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Physical Review A

Critical properties of two-level atom systems interacting with a radiation field

Giuseppe Di Liberti, Rosa Letizia Zaffino

Published Sep 22, 2004• 22 citations
Publisher

Abstract

The thermodynamic properties of $N$ 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.

DOI
10.1103/physreva.70.033808

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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.
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.
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 N-photon interaction of two two-level atoms with a single-mode quantized radiation field is studied. The general expression for the state vector of the atom–field system is presented with the probability amplitudes in analytic form. For the initial condition in which the atoms are both at their upper levels and the field is in a squeezed state, we study the time evolution of the atomic population inversions and that of the statistics of the field. We find that the ratio of the two coupling constants, the initial squeezing of the field, and N have significant effects on the features of the atoms and of the field.
In this thesis we present a theoretical analysis of some of the implications of the quantum nature of the electromagnetic field.In particular, we are interested in how atomic dynamics are modified by coupling to a quantized radiation field.In chapter two we review the Jaynes-Cummings model of atom-field interactions.This model consists of a single two-level atom coupled to a single mode of the radiation field.The evolution of the atom depends upon the initial quantum state of the field mode.Chapter three is a review of the quantum theory of dissipation.In chapter four we examine the properties of the thermofield states of the two-mode, quantized radiation fields.We investigate the use of these states as a representation of single mode thermal states.The thermo field representation is used to describe interactions involving thermal averages and in an analysis of two-mode squeezing in parametric amplifiers.Chapters five to seven are concerned with the Rydberg atom maser.In chapter five we review some of the properties of Rydberg atoms in cavities and consider the validity of the Jaynes-Cummings model as a description of the maser.In chapter six we discuss the modification of the atomic dynamics when more than one atom is present in the cavity.We show how the collective atomic inversion is influenced by many-atom effects, as well as the photon statistical effects present in the single atom Jaynes-CHAPTER
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.