Quantum Synchronization Blockade: Energy Quantization Hinders Synchronization of Identical Oscillators

Classically, the tendency towards spontaneous synchronization is strongest if the natural frequencies of the self-oscillators are as close as possible. We show that this wisdom fails in the deep quantum regime, where the uncertainty of amplitude narrows down to the level of single quanta. Under these circumstances identical self-oscillators cannot synchronize and detuning their frequencies can actually help synchronization. The effect can be understood in a simple picture: Interaction requires an exchange of energy. In the quantum regime, the possible quanta of energy are discrete. If the extractable energy of one oscillator does not exactly match the amount the second oscillator may absorb, interaction, and thereby synchronization, is blocked. We demonstrate this effect, which we coin quantum synchronization blockade, in the minimal example of two Kerr-type self-oscillators and predict consequences for small oscillator networks, where synchronization between blocked oscillators can be mediated via a detuned oscillator. We also propose concrete implementations with superconducting circuits and trapped ions. This paves the way for investigations of new quantum synchronization phenomena in oscillator networks both theoretically and experimentally.

Figure 1

(a) Illustration of an anharmonic oscillator level structure (grey) with nonlinear amplification (purple) and damping (green) tuned such that a particular Fock state (here, the state $|2⟩$) is stabilized. The relative thickness of the arrows indicates the transition rate; steady-state population probabilities are depicted in black. The corresponding classical system would have a continuous energy distribution, which is sketched in grey. (b) Implementation with an array of superconducting anharmonic oscillators driven by an amplification cavity (purple) and a damping cavity (green), see main text. (c) Implementation with trapped ions with excited states transition between ground state $|g⟩$ and excited states $|e⟩$, $|e^{\prime }⟩$ enabling, respectively, sideband cooling and sideband amplification of motion as depicted in (d).

Figure 2

Synchronization measure $S$ calculated using Eq. (6) as a function of the detuning $\mathrm{\Delta }$ between two oscillators. All other parameters are identical for both oscillators. Panel (a) shows the result of classical Monte Carlo simulations of Eq. (5), where the width of the line indicates the statistical error. Panel (b) shows results from the numerical steady-state solution of the quantum master equation (4). Classical parameters: ${\gamma }_T{n}_t=0.1\gamma$, $\sigma =0.2$, $n_{+}=2$, $n_{-}=3$, $K=2\gamma$, $V=0.1\gamma (\frac{1}{6},\frac{1}{4},\frac{3}{8},\frac{1}{2})$. Quantum parameters: $\sigma =0.2$, $n_{+}=2$, $n_{-}=3$, $K=10\gamma$, $V=\gamma (0.05,0.2,0.35,0.5)$. In both panels ${\gamma }_{1\pm }={\gamma }_{2\pm }=\gamma$. In first-order perturbation theory (dashed black line), the height of the maximal peak is proportional to the coupling $V$ [46]. Plotting $S$ in units of $V$ for the numerical results, the height of the peak decreases with increasing $V$, where higher-order effects play a role. The noise level is chosen such that the effect of the thermal noise in panel (a) is approximately as strong as the quantum noise in panel (b).

Figure 3

Plots of synchronization measure $S$ from Eq. 2. (a) Identical oscillators differing only in frequency. Inset shows cuts of $S$ scaled by $\sigma$ at $\sigma =0.2$, 0.4, 0.6, 0.8 in black, blue, red, green. (b) Identical oscillators differing only in amplitude. (c) Overview of these resonances in $\mathrm{\Delta }n$ and $\mathrm{\Delta }$. (d) Resonances as a function of the Kerr nonlinearity $K$. Parameters: In the upper panels $n_1^{+}\equiv 4$, $n_j^{-}=n_j^{+}+1$, $V=0.1\gamma$, $K=\gamma /{\sigma }_{\pm }$, ${\gamma }_{1\pm }={\gamma }_{2\pm }=\gamma$. In (a), $n_2^{+}=4$, and in (b), ${\omega }_1={\omega }_2$. In the lower panels, ${\sigma }_{\pm }=0.2$, and all other parameters are as above.

Figure 4

Synchronization measure $S$ for the network of three oscillators depicted in the inset of panel (c). In the upper panels, two identical oscillators are connected indirectly via an oscillator $C$ with detuning $\mathrm{\Delta }$, while the direct link $V_2=0$. In the lower panel, the direct link $V_2$ is also turned on and $\mathrm{\Delta }\equiv 2K$. In both panels, $n_1^{+}\equiv 1$, $n_j^{-}=n_j^{+}+1$, $K=10\gamma$, ${\gamma }_{1\pm }={\gamma }_{2\pm }=\gamma$. In both rows, the left panel shows synchronization between different oscillators and the right panel between identical oscillators. We conclude, from the upper panels (a) and (b), that identical oscillators can be synchronized indirectly via a detuned oscillator. The lower panels indicate that increasing the direct coupling can even decrease synchronization for a strong enough indirect link: In (c) the contour lines bend to the right, while in (d), a strip of suppressed synchronization appears.