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

TL;DR

This paper compares semi-classical and quantum models of three-level atoms interacting with a single electromagnetic mode, analyzing phase transitions and quantum statistics of the ground state, highlighting the limitations of semiclassical approximations.

Contribution

It provides an explicit analysis of quantum phase transition orders for different configurations and demonstrates how to correct semiclassical predictions using projection methods.

Findings

Quantum phase transition orders are explicitly determined for each configuration.

Ground state exhibits sub-Poissonian photon and excitation number statistics.

Semiclassical approximation fails to capture fluctuations, but can be corrected by state projection.

Abstract

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 $\cal{M}=\langle \bm{M} \rangle$ and photon number $n=\langle \bm{n} \rangle$ have an excellent correspondence as functions of the control parameters. We prove that the ground state of the collective regime obeys sub-Poissonian statistics for the ${\cal M}$ and $n$ distribution functions. Therefore, their corresponding fluctuations are not well described by the semiclassical approximation. We show that this can be corrected by projecting the variational state to a definite value of ${\cal M}$.