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Physical Review

Solution of a Multiatom Radiation Model Using the Bargmann Realization of the Radiation Field

William R. Mallory

Published Dec 25, 1969• 44 citations
Publisher

Abstract

A model for interacting single-mode electromagnetic radiation and matter consisting of $N$ two-level particles is solved exactly using a representation in which the creation and destruction operators for the field act by multiplication and differentiation, respectively. The model includes off-resonant interactions. Closed-form expressions can be given for some special cases. The solutions are transformed to the more familiar Fock and Schr\"odinger representations. Numerical results are given to indicate the behavior of the solutions. An equation for a generating function is derived which would give solutions of the (algebraic) matrix equation to which the problem has been reduced.

DOI
10.1103/physrev.188.1976

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A model for interacting single-mode electromagnetic radiation and matter consisting of NNN two-level particles is solved exactly using a representation in which the creation and destruction operators for the field act by multiplication and differentiation, respectively. The model includes off-resonant interactions. Closed-form expressions can be given for some special cases. The solutions are transformed to the more familiar Fock and Schr"odinger representations. Numerical results are given to indicate the behavior of the solutions. An equation for a generating function is derived which would give solutions of the (algebraic) matrix equation to which the problem has been reduced.
In Chap. 1 we introduced the Schrodinger equation and gave solutions for a number of simple cases. The general solution was then obtained by a linear superposition of these solutions. In our work, we are particularly concerned with the way in which atomic systems interact with electromagnetic radiation. These systems are still described by wave functions given by superpositions of simple solutions, but the expansion coefficients Cn (probability amplitudes) become functions of time. In this chapter we derive equations of motion for the expansion coefficients using Schrddinger’s equation and introduce the electric-dipole interaction energy. Then, specializing to a two-level system and a monochromatic electromagnetic field, we solve the equations of motion both by time-dependent perturbation theory and by the more exact Rabi method.
We investigate the long-time behavior of a model consisting of N two-level atoms in a lossless cavity. The Hamiltonian of our system contains the radiation oscillators in addition to the matter Hamiltonian and the usual ∫j·Adv interaction term. In order to treat the system perturbatively, it would be necessary to remove the tremendous degeneracy of the system. Since this is prohibitively difficult, and since we are interested in the long-time behavior of the system, we solve the quantum mechanical Liouville equation directly for a wide class of physically important initial distribution functions. We show the effective expansion parameter is γ̃Nγ̃ where γ̃ is a dimensionless atomic dipole moment and N is the number of atoms. In the lowest order we find the self-consistent field approximation. In the next order, particle-field correlations appear. We explicitly solve the equations of motion for the particle-field correlations in terms of the average quantities that appear in the self-consistent field approximation. We show the self-consistent field approximation consists of five first-order differential equations. Next we show the equations of motion for the density matrix of the system correct to order (γ̃Nγ̃)2 are equivalent to eight first-order differential equations. The three additional equations are needed to describe the three second moments of the density matrix of the electromagnetic field that appear in second order. Our lowest-order microscopic equations are equivalent to semiphenomenological theories and our higher-order equations contain only the measurable second-order moments of the electromagnetic field in addition to the variables that appear in semi-phenomenological theories.
The interaction is considered between NNN stationary two-level atoms and a radiation field described by a single cavity mode. The state vector for the complete system of atoms plus radiation is expressed as a linear superposition of states constructed from a product of photon states in the nnn representation and products of Pauli spin eigenstates describing all combinations of atoms in the lower and upper energy levels. Equations for the corresponding probability amplitudes are derived by substituting this superposition into the Schr\"odinger equation. The resulting equations are combined into bilinear form and phenomenological damping contributions are added. After the neglect of certain of the bilinear quantities, a master equation is derived which governs the probability Pmn{{P}_{m}}^{n}Pm​n of having mmm atoms in the lower level and nnn photons in the mode. This master equation takes account of multiple single-quantum absorption and emission processes but not of simultaneous multiple processes involving two or more atoms at a time. The equation which governs the expected number of photons 〈n〉〈n〉〈n〉 derived from the master equation bears a close resemblance to a rate equation. The effect of radiation loss from the cavity is incorporated into the master equation. Numerical calculations for a QQQ-spoiled laser show that the statistics of the number of photons in the mode bear a qualitative resemblance to Poisson statistics.
Abstract In quantum electrodynamics, the matter degrees of freedom are coupled to the radiation field through a local, gauge-invariant interaction density. Due to the linear structure of this coupling, the problem of constructing a complete formal representation of the reduced matter dynamics can be solved exactly if the electromagnetic radiation field is initially in a Gaussian state. This chapter combines field theoretical methods with superoperator and influence functional techniques to derive an exact, relativistic representation for the reduced density matrix of the matter degrees of freedom which completely describes the influence of the electromagnetic radiation field on the matter dynamics. Several applications are treated, such as the suppression of the quantum coherence of charged particles caused by the emission of bremsstrahlung and the decoherence of many-particle states.
A dynamical problem in quantum optics is considered to be that of solving Schrodinger's equation with the initial state of the radiation field as a coherent state By cairying out a particular unitary transformation it is shown how one can arrive at a representation of the radiation field operators which is particulaily convenient for Proc.esses in which the interaction with the material system only involves one photon or a few photons. It is shown in which sense this representation is a counterpart of the Fock representation. In the derivation of the representation, which is exact throughout, some features are brought out, which may be interpreted as being semi-classical.
A class of bound-state problems which represent the interaction between a system of N two-level atoms and an electromagnetic radiation via an m-photon process is studied, with m = 1; 2; 3;.... We also evaluate some nonlinear effects usually related with the polarisability of the medium and with the dependence on the intensity of the radiation field on the matter-radiation interaction. We obtain exact values for the eigenstates and eigenvalues for all values of total angular momentum of the system and all possible number m of photons involved in the interaction. We give explicit analytic expressions for small numbers of atoms and discuss some aspects of the composition of the eigenstates and of the spectra obtained in these cases.