Mean-field and fluctuation dynamics in off-resonant two-mode atom-field interactions
We study a two-level system coupled to two quantized electromagnetic modes within the Jaynes-Cummings framework. While the single-mode model is exactly solvable due to its conserved excitation number, yielding finite-dimensional invariant subspaces, the two-mode model extension presents a fundamental challenge: although the total excitation number remains conserved, each invariant subspace is infinite-dimensional, preventing a closed-form analytical solution. Our scheme separates the dynamics into a dominant, exactly solvable semiclassical component, the atom interacting with the mean fields of both modes, and treats the remaining quantum fluctuations through a sequence of unitary transformations that preserve essential quantum features. We validate our approach through direct comparison with numerical solutions, focusing on the non-resonant regime where multiple detunings give rise to rich interference effects and multi-timescale dynamics inaccessible to standard approximations. The method accurately reproduces atomic inversion, field observables, and fidelity over relevant timescales, while remaining computationally efficient.