Quantum Langevin theory for two coupled phase-conjugated electromagnetic waves

We provide a general macroscopic phenomenological formula of quantum Langevin equations for two coupled phase-conjugated electromagnetic fields with linear loss (gain) and complex nonlinear coupling coefficient. The macroscopic phenomenological formula is obtained from the coupling matrix to preserve the field commutation relations and correlations, which does not require knowing the microscopic details of light-matter interaction and internal atomic structures. To validate this phenomenological formula, we take spontaneous four-wave mixing in a double- (sic) four-level atomic system as an example to numerically confirm that our macroscopic phenomenological result is consistent with that obtained from the microscopic Heisenberg-Langevin theory. We find that a complex-valued nonlinear coupling coefficient can lead to noises even without linear gain or loss. Finally, we apply the quantum Langevin equations to study the effects of linear gain and loss, complex phase mismatching, as well as complex nonlinear coupling coefficient in entangled photon pair (biphoton) generation, particularly to their temporal quantum correlations.

Automated summary

We propose a macroscopic phenomenological formula for quantum Langevin equations describing two coupled phase-conjugated electromagnetic fields with linear loss (gain) and complex nonlinear coupling coefficient. This formula, obtained from the coupling matrix, preserves the field commutation relations and correlations without requiring knowledge of microscopic light-matter interaction and atomic structures. We validate the formula by numerically confirming its consistency with the microscopic Heisenberg-Langevin theory in the context of spontaneous four-wave mixing in a double-four-level atomic system. We also find that a complex-valued nonlinear coupling coefficient can induce noise even in the absence of linear gain or loss. Finally, we apply the quantum Langevin equations to study the effects of linear gain and loss, complex phase mismatching, and complex nonlinear coupling coefficient on entangled photon pair generation, particularly their temporal quantum correlations.