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Off-resonant transitions in the collective dynamics of multilevel atomic ensembles

We study the contributions of off-resonant transitions to the dynamics of a system of N multilevel atoms sharing one excitation and interacting with the quantized vector electromagnetic field. The Rotating Wave Approximation significantly simplifies the derivation of the equations of motion describing the collective atomic dynamics, but it leads to an incorrect expression for the dispersive part of the atom-atom interaction terms. For the case of two-level atoms and a scalar electromagnetic field, it turns out that the atom-atom interaction can be recovered correctly if integrals over the photon mode frequencies are extended to incorporate negative values. We explicitly derive the atom-atom interaction for multi-level atoms, coupled to the full vector electromagnetic field, and we recover also in this general case the validity of the results obtained by the extension to negative frequencies of the formulas derived with the Rotating Wave Approximation.