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Physical Review A

Thermodynamics of two-level atoms interacting with the continuum of electromagnetic field modes

Kazimierz Rza̧żewski, K. Wódkiewicz

Published May 1, 1976• 27 citations
Publisher

Abstract

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 ${A}^{2}$ 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 ${A}^{2}$ term, is absent.

DOI
10.1103/physreva.13.1967

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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.
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We study the structure of the phase diagram for systems consisting of two- and three-level particles dipolarly interacting with a one-mode electromagnetic field, inside a cavity, paying particular attention to the case of a finite number of particles, and showing that the divergences that appear in other treatments are a consequence of the mathematical approximations employed and can be avoided by studying the system in an exact manner quantum-mechanically or via a catastrophe formalism with variational trial states that satisfy the symmetries of the appropriate Hamiltonians. These variational states give an excellent approximation not only to the exact quantum phase space, but also to the energy spectrum and the expectation values of the atomic and field operators. Furthermore, they allow for analytic expressions in many of the cases studied. We find the loci of the transitions in phase space from one phase to the other, and the order of the quantum phase transitions are determined explicitly for each of the configurations, with and without detuning. We also derive the critical exponents for the various systems, and the phase structure at the triple point present in the Ξ -configuration of three-level systems is studied.