Fig. 1: Model and Sagnac-effect-induced nonreciprocal quantum synchronization.

a Schematic of hybrid quantum devices consisting of a clockwise-spinning silica microsphere (with angular velocity Ω) and a YIG sphere, both supporting phonons respectively excited by the radiation-pressure interaction g_a induced by circulating optical fields or by the magnetostrictive coupling g_m induced by microwave-driven magnons. The phonon-phonon coupling χ originates from direct physical contact between a spinning silica microsphere and a counter-rotating YIG sphere, both maintained at a constant angular velocity Ω. The clockwise-spinning silica microsphere is driven by light from the left or right of the fibre, resulting in a counterclockwise or clockwise optical mode, respectively. The externally applied magnetic field is aligned parallel to either one chosen direction (\({{{\mathcal{CD}}}}\), i.e., \(\left[110\right]\)) or the other direction (\({{{\mathcal{OD}}}}\), i.e., \(\left[100\right]\)) wrt the crystallographic axes of the YIG sphere, leading to a magnon-Kerr coefficient K < 0 or K > 0, respectively (see Supplementary Information). b Schematic of the coupling between magnons, phonons, and photons. c Sagnac-Fizeau shift σ_a versus Ω when light enters from the left or right of the fiber. d Quantum synchronization measure \({{{{\mathcal{S}}}}}_{{{{\mathcal{Q}}}}}\) versus the optical detuning Δ_a when light is injected from the left or right of the fibre. Here we set ω₁ = 2π × 10 MHz as a reference unit of frequency and choose experimentally feasible parameters^{32,33,43 − 46,61}: ω₂/ω₁ = 1.005, γ_j/ω₁ = 0.005, κ_a(m)/ω₁ = 0.15 (0.2), g_a(m)/ω₁ = 0.005, χ/ω₁ = 0.02, Δ_a(m)/ω₁ = − 1.005( − 1), \({\bar{n}}_{j}=0\), \({{{{\mathcal{E}}}}}_{a(m)}/{\omega }_{1}=35\), K = 0, and Ω = 6 kHz.