Quantum Langevin theory for two coupled phase-conjugated electromagnetic waves

While loss-gain-induced Langevin noises have been intensively studied in quantum optics, the effect of a complex-valued nonlinear coupling coefficient on two coupled phase-conjugated optical fields has never questioned before. Here, we provide general macroscopic phenomenological formula equations for with linear loss (gain) and complex coefficient. The is obtained from matrix to preserve field commutation relations correlations, which does not require knowing microscopic details light-matter interaction internal atomic structures. To validate this formula, take spontaneous four-wave mixing double-$\Lambda$ four-level system as an example numerically confirm that our result consistent Heisenberg-Langevin theory. Finally, apply study effects gain loss, phase mismatching, well entangled photon pair (biphoton) generation, particularly their temporal correlations.