organization discussed here, with single-photon atom-cavity Rabi coupling $g_0$. Generalizing the level structure above to an incoherent repumping scheme should enable superradiant umklapp lasing [18, 19, 20].

Figure 2

Cavity photon spectrum in one dimension. Left: Spectral function Eq. (2) in color scale. The narrow features are actually sharp $\delta$-function peaks, the lower of which moves toward $\omega =0$ for $\mathrm{\Omega }\to {\mathrm{\Omega }}_D$. Right: Spectral function at fixed coupling close to threshold, $\mathrm{\Omega }=0.98{\mathrm{\Omega }}_D$, for two different densities. Black dashed curve: Perfectly nested case in which the fermionic particle-hole continuum reaches down to zero frequency and ${\mathrm{\Omega }}_D=0$. Parameters are ${\mathrm{\Delta }}_c=-1.2E_R$, $N{g}_0^2/{\mathrm{\Delta }}_a=-0.05E_R$, $g_0/{\mathrm{\Delta }}_a=-0.1E_R$, $T=0$. Within the particle-hole continuum there is “broadband” emission for a range of frequencies; outside of the particle-hole continuum, the linewidth remains narrow.

Figure 3

Absorption (imaginary part of the particle-hole continuum, black box-shaped line) and refractive properties (real part of the particle-hole continuum, blue double-peaked line) of the coherent atomic Fermi medium in one dimension at $\mathrm{\Omega }=0.6{\mathrm{\Omega }}_D$. The lower sideband in Fig. 2 appears in the absorptionless “transparency window” for $\omega /E_R\lesssim 0.6$, for which the imaginary part is identically zero since here $k_F=0.2Q$ is away from perfect nesting. At perfect nesting $Q=2k_F$, the absorption reaches to $\omega =0$. The optical properties of the atomic Fermi medium can be tuned by the coherent drive $\mathrm{\Omega }$ highlighting connections to electromagnetically induced transparency [56].

Figure 4

Phase diagram for a one-dimensional Fermi gas in an optical cavity as a function of Fermi ($k_F$) over cavity ($Q=\sqrt{2m{E}_R}$) momentum versus pump amplitude $\mathrm{\Omega }$. Inset shows gap opening at the Fermi points. Temperatures are $k_{B}T=0$ (black dashed line), $k_{B}T=0.01E_R$ (blue solid line). At $Q=2k_F$ the system is perfectly nested and Peierls reconstruction into a superradiant Peierls insulator (SPI) sets in at relatively small $\mathrm{\Omega }$. Away from nesting the $Z_2$ charge symmetry breaking leads to a superradiant charge-ordered Fermi liquid (SCOFL). The red shaded area extending from the second-order transition line indicates the hysteresis region preceding the first-order phase transition from the Fermi liquid (FL) into the the SPI at $k_F/Q=1$; there the Fermi energy lies in the gap between the second and third cavity generated bands. The remaining parameters are ${\mathrm{\Delta }}_c=-0.2E_R$, $N{g}_0^2/{\mathrm{\Delta }}_a=-0.05E_R$, $g_0/{\mathrm{\Delta }}_a=-0.1$.

Figure 5

Phase diagram for a two-dimensional Fermi gas in an optical cavity for $Q_x=Q_y=\sqrt{2m{E}_R}$, $k_{B}T=0.01E_R$ (blue dashed line), $k_{B}T=0.05E_R$ (orange dashed line), and $k_{B}T=0.1E_R$ (red dotted line). Lower inset: Hysteresis region. The vertical line separates the two different superradiant regimes with topologically trivial Fermi surface and reconstructed Fermi surface, as illustrated in the two upper insets: left at $k_{F,\widehat{\mathbf{Q}}}({\mathrm{\Omega }}_D)=0.49Q$, $\alpha =0.1$, right at $k_{F,\widehat{\mathbf{Q}}}({\mathrm{\Omega }}_D)=0.53Q$, $\alpha =0.2$. Here the Fermi surface is shown in the repeated-zone scheme relative to the Brillouin zone $\mathcal{B}=(-Q_x/2<k_x<Q_x/2,-Q_y/2<k_y<Q_y/2)$. Purple lines delimit “electron” pockets with occupied levels while blue lines delimit hole pockets with empty levels. The vertical line separating the two superradiant phases is straight only close to ${\mathrm{\Omega }}_D$. The remaining parameters are as in Fig. 4.