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Physica Scripta

Quantum phase synchronization and non-Markovian relaxation dynamics of a driven two-level system in a hierarchical environment

Yan‐Ling Li, Duo Huang, Xing Mao Xiao

Published Sep 28, 2026• 0 citations

Abstract

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.

DOI: 10.1088/1402-4896/aead60

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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 study the quantum phase synchronization of a driven two-level system (TLS) coupled to a structured environment and demonstrate that quantum synchronization can be enhanced by the classical driving field. We use the Husimi

Q

function to characterize the phase preference and find the in-phase and anti-phase locking phenomenon in the phase diagram. Remarkably, we show that the in-phase classical driving enables a TLS to reach stable anti-phase locking in the Markovian regime. However, we find that the synergistic action of classical driving and non-Markovian effects significantly enhances the initial in-phase locking. By introducing the

S

function and its maximal value to quantify the strength of synchronization and sketch the synchronization regions, we observe the typical signatures of the hollowed Arnold tongue in the parameter regions of synchronization. In the hollowed Arnold tongue, the synchronization regions exist both inside and outside the tongue while unsynchronized regions only lie on the boundary line. We also provide an intuitive interpretation of the above results by using the quasimode theory.

We investigate how detuning and auxiliary qubits collaboratively enhance quantum synchronization in a dissipative multi-qubit system that is coupled to a structured reservoir. Our findings indicate that while detuning is ineffective in Markovian environments, it emerges as a powerful control parameter in the non-Markovian regime, where environmental memory facilitates long-lived phase coherence. It is shown that adding more auxiliary qubits amplifies this effect by strengthening the collective coupling and enhancing memory, resulting in robust phase locking within the system. Analysis using the Husimi Q-function, synchronization measures, and Arnold tongue structures reveals a detuning-induced enhancement of phase locking, which significantly improves stability compared to the resonance case. These results establish a cooperative control strategy where detuning actively engineers phases, while auxiliary qubits provide the necessary memory for sustained synchronization.

We investigate how detuning and auxiliary qubits collaboratively enhance quantum synchronization in a dissipative multi-qubit system that is coupled to a structured reservoir. Our findings indicate that while detuning is ineffective in Markovian environments, it emerges as a powerful control parameter in the non-Markovian regime, where environmental memory facilitates long-lived phase coherence. It is shown that adding more auxiliary qubits amplifies this effect by strengthening the collective coupling and enhancing memory, resulting in robust phase locking within the system. Analysis using the Husimi Q-function, synchronization measures, and Arnold tongue structures reveals a detuning-induced enhancement of phase locking, which significantly improves stability compared to the resonance case. These results establish a cooperative control strategy where detuning actively engineers phases, while auxiliary qubits provide the necessary memory for sustained synchronization.

Synchronizing a few-level quantum system is of fundamental importance to the understanding of synchronization in the deep quantum regime. We investigate quantum phase synchronization of a two-level system (qubit) driven by a semiclassical laser field, in the presence of a general non-Markovian dissipative environment. The phase preference of the qubit is demonstrated through Husimi Q-function, and the existence of a limit cycle is also shown in our system. Synchronization of the qubit is quantified using the shifted phase distribution. The signature of quantum phase synchronization viz the Arnold tongue is obtained from the maximal value of the shifted phase distribution. Two distinct types of qubit dynamics is considered depending on the reservoir correlation time being very short and a situation when bath correlation time is finite. In the Markov regime of the environment, the phase preference of the qubit goes away in the long time limit, whereas the long-time phase localization persists in the non-Markovian regime. We also plot the maximum of the shifted phase distribution in two ways: (a) by varying the detuning and laser driving strength, and (b) by varying the system-bath coupling and laser driving strength. Various system-environment parameters determine the synchronization regions and the qubit phase synchronization is shown to be enhanced in the non-Markov regime.

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.

We study phase-synchronization in a driven two-level system coupled to a non-Markovian bosonic reservoir. The dynamics is described by treating the system-bath coupling and the coherent drive without invoking the rotating-wave approximation, and simulated using the numerically exact hierarchical equations of motion. We observe that a robust phase-locking develops and that the corresponding synchronization measure rapidly acquires a finite value when the system is tuned to what we identify as a resonant-ratio condition, namely when the ratio between the drive amplitude and its frequency coincides with a zero of the Bessel function

J_0

. We provide an explanation for this phenomenon by means of a static approximation derived from a Fourier analysis of the periodically driven Hamiltonian.