Quantum Critical Regime in a Quadratically Driven Nonlinear Photonic Lattice

We study an array of coupled optical cavities in the presence of two-photon driving and dissipation. The system displays a critical behavior similar to that of a quantum Ising model at finite temperature. Using the corner-space renormalization method, we compute the steady-state properties of finite lattices of varying size, both in one and two dimensions. From a finite-size scaling of the average of the photon number parity, we highlight the emergence of a critical point in regimes of small dissipations, belonging to the quantum Ising universality class. For increasing photon loss rates, a departure from this universal behavior signals the onset of a quantum critical regime, where classical fluctuations induced by losses compete with long-range quantum correlations.

Figure 1

A pictorial sketch of the phase diagram of the quadratically driven dissipative Bose Hubbard model. For vanishing one-photon loss rate $\gamma$, when increasing the amplitude of the two-photon driving field $G$, the system undergoes a quantum phase transition. A quantum critical point (QCP) marks the transition between a paramagnetic (PM) phase, where the NESS in the thermodynamic limit has a definite parity, and a ferromagnetic (FM) phase, where the ${\mathbb{Z}}_2$ symmetry of the system is spontaneously broken. For finite values of the loss rate $\gamma$, in the vicinity of the QCP, a quantum critical regime (QCR) arises, where classical fluctuations act directly on the quantum-critical entangled state.

Figure 2

Computed values of the von Neumann entropy $S$ (a) and of the parity $\mathrm{\Pi }$ (b) as a function of $G/\gamma$ for the NESS of 2D lattices of different sizes. (c) Finite size scaling of the parity using the critical exponents of the quantum 2D transverse Ising model ($\beta =0.32642$ and $\nu =0.62997$). Inset: rescaled parity plotted vs $G/\gamma$. Parameters: $U/\gamma =40$ and $J/\gamma =20$.

Figure 3

(a) von Neumann entropy $S$ computed as a function of $G/\gamma$ for the NESS of 2D lattices of different sizes. (b) Finite size scaling analysis of the parity using the critical exponents of the quantum 2D transverse Ising model ($\beta =0.32642$ and $\nu =0.62997$). Inset: rescaled parity plotted vs $G/\gamma$. Parameters: $U/\gamma =20$ and $J/\gamma =10$.

Figure 4

(a) von Neumann entropy $S$ computed as a function of $G/\gamma$ for the NESS of 1D arrays of different lengths. (b) Finite size scaling analysis of the parity using the critical exponents of the quantum 1D transverse Ising model ($\beta =0.125$ and $\nu =1$). Inset: rescaled parity plotted vs $G/\gamma$. Parameters: $U/\gamma =100$ and $J/\gamma =50$.