Quantum-classical transition of correlations of two coupled cavities

We study the difference between quantum and classical behavior in a pair of nonidentical cavities with second-harmonic generation. In the classical limit, each cavity has a limit-cycle solution, in which the photon number oscillates periodically in time. Coupling between the cavities leads to synchronization of the oscillations and classical correlations between the cavities. In the quantum limit, there are quantum correlations due to entanglement. The quantum correlations persist even when the cavities are far off resonance with each other, in stark contrast with the classical case. We also find that the quantum and classical limits are connected by an intermediate regime of almost no correlations. Our results can be extended to a wide variety of quantum models.

Figure 1

Two cavities with nonlinear crystals inside are laser-driven and dissipate photons. They are coupled to each other due to overlap of their photonic wave functions.

Figure 2

Classical and quantum trajectories for one cavity, showing photon numbers of mode $a_1$ (thick black line) and mode $a_2$ (thin red line) over time. (a) Limit-cycle solution of the classical model for $E=8{\kappa }_1$, $\chi =0.8{\kappa }_1$. (b–d) Quantum trajectories for the same system, but as it becomes more quantum: (b) $E=8{\kappa }_1$, $\chi =0.8{\kappa }_1$; (c) $E=2{\kappa }_1$, $\chi =3.2{\kappa }_1$; and (d) $E={\kappa }_1$, $\chi =6.4{\kappa }_1$. In (d), the modes are antibunched with each other due to quantum correlations, despite appearing to be positively correlated in the plot. (e) Correlation $g_2(a_1,a_2)$ for one cavity, using color scale on right. The black line indicates the location of the Hopf bifurcation $E_c$. All plots use ${\kappa }_2=0.5{\kappa }_1$, ${\Delta }_1=0.5{\kappa }_1$, and ${\Delta }_2={\kappa }_1$.

Figure 3

(a) Eigenstates for one cavity. (b) Eigenstates for two identical cavities. When ${\Delta }_1<0$, there is antibunching between $a_1$ and $b_1$ (blue arrows). When ${\Delta }_1>0$, there is antibunching (red arrows).

Figure 4

Classical and quantum trajectories for two identical cavities, showing photon numbers of mode $a_1$ (thick black line) and mode $b_1$ (thin red line) over time. (a) Antiphase synchrony for ${\Delta }_1^a={\Delta }_1^b=0.5{\kappa }_1,{\Delta }_2={\kappa }_1$ in the classical model. (b) In-phase synchrony for ${\Delta }_1^a={\Delta }_1^b=-0.5{\kappa }_1,{\Delta }_2=-{\kappa }_1$ in the classical model. (c, d) Quantum trajectories for the same parameters as (a) and (b), respectively. All plots use $E=8{\kappa }_1$, $\chi =0.8{\kappa }_1$, ${\kappa }_2=0.5{\kappa }_1$, $V_1=0.2{\kappa }_1$, and $V_2=0$.

Figure 5

Phase diagram for two identical cavities as a function of ${\Delta }_1\equiv {\Delta }_1^a={\Delta }_1^b$ and ${\Delta }_2$. (a) Classical limit for $\chi =0.8{\kappa }_1$, ${\kappa }_2=0.5{\kappa }_1$, $V_1=0.2{\kappa }_1$, and $V_2=0$. For each value of ${\Delta }_1$ and ${\Delta }_2$, $E$ is set to $E_c+0.5{\kappa }_1$. (b) Quantum limit for the same parameters, except with $E\to 0$ and $\chi \to \infty$.

Figure 6

Correlation $g_2(a_1,b_1)$ for two nonidentical cavities vs the difference $\delta \equiv {\Delta }_1^a-{\Delta }_1^b$ for a fixed average ${\Delta }_1\equiv ({\Delta }_1^a+{\Delta }_1^b)/2$. (a) Classical limit for $E=10.5{\kappa }_1$, $\chi ={\kappa }_1$, ${\kappa }_2=0.5{\kappa }_1$, ${\Delta }_1=-3{\kappa }_1$, ${\Delta }_2=V_2=0$, and $V_1=0.01{\kappa }_1$. (b) Quantum limit for the same parameters, except with $E\to 0,\chi \to \infty$.

Figure 7

Quantum-classical transition for two identical cavities. For each value of $E$, $\chi$ is set so that $E\chi =6.4{\kappa }_1^2$. (a) ${\Delta }_1^a={\Delta }_1^b=0.5{\kappa }_1$, ${\Delta }_2={\kappa }_1$. (b) ${\Delta }_1^a={\Delta }_1^b=-0.5{\kappa }_1$, ${\Delta }_2=-{\kappa }_1$. Red arrows point to the classical values of $g_2(a_1,b_1)$. Dashed lines mark $g_2(a_1,b_1)=1$. Other parameters are ${\kappa }_2=0.5{\kappa }_1$, $V_1=0.2{\kappa }_1$, and $V_2=0$. The statistical uncertainty for each point is smaller than the marker size.