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Bulletin de la Classe des sciences

Comparison between kinetic theory and wave function formalism in interaction between a two level system and radiation

Françoise Hénin

Published Jan 1, 1968• 0 citations
Publisher

Abstract

The meaning of perturbed states in the kinetic theory is first discussed. It is shown that the canonical form of the density matrix for a particle in a perturbed level does not correspond to a pure case, in the bare particle representation. The problem of spontaneous emission by an atom in its unperturbed excited state is then discussed. We first show that the results of the kinetic theory can also be obtained by the level shift operator formalism if one goes beyond the usual approximation. It is also shown that these results give a modified Lorentzian shape. Finally, we consider the resonant scattering of a photon. The solution of the kinetic equation gives us the transition probability per unit time. The frequency dependence is given by a modified Lorentz function, analogous to that obtained for the spontaneous emission line.

DOI
10.3406/barb.1968.62294

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The meaning of perturbed states in the kinetic theory is first discussed. It is shown that the canonical form of the density matrix for a particle in a perturbed level does not correspond to a pure case, in the bare particle representation. The problem of spontaneous emission by an atom in its unperturbed excited state is then discussed. We first show that the results of the kinetic theory can also be obtained by the level shift operator formalism if one goes beyond the usual approximation. It is also shown that these results give a modified Lorentzian shape. Finally, we consider the resonant scattering of a photon. The solution of the kinetic equation gives us the transition probability per unit time. The frequency dependence is given by a modified Lorentz function, analogous to that obtained for the spontaneous emission line.
In this paper we calculate exactly the lineshape for a model of an excited two-level atom in interaction with a continuous spectrum of radiation for the problem of spontaneous emission. Specifically, for the case of a d=1 radiation field, we use the exact results reported in our earlier work [J. Math. Phys. 14, 414, 423 (1973)] for the probability ρ(τ) of the atom’s being in the excited state at time τ to obtain an analytic expression for the lineshape of the emitted radiation (i.e., the photon). We also calculate the lineshape using two Wigner–Weisskopf style approximations and the results, for a given choice of coupling function, are compared numerically with the profile generated using the exact solution. These comparisons show convincingly the success with which these two approximations can be expected to reproduce the qualitative and quantitative features of the exact lineshape for values of the coupling constant α ranging between 0.01 and 0.3. Finally, we calculate explicitly the scattering operator corresponding to the above spontaneous emission problem and work out exactly the probability of forward and backward scattering of a photon by the atom at and away from resonance.
An improved form of semiclassical radiation theory is developed which includes the effect of the atom's radiation field back on the atom. This formalism is applied to the problem of a single "two-level atom" interacting with a monochromatic field. The resulting equations are solved without resorting to time-dependent perturbation theory, and are found to predict the behavior of the system over times long compared with the lifetime for spontaneous transitions. Not only stimulated emission and absorption, but also spontaneous emission with the proper Einstein AAA coefficient, and a frequency shift which agrees at least semiquantitatively with the Lamb shift are described. In addition, several nonlinear effects involving the interference between spontaneous and stimulated radiation are described, and new experiments which might detect such effects are suggested.
An improved form of semiclassical radiation theory is developed which includes the effect of the atom's radiation field back on the atom. This formalism is applied to the problem of a single two-level atom interacting with a monochromatic field. The resulting equations are solved without resorting to time-dependent perturbation theory, and are found to predict the behavior of the system over times long compared with the lifetime for spontaneous transitions. Not only stimulated emission and absorption, but also spontaneous emission with the proper Einstein AAA coefficient, and a frequency shift which agrees at least semiquantitatively with the Lamb shift are described. In addition, several nonlinear effects involving the interference between spontaneous and stimulated radiation are described, and new experiments which might detect such effects are suggested.
A modified perturbation theory is developed for the calculation of the time-dependent wave function describing the propagation of an atom in a two-mode laser field. With the mode frequencies nearly equal, a first approximation is obtained that reproduces and extends known results, for both resonant and nonresonant transitions. This lowest-order solution sums terms that describe the absorption of a photon from one mode and emission into the other, leaving the total number of photons unchanged and shifting the momentum of the atom accordingly. Corrections are generated by expansion of a resolvent from which the nearly degenerate states have been projected out. These approximate solutions are used to construct asymptotic states in a formulation of atom-atom scattering in the two-mode field. In close analogy with the soft-photon approximation, a ``soft-pair'' approximation is developed based on the dominance of those asymptotic interactions in which the transfer of energy from the field to the atom is small, allowing one to relate the scattering amplitude in the presence of the field to the field-free amplitude. When the atomic ground state and an excited state are closely coupled by the field, the asymptotic wave function can have mixed parity allowing for long-range contributions to the effective potential. In addition to an enhancement of forward-scattering probabilities, this leads to a threshold anomaly that is analyzed here, in the resonant case, with the aid of a modified effective-range theory.