Oscillation collapse in coupled quantum van der Pol oscillators

The classical self-oscillations can collapse merely due to their mutual couplings. We investigate this oscillation collapse in quantum van der Pol oscillators. For a pair of quantum oscillators, the steady-state mean phonon number is shown to be lower than in the corresponding classical model with a Gaussian white noise that mimics quantum noise. We further show within the mean-field theory that a number of globally coupled oscillators undergo a transition from the synchronized periodic motion to the collective oscillation collapse. A quantum many-body simulation suggests that the increase in the number of oscillators leads to a lower steady-state mean phonon number, bounded below by the mean-field result.

Figure 1

(a) Mean phonon number $\langle a_1^{†}{a}_1\rangle =\langle a_2^{†}{a}_2\rangle$ and (b) Mandel $Q$ parameter $Q_1=Q_2$, in the steady state of the master equation (2) for $\kappa /G=0.2$. Solid line and curve denote the classical boundaries between the periodic synchronized motion and the oscillator death. Dashed line is the classical boundary of the phase synchronization derived by the neglect of the amplitude dynamics. Insets of (a) show the Wigner functions $W\left({\alpha }_1\right)=W\left({\alpha }_2\right)$ for $V/G=8,\mathrm{\Delta }/G=1$ (upper panel) and for $V/G=3,\mathrm{\Delta }/G=5$ (lower panel). Both insets show the same range $-5\le \mathrm{Re}\left({\alpha }_j\right),\mathrm{Im}\left({\alpha }_j\right)\le 5$.

Figure 2

The mean phonon number of the individual quantum vdP oscillator (solid curve), the squared amplitude of the noiseless classical oscillator (dashed), and the average of the squared amplitude of the noisy classical oscillator (dots), in the steady state for fixed $V/G=10$ and $\kappa /G=0.05$.

Figure 3

(a) Mean phonon number per oscillator ${\overline{n}}_{\mathrm{mf}}$ in the steady state obtained from Eq. (6) for $N=100$ and $\kappa /G=1$. The value of $A$ is nonzero in the left region of the solid curve, and it is zero in the right region. Dashed curve corresponds to the classical boundary given by Eq. (8). (b) Mean phonon number ${\overline{n}}_{\mathrm{mf}}$ for a fixed $V/G=7$ for $\kappa /G=0.5,0.75,1.0,1.5$. (c) The difference between the (noiseless) classical and quantum critical points ${\mathrm{\Delta }}_c^{\left(\mathrm{cl}\right)}-{\mathrm{\Delta }}_c^{\left(\mathrm{mf}\right)}$ versus $\kappa /G$ for $V/G=7$.

Figure 4

Size effects on the oscillation collapse. Dots denote the mean phonon number per oscillator $\overline{n}$ obtained from Eq. (4), and solid line denotes the mean-field result for $\kappa /G=100$. The coupling strength and the frequency mismatch are fixed as $V/G=5$ and $\mathrm{\Delta }/G=10$ at which the mean-field theory predicts the oscillation collapse.