Quantum synchronization in an optomechanical system based on Lyapunov control

We extend the concepts of quantum complete synchronization and phase synchronization, which were proposed in A. Mari et al., Phys. Rev. Lett. 111, 103605 (2013), to more widespread quantum generalized synchronization. Generalized synchronization can be considered a necessary condition or a more flexible derivative of complete synchronization, and its criterion and synchronization measure are proposed and analyzed in this paper. As examples, we consider two typical generalized synchronizations in a designed optomechanical system. Unlike the effort to construct a special coupling synchronization system, we purposefully design extra control fields based on Lyapunov control theory. We find that the Lyapunov function can adapt to more flexible control objectives, which is more suitable for generalized synchronization control, and the control fields can be achieved simply with a time-variant voltage. Finally, the existence of quantum entanglement in different generalized synchronizations is also discussed.

Figure 1

Diagram of an optomechanical system corresponding to our model. Two oscillators are placed at the wave nodes of a Fabry-Pérot cavity, and they couple with the cavity field via linear optomechanical interactions. Their origins are set at the equilibrium positions.

Figure 2

Evolution of the expectation values (blue dashed and red solid lines), the control field, and the errors (the black solid and green dashed lines indicate that the control field is imposed or removed, respectively). Parts (a) and (b) correspond to the momentum and position operators of each oscillator, respectively. Here we set $\mathrm{\Delta }=1$ as a unit, and the other parameters are ${\omega }_{m1}=1, {\omega }_{m2}=1.005, g_1=0.008, g_2=0.005, E=10, \kappa =0.15, \gamma =0.005$, and ${\overline{n}}_b=0.05$. For the control field, the parameters are taken as $k=2$ and $c_{-}=3$. (c) The limit cycles of two oscillators in phase space. (d) The robustness of our controlled system. Here none of the simulations except for $R\left(\sigma \right)$ contain any noise. The bottom inset in (d) is the contrast of control fields without (red, dark) and with noise (yellow, pale), and the upper inset in (d) is the synchronization errors $p_{-}^{\prime }$ (black solid) and $q_{-}^{\prime }$ (green dashed) when the control field has noise. The horizontal axes of all the insets are the time $t$.

Figure 3

(a) Evolutions of the modified synchronization measure $S_g^{\prime }$. The inset in (a) shows the evolutions of the synchronization measure $S_g$. (b) Time-averaged synchronization measures with varied bath temperature. Blue (solid) lines correspond to the imposed control field, and red (dotted) lines denote that the case control field disappears. In these simulations we set $T=200$, and other parameters are the same as those in Fig. 2.

Figure 4

Evolutions of the expectation values (blue dashed and red solid lines), the control field, and the errors (black solid and green dashed lines indicate that the control field is imposed or removed, respectively). Parts (a) and (b) correspond to the momentum and position operators of each oscillator, respectively. (c) The limit cycles of two oscillators in phase space. This phase diagram corresponds to the curve segment within the black dotted box in (b). (d) The robustness of the control system. Here we set $\tau =5, C_M=1$, and the other parameters are the same as those in Fig. 2. The bottom inset in (d) is the contrast of the control field without (red, dark) and with noise (yellow, pale), and the upper inset in (d) is the synchronization errors $p_{-}^{\prime }$ (black solid) and $q_{-}^{\prime }$ (green dashed) when the control field has noise. The horizontal axes of the insets in (a), (b), and (d) are the time $t$. Here we define the moment imposing the control field as $t=0$.

Figure 5

(a) Evolutions of the modified synchronization measure. (b) Time-averaged synchronization measures with varied bath temperature. Blue (solid) lines correspond to the imposed control field, and red (dotted) lines indicate that the control field disappears. Here all parameters are the same as those in Fig. 3.

Figure 6

The largest Lyapunov exponents of the errors with varied $c_{-}$ (a) and $\tau$ (b) corresponding to constant error synchronization and time delay synchronization, respectively. Here all the parameters except $c_{-}$ and $\tau$ are the same as those in Fig. 2. The inset in (b) shows the largest Lyapunov exponents of the errors with varied $C_M$ under $\tau =5$.

Figure 7

(a) The maximum negativities of the errors with varied $c_{-}$ (a) and $\tau$ and $C_M$ (b) corresponding to constant error synchronization and time-delay synchronization, respectively. Here all parameters except $c_{-}, \tau$, and $C_M$ are the same as those in Fig. 2. (b) Time-averaged synchronization measures with varied bath temperature. Here the black (dark) lines correspond to complete synchronization (solid line) and phase synchronization (dotted line) in Ref. [1], respectively. The yellow (pale) lines correspond to our constant error synchronization (dotted line) and time-delay synchronization (solid line).

Figure 8

Comparison of two computation methods corresponding to the linear mean-field approximation and the nonlinear quantum master equation. (a) Evolutions of expectation values $q_1$. (b) Evolutions of $S_g^{\prime }$. (c) Relative error as a function of $\kappa$ under $g_1=0.008$. The inset in (c) shows the relative error as a function of $g_1$ under $\kappa =0.15$. The other parameters are the same as those in Fig. 2.