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Journal of Physics A Mathematical Nuclear and General

On the quantum and semi-classical theories of a two-level atom interacting with a single-field mode

S. Swain

Published Dec 1, 1973• 16 citations
Publisher

Abstract

Formally exact solutions to the quantum and semi-classical models of a single atom interacting with a single-field mode in the dipole approximation are compared. It is shown that the quantum results tend to the semi-classical ones in the limit n to infinity , in disagreement with the results of Chang and Stehle (1971).
DOI
10.1088/0305-4470/6/12/003

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The Rabi model describes a single mode electromagnetic field interacting with a two-level atom. As is well-known, the quantum treatment and the semi-classical treatment (where the field is treated classically rather than quantum mechanically) give qualitatively different results. In this paper, we analyse the model using a different semi-classical approximation based on Bohmian mechanics. In this approximation, the back-reaction from the two-level atom onto the classical field is mediated by the Bohmian configuration of the two-level atom. We find that the Bohmian semi-classical approximation gives results comparable to the usual mean field one for the transition between ground and first excited state. However it does not always perform well when considering a higher excited state. Both semi-classical approximations tend to reproduce the collapse of the population inversion, but fail to reproduce the revival, which is characteristic of the full quantum description.
The author attempts to show that, in contradiction with work by Chang and Stehle, the semiclassical description of RF resonance experiments is valid. The density matrix equation of motion is set up for a two-level atom interacting with a one-mode quantized field. The terms giving the semiclassical description are separated and the contributions arising from the rest are carefully evaluated. Corrections to the semiclassical treatment are found to be of order n 0 -2 , where n 0 is the average number of RF photons. In practice such corrections are negligible.
We consider N a three-level atoms (or systems) interacting with a one-mode electromagnetic field in the dipolar and rotating wave approximations. The order of the quantum phase transitions is determined explicitly for each of the configurations Ξ, Λ and V , with and without detuning. The semi-classical and exact quantum calculations for both the expectation values of the total number of excitations and photon number have excellent correspondence as functions of the control parameters. We prove that the ground state of the collective regime obeys sub-Poissonian statistics for the and n distribution functions. Therefore, their corresponding fluctuations are not well described by the semi-classical approximation. We show that this can be corrected by projecting the variational state to a definite value of .