Adiabatic elimination in the presence of multiphoton transitions in atoms inside a cavity

TL;DR

This paper introduces a new adiabatic elimination method for atoms in cavities with multiphoton transitions, applicable even without Raman-like resonance conditions, advancing the theoretical tools for cavity quantum electrodynamics.

Contribution

It generalizes existing adiabatic elimination techniques to intracavity systems with multiphoton processes, not limited by resonance conditions, using a Heisenberg picture approach.

Findings

Derives effective Hamiltonians for multiphoton transitions in cavities.

Works without requiring Raman-like resonance conditions.

Applicable to a broader range of intracavity atomic systems.

Abstract

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.