  Mean-field treatment (MFT) is frequently applied to approximately predict the
dynamics of quantum optics systems, to simplify the system Hamiltonian through
neglecting certain modes that are driven strongly or couple weakly with other
modes. While in practical quantum systems, the quantum correlations between
different modes might lead to unanticipated quantum effects and lead to
significantly distinct system dynamics. Here, we provide a general and
systematic theoretical framework based on the perturbation theory in company
with the MFT to capture these quantum effects. The form of nonlinear
dissipation and parasitic Hamiltonian are predicted, which scales inversely
with the nonlinear coupling rate. Furthermore, the indicator is also proposed
as a measure of the accuracy of mean-field treatment. Our theory is applied to
the example of quantum frequency conversion, in which mean-field treatment is
commonly applied, to test its limitation under strong pump and large coupling
strength. The analytical results show excellent agreement with the numerical
simulations. Our work clearly reveals the attendant quantum effects under
mean-field treatment and provides a more precise theoretical framework to
describe quantum optics systems.
