Dissipation-Assisted Prethermalization in Long-Range Interacting Atomic Ensembles

We theoretically characterize the semiclassical dynamics of an ensemble of atoms after a sudden quench across a driven-dissipative second-order phase transition. The atoms are driven by a laser and interact via conservative and dissipative long-range forces mediated by the photons of a single-mode cavity. These forces can cool the motion and, above a threshold value of the laser intensity, induce spatial ordering. We show that the relaxation dynamics following the quench exhibits a long prethermalizing behavior which is first dominated by coherent long-range forces and then by their interplay with dissipation. Remarkably, dissipation-assisted prethermalization is orders of magnitude longer than prethermalization due to the coherent dynamics. We show that it is associated with the creation of momentum-position correlations, which remain nonzero for even longer times than mean-field predicts. This implies that cavity cooling of an atomic ensemble into the self-organized phase can require longer time scales than the typical experimental duration. In general, these results demonstrate that noise and dissipation can substantially slow down the onset of thermalization in long-range interacting many-body systems.

Figure 1

(a) Atoms interact with the standing-wave mode of a cavity and are transversally driven by a laser. The laser amplitude ($\mathrm{\Omega }$) and/or frequency (${\mathrm{\Delta }}_c$) are suddenly quenched across the threshold, above which the atoms organize in regular spatial patterns at the steady state. The coherent scattering amplitude per atom, $S$, is tuned by the laser, $S\propto \mathrm{\Omega }$, and the resonator dissipates photons at rate $\kappa$. (b) Phase diagram of the second-order self-organization transition as a function of $\overline{n}$ (proportional to $S^2$) and ${\mathrm{\Delta }}_c/\kappa$ (that determines the asymptotic temperature). The black line separates the homogeneous phase (with order parameter $\mathrm{\Theta }=0$) from the self-organized one (with $\mathrm{\Theta }\to \pm 1$). The red dashed lines $A$ and $B$ illustrate the initial and final values, respectively, of the sudden quenches we analyze.

Figure 2

Numerical simulation of the dynamics following a sudden quench along path $A$ using the SDE [31]. At $t=0$, the atoms are in the stationary state of Eq. (1) for ${\overline{n}}_i=0.01{\overline{n}}_c$ with ${\mathrm{\Delta }}_c=-\kappa$, and $\overline{n}$ is quenched to the value ${\overline{n}}_f>{\overline{n}}_c$ [see the legend in (a)]. (a) The modulus of the order parameter $⟨|\mathrm{\Theta }|⟩$ and (b) the single-particle kinetic energy $⟨p^2/(2m)⟩$ (in units of $\hslash {\omega }_r$) as a function of time (in units of ${\kappa }^{-1}$) for $N=50$. The corresponding insets display the time evolution of the relative localization $\delta \mathrm{\Theta }/⟨|\mathrm{\Theta }|⟩$, where $\delta \mathrm{\Theta }=\sqrt{⟨{\mathrm{\Theta }}^2⟩-⟨|\mathrm{\Theta }|{⟩}^2}$, and of the kurtosis $\mathcal{K}$. The initial values $⟨|\mathrm{\Theta }|{⟩}_{t=0}\simeq 1/\sqrt{\pi N}\approx 0.08$ in (a) are due to finite $N$ [31]. Here, $\kappa \approx 390{\omega }_r$ and $N|U_0|=0.05\kappa$. The three relaxation stages are indicated by the labels (i), (ii), and (iii).

Figure 3

Dynamics following a sudden quench along path $A$ with ${\overline{n}}_f=2{\overline{n}}_c$ and $N=200$. At $t=0$, the atoms are in the stationary state of Eq. (1) for ${\overline{n}}_i=0.01{\overline{n}}_c$ and ${\mathrm{\Delta }}_c=-\kappa$. Subplot (a) compares the evolution of $⟨|\mathrm{\Theta }|⟩$ and $\mathcal{K}$ (inset) obtained by integrating Eq. (1) (black line) with the one found after setting $\mathrm{\Gamma }=U_0=0$ (blue line). The red line is the fit obtained by a stability analysis of the homogeneous Vlasov solution, the dashed-dotted line by a mean-field model (see Supplemental Material [42]). (b) Time evolution of the QSS observable ${\varphi }_{11}$ [Eq. (3)] corresponding to the curves in (a).

Figure 4

Relaxation after a sudden quench along path $A$ for the parameters of Fig. 3 (${\overline{n}}_f=2{\overline{n}}_c$) and for different $N$. Inset: Time evolution of $⟨{\mathrm{\Theta }}^2⟩$ for $N=50$ (gray) and $N=200$ (black). The dashed lines are the prediction of the MF model (see Supplemental Material [42]). The horizontal dashed line indicates where $⟨{\mathrm{\Theta }}^2⟩$ has reached 60% of its stationary value and identifies the corresponding time $t_N^{*}$. Onset: Time $t_N^{*}$ (in units of ${\kappa }^{-1}$) as a function of $N$ for the full FPE (squares) and MF (stars). The lines are the corresponding linear fits in the log-log plot.