Mediated Transmission of Quantum Synchronization in Star Networks

TL;DR

This work investigates mediated quantum synchronization in a star network of spin-1 oscillators using a Lindblad master equation, revealing how a detuned or dissipatively imbalanced hub can mediate coherence between leaves via competing $1:1$ and $2:1$ phase-locking channels. The authors define a phase-space synchronization measure $S_2(\phi)$ incorporating first- and second-harmonic correlations and show regimes of remote synchronization and quasi-explosive synchronization, with transitions controlled by coupling strength, detuning $\Delta$, and dissipation asymmetry. In identical networks, leaves synchronize remotely through a blockaded hub, while symmetry-breaking dissipation or hub detuning can produce Arnold-tongue–like behavior and detuning-induced transitions that mirror or extend classical star-network dynamics. Overall, the results illuminate rich, tunable quantum synchronization transmission and point toward scaling to larger quantum networks with engineered dissipation and detuning.

Abstract

Synchronization transmission describes the emergence of coherence between two uncoupled oscillators mediated by their mutual coupling to an intermediate one. In classical star networks, such mediated coupling gives rise to remote synchronization--where nonadjacent leaf nodes synchronize through a nonsynchronous hub--and to explosive synchronization, characterized by an abrupt collective transition to coherence. In the quantum regime, analogous effects can arise from the interplay between 1:1 phase locking and 2:1 phase-locking blockade in coupled spin-1 oscillators. In this work, we investigate a star network composed of spin-1 oscillators. For identical oscillators, symmetric and asymmetric dissipation lead to distinct transmission behaviors: remote synchronization and quasi-explosive synchronization appear in different coupling regimes, a phenomenon absent in classical counterparts. For nonidentical networks, we find that at large detuning remote synchronization emerges in the weak-coupling regime and evolves into quasi-explosive synchronization as the coupling increases, consistent with classical star-network dynamics. These findings reveal the rich dynamical characteristics of mediated quantum synchronization and point toward new possibilities for exploring synchronization transmission in larger and more complex quantum systems.

Figures (6)

• Figure 1: (a) Schematic of the star network. The central oscillator $0$ represents the hub, and the outer oscillators $1$–$N$ represent the leaves. Each leaf is coupled to the hub with coupling strength $V$. Every oscillator consists of three spin-1 states $\ket{0}$, $\ket{1}$, and $\ket{2}$, with transition frequency $\omega_i$. Dissipation occurs toward the intermediate state $\ket{1}$, characterized by the gain and damping rates $\gamma_i^{g}$ and $\gamma_i^{d}$, respectively. (b), (c) Phase distributions $S_2(\phi_{01})$ for two coupled spin-1 oscillators, illustrating the two distinct phase-locking regimes. (b) $1\!:\!1$ phase locking for $\gamma_0^d = \gamma_1^g = 0.1\,\gamma_0^g = 0.1\,\gamma_1^d$. (c) $2\!:\!1$ phase locking for $\gamma_0^g = \gamma_1^g = 0.1\,\gamma_0^d = 0.1\,\gamma_1^d$. In both cases, the coupling strength is $V = 0.05\,\gamma_1^d$.
• Figure 2: (a) Schematic of the star network with identical hub and leaf oscillators ($N=4$). (b) Effective synchronization measures $\mathcal{S}_{01}$ (hub-leaf) and $\mathcal{S}_{12}$ (leaf-leaf) under symmetric gain and damping rates ($\gamma^{g} = \gamma^{d}$). $\mathcal{S}_{01}$ remains zero, whereas $\mathcal{S}_{12}$ increases monotonically with the coupling strength. (c) Phase distributions $S_2(\phi_{01})$ and $S_2(\phi_{12})$ at $V = 0.2\,\gamma^{d}$ corresponding to panel (b). $S_2(\phi_{01})$ exhibits two maxima, while $S_2(\phi_{12})$ shows a single maximum.
• Figure 3: (a) Effective synchronization measures $\mathcal{S}_{01}$ (hub-leaf) and $\mathcal{S}_{12}$ (leaf-leaf) for gain and damping rates $\gamma^{g} = 0.1\,\gamma^{d}$. Both measures exhibit nonmonotonic behavior: $\mathcal{S}_{01}$ first increases and then decreases to zero, while $\mathcal{S}_{12}$ shows a similar trend, reaching its maximum simultaneously with $\mathcal{S}_{01}$ before decreasing and increasing again at stronger coupling. (b), (c) Phase distributions $S_2(\phi_{01})$ and $S_2(\phi_{12})$ for $V = 0.2\,\gamma^{d}$ and $V = 0.05\,\gamma^{d}$, respectively, under the same conditions as in panel (a). In both cases, $S_2(\phi_{12})$ displays a single maximum, whereas $S_2(\phi_{01})$ exhibits two maxima at $V = 0.2\,\gamma^{d}$ and a single maximum at $V = 0.05\,\gamma^{d}$. (d), (e) First- and second-order contributions $S_2^{(1)}(\phi_{01})$ and $S_2^{(2)}(\phi_{01})$ for $V = 0.05\,\gamma^{d}$ and $V = 0.2\,\gamma^{d}$, respectively. At $V = 0.05\,\gamma^{d}$, the first-order term $S_2^{(1)}(\phi_{01})$ dominates, whereas at $V = 0.2\,\gamma^{d}$, the first- and second-order components become comparable.
• Figure 4: Effective synchronization measures $\mathcal{S}_{01}$ and $\mathcal{S}_{12}$ evaluated at two coupling strengths: (a) $V = 0.05\,(\gamma^{g} + \gamma^{d})$; (b) $V = 0.2\,(\gamma^{g} + \gamma^{d})$.
• Figure 5: (a)–(e) Effective synchronization measures $\mathcal{S}_{01}$ and $\mathcal{S}_{12}$ for a star network with one detuned hub and four identical leaves ($N=4$). The inset in panel (a) illustrates the network configuration. (a), (b) Symmetric dissipation, $\gamma_0^g = \gamma_N^g = \gamma_0^d = \gamma_N^d = \gamma$. $\mathcal{S}_{01}$ remains zero for all detunings and coupling strengths, whereas $\mathcal{S}_{12}$ decreases with detuning and increases with coupling. (c)-(e) Asymmetric dissipation, $\gamma_0^g = \gamma_N^g = 0.1\,\gamma_0^d = 0.1\,\gamma_N^d = 0.1\,\gamma$. Under resonance, $\mathcal{S}_{01}$ first increases and then drops to zero, while under detuning it decreases to a finite value. $\mathcal{S}_{12}$ exhibits a similar nonmonotonic behavior, increasing, then decreasing, and gradually rising again near resonance. (e) Same asymmetric dissipation as in panels (c),(d), with fixed detuning $\Delta = 4\,\gamma$.