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Physical Review

Exact Solution for an N -Molecule—Radiation-Field Hamiltonian

M. T. Tavis, F.W. Cummings

Published Jun 10, 1968• 1,564 citations
Publisher

Abstract

The exact solution for a problem of $N$ 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.

DOI
10.1103/physrev.170.379

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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.
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.
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.
A class of quasi-exact solutions of the Rabi Hamiltonian, which describes a two-level atom interacting with a single-mode radiation field via a dipole interaction without the rotating-wave approximation, are obtained by using a wavefunction ansatz. Exact solutions for part of the spectrum are obtained when the atom-field coupling strength and the field frequency satisfy certain relations. As an example, the lowest exact energy level and the corresponding atom-field entanglement at the quasi-exactly solvable point are calculated and compared to results from the Jaynes–Cummings and counter-rotating cases of the Rabi Hamiltonian.