A semi-classical versus quantum description of the ground state of three-level atoms interacting with a one-mode electromagnetic field

We consider N-a three-level atoms (or systems) interacting with a one-mode electromagnetic field in the dipolar and rotating wave approximations. The order of the quantum phase transitions is determined explicitly for each of the configurations Xi, Lambda and V, with and without detuning. The semi-classical and exact quantum calculations for both the expectation values of the total number of excitations M = < M > and photon number n = < n > have excellent correspondence as functions of the control parameters. We prove that the ground state of the collective regime obeys sub-Poissonian statistics for the M and n distribution functions. Therefore, their corresponding fluctuations are not well described by the semi-classical approximation. We show that this can be corrected by projecting the variational state to a definite value of M.