  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 [PhysRevLett.132.093602] 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.
