Effect of the Atomic Dipole-Dipole Interaction on the Phase Diagrams of Field-Matter Interactions I: Variational procedure

TL;DR

This paper investigates how atomic dipole-dipole interactions influence the quantum phase diagrams of multi-level atomic systems coupled with electromagnetic fields, using a variational approach within second quantization.

Contribution

It introduces a variational method to analyze the impact of dipole-dipole interactions on quantum phase transitions in multi-level atom-field systems.

Findings

Identifies quantum phase transitions in 2- and 3-level systems with dipole interactions.

Classifies the order of phase transitions using Ehrenfest criteria.

Proposes new procedures for undetermined transition cases.

Abstract

We establish, within the second quantization method, the general dipole-dipole Hamiltonian interaction of a system of $n$-level atoms. The variational energy surface of the $n$-level atoms interacting with $\ell$-mode fields and under the Van Der Waals forces is calculated with respect the tensorial product of matter and electromagnetic field coherent states. This is used to determine the quantum phase diagram associated to the ground state of the system and quantify the effect of the dipole-dipole Hamiltonian interaction. By considering real induced electric dipole moments, we find the quantum phase transitions for $2$- and $3$-level atomic systems interacting with $1$- and $2$- modes of the electromagnetic field, respectively. The corresponding order of the transitions is established by means of Ehrenfest classification; for some undetermined cases, we propose two procedures: the difference of the expectation value of the Casimir operators of the $2$-level subsystems, and by maximizing the Bures distance between neighbor variational solutions.