Minimal model for spontaneous quantum synchronization

We show the emergence of spontaneous synchronization between a pair of detuned quantum oscillators within a harmonic network. Our model does not involve any nonlinearity, driving, or external dissipation, thus providing the simplest scenario for the occurrence of local coherent dynamics in an extended harmonic system. A sufficient condition for synchronization is established by building upon the Rayleigh normal mode approach to vibrational systems. Our results show that mechanisms favoring synchronization, even between oscillators that are not directly coupled to each other, are transient energy depletion and crosstalk. We also address the possible buildup of quantum correlations during synchronization and show that indeed entanglement may be generated in detuned systems, starting from uncorrelated states and without any direct coupling between the two oscillators.

Figure 1

Synchronization measure $C\left(t\right)$ and dynamics for ${\omega }_2/{\omega }_1=1.1, \lambda =0.5{\omega }_1^2$, and $x_1\left(0\right)=0.14{\omega }_1^{-1/2}$. The other parameters are $x_2\left(0\right)=1.4{{\omega }_1}^{-1/2}, M=300, {\mathrm{\Omega }}_0=0.4{\omega }_1, g=1.2{\omega }_1^2$, and $K=0.2{\omega }_1^2$. In the lower panel the black line denotes the function $\langle q_1\left(t\right)\rangle$ and the red one is for $\langle q_2\left(t\right)\rangle$. Here $\langle q_{1,2}\rangle$ are in units of ${\omega }_1^{-1/2}$ and time is in units of ${\omega }_1^{-1}$.

Figure 2

Shown in the top plot is the synchronization measure $C\left(t\right)$ as a function of time for two different values of the coupling $K=0.1$ (solid black) and 0.8 (dotted red), in units of ${\omega }_1^2$. The other parameters are fixed: ${\omega }_2/{\omega }_1=1.1, \lambda =0, M=300, {\mathrm{\Omega }}_0=0.4{\omega }_1$, and $g=1.2{\omega }_1^2$. In the inset we show a schematic representation of the considered configuration. The bottom plot shows the dynamics of the two oscillators plugged into the first node of the chain, with $K=0.1$ (left) and $K=0.8$ (right), in units of ${\omega }_1^2$. Here $\langle x_{1,2}\rangle$ are in units of ${\omega }_1^{-1/2}$ and time in units of ${\omega }_1^{-1}$.

Figure 3

(a) Dynamics, (b) synchronization measure $C\left(t\right)$, and (c) enlarged view of the dynamics for two uncoupled oscillators ($\lambda =0$), with detuning $\delta \omega /{\omega }_1=0.1$ coupled to the extremities of the oscillator chain with coupling constant $K=0.2{\omega }_1^2$ and $M=300, {\mathrm{\Omega }}_0=0.4{\omega }_1$, and $g=1.2{\omega }_1^2$. The inset in (a) shows a schematic representation of the configuration under analysis. (b) Synchronization measure for two different initial conditions: symmetric $x_1\left(0\right)=x_2\left(0\right)=1.4{\omega }_1^{-1/2}$ and $p_1\left(0\right)=p_2\left(0\right)=0$ (solid black line) and asymmetric $x_1\left(0\right)=x_2\left(0\right)=2{\omega }_1^{-1/2}$ and $p_1\left(0\right)=0, =p_2\left(0\right)=10\sqrt{{\omega }_1}$ (dashed red line). (c) Dynamics of the two oscillators $\langle x_1\left(t\right)\rangle$ (solid black line) and $\langle x_2\left(t\right)\rangle$ (dashed red line) for a time window where synchronization appears. The background colors in (a) and (b) are inserted as a guide for the eye to mark the three regions of independent dissipation (pink), crosstalk (blue), and revival (light blue). Here $\langle x_{1,2}\rangle$ are in units of ${\omega }_1^{-1/2}$ and time is in units of ${\omega }_1^{-1}$.

Figure 4

Shown from top to bottom are quadratures, mutual information and entanglement, and the synchronization factor, starting from a separable squeezed state with squeezing parameters $r_1=r_2=2$, frequencies ${\omega }_2/{\omega }_1=1.2$, and coupling $\lambda =0$. The two oscillators are strongly coupled to the reservoir $K=0.8{\omega }_1^2$ and the bath parameters are the same as in previous figures. Here $\langle x_{1,2}^2\rangle$ are in units of ${\omega }_1^{-1}$ and time is in units of ${\omega }_1^{-1}$.

Figure 5

Show from top to bottom are the position variances, mutual information and entanglement, and synchronization factor, starting from a separable squeezed state with squeezing parameters $r_1=r_2=2$, frequencies ${\omega }_2/{\omega }_1=1.2$, and coupling $\lambda =0$. The coupling to the reservoir is $K=0.2{\omega }_1^2$ and the bath parameters are the same as in previous figures. Here $\langle x_{1,2}^2\rangle$ are in units of ${\omega }_1^{-1}$ and time is in units of ${\omega }_1^{-1}$.

Figure 6

Density plot of the synchronization measure $C\left(t\right)$ for a 300-site chain as a function of the plugging site $m$ and the time $t$. The parameters are set as ${\omega }_2/{\omega }_1=1.1, \lambda =0.5{\omega }_1^2, K=0.06{\omega }_1^2, {\mathrm{\Omega }}_0=0.4{\omega }_1$, and $g=1.2{\omega }_1^2$.