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Physical Review A

Soluble one-dimensional semiclassical model for spontaneous emission

R. C. T. da Costa, Gustavo Perez Munguia

Published Nov 1, 1976• 1 citation
Publisher

Abstract

The radiation emitted by plane of excited two-level atoms is exactly calculated in the semiclassical approximation. The well-known hyperbolic-secant solution is obtained in the limit case of small plane densities.
DOI
10.1103/physreva.14.1745

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Some generalizations are made onthe spontaneous emission by a plane of excited atoms, described by two leveI atom-model, in the ~1=1, ~m=1, transition and using the semiclassical radiation approximation -both discussed in the text.lnitially,theradiation rate of an infinite plane of excited atoms is investigated, using ~1=O, ~m=O, transition.It is shown that we can observe a limit solution depending on the coupling hetween field ànd matter.We simulate the effect of the emitter thickness,heretofore neglected, by using two infinite parallel planes of atoms, separated by a finite distance.Theauto-consistent set of eguation for the interaction between the radiant energy and matter was established, in the general case, when 61=1, 6m=1.Westill discussed two extreme situations: a) the planes are coincident and b) they are infinitely separated.We investigate, in some cases, by means of a computer , the influence of plane separation, the phase ~difference between electric dipoles and lnicial.energy•of each plane in the spontaneous emission behaviour when the field-matter coupling is weak.íNDICE Capítulo I

•Apr 20, 1983•0 citations
NNN two-level "atoms" are considered in interaction with a single-mode resonant electromagnetic field. The exact solution is given nonrelativistically for all times for the case of spontaneous emission, when only one atom is initially excited. The solution is given for the general case of the NNN atoms in inequivalent mode positions.
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.
Spontaneous emission from a system of NNN identical two-level atoms is considered using a master equation recently derived by the author. The master equation describing the time evolution of the phase-space distribution function associated with the reduced density operator of the atomic system is obtained. This master equation, which is of the type of a Fokker-Planck equation, is used to derive the equation of motion for the mean values of various atomic operators characterizing the physical properties of the system. This leads to a hierarchy of equations, which is decoupled by making a suitable approximation. The intensity of the spontaneously emitted radiation is then calculated. Next, the spontaneous emission from geometrically small systems is considered. For this case, the master equation is solved exactly, and an exact expression for the radiation rate is obtained. The exact solution of the master equation is also used to calculate the normally ordered correlation functions for the electric field. Section V deals with the spontaneous emission from a system of harmonic oscillators, the size of the system being small compared to a wavelength. The master equation for this problem is also solved exactly, and it is shown that this system also leads to superradiant emission in some cases, e.g., if all the oscillstors are excited initially to some coherent state ∣z0〉|{z}_{0}〉∣z0​〉.
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