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Journal of Physics A Mathematical and General

A comparison of quantum and semi-classical theories of the interaction between a two-level atom and the radiation field

Thomas Von Foerster

Published Jan 1, 1975• 76 citations
Publisher

Abstract

The interaction of a two-level atom with a single mode of the radiation field is considered both in a fully quantized and a semi-classical theory. With only the dipole and rotating-wave approximations the Liouville equations are solved to give the exact behaviour of the diagonal elements of the density operator and of their semi-classical analogue. The results suggest experimentally verifiable differences between the two theories.

DOI
10.1088/0305-4470/8/1/015

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The interaction of a two-level atom with a single mode of the radiation field is considered both in a fully quantized and a semi-classical theory. With only the dipole and rotating-wave approximations the Liouville equations are solved to give the exact behaviour of the diagonal elements of the density operator and of their semi-classical analogue. The results suggest experimentally verifiable differences between the two theories.
In the preceding chapters concerning the interaction of a radiation field with matter, we assumed the field to be classical. In many situations this assumption is valid. There are, however, many instances where a classical field fails to explain experimentally observed results and a quantized description of the field is required. This is, for example, true of spontaneous emission in an atomic system which was described phenomenologically in Chapter 5. For a rigorous treatment of the atomic level decay in free space, we need to consider the interaction of the atom with the vacuum modes of the universe. Even in the simplest system involving the interaction of a single-mode radiation field with a single two-level atom, the predictions for the dynamics of the atom are quite different in the semiclassical theory and the fully quantum theory. In the absence of the decay process, the semiclassical theory predicts Rabi oscillations for the atomic inversion whereas the quantum theory predicts certain collapse and revival phenomena due to the quantum aspects of the field. These interesting quantum field theoretical predictions have been experimentally verified. In this chapter we discuss the interaction of the quantized radiation field with the two-level atomic system described by a Hamiltonian in the dipole and the rotating-wave approximations. For a single-mode field it reduces to a particularly simple form. This is a very interesting Hamiltonian in quantum optics for several reasons. First, it can be solved exactly for arbitrary coupling constants and exhibits some true quantum dynamical effects such as collapse followed by periodic revivals of the atomic inversion.
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.
This chapter considers an atom as a quantum mechanical system. But the electromagnetic radiation will be treated as classical waves described by the Maxwell equations. The treatment is called semi-classical. Quantum mechanics describes an atom as a quantized system with a set of discrete energy levels. Each energy level is designated by a set of quantum numbers like the principal quantum of the Rydberg level and the quantum numbers representing the orbital, spin and total angular momentum. A molecule will have a different set of quantum numbers arising from its rotational and vibrational motions. Atoms in a solid have energy levels in the form of a band. The observed spectrum due to spontaneous emission from a collection of atoms is not monochromatic, but it is spread over a certain range of frequencies. The absorption spectrum also shows that the atom does not absorb at a single frequency, but it is spread over a band of frequencies around the resonance frequency.
We study the interaction between a two-level system and a harmonic oscillator without the rotating-wave approximation (RWA). In quantum optics, this model (sometimes called the spin-boson model) is traditionally thought of as a two-Ievel atom coupled to a single-mode radiation field, but it could also describe (1) a two-level ion in a trap coupled to the motion of its own center of mass or (2) a two-level particle in a solid coupled to a phonon mode. For sufficiently strong coupling, this model is known to exhibit chaos in a variety of semi-classical approximations.1