Gaussian trajectory approach to dissipative phase transitions: The case of quadratically driven photonic lattices

We apply the Gaussian trajectories approach to the study of the critical behavior of two-dimensional dissipative arrays of nonlinear photonic cavities, in the presence of two-photon driving and in regimes of sizable loss rates. In spite of the highly mixed character of the density matrix of this system, the numerical approach is able to provide precise estimations of the steady-state expectation values, even for large lattices made of more than 100 sites. By performing a finite-size scaling of the relevant properties of the steady state, we extrapolate the behavior of the system in the thermodynamic limit and we show the emergence of a second-order dissipative phase transition, belonging to the universality class of thermal Ising model. This result indicates the occurrence of a crossover when the loss rate is increased from the weak-loss limit, in which the phase transition belongs to the universality class of the quantum Ising model.

Figure 1

The steady-state relative occupation of the $k=0$ mode increases as a function $G/\gamma$, as computed in lattices of different size. Results are obtained from 1000 trajectories (500 for the $12\times 12$ lattice), each evolved over a total time $t=100{\gamma }^{-1}$. For each quantum trajectory, the vacuum is chosen as initial state and the two-photon driving is switched on slowly to the desired value, in order to reduce the formation of defects which would appear from strong quenching [70]. The steady-state expectation values are computed averaging the results obtained in a time interval from a size-dependent initial time $t_{\text{relax}}$ to the final time $t_{\mathrm{fin}}=100{\gamma }^{-1}$.

Figure 2

[(a)–(d)] Time evolution of the order parameter $\overline{\alpha }$ along a single Gaussian trajectory in a $6\times 6$ lattice, for different values of the driving amplitude: (a) $G/\gamma =0.7$, (b) $G/\gamma =0.8$, (c) $G/\gamma =0.9$, and (d) $G/\gamma =1.0$. (e) Probability distribution $p\left(\overline{\alpha }\right)$ of the order parameter for different values of $G$. For each value of $G$, we consider 1000 trajectories each evolved over a time $t=400$ and we collect the data for $\overline{\alpha }\left(t\right)$ for $t\ge 20$, i.e., for long times in which the density matrix of the dissipative system has reached the steady state. The distribution $p\left(\overline{\alpha }\right)$ is obtained as the histogram of the collected data. As $G$ increases, a qualitative transition from monomodal to bimodal distribution is evident.

Figure 3

Binder cumulant $U_L$ as a function of the two-photon driving amplitude $G$, for different lattice sizes $L$. A crossing of the curves for different lattice sizes can be seen, signaling the emergence of a critical point at $G=G_c$ (the difference $2/3-U_L$ is shown on a logarithmic scale). Inset: The curves for different sizes show an universal behavior when plotted as a function of $(G-G_c)L^{1/\nu }$, with $\nu =1$ being the critical exponent for the correlation length in the classical 2D Ising model.