Quantum correlations and mutual synchronization

We consider the phenomenon of mutual synchronization in a fundamental quantum system of two detuned quantum harmonic oscillators dissipating into the environment. We identify the conditions leading to this spontaneous phenomenon, showing that the ability of the system to synchronize is related to the existence of disparate decay rates and is accompanied by robust quantum discord and mutual information between the oscillators, preventing the leak of information from the system.

Figure 1

(a) $\langle x_1^2\left(t\right)\rangle /\mathcal{S}$ (light color) and $\langle x_2^2\left(t\right)\rangle /\mathcal{S}$ (black) normalized with the shot noise $\mathcal{S}$, for ${\omega }_2=1.4{\omega }_1$ and $\lambda =0.7{\omega }_1^2$ starting from squeezed vacuum states for a CB and (b) synchronization $C_{\langle x_1^2\rangle \langle x_2^2\rangle }(t,15)$ for a CB (dark color) and SB (light color) for temperature $T=10{\omega }_1$ (in natural units). The insets show synchronization values $|C_{\langle x_1^2\rangle \langle x_2^2\rangle }|$ varying ${\omega }_2/{\omega }_1$ and $\lambda /{\omega }_1^2$ at $t=300$. Time $t$ is scaled with ${\omega }_1$ (and is therefore adimensional) and $\gamma =0.01{\omega }_1$ here and in the following. The initial state is separable with squeezing parameters 2 and 4, respectively, in the two oscillators.

Figure 2

(a) Eigenvalues ${\mu }_i$ in the complex plane for a CB for ${\omega }_2/{\omega }_1=1.31$ and increasing the coupling from $\lambda =0$ [circles with $\mathrm{Re}\left({\mu }_i\right)\sim -0.01$] to $\lambda =0.9{\omega }_1^2$ in the direction of darker colors. (b) Ratio between minimum and maximum eigenvalue $\mathrm{Re}\left({\mu }_m\right)/\mathrm{Re}\left({\mu }_M\right)$ for a CB.

Figure 3

(a) Mutual information and (b) discord on a logarithmic scale for a common bath and separate baths. Both the exact time evolutions and the filtered ones (Gaussian filter) are shown. The parameters are ${\omega }_2=1.05{\omega }_1$ and $\lambda =0.3{\omega }_1^2$.

Figure 4

Evolution of the discord for a CB and the parameters ${\omega }_2/{\omega }_1=1.05\mathrm{and}\lambda =0.8{\omega }_1^2$ (dashed line $A$) and ${\omega }_2/{\omega }_1=1.4\mathrm{and}\lambda =0.3{\omega }_1^2$ (solid line $B$). The inset represents the quantum discord at $t=300$ for a CB.

Figure 5

(a) Synchronization indicator and (b) decay of quantum correlations for different initial conditions in the case of a common bath: separable squeezed state with squeezings $r_1=2$ and $r_2=4$ (solid line), separable vacuum state (dotted line), and entangled two-mode squeezed state (dashed line) with squeezing $r=2$. Other parameters are ${\omega }_2/{\omega }_1=1.1\mathrm{and}\lambda =0.8{\omega }_1^2$

Figure 6

Rates ${\tilde{\Gamma }}_{++}^{\mathrm{CB}}$ and ${\tilde{\Gamma }}_{--}^{\mathrm{CB}}$, from Eq. (3), and $({\tilde{\Gamma }}_{++}^{\mathrm{CB}}+{\tilde{\Gamma }}_{--}^{\mathrm{CB}})/2$ (dots) compared with the (three different) real parts of the dynamical eigenvalues $\mathrm{Re}\left({\mu }_i\right)$ (continuous line) in the case of a common bath for ${\omega }_2/{\omega }_1=1.31$.

Figure 7

(a) Synchronization and (b) discord obtained from the complete master equation (A4) compared with the values obtained after the rotating-wave approximation, as described in the text in the case of common bath, for ${\omega }_2/{\omega }_1=1.4\mathrm{and}\lambda =0.7{\omega }_1^2$.