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Soviet Journal of Quantum Electronics

Model of resonant interaction between long-wavelength photons and two-level atoms in solids

Igor K. Kudryavtsev, A N Meleshko, A. S. Shumovskiǐ

Published Dec 31, 1979• 0 citations
Publisher

Abstract

A modification of the Dicke model is proposed in which allowance is made for direct interaction between atoms polarized by the electromagnetic field and their thermal vibrations in a crystal. It is shown that depending on the parameters, both second- and first-order phase transitions are possible in this system. Rigorous equations are obtained for the order parameter of the atomic subsystem, the difference between the level populations, and the collective precession of the dipole moments.

DOI
10.1070/qe1979v009n12abeh009811

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A modification of the Dicke model is proposed in which allowance is made for direct interaction between atoms polarized by the electromagnetic field and their thermal vibrations in a crystal. It is shown that depending on the parameters, both second- and first-order phase transitions are possible in this system. Rigorous equations are obtained for the order parameter of the atomic subsystem, the difference between the level populations, and the collective precession of the dipole moments.
The authors consider a system comprised of a collection of identical two-level atoms interacting with the electromagnetic field in the dipole approximation and with an externally applied steady-state coherent driving field. The atoms are considered to interact with each other only via the electromagnetic field and are assumed to be contained within a volume much smaller than a resonance wavelength. The system is treated as a quantum-statistical ensemble in the rotating frame, rotating at the carrier frequency of the externally applied coherent field. The Hamiltonian in this frame is explicitly time independent and the exact analog of a spin temperature is defined. The thermodynamic-Green's-function method is used to determine the thermodynamic equilibrium properties of the system in the rotating frame. This results in a nonlinear relation between the applied field and the field in the volume containing the atoms. The expression is a nonlinear function of the effective or "spin" temperature as well, and has the form of a first-order phase transition with the macroscopic atomic polarization as order parameter. In the low-"spin"-temperature limit, and for perfect tuning, the expression reduces to the exact form of the nonlinear relation for optical bistability derived by Bonifacio and Lugiato and relating the incident and transmitted fields for atoms in a ring cavity in the mean-field approximation. The results contain absorptive as well as dispersive contributions. These results predict essentially different behavior as a function of off tuning compared with behavior predicted by others using statistical steady-state models far from thermodynamic equilibrium. In the limit of zero value for the applied field amplitude, the results reduce to the conditions for the well-known second-order superradiant" phase transition. The condition for the existence of the second-order phase transition is shown to depend on the cavity tuning and the photon escape rate. The authors also show why, in light of the experiments in optical bistability, this second-order phase is difficult to observe.
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.
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.
By using the thermodynamic Green's function method, the authors study the phase transition of a collection of two- as well as three-level atoms interacting with the electromagnetic field and contained within a volume much smaller than the smallest resonance wavelength (Dicke model). They show for the case of three-level atoms the existence of two critical temperatures where a second-order phase transition takes place. The upper critical temperature is determined by the largest of the three coupling constants in the three-level model, whereas the lower critical temperature is a function of the remaining two coupling constants. It is also shown that for a collection of two- as well as three-level atoms, the critical temperature depends upon the number of atoms contained within lambda r 3 ( lambda r is a resonance wavelength) rather than the density as has previously been suggested. Also, using the Green's function method, we show the formal equivalence of the Dicke model in thermodynamic equilibrium with the BCS model of superconductivity.