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Physical Review

Approximate Solutions for an N -Molecule-Radiation-Field Hamiltonian

M. T. Tavis, F.W. Cummings

Published Dec 10, 1969• 304 citations
Publisher

Abstract

Three approximation schemes are discussed and compared with the exact solutions for a Hamiltonian describing the interaction of a single-mode quantized radiation field and $N$ two-level "molecules," each resonant with the mode frequency. In particular, the approximation of treating the molecules as uncorrelated is found to be accurate when the system energy is several times larger than the cooperation number of the molecular system.

DOI
10.1103/physrev.188.692

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Three approximation schemes are discussed and compared with the exact solutions for a Hamiltonian describing the interaction of a single-mode quantized radiation field and NNN two-level "molecules," each resonant with the mode frequency. In particular, the approximation of treating the molecules as uncorrelated is found to be accurate when the system energy is several times larger than the cooperation number of the molecular system.
The exact solution for a problem of NNN identical two-level molecules interacting through a dipole coupling with a single-mode quantized radiation field at resonance is given. Approximate expressions for the eigenvectors and eigenvalues for the ground and low-lying excited states, as well as the most highly excited states, are developed and compared with the exact results.
In this dissertation a simple Hamiltonian for a system of inter-acting molecules and radiation field is developed from a model of N Two-Level Molecules interacting, via a dipole approximation, with a single mode, quantized radiation field. The exact eigenvalues and eigenvectors of this Hamiltonian are compared with analytical results obtained from several different approximation schemes applied to the simple model. The consequence of this comparison is the development of validity criteria for the utilization of these approximation schemes to more difficult problems. For example, an analysis done here indicates that using an independent molecule approximation in an explanation of the "build up" of coherent radiation in a gas laser (maser) may be inappropriate.
We consider the description of the quantum dynamics of a two-level system interacting with a single Glauber coherent mode of the radiation field. An exact solution (within the rotating wave and electric dipole approximations) is provided for both the time-independent and time-dependent Green’s functions for the coupled matter–radiation system. The atom (molecular) transition probabilities are shown to be Poisson averages of the corresponding semiclassical problem when the latter is likewise treated in the rotating wave approximation. At low radiation intensities, the quantum and semiclassical results differ, but at high intensities, they become the same when the average photon number n0 in the field satisfies the condition n01/2≫1. A similar behavior is displayed for the expectation value of the electric field of the coherent mode in the presence of the atom. The spectrum of the radiation, resonantly scattered by the atom, is shown to coincide with the results of Mollow.
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 interaction of a resonant radiation field with two-level atoms (in the long-wavelength limit) is described by a Hamiltonian trilinear in the boson operators. It is shown that the dynamics of the system can be determined nearly exactly if either at least one of the three modes is strongly populated for all times or any two modes are separately strong in two complementary time domains. Explicit results are given in two important specific cases: (i) pumping of a ground-state atomic system by a coherent radiation field and (ii) coherent spontaneous emission. The results are compared with previously known approximate results. Our results are also applicable to certain parametric processes such as parametric amplification, frequency conversion, Raman and Brillouin scattering, etc., where an identical trilinear Hamiltonian is used to describe these processes.