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Instantaneous phase synchronization of two decoupled quantum limit-cycle oscillators induced by conditional photon detection

We show that conditional photon detection induces instantaneous phase synchronization between two decoupled quantum limit-cycle oscillators. We consider two quantum van der Pol oscillators without mutual coupling, each with an additional linearly coupled bath, and perform continuous measurement of photon counting on the output fields of the two baths interacting through a beam splitter. It is observed that in-phase or anti-phase coherence of the two decoupled oscillators instantaneously increases after the photon detection and then decreases gradually in the weak quantum regime or quickly in the strong quantum regime until the next photon detection occurs. In the strong quantum regime, quantum entanglement also increases after the photon detection and quickly disappears. We derive the analytical upper bounds for the increases in the quantum entanglement and phase coherence by the conditional photon detection in the quantum limit.