Quantum Synchronization of a Driven Self-Sustained Oscillator

Synchronization is a universal phenomenon that is important both in fundamental studies and in technical applications. Here we investigate synchronization in the simplest quantum-mechanical scenario possible, i.e., a quantum-mechanical self-sustained oscillator coupled to an external harmonic drive. Using the power spectrum we analyze synchronization in terms of frequency entrainment and frequency locking in close analogy to the classical case. We show that there is a steplike crossover to a synchronized state as a function of the driving strength. In contrast to the classical case, there is a finite threshold value in driving. Quantum noise reduces the synchronized region and leads to a deviation from strict frequency locking.

Figure 1

Classical phase-space trajectory (black solid line), Wigner function, and the corresponding phase probability histograms for $\mathrm{\Omega }/{\gamma }_1=1$ and ${\gamma }_2/{\gamma }_1=0.1$. (a),(e) $\mathrm{\Delta }/{\gamma }_1=16$, (b),(f) $\mathrm{\Delta }/{\gamma }_1=0.6$, (c),(g) $\mathrm{\Delta }/{\gamma }_1=0.1$, and (d),(h) $\mathrm{\Delta }/{\gamma }_1=0$.

Figure 2

Classical phase-space trajectory (black solid line), Wigner function, and the corresponding phase probability histograms for $\mathrm{\Omega }/{\gamma }_1=3$ and ${\gamma }_2/{\gamma }_1=3$. (a),(e) $\mathrm{\Delta }/{\gamma }_1=100$, (b),(f) $\mathrm{\Delta }/{\gamma }_1=12$, (c),(g) $\mathrm{\Delta }/{\gamma }_1=0.1$, and (d),(h) $\mathrm{\Delta }/{\gamma }_1=0$.

Figure 3

Normalized classical (full lines) and quantum (dashed lines) spectrum. For $\mathrm{\Delta }/{\gamma }_1=0.7$ (blue) both the quantum and classical model show a peak at a frequency smaller than the detuning (indicated by vertical black lines). For $\mathrm{\Delta }/{\gamma }_1=0.5$ (red) the classical model is frequency locked to the drive, while the quantum model is not. The other parameters are $\mathrm{\Omega }/{\gamma }_1=1$ and ${\gamma }_2/{\gamma }_1=0.1$.

Figure 4

Observed frequency ${\omega }_{\mathrm{obs}}$ vs detuning $\mathrm{\Delta }$ for ${\gamma }_2/{\gamma }_1=0.1$. Blue: undriven case. Black: classical model. Red: quantum model. In (a), for $\mathrm{\Omega }/{\gamma }_1=0.1$, the quantum model shows weak frequency entrainment. Inset: difference between undriven and the quantum mechanical case. In (b), for $\mathrm{\Omega }/{\gamma }_1=1$, the quantum model shows strong frequency entrainment and a tendency towards frequency locking. Inset: zoom-in for the quantum case at small $\mathrm{\Delta }$.

Figure 5

Observed frequency ${\omega }_{\mathrm{obs}}$ vs driving $\mathrm{\Omega }$ in the quantum model for various values of ${\gamma }_2/{\gamma }_1$ and $\mathrm{\Delta }/{\gamma }_1=0.01$. A steplike crossover from weak to strong entrainment can be observed. At larger ${\gamma }_2/{\gamma }_1$ the step shifts to larger driving strengths. Below threshold, increasing ${\gamma }_2/{\gamma }_1$ reduces frequency entrainment, i.e., inhibits synchronization. Right inset: Observed frequency at larger driving. Increasing ${\gamma }_2/{\gamma }_1$ leads to increased frequency entrainment, i.e., improved synchronization. Left inset: The observed frequency ${\omega }_{\mathrm{obs}}$ for ${\gamma }_2/{\gamma }_1\to \infty$ (black line) vanishes like $(\mathrm{\Omega }/{\gamma }_1{)}^{-2}$ (gray dashed line).

Figure 6

Observed frequency ${\omega }_{\mathrm{obs}}$ vs nonlinear damping rate ${\gamma }_2$ in the quantum model for various driving strengths $\mathrm{\Omega }$ and $\mathrm{\Delta }/{\gamma }_1=0.01$. Increasing the driving strength is seen to enhance frequency entrainment. For weak (strong) driving and at ${\gamma }_2/{\gamma }_1\lesssim 1$, frequency entrainment is reduced (enhanced) by increasing ${\gamma }_2/{\gamma }_1$. Squares on the right are obtained analytically in the limit ${\gamma }_2/{\gamma }_1\to \infty$.