Effect of Active Photons on Dynamical Frustration in Cavity QED

We study the far-from-equilibrium dynamical regimes of a many-body spin-boson model with disordered couplings relevant for cavity QED and trapped ion experiments, using the discrete truncated Wigner approximation. We focus on the dynamics of spin observables upon varying the disorder strength and the frequency of the photons, finding that the latter can considerably alter the structure of the system’s dynamical responses. When the photons evolve at a similar rate as the spins, they can induce qualitatively distinct frustrated dynamics characterized by either logarithmic or algebraically slow relaxation. The latter illustrates resilience of glassylike dynamics in the presence of active photonic degrees of freedom, suggesting that disordered quantum many-body systems with resonant photons or phonons can display a rich diagram of nonequilibrium responses, with near future applications for quantum information science.

Figure 1

Qualitative portrait of the dynamical responses at large $\alpha =N_b/N_s$ and for an initial state with spins polarized in the $\widehat{x}$ direction. The axes give the photons frequency $\omega h^{-1}$ and disorder strength $\sigma h^{-1/2}$ and with scales set by the transverse field $h$. At $\sigma h^{-1/2}>1$ and for large $\omega h^{-1}$, the spin magnetization experiences logarithmic relaxation due to the fast photons mediating a long-range frustrated interaction. Upon reducing $\omega /h$, photons become active and the slow relaxation of the spins follows an algebraic decay. Large values of $\sigma$ “freeze” the dynamics of spins and their relaxation becomes critically slow for our simulations. For weak disorder, we observe dynamical paramagnetism and a photon-assisted relaxation mechanism: the asymptotic transverse magnetization crosses over from a vanishing to a finite value upon reducing $\omega /h$.

Figure 2

Left: effect of photon frequency on frustrated dynamics. System’s parameters are $\alpha =10$, $N_t=1000$, and $\sigma /\sqrt{h}=2$; $\omega /h$ varies from 3.6 (dashed, bright green) to 10 (solid, dark blue), deep in the regime of validity of adiabatic elimination. Logarithmic relaxation is marked by black lines for intermediate values of photon frequencies; for $\omega /h>7.5$ relaxation is slower. Here dynamics are beyond the validity of the perturbative regime which holds up to $t\lesssim t_h\approx 2/h$. At a second timescale $t_{\omega }$ (marked by black crosses), logarithmic relaxation crosses over into a qualitatively different dynamical regime. Center: dynamics beyond adiabatic elimination. The figure shows power law relaxation for $\omega /h=1$, $\alpha =10$, $N_t=1000$, and $\sigma /\sqrt{h}$ varying from 2.1 (dashed, bright green) to 5 (solid, dark blue). The power law relaxation $t^{-\gamma }$ (with $\gamma \approx 0.75$) starts after $t_h$ when perturbation theory breaks down (marked by the black dots); for the parameters considered in this panel, $t_{\omega }$ occurs before $t_h$. Right: photon-assisted relaxation in the dynamical paramagnet. Dynamics of ${\sigma }^z$ after a quench into the paramagnetic phase for $\alpha =10$, $\sigma /\sqrt{h}=0.7$ and with $\omega /h$ shown in the legend.

Figure 3

Left: diagram of dynamical responses as a function of $\alpha$ and ${\omega }^{-1}$ for $\sigma >{\sigma }_c$. The critical value of $\alpha$ in general depends on $\sigma$, the value ${\alpha }_c\approx 0.37$ shown is for $\sigma /\sqrt{h}=5$. Center: transition to exponential relaxation at $\alpha \lesssim 0.37$ (for $\sigma /\sqrt{h}=5$ and $\omega /h=1$). Right: transition from subexponential to exponential relaxation for $\alpha \lesssim {\alpha }_c\approx 3$ (here $\sigma /\sqrt{h}=2$ and $\omega /h=10$). System’s parameters in all plots are $N_t=640$; straight black lines mark the regimes of exponential relaxation in the central and right panels (the sampling error in ${\sigma }^x$ is around $\approx {10}^{-2}$, and we therefore do not display dynamics when ${\sigma }^x$ is below such threshold). In the SM [91], we plot both figures in log-log scale.