Quantum limit cycles and synchronization from a measurement perspective

TL;DR

The paper addresses how quantum limit cycles and synchronization can be understood from a measurement perspective by analyzing quantum van-der-Pol oscillators and spin-$1/2$ systems under continuous heterodyne detection. It shows that quantum trajectories reveal persistent limit-cycle dynamics and synchronization akin to classical noisy oscillators, while ensemble descriptions may obscure these features. By employing Husimi-Q phase-space representations and heterodyne currents, the work connects theoretical synchronization measures to experimentally accessible observables, enabling phase locking and frequency entrainment diagnostics in both oscillator and spin systems. The approach highlights how the type of measurement shapes the visibility of limit-cycle behavior and offers a practical framework for exploring quantum synchronization in experiments with superconducting qubits, cold atoms, and related platforms.

Abstract

Limit-cycle oscillators are the basic building blocks for synchronization; yet, the notion of a quantum limit cycle has remained unclear. Here, we study quantum limit cycles and synchronization in the presence of continuous heterodyne measurement. The resulting quantum trajectories, i.e., time evolutions of the quantum state conditioned on the measurement outcome, make the quantum limit cycles apparent. We focus on the paradigmatic model of the quantum van-der-Pol oscillator and on two-level systems. Our work provides insights into limit cycles in quantum systems, emphasizing their similarity to classical limit cycles subject to noise. Additionally, we connect theoretical measures of quantum synchronization to quantities experimentally accessible via heterodyne detection.

Figures (13)

• Figure 1: Four trajectories of a van-der-Pol oscillator, \ref{['eq:vdP']}, with $\kappa_1 = \kappa_2 = \omega / 2$. The red circle shows the limit cycle to which all trajectories converge.
• Figure 2: Limit-cycle oscillations in the presence of noise. The blue lines show $x=\Re[\alpha]$ for ten different trajectories, all starting with the same initial condition. One of them is highlighted in a darker blue. The black line is obtained by averaging over 16 000 trajectories. The red line shows a trajectory in the absence of noise. Parameters: $\kappa_1 = \kappa_2 = \omega / 2 = 10\sigma^2$.
• Figure 3: Time evolution of the classical vdP oscillator shown by the probability distributions $P(x,p)$ (top row) and $P(\phi)$ (bottom row). In the top row, the grayscale indicates the value of $P(x,p)$. The distributions are obtained by sampling 16 000 trajectories and counting the occurrences per $(x,p)$ or per $\phi$. The blue line in the top left panel shows one example of a trajectory. Parameters: $\kappa_1 = \kappa_2 = \omega / 2 = 10 \sigma^2$.
• Figure 4: Phase locking of classical vdP oscillators. (a) Arnold tongue. The blue dashed line indicates the synchronization transition $V = \abs{\delta}$. The grayscale shows the maximum of $P(\phi_{AB})$, a measure for synchronization in the presence of noise. (b) Distribution $P(\phi_{AB})$ of the phase difference in the long-time limit in the presence of noise and detuning, $\sigma^2=\delta$. The blue and red ticks indicate the value of the phase $\phi_{AB}^\mathrm{f}=\arcsin{\delta/V}$ in the absence of noise.
• Figure 5: Frequency entrainment of classical vdP oscillators. (a) Observed frequency difference as a function of detuning. The dashed line shows the identity line for reference. The black line shows the noiseless case ($\sqrt{\delta^2/V^2-1}$). The blue and red lines show the observed frequency for two different noise strengths. Although difficult to see, the observed frequency is not exactly zero for any $\sigma^2 > 0$ and $\abs{\delta}>0$. (b) Spectra of the two phase oscillators for $\sigma^2 = \delta/5$, as defined in \ref{['eq:spectra_classical']} normalized to their maximum value. Each spectrum is averaged in bins of width $\omega/\delta = 0.05$.