  Various approaches have been used in the literature for eliminating
nonresonant levels in atomic systems and deriving effective Hamiltonians.
Important among these are elimination techniques at the level of probability
amplitudes, operator techniques to project the dynamics on to the subspace of
resonant levels, Green's function techniques, the James' effective Hamiltonian
approach, etc. None of the previous approaches is suitable for deriving
effective Hamiltonians in intracavity situations. However, the James' approach
does work in the case of only two-photon transitions in a cavity. A
generalization of the James' approach works in the case of three-photon
transitions in a cavity, but only under Raman-like resonant conditions. Another
important approach for adiabatic elimination is based on an adaptation of the
Markov approximation well-known in the theory of system-bath interactions.
However, this approach has not been shown to work in intracavity situations. In
this paper, we present a method of adiabatic elimination for atoms inside
cavities in the presence of multiphoton transitions. We work in the Heisenberg
picture, and our approach has the advantage that it allows one to derive
effective Hamiltonians even when Raman-like resonance conditions do not hold.
