Quantum kinetics of ultracold fermions coupled to an optical resonator

We study the far-from-equilibrium statistical mechanics of periodically driven fermionic atoms in a lossy optical resonator. We show that the interplay of the Fermi surface with cavity losses leads to subnatural cavity linewidth narrowing, squeezed light, and nonthermal quantum statistics of the atoms. Adapting the Keldysh approach, we set up and solve a quantum kinetic Boltzmann equation in a systematic $1/N$ expansion with $N$ the number of atoms. In the strict thermodynamic limit $N,V\to \infty ,N/V=\text{const.}$ we find that the atoms (fermions or bosons) remain immune against cavity-induced heating or cooling. At next-to-leading order in $1/N$, we find a “one-way thermalization” of the atoms determined by cavity decay. In absence of an equilibrium fluctuation-dissipation relation, the long-time limit $\Delta t\to \infty$ does not commute with the thermodynamic limit $N\to \infty$, such that for the physically relevant case of large but finite $N$, the dynamics ultimately becomes strongly coupled, especially close to the superradiance phase transition.

Figure 1

Illustration of $N$ noninteracting atoms with two internal electronic levels trapped inside a cavity. The atoms are driven periodically by a pump laser and scatter photons from the pump laser into the cavity and dissipate into the environment with decay rate $\kappa$.

Figure 2

Steady-state distribution [Eq. (2)] for a fermionic gas coupled to a cavity mode decaying with $\kappa =2E_R$. For increasing values of the detuning, the distribution tends to a thermal Fermi-Dirac distribution with the effective temperature Eq. (3).

Figure 3

(Left) Damping rate (minus the imaginary part of the eigenmode) of the lower polariton collective mode as a function of the two-photon atom-cavity coupling, for increasing values of the cavity decay rate. (Inset) Highest damping rate as a function of the ratio of cavity decay to detuning. The damping rate increases with increasing coupling, up to a maximum value (coinciding with the vanishing of the real part of the eigenmode), after which it decreased to finally vanish at the superradiant threshold. Initially, the role of cavity decay is to increase the polariton damping as long as it is of the order of the recoil energy $E_R$. Beyond this value, we reach the bad-cavity regime where the polariton damping rate decreases with cavity decay. This decrease is restricted to values of the coupling up to the point where the soft mode becomes purely dissipative, marked by the maximum in the damping as a function of the coupling. There is no additional polariton damping source apart from cavity decay since for the degenerate Fermi gas Landau damping is suppressed for frequencies outside the particle-hole continuum. (Right) Two examples of the intracavity photon spectral density inside and outside the bad-cavity regime. Here we chose an almost resonant case ${\delta }_c=1.2E_R$. Away from resonance ${\delta }_c\gg E_R$ the polariton damping is even less affected by cavity decay but the qualitative behavior above remains. We chose a low density regime $k_F=0.2Q$ and a volume $VQ=140$ in $d=1$.

Figure 4

(Left) Comparison of the equal-time quadrature variance at coupling $\lambda =0.9{\lambda }_{\mathrm{sr}}$ for three different systems: $1d$ Fermi gas at $T=0$ (black solid line), spins [40] (red dashed line), $3d$ Bose gas $T=1.1T_{\mathrm{bec}}$ (blue dash-dotted line). Parameters are $\kappa =0.2E_{\mathrm{R}}$ and ${\delta }_{\mathrm{c}}=1.2E_{\mathrm{R}}$ with a large density of the $1d$ Fermi gas: $k_{\mathrm{F}}=6.2Q$ and $VQ=140$. The Fermi gas induces squeezing of the quadrature variance below the vacuum shot noise (black dashed line). (Right) Corresponding behavior of the photonic dispersion in the atomic medium, which corresponds to the effective frequency-dependent driving strength of the cavity mode, achieved by coherently driving the atoms. Squeezing is achieved at high fermionic density because the frequency dependence of the dispersion in the atomic medium is suppressed due to Pauli principle. This induces an effective flat driving strength, analog to the one employed to achieve squeezing in an optical parametric oscillator [48].

Figure 5

Atomic self-energy insertions up to second order in the cavity field $a_{cl,q}$. The solid lines represent the bare (matrix) atom propagator, while the dotted ones represent the bare (matrix) photon propagator. The coupling constants are $U_0=g_0^2/{\Delta }_{\mathrm{a}}$ and $\lambda =g_0\Omega /{\Delta }_{\mathrm{a}}$. In a double expansion in $N$ and sufficiently far from superradiant threshold, these diagrams are evaluated with bare photon propagators (see Appendix pp1).

Figure 6

Polarization bubble diagram of atomic density fluctuations that modify the cavity photon dispersion and decay.

Figure 7

Behavior of the soft polariton mode for a $d=1$ Fermi gas coupled to a cavity mode with ${\delta }_c=1.2E_R$ and $\kappa =1E_R$. The position of the particle-hole continuum is set by $k_F=0.2Q$. We also choose a volume $VQ=140$ in $d=1$.

Figure 8

Comparison of the cavity photon correlation spectrum upon approaching threshold (from left to right) for small densities (top row) to large densities (bottom row). The fermionic particle-hole continuum is visible between the polariton peaks in the bottom row. (Subnatural) Cavity-linewidth narrowing is most pronounced in the top row upon approaching threshold. Parameters are ${\delta }_c=4.2E_R,\kappa =1E_R$, and $VQ=140$ in $d=1$.