Genuine Quantum Signatures in Synchronization of Anharmonic Self-Oscillators

We study the synchronization of a Van der Pol self-oscillator with Kerr anharmonicity to an external drive. We demonstrate that the anharmonic, discrete energy spectrum of the quantum oscillator leads to multiple resonances in both phase locking and frequency entrainment not present in the corresponding classical system. Strong driving close to these resonances leads to nonclassical steady-state Wigner distributions. Experimental realizations of these genuine quantum signatures can be implemented with current technology.

Figure 1

Van der Pol self-oscillator with Kerr anharmonicity. (a) Lowest-lying energy levels. The Kerr anharmonicity $K$ leads to a level spacing ${\omega }_m+(2n+1)K$ increasing with excitation number $n$. In this figure, the coherent drive (green arrows) is resonant with the transition between the states $|1⟩$ and $|2⟩$. The wiggly lines denote two incoherent processes: linear (one-phonon) antidamping with rate ${\gamma }_1$ (red arrows) and nonlinear (two-phonon) damping with rate ${\gamma }_2$ (blue arrows). (b) The amplitude-dependent damping rate $\mathrm{\Gamma }$ (blue solid line) and amplitude-dependent diffusion constant $D$ (green dashed line) in the semiclassical equation (3) as a function of the amplitude $A$. In the limit of large amplitude $A$, the radial Wigner density $W(A)$ is a Gaussian with variance ${\sigma }_A^2=3/8$ around the zero of $\mathrm{\Gamma }$, i.e., $\mathrm{\Gamma }(A)=0$.

Figure 2

(a) Phase-locking measure $|S|$ for forced synchronization and corresponding Wigner distributions. The phase-locking behavior for the quantum system (black solid line) described by $\mathrm{\Lambda }$ [defined in Eq. (2)] can be understood with our perturbative expression (8) (red bold dotted line). For the parameters of this plot (${\gamma }_2/{\gamma }_1=7$, $E/{\gamma }_1=2.25$, and $K/{\gamma }_1=50$), approximately three energy levels have a significant occupation, so that two resonances are possible. The blue dashed line shows the results of the corresponding semiclassical model ${\mathrm{\Lambda }}_c$ [defined in Eq. (3)], for which there is only one resonance as expected. The time-independent steady-state Wigner distributions for the parameters at the two peaks (b),(d) and the minimum (c) illustrate the quantum phase-locking behavior of $\mathrm{\Lambda }$.

Figure 3

Global behavior of the phase-locking measure $|S|$ for the steady state of $\mathrm{\Lambda }$ (a),(c),(e) and ${\mathrm{\Lambda }}_c$ (b),(d),(f). In (a) and (b) $|S|$ is plotted as a function of $\mathrm{\Delta }$ and ${\gamma }_2$ for $E=2.25{\gamma }_1$ and $K=50{\gamma }_1$, in (c) and (d) as a function of $\mathrm{\Delta }$ and $E$ for ${\gamma }_2=5{\gamma }_1$ and $K=50{\gamma }_1$, and in (e) and (f) as a function of $\mathrm{\Delta }$ and $K$ for ${\gamma }_2=5{\gamma }_1$, $E=2.25{\gamma }_1$, and $K_{\max }=50{\gamma }_1$.

Figure 4

(a) Power spectrum $P(\omega )$ of the quantum Van der Pol–Kerr oscillator as a function of detuning $\mathrm{\Delta }$ for ${\gamma }_2=0.8{\gamma }_1$, $E=4.5{\gamma }_1$, and $K=25{\gamma }_1$. The red solid line indicates the maximum of the power spectrum, and the black dashed line the detuning $\mathrm{\Delta }$ of the external drive. Around $\mathrm{\Delta }\approx K$ and $\mathrm{\Delta }\approx 3K$, the two lines match, indicating a resonance in the frequency entrainment. At $\mathrm{\Delta }=5K$, the spectrum shows a third, smaller response. The steady state for these parameters is characterized by a Wigner distribution with negative density; see (b). The inset shows the Fock state probabilities $P(n)$ in the presence (right blue bars) and absence of the coherent drive (left black bars).