Entanglement tongue and quantum synchronization of disordered oscillators

We study the synchronization of dissipatively coupled van der Pol oscillators in the quantum limit, when each oscillator is near its quantum ground state. Two quantum oscillators with different frequencies exhibit an entanglement tongue, which is the quantum analog of an Arnold tongue. It means that the oscillators are entangled in steady state when the coupling strength is greater than a critical value, and the critical coupling increases with detuning. An ensemble of many oscillators with random frequencies still exhibits a synchronization phase transition in the quantum limit, and we analytically calculate how the critical coupling depends on the frequency disorder. Our results can be experimentally observed with trapped ions or neutral atoms.

Figure 1

(a) Arnold tongue for phase locking of two classical oscillators. $V$ is the coupling strength, and $\Delta$ is the difference of the intrinsic frequencies. (b) Entanglement tongue for two oscillators in the quantum limit. Concurrence is plotted using color scale on right. The dashed line marks the edge of the entangled region.

Figure 2

Two-mode Wigner function in (a) the quantum limit (${\kappa }_2\to \infty$) and (b) the classical limit (${\kappa }_2\to 0$), plotted as a function of phase difference with $V=3{\kappa }_1$: $\Delta =0$ (solid, black line) and $\Delta =4{\kappa }_1$ (dashed, red line).

Figure 3

Phase diagram for uniform frequency disorder, showing critical coupling $V_c$ vs. disorder width $\Gamma$. (a) Quantum model with ${\kappa }_2=100{\kappa }_1$ using Eq. (54). (b) Classical model using Eq. (58).

Figure 4

Phase diagram for Lorentzian frequency disorder, showing critical coupling $V_c$ vs. disorder width $\Gamma$. (a) Quantum model with ${\kappa }_2=100{\kappa }_1$. The solid line is without a cutoff [Eq. (56)]. The dashed line is with a cutoff at $\left|\omega \right|=100\Gamma$. (b) Classical model using Eq. (59).

Figure 5

Synchronization phase transition found by simulating mean-field Eqs. (27, 28, 29, 30, 31, 32) with $N=3000$ and ${\kappa }_2=100{\kappa }_1$. Order parameter $\left|A\right|$ is plotted as a function of coupling strength $V$ for different frequency distributions. (a) Delta-function distribution (identical oscillators). (b) Uniform distribution with $\Gamma =20{\kappa }_1$. (c) Lorentzian distribution with $\Gamma =0.7{\kappa }_1$ and cutoff at $\left|\omega \right|=100\Gamma$. Arrows point to the critical coupling predicted by Eq. (46).