Quantum spin van der Pol oscillator -- a spin-based limit-cycle oscillator exhibiting quantum synchronization

TL;DR

This work introduces a quantum spin van der Pol oscillator as a spin-based model of limit-cycle dynamics that reduces to the quantum optical vdP oscillator in the high-spin limit $J \rightarrow \infty$, i.e., the Stuart-Landau normal form $\dot{u} = (\gamma_1/2 - i\omega)u - 2J\gamma_2 u|u|^2$. It uses a spin-coherent-state framework and a Lindblad master equation with negative damping $\mathcal{D}[J'_{+}]$ and nonlinear damping $\mathcal{D}[(J'_{-})^2]$ to demonstrate stable limit cycles, frequency entrainment to external drives, and synchronization phenomena. The study reports mutual synchronization and entanglement tongues for two dissipatively coupled spin-$1$ oscillators, and a Kuramoto-like collective synchronization transition in globally coupled networks, with clear signatures even at the smallest spin $J=1$. The results provide a spin-based platform for analyzing quantum synchronization and offer pathways toward experimental realization in atomic ensembles or trapped-ion systems, linking spin dynamics to the broader quantum synchronization literature.

Abstract

We introduce a quantum spin van der Pol (vdP) oscillator as a prototypical model of quantum spinbased limit-cycle oscillators, which coincides with the quantum optical vdP oscillator in the high-spin limit. The system is described as a noisy limit-cycle oscillator in the semiclassical regime at large spin numbers, exhibiting frequency entrainment to a periodic drive. Even in the smallest spin-1 case, mutual synchronization, Arnold tongues, and entanglement tongues in two dissipatively coupled oscillators, and collective synchronization in all-to-all coupled oscillators are clearly observed. The proposed quantum spin vdP oscillator will provide a useful platform for analyzing quantum spin synchronization.

Figures (6)

• Figure 1: Mapping of the point $|{\theta, \phi}\rangle$ on the unit sphere characterizing a spin coherent state to the point $(u, u^*)$ on the complex plane.
• Figure 2: Limit-cycle oscillations of the quantum spin vdP oscillators with different spin numbers $J$. (a-c): $\bar{Q}(u, u^*)$ functions on the complex plane with the corresponding classical limit cycles of Eq. (\ref{['eq:qsvdp_ctraj']}) (red-thin lines). (d-f): $Q(\theta, \phi)$ functions on the unit sphere. (g-i): Elements of the steady-state density matrices $\rho_{ss}$. The spin numbers and other parameters are (a, d, g) $J = 40$ and $(\omega, \gamma_2)/\gamma_1 = (-0.1, 0.05)$, (b, e, h) $J = 20$ and $(\omega, \gamma_2)/\gamma_1 = (-0.1, 0.5)$, and (c, f, i) $J = 1$ and $(\omega, \gamma_2)/\gamma_1 = (-0.1, 100)$, where $\gamma_1 = 1$.
• Figure 3: Frequency entrainment of a quantum spin vdP oscillator to a periodic external drive. (a-c): Observed frequencies $\omega_{obs}$. The inset at top left in each figure displays the power spectrum when $\Delta = 0.1$. The inset at bottom right in each figure highlights the area with small detuning parameter $\Delta$. (d-f): Order parameters ${|S_{1}|}$ on the $\Delta-E$ plane, (g-i): $\bar{Q}(u, u^*)$ functions on the complex plane with stable limit-cycle solutions of Eq. (\ref{['eq:qsvdp_ctraj']}) (red-thin lines). The parameters are (a, d, g) $J = 40$ and $\gamma_2/\gamma_1 = 0.05$, (b, e, h) $J = 20$ and $\gamma_2/\gamma_1 = 0.5$, (c, f, i) $J = 1$ and $\gamma_2/\gamma_1 = 100$, (a, b, c) $E/\gamma_1 = 1$, and (g, h, i) $(E, \Delta)/\gamma_1 = (1, 0.1)$, where $\gamma_1 = 1$.
• Figure 4: Frequency synchronization of two dissipatively coupled quantum spin vdP oscillators. The inset displays the power spectra with $\Delta_{12} = 20, V= 30$. (a): Dependence of the observed detuning $\Delta_{obs}$ on the parameter $\Delta_{12}$. (b, c): Dependence of (b) the order parameter ${|S_{12}|}$ and (c) the negativity $\mathcal{N}$ on the parameters $V$ and $\Delta_{12}$. The other parameters are $J = 1$ and $\gamma_2/\gamma_1 = 100$. We set $\gamma_1 = 1$.
• Figure 5: Collective synchronization of globally coupled quantum spin vdP oscillators. Dependence of the phase coherence ${|S_1|}$ on the coupling strength $K$. The parameters are $J = 1$ and $\omega= -20$. We set $\gamma_1 = 1$.