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Physical Review A

Analyzing quantum synchronization through Bohmian trajectories

Wenlin Li

Published Aug 15, 2022• 21 citations
Publisher

Abstract

We analyze quantum effects involved in continuous variable synchronization between two self-sustaining resonators in the framework of Bohmian mechanics. Bohmian trajectories provide visual descriptions of significant nonclassical dynamics, which allow us to characterize the level of quantum synchronization more intuitively compared with only utilizing some designed synchronization measures. It is found that in the quantum limit, the Bohmian trajectories will deviate from their corresponding semiclassical limit cycle behavior after they are distorted by the quantum potential. We explore the roles of zero-point fluctuation, superposition, and nonlocal correlation in synchronization dynamics, corresponding to the oscillators in a coherent state, superposition state, and entanglement state, respectively. We also explore the influence of squeezed Hamiltonian on spontaneous synchronization.

DOI
10.1103/physreva.106.023512

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We analyze quantum effects involved in continuous variable synchronization between two self-sustaining resonators in the framework of Bohmian mechanics. Bohmian trajectories provide visual descriptions of significant nonclassical dynamics, which allow us to characterize the level of quantum synchronization more intuitively compared with only utilizing some designed synchronization measures. It is found that in the quantum limit, the Bohmian trajectories will deviate from their corresponding semiclassical limit cycle behavior after they are distorted by the quantum potential. We explore the roles of zero-point fluctuation, superposition, and nonlocal correlation in synchronization dynamics, corresponding to the oscillators in a coherent state, superposition state, and entanglement state, respectively. We also explore the influence of squeezed Hamiltonian on spontaneous synchronization.
It is well known from classical physics that weakly coupled self-sustained oscillators may spontaneously lock their phases. Just like classical synchronization is known to break down due to noise induced phase slips, we show here how the synchronization of continuous variable quantum systems breaks down by proliferation of quantum phase slips. Within a Keldysh path integral formulation of limit cycles, we analyze the phase dynamics and show how, in spite of strong phase correlations, quantum phase slips degrade the actual phase locking. This approach also allows us to address non-Markovian effects on the synchronization of limit cycles, which we illustrate explicitly for superconducting resonators coupled via a voltage biased double quantum dot.
Synchronization is a collective phenomenon in which interacting nonlinear oscillators develop a persistent relation between their phases and frequencies. In quantum systems, its description requires connecting the classical dynamics of limit-cycle oscillators with quantum fluctuations, dissipation, and the notion of phase. In this tutorial review, we offer a systematic framework for analyzing synchronization in classical and quantum dynamical systems. Starting from amplitude-phase dynamics, we introduce successively topics of relevance, such as phase locking, the Adler equation, Arnold tongues, and the Kuramoto model, and use these concepts to analyze coupled oscillators and their stability. We then formulate the corresponding description for quantum limit-cycle oscillators using Lindblad master equations and phase-space methods, with explicit examples of two and many coupled oscillators. We further discuss noise, synchronization measures, and the extreme quantum regime, including synchronization in spin systems. The emphasis throughout is on analytical derivations and a practical methodology for identifying synchronized states, determining their stability, and characterizing synchronization in quantum systems. The review is designed as a self-contained guide from which we hope that students can learn the essential concepts and techniques, while researchers can readily find the analytical tools and methodological insights needed to approach new synchronization problems.
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.
Can synchronization -- the widespread spontaneous emergence of coordinated dynamics in classical nonlinear systems -- also occur in the quantum regime? This question has recently sparked intense research into collective behavior and temporal self-organization in quantum systems. Typical quantum features such as the linearity of time evolution and the presence of quantum noise seem to hinder the appearance of synchrony at the microscopic level. At the same time, quantum coherence and quantum correlations may provide novel mechanisms for enhancing synchronization beyond its classical counterpart. We here survey recent theoretical and experimental advances in quantum synchronization, ranging from the characterization of synchronous oscillations and genuinely nonclassical forms of synchrony to many-body synchronization on quantum networks.
It is desirable to observe synchronization of quantum systems in the quantum regime, defined by the low number of excitations and a highly nonclassical steady state of the self-sustained oscillator. Several existing proposals of observing synchronization in the quantum regime suffer from the fact that the noise statistics overwhelm synchronization in this regime. Here, we resolve this issue by driving a self-sustained oscillator with a squeezing Hamiltonian instead of a harmonic drive and analyze this system in the classical and quantum regime. We demonstrate that strong entrainment is possible for small values of squeezing, and in this regime, the states are nonclassical. Furthermore, we show that the quality of synchronization measured by the FWHM of the power spectrum is enhanced with squeezing.
Understanding the origin of phase synchronization between quantum self-sustained oscillators has garnered significant interest in recent years. In this work, we study phase synchronization in three settings: between two continuous-variable oscillators, between two arbitrary quantum spins, and within a hybrid setup involving a spin and an oscillator. We derive a simple and general condition on the elements of the joint density matrix that must be satisfied for them to contribute to the relative phase distribution. In particular, we identify the subset of coherence elements in the joint density matrix that serve as key resources for enabling quantum phase synchronization. Our theory is validated against the previously proposed interaction models known to induce synchronization between the self-sustained oscillators. Moreover, our approach offers valuable insights into the relationship between phase synchronization and various information-theoretic measures.