Disordered driven coupled cavity arrays: Nonequilibrium stochastic mean-field theory

We study the interplay of disorder with pumping and decay in coupled qubit-cavity arrays, the Jaynes-Cummings-Hubbard model. We find that relatively weak disorder can wash out the bistability present in the clean pumped system and that moreover the combination of disorder in on-site energies and decay can lead to effective phase disorder. To explore these questions, we present a nonequilibrium generalization of stochastic mean-field theory, providing a simple tool to address such questions. This technique is developed for rather general forms of light-matter coupling, driving, dissipation, and on-site disorder, making it applicable to a wide range of systems.

Figure 1

Probability distribution of $\left|\psi \right|$ as a function of pump frequency for $J<J_c$. The dotted (cyan [light gray]) line indicates the value of $\left|\psi \right|$ in the clean limit, and the color map shows the probability distribution of $\left|\psi \right|$ for Gaussian disorder of variance ${\sigma }_{\epsilon }/g=0.002g$. The dash-dotted (gray) line indicates the approximate solution to the clean case given in Eq. (6). Arrows mark the values of pump frequency at which the full probability distribution of complex $\psi$ is shown in Fig. 2. Other parameters are $f=\kappa =\gamma =0.005g,J/g=0.020$, and a geometry with $z=2$ is assumed.

Figure 2

Probability distribution of the complex observable $\psi$ at four different pump frequencies for $J<J_c$. The blue (gray) crosses indicate the value of $\psi$ found for the clean case, and the color map shows the probability distribution with ${\sigma }_{\epsilon }/g=0.002$. All parameters are as in Fig. 1.

Figure 3

Probability distribution of $\left|\psi \right|$ as a function of pump frequency for $J>J_c$. Lines and parameters are as for Fig. 1, except $J/g=0.04$ in this case.

Figure 4

Probability distribution of the complex observable $\psi$ at nine different pump frequencies for $J>J_c$. All parameters are as in Fig. 3.