Enhanced Magnon Synchronization in Coupled WGM Optomagnonic Resonators with Phase-Dependent Photon Hopping

Technical Summary: Enhanced Magnon Synchronization in Coupled WGM Optomagnonic Resonators with Phase-Dependent Photon Hopping

Problem Statement
The paper addresses the challenge of achieving and controlling quantum synchronization between two spatially separated magnon modes. While synchronization is a fundamental collective phenomenon, extending it to the quantum regime is constrained by the Heisenberg uncertainty principle. The authors focus on hybrid cavity optomagnonic systems, specifically utilizing two optical whispering-gallery-mode (WGM) resonators, each coupled to a yttrium iron garnet (YIG) sphere. The core problem is to determine how an indirect interaction, mediated by phase-dependent single-photon hopping between the optical cavities, can be engineered to synchronize the dynamics of distant magnon modes (specifically the Kittel mode) and how this synchronization is affected by thermal fluctuations.

Methodology
The study employs a theoretical framework based on quantum Langevin equations (QLEs) to model the dynamics of the coupled system.

• System Model: The Hamiltonian includes terms for two optical cavities, two magnon modes, local optomagnonic interactions (via the Faraday effect), and a non-local coupling term representing single-photon hopping with a controllable phase factor $\theta$.
• Linearization: The system is analyzed in the low-excitation regime using the Holstein-Primakoff transformation to map spin operators to bosonic magnon operators. The dynamics are linearized around a semi-classical steady state to separate mean values from quantum fluctuations.
• Covariance Matrix Formalism: The quantum fluctuations are characterized using a covariance matrix, which evolves according to a Lyapunov equation. This formalism allows for the calculation of steady-state synchronization measures.
• Synchronization Metrics: The authors quantify synchronization using three specific measures derived from the covariance matrix:
  1. Complete Synchronization ($S_c$): Based on the collective error quadratures of position and momentum.
  2. $\phi$-Synchronization ($S_\phi$): Based on phase-rotated quadratures.
  3. Quantum Phase Synchronization ($S_p$): A measure of phase locking.
• Numerical Simulation: The authors solve the linearized equations numerically, varying the photon-hopping phase ($\theta$), hopping strength ($J$), and thermal occupation number ($n_{th}$) to observe their effects on synchronization dynamics.

Key Contributions

• Phase-Controlled Synchronization: The paper demonstrates that the phase of the photon-hopping term ($\theta$) acts as a critical control parameter. By varying $\theta$ from $0$ to $\pi$, the system transitions from weakly correlated motion to a highly synchronized state.
• Indirect Interaction Mechanism: It establishes that phase-dependent photon hopping provides an effective indirect interaction channel between distant magnon modes, mediated by the optical fields, without requiring direct magnon-magnon coupling.
• Quantitative Analysis of Noise: The study systematically quantifies the degradation of synchronization due to thermal fluctuations, showing that while thermal noise suppresses coherent correlations, the phase-dependent hopping mechanism remains robust enough to maintain a residual degree of synchronization.

Results

• Trajectory Evolution: Numerical results show that at $\theta = 0$, the magnon trajectories exhibit significant phase and amplitude mismatches. As $\theta$ increases to $\pi$, the position and momentum quadratures of the two magnon modes nearly overlap, and their limit-cycle trajectories in phase space merge into a single orbit.
• Synchronization Measures: The synchronization measures ($S_c$, $S_\phi$, $S_p$) increase significantly as $\theta$ approaches $\pi$. The steady-state values of these measures are highest at $\theta = \pi$, indicating optimal synchronization.
• Role of Hopping Strength: Increasing the photon-hopping strength ($J$) monotonically enhances all synchronization measures, eventually leading to saturation where further increases yield diminishing returns.
• Impact of Thermal Noise: Introducing thermal occupation ($n_{th} = 5$) significantly reduces the magnitude of synchronization measures compared to the zero-temperature case ($n_{th} = 0$). However, the synchronization remains finite, and the trend of improvement with increasing $J$ and optimal $\theta$ persists.
• Error Reduction: The synchronization error ($q_1 - q_2$) is substantially reduced when $\theta = \pi$, demonstrating that the phase-dependent interaction effectively suppresses the dynamical mismatch between the oscillators.

Significance and Claims
The authors claim that their results demonstrate that the phase of the photon-hopping factor offers a "simple and effective approach" for controlling synchronization dynamics in WGM-based coupled cavity optomagnonic systems. The work provides a theoretical route toward the "coherent control of collective magnon dynamics" in such quantum optomagnonic devices.

The paper asserts that the proposed scheme is experimentally feasible using current technology, citing existing experimental achievements in WGM resonators coupled to YIG spheres, achievable coupling strengths, and demonstrated phase-control techniques. The authors suggest that these findings could be useful for "synchronization-assisted quantum control and coherent information transfer in magnon-based quantum networks and other hybrid quantum technologies," though they frame this as a potential application of the demonstrated control mechanism rather than a realized experimental outcome within the study.