Langevin Quantum theory for coupled phase-conjugated electromagnetic fields

Quantum noise is required for an open quantum system including gain and loss to fully describe the system's quantum behaviors. While loss-gain-induced Langevin noises have been intensively studied in quantum optics, the effect of a complex-valued nonlinear coupling coefficient on the noises of two coupled phase-conjugated optical fields has never been questioned before. Here, we formulate general macroscopic quantum Langevin equations from the coupling matrix by preserving commutation relations and correlations of the two coupled optical fields. We take spontaneous four-wave mixing in a four-level atomic system as an example to confirm that our macroscopic result is consistent with that obtained from the microscopic Heisenberg theory. We find that besides gain and loss, the complex-valued nonlinear coupling coefficient also results in Langevin noises. We further apply the quantum Langevin theory to study the effects of linear loss and gain, as well as complex nonlinear coupling coefficients in quantum correlations of entangled photon pairs. Our macroscopic phenomenological method to solve quantum Langevin coupled equations may have applications in two-mode squeezing, parametric oscillation, and other quantum light state generation.