Collective phases of strongly interacting cavity photons

We study a coupled array of coherently driven photonic cavities, which maps onto a driven-dissipative XY $\text{spin-}\frac{1}{2}$ model with ferromagnetic couplings in the limit of strong optical nonlinearities. Using a site-decoupled mean-field approximation, we identify steady-state phases with canted antiferromagnetic order, in addition to limit cycle phases, where oscillatory dynamics persist indefinitely. We also identify collective bistable phases, where the system supports two steady states among spatially uniform, antiferromagnetic, and limit cycle phases. We compare these mean-field results to exact quantum trajectory simulations for finite one-dimensional arrays. The exact results exhibit short-range antiferromagnetic order for parameters that have significant overlap with the mean-field phase diagram. In the mean-field bistable regime, the exact quantum dynamics exhibits real-time collective switching between macroscopically distinguishable states. We present a clear physical picture for this dynamics and establish a simple relationship between the switching times and properties of the quantum Liouvillian.

Figure 1

Mean-field phase diagram for $J/\gamma =10$ in the hard-core $(U\to \infty )$ limit. Represented are dark $\left(U_1\right)$ and bright $\left(U_2\right)$ uniform states, canted antiferromagnetic states (AF), frustrated AF states (f-AF), and limit cycles (LC). Regions with double labeling exhibit bistability between the indicated states. Discontinuous transitions are indicated by thick black lines. Inset: Limit cycle for $\mu /\gamma =2.5$ and $\mathrm{\Omega }/\gamma =6.5$ projected onto the Bloch sphere. Red (blue) lines represent the dynamics of cavities in the A (B) sublattice.

Figure 2

$\mathcal{N}=12$ cavities with periodic boundary conditions and nearest-neighbor couplings for $J/\gamma =10$. Black lines show the mean-field phase diagram boundaries. (a) Nearest-neighbor part of the ${\widehat{\sigma }}_i^y$ correlation function, ${\mathrm{\Sigma }}_1^y$. The red region indicates antiferromagnetic correlations. (b) $\delta N^2/N$ (see text), which is strongly enhanced in the presence of collective bistable switching. Inset in (a): ${\mathrm{\Sigma }}_i^y$ for $\mu /\gamma =10$ and $\mathrm{\Omega }/\gamma =6$ (12), exhibiting short-range incommensurate spin density wave (antiferromagnetic) order.

Figure 3

All panels correspond to $\mathcal{N}=12$ cavities and $\mu /\gamma =-2.5$. (a) Example quantum trajectory for $\mathrm{\Omega }/\gamma =2.5$, in the bistable regime, showing collective switching between dark (white) and bright (blue) states. (b) Mean-field calculation of total photon number, showing collective bistability. Solid (dashed) black lines correspond to stable (unstable) solutions of the mean-field equations. (c) Total photon number for the quantum trajectory shown in (a).

Figure 4

(a) Liouvillian gap $\mathrm{\Gamma }/\gamma$ for $\mathcal{N}=20$ cavities with $J/\gamma =10$ and infinite-range coupling in the hard-core $(U\to \infty )$ limit. The solid black line shows the mean-field bistable phase boundary for spatially uniform states. (b) Liouvillian gap for $\mu /\gamma =-5$. The solid dark (light) blue line shows the gap calculated by diagonalizing the Liouvillian for $\mathcal{N}=12$ (8) cavities. Circles (squares) show the gap extracted from quantum trajectory simulations (see text); error bars represent the standard error.

Figure 5

(a) Gutzwiller mean-field result for $({\sigma }_0^z+1)/2$ (equivalent to the local photon number) for $J/\gamma =10$ and $\mu /\gamma =-5$. Solid black lines represent steady-state solutions that are stable at $k=0$; the dashed black line represents a solution of the mean-field equations that is unstable at $k=0$. (b) Real part of the linear stability eigenvalues as a function of the wave number $k$. At values of $k$ for which $\mathrm{Re}[\omega /\gamma ]>0$ there is an instability of the uniform steady state.