Mutual information as an order parameter for quantum synchronization

Spontaneous synchronization is a fundamental phenomenon, important in many theoretical studies and applications. Recently, this effect has been analyzed and observed in a number of physical systems close to the quantum-mechanical regime. In this work we propose mutual information as a useful order parameter which can capture the emergence of synchronization in very different contexts, ranging from semiclassical to intrinsically quantum-mechanical systems. Specifically, we first study the synchronization of two coupled Van der Pol oscillators in both classical and quantum regimes and later we consider the synchronization of two qubits inside two coupled optical cavities. In all these contexts, we find that mutual information can be used as an appropriate figure of merit for determining the synchronization phases independently of the specific details of the system.

Figure 1

Synchronization analysis for the steady state of the two-VdPs model (2): (a),(b),(c) Mutual information; (d),(e),(f) semiclassical synchronization measure defined in Eq. (4). The quantumness parameter for (a),(d) is ${\kappa }_2/{\kappa }_1=0.1$; for (b),(e) is ${\kappa }_2/{\kappa }_1=10$; and for (c),(f) is ${\kappa }_2/{\kappa }_1=100$. $\Delta$ (units of ${\kappa }_1)$ is the detuning of VdPs and $g$ (units of ${\kappa }_1)$ is the coupling strength.

Figure 2

$a_1,a_2$ and $q_1,q_2$ are the optical modes and qubits in the first and second cavities, respectively. $E$ is a driving laser amplitude which is applied to the first cavity. Photons can coherently hop from one cavity to the other with a rate $g$.

Figure 3

Mutual information of the steady state of the two qubits model as a function of the optical coupling constant $g$ (units of $\mu )$ and detuning $\Delta$ (units of $\mu )$. The other parameters are ${\omega }_1=10,\kappa =0.05,E=3$, and $\mu =1$.

Figure 4

(a),(b),(c) Simulation of phase-locking measure $s_p$ defined in Eq. (8) as a function of time (in units of $\tau =2\pi /{\omega }_1)$. (d),(e),(f) Mutual information of qubits as a function of time (in units of $\tau )$. The parameters are (a),(d) ${\omega }_1=10,{\omega }_2=10,g=0.5,\mu =1,\kappa =0.1,E=3$; (b),(e) ${\omega }_1=10,{\omega }_2=10,g=-0.5,\mu =1,\kappa =0.1,E=3$; (c),(f) ${\omega }_1=10,{\omega }_2=20,g=0.5,\mu =1,\kappa =0.1,E=3$.

Figure 5

(a) Classical information and (b) quantum discord between the two qubits as a function of the optical coupling constant $g$ (units of $\mu )$ and detuning $\Delta$ (units of $\mu )$. The other parameters are as in Fig. 3.