Quantum effects beyond mean-field treatment in quantum optics

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.