Lie-algebraic quantum phase reduction based on heterodyne detection

Measurement backaction inherently alters observed dynamics in quantum physics. In the realm of quantum synchronization, this backaction induces a phase bias, making the assessment of synchronization critically dependent on the choice of the observables. In this study we extend the quantum phase reduction approach [W. Setoyama and Y. Hasegawa, Phys. Rev. Lett. 132, 093602 (2024)] into heterodyne detection, offering a comprehensive theoretical framework for analyzing quantum synchronization dynamics through uniform continuous measurement over all possible quadrature observables. This method averages out the backaction, allowing for unbiased evaluation of synchronization between quantum oscillators while avoiding measurement-induced phase bias. Furthermore, by defining the phase and limit-cycle solution independently of specific observables, our proposed method consistently adapts to the scenario where the observables are freely modified during the time evolution. Through simulations of noise-induced synchronization, our method reveals that the number of phase clusters between oscillators is restricted by their bosonic levels.