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Physical Review A

Dressed-state-mediated synchronization in coupled quantum oscillators

B. Vigneshwar, V. K. Chandrasekar, D. V. Senthilkumar

Published Sep 25, 2025• 4 citations
Publisher

Abstract

We investigate quantum synchronization in coupled self-sustained oscillators interacting via bosonic exchange. Using a phase-distribution-based synchronization measure, we identify bistable phase locking between 0 and $\ensuremath{\pi}$ and uncover an expanded Arnold-tongue structure under finite detuning, indicative of enhanced quantum phase locking. Power spectral analysis reveals dual-frequency locking associated with dressed-state formation, a uniquely quantum effect. This spectral splitting persists even without detuning and increases with coupling strength, signaling coherent hybridization of oscillator modes. A comparison with a classical approximation highlights key differences, including the absence of spectral doublets in the classical regime. Finally, we show that increasing nonlinear damping induces a sharp transition in the preferred phase difference and leads to decoherence through spectral peak broadening, underscoring the critical role of dissipation in shaping quantum synchronization dynamics.

DOI
10.1103/k5ry-bj1g

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We investigate quantum synchronization in coupled self-sustained oscillators interacting via bosonic exchange. Using a phase-distribution-based synchronization measure, we identify bistable phase locking between 0 and \ensuremathπ\ensuremath{\pi}\ensuremathπ and uncover an expanded Arnold-tongue structure under finite detuning, indicative of enhanced quantum phase locking. Power spectral analysis reveals dual-frequency locking associated with dressed-state formation, a uniquely quantum effect. This spectral splitting persists even without detuning and increases with coupling strength, signaling coherent hybridization of oscillator modes. A comparison with a classical approximation highlights key differences, including the absence of spectral doublets in the classical regime. Finally, we show that increasing nonlinear damping induces a sharp transition in the preferred phase difference and leads to decoherence through spectral peak broadening, underscoring the critical role of dissipation in shaping quantum synchronization dynamics.
We report a normal-mode induced synchronization blockade in coupled quantum van der Pol oscillators under the influence of external drive. In this mechanism, the coupling hybridizes the oscillator modes into spectrally split normal modes. The destructive interference between the transitions to these modes blocks synchronization. We find that this blockade can be controlled simply by tuning the coupling strength and detuning allowing dynamic manipulation of quantum synchronization through collective mode dynamics. We analyze the phase-locking behaviour using perturbation analysis. Further, by deriving steady-state probability amplitudes we show how the energy redistribution and spectral splitting forms the basis of the blockade. Our results might provide new insights into how synchronization can be controlled in quantum systems.
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.
Spontaneous synchronization between coupled periodic systems occur in a wealth of classical physical setups. Here, we show theoretically that the phase of two distinct quantum harmonic oscillators spontaneously synchronize when they are strongly coupled to a common bosonic quantum dissipative environment at zero temperature, in the absence of any driving mechanism. To do so, we compute the dynamics of the oscillators with an exact master equation obtained from a path integral formalism. Above some value of the system-environment coupling strength, we observe numerically a strongly reduced damping and a frequency locking in the dynamics of the oscillators. Beyond the synchronization mechanism, we also describe the rich phenomenology of the dynamics in the model and notably identify the additional emergence of a regime with long-lived oscillations.
Abstract We investigate forced quantum phase synchronization and transient relaxation dynamics of a driven two-level system (TLS) coupled to a hierarchical structured environment arranged in a wheel-graph, where a primary cavity directly couples to the TLS and interacts uniformly with a closed ring of N nearest-neighbor-coupled auxiliary cavities. Employing the dressed-state secular representation and conditional non-Hermitian Hamiltonian in the dressed-state subspace, we characterize the phase-locking behavior using the Husimi Q -function and the synchronization measure S . In the steady-state regime, the TLS exhibits anti-phase locking relative to the coherent drive, governed by the dynamic competition between detuning-induced phase drift and drive-induced phase torque, which naturally delineates a quantum analogue of the classical Arnold tongue. Crucially, we identify an environmental dimensionality decoupling effect, wherein the steady-state phase synchronization remains invariant to the auxiliary reservoir size N . In contrast, the transient relaxation time toward synchronization displays a non-monotonic dependence on the primary–auxiliary coupling strength κ . Below a critical threshold κ crit , the auxiliary array acts as a broadband Markovian reservoir, accelerating phase locking via Purcell-like dissipation. Above this threshold, mode hybridization triggers non-Markovian information backflow, which prolongs transient oscillations and delays synchronization. Finally, we discuss an experimentally feasible implementation based on superconducting circuit quantum electrodynamics. These results provide insights into the mechanism governing steady-state quantum phase synchronization in structured environments and offer practical guidance for reservoir engineering in quantum technologies.
We investigate how Kerr nonlinearity modifies quantum synchronization in a squeezed quantum van der Pol oscillator. We show that the Kerr interaction produces an amplitude-dependent frequency shift that drives a saddle-node bifurcation, transforming the classical phase-space structure from bistable to monostable dynamics. In the quantum regime, this transition manifests as systematic frequency pulling and spectral broadening, while the steady-state Wigner function reveals a continuous correspondence between the quantum state and the semiclassical attractor despite finite quantum fluctuations. By constructing global synchronization phase diagrams in the squeezing--Kerr parameter space, we uncover a remarkably linear dependence of the critical squeezing strength required to maintain phase locking on the Kerr nonlinearity. We further demonstrate that the synchronization boundary does not coincide with the crossover between super- and sub-Poissonian photon statistics, showing that synchronization and photon-number statistics characterize distinct aspects of the quantum steady state. These results provide quantitative design principles for controlling quantum synchronization through Kerr nonlinearity, with potential relevance to trapped-ion, superconducting-circuit, and optomechanical platforms.
This study explores the dual role of a squeezed reservoir in controlling the quantum phase synchronization of a driven two-level system. We first demonstrate, through a Liouvillian eigen-spectrum analysis, that the squeezed reservoir can induce a stable limit cycle, transforming the passive TLS into a genuine self-sustained oscillator. This enables a qualitative transition from a weak ``forced response" to a robust, high-quality synchronization (or entrainment). This enhancement is characterized not only by a greater degree of phase locking but also by an increased frequency selectivity, manifested as a narrower Arnold tongue. More strikingly, we reveal that the squeezing angle acts as a control parameter to actively suppress synchronization. By tuning this angle, the reservoir can drive the system into a classical mixed state, inducing a quantum synchronization blockade via the quenching of steady-state coherence. Our findings establish squeezed-reservoir engineering as a versatile strategy for actively modulating quantum synchronization, with feasible implementations in circuit quantum electrodynamics.