Fig. 3: Imperfection-tolerant quantum synchronization.

a \({{{{\mathcal{S}}}}}_{{{{\mathcal{Q}}}}}\) versus the mass ratio ρ = m_j/m₀ and K. b \({{{{\mathcal{S}}}}}_{{{{\mathcal{Q}}}}}\) versus the phonon decay rate γ_j and K. Although the masses (decay rates) of the two microspheres generally differ, for simplicity, we assume equal masses (decay rates) in our simulations. c In the magnon-Kerr-off regime (K = 0), \({{{{\mathcal{S}}}}}_{{{{\mathcal{Q}}}}}\) versus Δ_m when thermal phonon numbers \({\bar{n}}_{j}=0\), 1, and 10. d For \({\bar{n}}_{j}=10\), \({{{{\mathcal{S}}}}}_{{{{\mathcal{Q}}}}}\) versus Δ_m in both magnon-Kerr-off and magnon-Kerr-on (K/ω₁ = −2 × 10⁻⁵) regimes. e Density plot of the quantum-synchronization-revival factor Λ versus \({\bar{n}}_{j}\) and K. f Symmetric and asymmetric couplings. Effective coupling G_m(a) (see Supplementary Information) of the magnon (photon) and first (second) phonon mode versus the magnon-Kerr strength K. The magnetic field entering from the \({{{\mathcal{CD}}}}\) results in symmetric coupling, corresponding to enhancing the resilience against both thermal fluctuations and random fabrication imperfections of the resonator; while the magnetic field injected from the \({{{\mathcal{OD}}}}\) leading to asymmetric coupling, corresponding to reducing the resonator resilience. g Broad applicability and universality of our model. The proposed framework naturally extends to a range of nonreciprocal quantum phenomena, including nonreciprocal quantum synchronization, nonreciprocal quantum squeezing, nonreciprocal quantum entanglement, and nonreciprocal topological phononics. Here we set Ω = 0 and other parameters are the same as those in Fig. 1.