Mirror symmetry in the energy spectra of n-level systems

The energy spectrum of a system of $N_a$ atoms of $n$ levels interacting with a one-mode electromagnetic field is studied in the dipole and rotating wave approximations. We find that, under the resonant condition, it exhibits a mirror symmetry with respect to the energy $E=M$ where $M$ the total number of excitations. Thus, for any eigenstate $|\psi_M^{+}\rangle$ with energy $E=M+{\cal E}$ there exists a related eigenstate $|\psi_M^{-}\rangle$ with energy $E=M-{\cal E}$ via the unitary parity operator in the number of photons . This is independent of the dipolar coupling between the levels. We give explicit examples for $3$-level systems.