Time evolution of pertinent operators in the Rabi Hamiltonian and its rotating wave approximation (RWA) version, the Jaynes-Cummings model (JCM), in the Heisenberg picture, gives systems of nonlinear differential equations (NDEs). Considering well localized atom, the mean field theory (MFT) was applied to replace the operators by equivalent expectation values. The Rabi model was reduced to a fourth orders NDE describing atoms position. Solution by the harmonic balance method (HBM) showed good accuracy and consistency to the numerical results, which introduces it as a useful tool in the quantum dynamics studies. The NDEs describing the JCM in the Heisenberg picture structurally prevent applying the MFT and shows inconsistency to the Ehrenfest’s theorem, contrary to the Rabi model.
