Properties and relative measure for quantifying quantum synchronization

Although quantum synchronization phenomena and corresponding measures have been widely discussed recently, it is still an open question how to characterize directly the influence of nonlocal correlation, which is the key distinction for identifying classical and quantum synchronizations. In this paper, we present basic postulates for quantifying quantum synchronization based on the related theory in Mari's work [Phys. Rev. Lett. 111, 103605 (2013)], and we give a general formula of a quantum synchronization measure with clear physical interpretations. By introducing Pearson's parameter, we show that the obvious characteristics of our measure are the relativity and monotonicity. As an example, the measure is applied to describe synchronization among quantum optomechanical systems under a Markovian bath. We also show the potential by quantifying generalized synchronization and discrete variable synchronization with this measure.

Figure 1

(a) Schematic diagram of an optomechanical system: a mechanical resonator is coupled with a Fabry-Pérot cavity through the nonlinear radiation pressure force of a quantized optical mode. (b) Two kinds of coupling structures. Here, each node represents an optomechanical system, and the lines denote phonon tunnelings.

Figure 2

(a), (b), and (c) Evolutions of $q_1, \langle {\delta }^2{q}_1\rangle$, and ${\mathcal{S}}_c$ based on the mean-field approximation (yellow dashed line) and the stochastic dynamic method (black solid line), respectively. The blue dashed lines on the right side of (a) present the 100 stochastic trajectories. Here, we set ${\omega }_{m1}=1$ as a unit, and other parameters are ${\omega }_{m2}=1, {\mathrm{\Delta }}_j=1, g=0.003, E=10, \mu =0.02, {\overline{n}}_b=0, \kappa =0.15$, and $\gamma =0.005$. In this simulation, the stochastic dynamics (black solid lines) are obtained based on 10 000 calculations of the stochastic Langevin equations.

Figure 3

(a) and (b) Comparisons of Gaussian fidelity, local measure [blue marks in (a)], Mari's measure, and our measure with varied interaction intensity $\mu$ and bath phonon number ${\overline{n}}_b$. Parts (a) and (b) are obtained, respectively, by 5000 and 3000 calculations of the stochastic Langevin equations. Here, the black (dark) solid lines denote Gaussian fidelity, and black (dark) and yellow (pale) dashed lines are our measures ${\mathcal{L}}_c$ and ${\mathcal{L}}_c^{\prime }$. The yellow (pale) circles and lines are normalized Mari's measure ${\mathcal{S}}_c/\text{max}\left\{S_c\right\}$ calculated by stochastic Langevin equations and the mean-field method, respectively. The local measure and fidelity are calculated by the mean-field method. (c) and (d) Synchronizations corresponding to the same value of Mari's measure. The blue dashed lines are the 100 stochastic trajectories of $q_1$ and $\text{Er}=q_1-q_2$, and the black lines denote $\langle q_2\rangle$ and $\langle \text{Er}\rangle$. Here we set $\mathrm{\Delta }t=0.1$ for ${\mathcal{L}}_c$ and $g=0$ in (c,d), and the other parameters are the same as those in Fig. 2.

Figure 4

(a) Comparisons of Mari's measure (${\mathcal{S}}_c/\text{max}\left\{S_c\right\}$) and our measure with varied classical amplitudes. Here we set $\langle {\widehat{b}}_j\rangle \left(0\right)=20A_m$. (b) Comparisons of our measure ${\mathcal{L}}_c$ with a different time window $\mathrm{\Delta }t$. The parameters in (a) and (b) are the same as those in Figs. 3 and 2, respectively.

Figure 5

Coordinates (a), momenta (b), and errors (c) of three coupled optomechanical systems. In (a) and (b), the black (dark, solid) lines correspond to system 1, and the yellow (pale, dashed) lines denote system 2. The subfigures on the right side are the corresponding local magnifications in which the blue circles correspond to system 3. Here we set $g=0.005, {\omega }_s=1, {\omega }_1=1.03, {\omega }_2=1.02$, and ${\omega }_3=1.01$, which leads to ${\mu }_1=0.02$ and ${\mu }_2=0.01$. The other parameters are the same as those in Fig. 2. In this simulation, the stochastic dynamics are obtained by 1000 calculations of the stochastic Langevin equations.

Figure 6

Comparisons of Mari's measure ${\mathcal{S}}_c$ (a) and our measures ${\mathcal{L}}_c$ (b) and ${\mathcal{L}}_c^{\prime }$ (c). Here the black (dark, solid) lines denote the measure between systems 1 and 2, and the yellow (pale, dashed) lines present the measure between systems 1 and 3. The parameters are the same as those in Fig. 5. In (a), the time window of the measure is set as $\mathrm{\Delta }t=10$.