The conservative system of N atoms coupled with one photon

Abstract In this work the conservative system of N identical two-level atoms coupled with a quantized electromagnetic field, prepared via a single photon Fock state, is investigated. The non-zero initial detuning of the Fock state is under consideration. The present paper reveals and investigates the certain fundamental problems concerning the application of the dipole approximation for the “interaction” of the quantized electromagnetic field with the many-body system. Here, we do not restrict ourselves to any boundary conditions and do not apply the “Weisskopf–Wigner” and “Markovian” approximations. We developed technique to estimate the most “problematic” integral expressions appearing in the quantum optic theory of a photon scattering by a polyatomic system. The Laplace transformation was applied in order to solve the corresponding N -particle evolution equations. It was shown, the dipole approximation, that is determined by a gauge non-invariant Hamiltonian, can result in certain irreversible processes during the system evolution with time. Namely, possible collective (coherent) modes for the probability amplitudes decay with time. At the same time, the mode corresponding to the field detuning frequency does not vanish. However, the ability to decay vanishes for the collective (coherent) modes, when the used inverse Laplace transformation undergoes a commutation in the specific limits of the problem. The investigated here structure of the solutions for the evolution equations therefore raises the fundamental problem about the correct representation of a damping phenomena in the commonly accepted approximations. The corresponding N -particle state amplitudes are calculated for the several space configurations of the closed conservative system. The elaborated theory allows to describe analytically the evolution of the system components with time for any space configurations, including the cases with relative distances between atoms shorter or larger than the emission wavelength.