Spontaneous Macroscopic Quantum Synchronization in an Ensemble of Two-level Systems

TL;DR

The paper addresses spontaneous macroscopic quantum synchronization in an all-to-all coupled ensemble of two-level systems by deriving a nonlinear quantum master equation and solving it analytically to reveal limit-cycle synchronization. It employs Bloch-sphere dynamics to show how phase-dependent interactions and dissipation cooperate to produce stable synchronization, and it maps a phase diagram in coupling strength $V$ and gain-to-damping ratio. A general analytical framework yields the synchronization frequency $\omega_{sync}$ as a function of the interaction phase $\theta$, with explicit results for $\theta=\pm \frac{\pi}{2}$ and a self-consistent limit-cycle description for arbitrary $\theta$. The work extends the analysis to two groups with detuning, demonstrating full and partial inter-group synchronization whose locking range can be engineered by tuning interaction phases, revealing an Adler-type regime and Arnold tongue structure. Overall, the study establishes TLS ensembles as a minimal, controllable platform for spontaneous quantum synchronization with a clear analytic handle on the synchronization frequency and stability.

Abstract

Spontaneous macroscopic quantum synchronization is an emergent phenomenon where an ensemble of quantum oscillators achieves global phase coherence through the interplay of interaction and dissipation. To illuminate this phenomenon, we study an ensemble of two-level systems (TLS) and establish its associated nonlinear quantum master equation, for which self-consistent analytical solutions of quantum synchronization can be obtained. The trajectories on the Bloch sphere vividly illustrate how dissipation and interaction drive the system toward a synchronized state. We present a phase diagram for macroscopic synchronization as a function of interaction strength and the gain-to-damping ratio. Furthermore, we demonstrate full synchronization and partial synchronization between two groups of TLS with different natural frequencies. This work establishes ensemble of TLS as a remarkable system for understanding spontaneous quantum synchronization.

Figures (10)

• Figure 1: Blue streamlines represent the flow structure in the $(m_x,m_y,m_z)$ space, while the red trajectory traces evolution from the initial state (the red dot) $\bm{m}_0=(-0.5,0.4,0.1)$ under the parameter ratio $\gamma_+/\gamma_-=0.2$. All trajectories ultimately flow to the stable point $\bm{m}_s=(0,0,\tfrac{3}{4})$.
• Figure 2: Dynamics of coherent rotation in (a) and dissipation in (b). The red trajectories trace evolution from the initial state $\bm{m}_0=(-0.5,0.4,0.1)$ under the same parameter as in Fig. \ref{['fig1']}
• Figure 3: Flows generated by evolutions of interaction Hamiltonians at distinct phase angles (a) $\theta=0$, (b) $\theta=-\pi/2$, and (c) $\theta=\pi/2$. The red trajectory traces evolution from the initial state $\bm{m}_0=(-0.5,0.4,0.1)$ under the same parameter and $V/(\gamma_++\gamma_-)=1$.
• Figure 4: Illustration of the appearance of quantum synchronization or not due to the interplay among coherent rotation, interaction and dissipation. (a)-(d) represent a limit cycle produced in the upper hemisphere under a set of parameters, specifically $\theta=\pi/2$ and $\gamma_+ / \gamma_-=5$; (e)-(h) represent a limit cycle produced in the upper hemisphere under a set of parameters, specifically $\theta=-\pi/2$ and $\gamma_+ / \gamma_-=0.2$; (i)-(l) represent a set of parameters that cannot produce limit cycle. The red trajectory traces evolution from the initial state $\bm{m}_0$ from Fig. \ref{['fig1']} and $V/(\gamma_++\gamma_-)=1$.
• Figure 5: Synchronization transition in an ensemble of two-level quantum oscillators. (a) The trajectories of quantum oscillator below (blue) and above (red) the critical coupling strength $V_c$. (b) Time evolution of the amplitude $\langle\sigma^+\rangle$ below and above the critical coupling strength $V_c$. Parameters in (a) and (b): Initial state $\bm{m}_0=(-0.5,0.4,0.1)$, $V/(\gamma_++\gamma_-)=1$ and $\gamma_+ / \gamma_-=5$. (c) Gray-scale image of order parameter $\left|\langle\sigma^+\rangle\right|_t$ for different phases of the interaction as a function of the coupling strength $V$ and the logarithmically scaled ratio $\gamma_+/\gamma_-$, with $\theta=\pi/2$ on the left and $\theta=-\pi/2$ on the right. The red dashed line displays the corresponding critical coupling strength obtained from a stability analysis.