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Soviet Journal of Quantum Electronics

Phase transition in a system of two-level atoms interacting with an electromagnetic field

Vladimir I Emel'yanov, Yu. L. Klimontovich

Published Apr 30, 1976• 2 citations
Publisher

Abstract

An analysis is made of the possibility of a phase transition in a system of two-level atoms interacting via a transverse electromagnetic field. The analysis is based on the total (including the polarization energy) Hamiltonian. It is shown that if allowance is made for the difference between the field acting on an atom and the Maxwellian field in the spectrum of collective (polariton) excitations, soft modes may arise and at a temperature T < T cr a phase transition may occur and a macroscopic polarization may appear at zero frequency.

DOI
10.1070/qe1976v006n04abeh011177

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An analysis is made of the possibility of a phase transition in a system of two-level atoms interacting via a transverse electromagnetic field. The analysis is based on the total (including the polarization energy) Hamiltonian. It is shown that if allowance is made for the difference between the field acting on an atom and the Maxwellian field in the spectrum of collective (polariton) excitations, soft modes may arise and at a temperature T < T cr a phase transition may occur and a macroscopic polarization may appear at zero frequency.
We study the thermodynamic properties of two-level atoms interacting with all the modes of the electromagnetic field. We show that the presence of the A2{A}^{2}A2 term in the interaction Hamiltonian has two important consequences: The divergence difficulties noted previously for the all-mode problem are overcome, so that a formal thermodynamic limit exists; and the phase transition, which occurs in the single-mode case without the A2{A}^{2}A2 term, is absent.
We apply the energy surface method to study a system of Na{N}_{a}Na​ three-level atoms interacting with a one-mode radiation field in the $\ensuremath{\Xi}$, $\ensuremath{\Lambda}$, and V\mathsf{V}V configurations. We obtain an estimation of the ground-state energy, the expectation value of the total number of excitations, and the phase diagram of the model in the interaction parameter space and compare the results with the exact solutions. The phase diagram separates the behavior of, in general, a dominant single state without excitations (M=0M=0M=0) from a collective state with excitations ($M\ensuremath{\ne}0$). We have first- and second-order phase transitions, except for the V\mathsf{V}V configuration, which only presents second-order phase transitions.
The atom - electromagnetic field interaction is studied in the Dicke model, wherein a single field mode is interacting with a collection of two level atoms at thermal equilibrium. It is found that in the superradiant phase of the system, wherein the Bose-Einstein condensation of photons takes place, the notion of photon as an elementary electromagnetic excitation ceases to exist. The phase and intensity excitations of the condensate are found to be the true excitations of electromagnetic field. It is found that in this phase, the atom interacts with these excitations in a distinct coherent transition process, apart from the known stimulated emission/absorption and spontaneous emission processes. In the coherent transition it is found that while the atomic state changes in course of the transition process, the state of electromagnetic field remains unaffected. It is found that the transition probability of such coherent transition process is macroscopically large compared to other stimulated emission/absorption and spontaneous emission processes.
A semiclassical analysis is presented to determine the quantum phase transition from single to collective regimes in three-level atoms in the presence of a radiation field. The energy surfaces of the Ξ, V, and Λ configurations are constructed by taking the expectation value with respect to U(3) coherent states that carry the totally symmetric representation, determined by the total number of atoms. The corresponding stability and equilibrium properties are calculated by means of the catastrophe theory, discovering the bifurcation and Maxwell sets. We establish that the atoms with Λ and Ξ configurations have the presence of double points, i.e., there are two independent quantum states with the same energy that can be obtained depending on the values of the dipolar strengths of the interaction between the atoms and the radiation field. Additionally, the Ξ configuration exhibits a fixed triple point.