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Europhysics Letters (EPL)

Equilibrium Thermodynamics of Matter Interacting with the Quantized Radiation Field

Wojciech Cegła, John T. Lewis, G. A. Raggio

Published Sep 1, 1987• 4 citations
Publisher

Abstract

The thermodynamic properties of a system of two-level atoms interacting with a quantized radiation field are investigated. We allow from the outset an infinite number of photon modes; we calculate the free-energy density in the thermodynamic limit exactly and rigorously. A second-order phase transition is evident from the expression for the free-energy density. We investigate it in detail by examining the distribution of the mean photon number density as a function of photon frequency. We use the Lieb-Berezin inequalities combined with the large-deviation principle.

DOI
10.1209/0295-5075/4/5/002

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The thermodynamic properties of a system of two-level atoms interacting with a quantized radiation field are investigated. We allow from the outset an infinite number of photon modes; we calculate the free-energy density in the thermodynamic limit exactly and rigorously. A second-order phase transition is evident from the expression for the free-energy density. We investigate it in detail by examining the distribution of the mean photon number density as a function of photon frequency. We use the Lieb-Berezin inequalities combined with the large-deviation principle.
The thermodynamic properties of several systems of multilevel atoms interacting with a quantized radiation field are investigated. We allow a quantum-mechanical treatment of the translational degrees of freedom and do not require the rotating-wave approximation. In the finite-photon-mode case one can calculate the free energy per atom in the thermodynamic limit exactly and rigorously. In the infinite-mode case we only get upper and lower bounds, but these are sufficient to give conditions for thermodynamic stability and instability. The kind of phase transition previously found by us for the one-mode Dicke model with the rotating-wave approximation persists in the general multimode case.
A system of NNN two-level atoms interacting with a quantized field, the so-called Dicke model of superradiance, is studied. By making use of a set of Glauber's coherent states for the field, the free energy of the system is calculated exactly in the thermodynamic limit. The results agree precisely with those obtained by Hepp and Lieb, who studied the same model using a different method. The exhibition of a phase transition of the system is presented mathematically in an elementary manner in our approach. The generalization to the case of finitely many radiation modes is also presented.
A non-relativistic quantum theory of matter-photon interaction is formulated, within the framework of the dipole approximation, by employing a canonical transformation to diago­ nalize the A' term plus the free photon Hamiltonian. The matter-photon interaction is there­ by expressed rigorously in a renormalized A·p form. Using this form for the Dicke model of two-level atoms interacting with a single-mode radiation field, we examine a possibility of the occurrence of stationary coherent states which could arise for a photon and atomic polarization as the ground state of the system, deducing a condition for such coherent states. The condition, expressed as an inequality which imposes a stable nontrivial Bloch angle of the uniform atomic polarization, is identical with that for the occurrence of a second order phase transition, and indicates that the non-existence of a second order phase transition pointed out by Rza2ewski et al. io; in fact a consequence of the present treatment of the A'­ term renormalization. However, a fulfillment of the condition is recovered, if exchange­ type atom-atom interaction is taken into account in the matter system. We also present a method to construct an effective Hamiltonian of atom-atom interaction which is equivalent to the above Dicke model in the thermodynamic limit.
The thermodynamic properties of NNN two-level atoms coupled to radiation field are analyzed taking into account the diamagnetic term by a slight modification to the usual approach to quantizing the electromagnetic field. Following this procedure, we are able to solve the discrepancy originating in the use of different states as a basis for expressing the state of radiation field. We develop a perturbative expansion of partition function and a simple analytic expression is found for a high coupling constant. We show that, in this limit, the interaction of independent atomic spins with a single photon mode can be interpreted as an effective spin-spin interaction of long range nature. Even in this approach, if the diamagnetic term is taken into account, a phase transition at nonzero temperature cannot occur due to sum-rule arguments.
The quantum mechanical analysis for a non-resonant case of a single radiation mode interacting with a two-level atom in a cavity is studied. Expressions for the mean value of photon number and the time evolution of the atomic motion are given. The frequency of exchange energy between the atomic levels and the radiation field systems for non- resonant and resonant interactions is obtained. Results for all initial states of the atom-radiation field are presented. Results for resonant and non-resonant interactions are compared.
The influence of quantum interference between different decay channels of three-level atoms on various characteristics of matter and radiation is considered for the case of thermodynamic equilibrium of atoms in a photon gas. It is shown that for rare gases when collisions between atoms are not important, the direct consequence of such an interference is a spontaneously induced atomic coherence between two upper levels. Therefore, for an accurate description of such atoms and radiation in thermodynamic equilibrium with each other it is not sufficient to specify only Boltzmann exponents, which correspond to the diagonal elements of the density matrix. The reason for this is that the conventional Boltzmann description is applied to infinitely sharp eigenstates of the unperturbed Hamiltonian. As we show, it becomes essential to take into account nonzero width of the levels if the levels are relatively close to each other. This produces nonzero nondiagonal elements of the density matrix that modify the emissivity of the equilibrium medium and lead to a zero point at a certain frequency as well as to an enhanced intensity in the red wing at the corresponding temperature. It is shown that the occupation number of the photons is not changed and obeys the equilibrium Planck distribution.