Quantum Langevin theory for two coupled phase-conjugated electromagnetic waves

TL;DR

This paper develops a macroscopic quantum Langevin framework for two coupled phase-conjugated optical fields with complex nonlinear coupling, validated through atomic system simulations, and explores their effects on entangled photon pair generation.

Contribution

It introduces a general phenomenological quantum Langevin model for coupled phase-conjugated fields with complex nonlinear coupling, without requiring microscopic details.

Findings

The model preserves field commutation relations and correlations.

Numerical validation matches microscopic Heisenberg-Langevin results.

Analyzes effects on biphoton temporal quantum correlations.

Abstract

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 provide a general macroscopic phenomenological formula of quantum Langevin equations for two coupled phase-conjugated 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-$\Lambda$ 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. 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.