A general interpretation of nonlinear connected time crystals: quantum self-sustaining combined with quantum synchronization

Abstract

Although classical nonlinear dynamics suggests that sufficiently strong nonlinearity can sustain oscillations, quantization of such model typically yields a time-independent steady state that respects time-translation symmetry and thus precludes time-crystal behavior. We identify dephasing as the primary mechanism enforcing this symmetry, which can be suppressed by intercomponent phase correlations. Consequently, a sufficient condition for realizing a continuous time crystal is a nonlinear quantum self-sustaining system exhibiting quantum synchronization among its constituents. As a concrete example, we demonstrate spontaneous oscillations in a synchronized array of van der Pol oscillators, corroborated by both semiclassical dynamics and the quantum Liouville spectrum. These results reduce the identification of time crystals in many-body systems to the evaluation of only two-body correlations and provide a framework for classifying uncorrelated time crystals as trivial.

Figures (4)

• Figure 1: (a): Schematic of the studied time crystal. The crystal surface is modeled as a 2D array of van der Pol oscillators, while the bulk serves as a common reservoir. Long-range interactions among surface oscillators emerge through their coupling to this shared reservoir. (b): Mechanism by which a classical self-sustained oscillator restores time-translation symmetry upon quantization. Every point on the classical limit cycle is a steady state, so phase noise diffuses freely, spreading the probability distribution along the trajectory. The blue region depicts the probability distribution, with the blue dot marking the mean value of the field operator ($\langle \hat{a}\rangle$). (c): Synchronization-induced breaking of time-translation symmetry. Left: Phase diffusion in the same direction preserves synchronization, yielding a valid steady state and thus time-translation symmetry. Right: Diffusion in opposite directions disrupts synchronization; the synchronization mechanism suppresses this process, breaking time-translation symmetry.
• Figure 2: (a) Time evolution of the order parameter $r$ and its amplitude for various $N$. The inset shows the amplitude on a logarithmic scale.(b) Fourier transform of the order parameter, with $N$ values color-coded as in (a). (c) and (d) Phase fluctuations and effective amplitude dissipation rate versus $1/N$. Here, we set $\omega_m = 1$ as the unit, and the other parameters are $\kappa_1=0.1$, $\kappa_2 = 0.005$, and $\mu=0.3$ and $\gamma=0$. Panels (a) and (b) derive from $100,000$ stochastic simulations using the Langevin equation \ref{['eq:op classical QLE']}; panels (c) and (d) from $50,000$.
• Figure 3: (a): Multi-body synchronization measure $\mathcal{S}_c(N,t)$ as a function of $1/N$. The inset shows its time evolution for $N=2$, $5$, and $50$, with color coding as in Fig. \ref{['fig:2']}. (b) and (d): Phase-space probability distributions for oscillator 1 for $N=2$ and $5$. (c) and (e) Corresponding probability distributions of error-mode which is defined in Eq. \ref{['eq:Sc N']}. Here we set $\omega_m t=10000$ and all parameters match those in Fig. \ref{['fig:2']}.
• Figure 4: The eigenvalues $\lambda$ of the Liouvillian. The insets show an enlargement over the eigenvalues with the largest real part. The parameters here is $\kappa_2 = 0.2$, and the other parameters match those in Fig. \ref{['fig:2']}.