  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.
