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Journal of Physics A Mathematical Nuclear and General

Perturbation theory of super-radiance. II. Cooperative and non-cooperative level shifts

R Saunders, R. Bullough

Published Sep 1, 1973• 12 citations
Publisher

Abstract

For pt. I, see ibid., vol. 6, 1348 (1973). The authors extend the perturbation theory of super-radiant emission to derive a number of results on the level shifts of n atom systems. These also divide into 'coherent' and 'incoherent' parts. The incoherent part divides further into a generalized Lamb shift which is not cooperative, and an interatomic term which is. The former proves to be the only shift is not cooperative, and an interatomic term which is. The former proves to be the only shift which can depend on the presence of ambient free photons. These results for the generalized Lamb shift support the view that a dynamical neoclassical theory of the Lamb shift is unacceptable. The theory is compared with the pseudoboson theories of linear dielectrics. The pseudoboson theory of an inverted dielectric (the 'amplifier') should be intrinsically unstable unlike the pseudoboson theory of the refractive index which is concerned with the 'attenuator'.

DOI
10.1088/0305-4470/6/9/011

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For pt. I, see ibid., vol. 6, 1348 (1973). The authors extend the perturbation theory of super-radiant emission to derive a number of results on the level shifts of n atom systems. These also divide into 'coherent' and 'incoherent' parts. The incoherent part divides further into a generalized Lamb shift which is not cooperative, and an interatomic term which is. The former proves to be the only shift is not cooperative, and an interatomic term which is. The former proves to be the only shift which can depend on the presence of ambient free photons. These results for the generalized Lamb shift support the view that a dynamical neoclassical theory of the Lamb shift is unacceptable. The theory is compared with the pseudoboson theories of linear dielectrics. The pseudoboson theory of an inverted dielectric (the 'amplifier') should be intrinsically unstable unlike the pseudoboson theory of the refractive index which is concerned with the 'attenuator'.
Perturbation theory is used to show that that super-radiance as envisaged by Dicke, in which the radiation rate goes as N 2 , the square of the number of atoms, can be interpreted in the same terms as classical diffraction theory. The rate goes as N 2 from samples small in three dimensions compared with a wavelength but goes essentially as N from larger samples. There is a natural distinction between coherent and incoherent contributions to the total radiation rate from larger samples prepared (supposedly by an incident pulse) in Dicke states phased with a vector k.
A landmark in the development of quantum electrodynamics was the discovery that emission–reabsorption of virtual photons modifies the value of energy levels in an atom from those computed by using Dirac’s equation (Lamb shift). An early result of statistical quantum electrodynamics was that the exchange of virtual photons in an ensemble of identical atoms leads as well to a change in the frequency of the radiation emitted from this system (cooperative Lamb shift). Dicke’s discovery that coherence effects lead to the shortening of the emission lifetime from a small sample by a factor equal to the number of atoms in the ensemble (superradiance or cooperative decay rate) was an early landmark in quantum optics. Both cooperative decay rate and cooperative Lamb shift were shown to have the same physical origin—the exchange of virtual photons, a process described by the Lienard–Wiechert dipole–dipole interaction. This effective potential is the kernel of the integral equation describing the dynamics of the system. This complex long-range kernel gives, for both cooperative quantities, strong dependence on the geometry of the atomic cloud. I summarize the known expressions for the initial cooperative decay rate and the cooperative Lamb shift in different geometries. The results for both the scalar photon and the vector photon (electrodynamics) theories for experimentally realizable systems of either uniform or phased polarization are given.
A natural all-order decorrelation procedure is described for the interaction of a single two-level atom with all modes of the radiation field which replaces the semi-classical boson approximation for the atom by an exact fermion treatment. In consequence the vacuum Lamb shift appears as twice the Bethe shift for each level separately instead of the limiting neo-classical shift obtained in the Bose theory. Stimulated emission and absorption modify the Einstein A-coefficient and the vacuum shift is extended to include a field dependent shift. The all-order decorrelated perturbation theory yields an equation of motion for the single-atom dipole moment which includes those reported from operator radiation reaction theory recently; but it also generalizes these by including the effects of stimulated processes. The method is extended to a many-atom case in which it becomes necessary to modify the procedure slightly.
This chapter considers super-radiance on an elementary level in the sense that the main notions and concepts of the phenomenon will be treated as they evolved in the course of the early studies starting with the classical works by R. H. Dicke. It considers super-radiance from a system with dimensions small compared with the wavelength of the emission that is the Dicke model. The chapter discusses the results to be obtained are also applicable to the single-mode super-radiance of an extended system. It then considers mainly the cooperative spontaneous emission of a collection of N identical atoms confined within a small volume. If the audience wish to investigate the super-radiance of an extended system, the dimensions of which exceed the emission wavelength, the semiclassical method is more convenient. In essence, the semiclassical approach discussed in the chapter combines the quantum mechanical treatment of the atomic system that uses the density matrix formalism with the classical treatment of the radiation field.
For pt.I see ibid., vol.18, p.3811 (1985). This paper is the second of a theoretical study of subradiance, i.e. of the cooperative inhibition of spontaneous emission by a destructive interatomic interference, a phenomenon which has recently been observed. It is devoted to the particular case of atomic systems with degenerate transitions, i.e. with several transitions of the same frequency and the same polarisation. The interatomic interference which then occurs between the different transitions is responsible, when destructive, for a new sort of subradiance. The phenomenon is studied for a few typical level configurations, using both quantum mechanical and semiclassical models. The first example concerns a four-level configuration with two transitions of the same frequency and polarisation. The other examples concern three-level configurations with a degenerate transition; in the case of the 'degenerate' cascade, new superradiant beats and subradiance effects are predicted.