Prethermalization of Atoms Due to Photon-Mediated Long-Range Interactions

Atoms can spontaneously form spatially ordered structures in optical resonators when they are transversally driven by lasers. This occurs when the laser intensity exceeds a threshold value and results from the mechanical forces on the atoms associated with superradiant scattering into the cavity mode. We treat the atomic motion semiclassically and show that, while the onset of spatial ordering depends on the intracavity-photon number, the stationary momentum distribution is a Gaussian function whose width is determined by the rate of photon losses. Above threshold, the dynamics is characterized by two time scales: after a violent relaxation, the system slowly reaches the stationary state over time scales exceeding the cavity lifetime by several orders of magnitude. In this transient regime the atomic momenta form non-Gaussian metastable distributions, which emerge from the interplay between the long-range dispersive and dissipative mechanical forces of light. We argue that the dynamics of self-organization of atoms in cavities offers a test bed for studying the statistical mechanics of long-range interacting systems.

Figure 1

Atoms in a standing-wave cavity and driven by a transverse laser can spontaneously form ordered patterns when the laser intensity $\mathrm{\Omega }$ exceeds the rate of photon losses, here due to cavity decay at rate $\kappa$. In this regime the atoms experience a long-range interaction mediated by the cavity photons and their motion becomes strongly correlated.

Figure 2

(a) Distribution $P(\mathrm{\Theta })$ of the magnetization $\mathrm{\Theta }$ at steady state for $\overline{n}/{\overline{n}}_c=0.1$, 0.9, 1, 1.1, 4 (see box for color code). (b) Typical trajectories at the asymptotics for $N=200$ atoms are shown in (b) as a function of time (in units of $1/\kappa$) and for $\overline{n}/{\overline{n}}_c=0.1$, 1, 1.1, 4. Note that the cavity field amplitude is proportional to $\mathrm{\Theta }$. (c) Mean intracavity-photon number as a function of time for the trajectory at $\overline{n}=1.1{\overline{n}}_c$. (d) Spectrum $S(\omega )$ of the intensity of the emitted light (in arbitrary units) as a function of $\omega$ (in units of $\kappa$) for $\overline{n}/{\overline{n}}_c=0.1$, 0.9, 1, 1.1 (from top to bottom). (e) $g^{(2)}(0)$ as a function of $\overline{n}$ for different atom numbers. The dots correspond to numerical results obtained by integrating the SDE. The cavity parameters are rescaled with $N$ so that ${\overline{n}}_c$ is independent on $N$ and finite (see [21]). The atomic transition is the $D_2$ line of ${}^{85}\mathrm{Rb}$ at half linewidth $\gamma =2\pi \times 3\mathrm{MHz}$. The laser detuning from the atomic frequency is ${\mathrm{\Delta }}_a=-500\gamma$. Here, ${\mathrm{\Delta }}_c=-\kappa$ with $\kappa =0.5\gamma$.

Figure 3

Dynamics of the order parameter above threshold: (a) $\mathrm{\Theta }$ as a function of time (in units of ${\kappa }^{-1}$) for $N=200$ atoms and 500 trajectories at $\overline{n}=4{\overline{n}}_c$ and ${\mathrm{\Delta }}_c=-\kappa$, for an initially spatially uniform distribution at temperature $k_{B}T=\hslash \kappa /2$. $P(\mathrm{\Theta })$ at the transient and at the asymptotics is shown in panel (b). The position (c) and momentum (d) distributions are displayed at the times indicated by (1), (2), and (3) in panel (a). The dashed lines correspond to the initial distributions [which overlaps to (1) in (c)]. (e) Intensity-intensity correlations of the light at the cavity output, $g^{(2)}(\tau )$ as a function of $\tau$ (in units of ${\kappa }^{-1}$) for $\overline{n}/{\overline{n}}_c=0.1$, 0.9, 1, 1.1, 4, (same color code as in Fig. 2) evaluated after the system has reached the stationary state.