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Physical Review A

Non-Dicke decay in a small spherical sample with radially varying density

Richard M. Friedberg, Jamal T. Manassah

Published Jan 23, 2012• 11 citations
Publisher

Abstract

It is a familiar fact that, in an isolated sphere or ellipsoid of uniform polarization density, the electrostatic field also is uniform. Because of this, the state of uniform polarization in a spherical sample of uniformly distributed two-level atoms is an eigenmode of the coherent decay process in the limit of a small radius compared to the resonant wavelength of a single atom. Consequently, in this special geometry, the Dicke picture of uniform exponential decay should hold. In nonspheroidal geometries or in spherical geometries with nonuniform atomic density, the decay should be more complicated. Here, we find the characteristic equations that determine the eigenmodes of the Lienard-Wiechert interaction for a partly hollowed sphere of identical two-level atoms in two different radial configurations. We show that the Dicke picture for emission from a coherently prepared sample does not correctly describe the system's dynamics even when we take the radius of the sphere to be much smaller than the radiation wavelength.

DOI
10.1103/physreva.85.013834

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It is a familiar fact that, in an isolated sphere or ellipsoid of uniform polarization density, the electrostatic field also is uniform. Because of this, the state of uniform polarization in a spherical sample of uniformly distributed two-level atoms is an eigenmode of the coherent decay process in the limit of a small radius compared to the resonant wavelength of a single atom. Consequently, in this special geometry, the Dicke picture of uniform exponential decay should hold. In nonspheroidal geometries or in spherical geometries with nonuniform atomic density, the decay should be more complicated. Here, we find the characteristic equations that determine the eigenmodes of the Lienard-Wiechert interaction for a partly hollowed sphere of identical two-level atoms in two different radial configurations. We show that the Dicke picture for emission from a coherently prepared sample does not correctly describe the system's dynamics even when we take the radius of the sphere to be much smaller than the radiation wavelength.
Radiation by the atoms of a resonant medium is a cooperative process in which the medium participates as a whole. In two previous papers we treated this problem for the case of a medium having slab geometry, which, under plane-wave excitation, supports coherent waves that propagate in one dimension. We extend the treatment here to the three-dimensional problem, focusing principally on the case of spherical geometry. By regarding the radiation field as a superposition of electric and magnetic multipole fields of different orders, we express it in terms of suitably defined scalar fields. The latter fields possess a sequence of exponentially decaying eigenmodes corresponding to each multipole order. We consider several examples of spherically symmetric initial excitations of a sphere. Small uniformly excited spheres, we find, tend to radiate superradiantly, while the radiation from a large sphere with an initially excited inner core exhibits temporal oscillations that result from the participation of a large number of coherently excited amplitudes in different modes. The frequency spectrum of the emitted radiation possesses a rich structure, including a frequency gap for large spheres and sharply defined and closely spaced peaks caused by the small frequency shifts and even smaller decay rates characteristic of the majority of eigenmodes.
The coherent radiation of N atoms that are excited by an external field is studied close to a resonant transition. The resonant atoms are uniformly distributed over a sphere and the external field is a pulse that propagates in a fixed direction. The linear classical continuum theory is used for the above configuration. Exact expressions are obtained for the polarization, for the fields everywhere in space, and for the energy radiated and absorbed by the resonant atoms. Two equivalent methods for the derivation of the exact expression of the polarization are given, i.e., either by solving Maxwell’s equations and boundary conditions, or by solving an integral equation inside the sample. The equivalence of the two methods, in the general case of an arbitrary external field, has been proved elsewhere. Approximate expressions are also obtained for the physical quantities mentioned above in the extreme cases of a dilute and a dense resonant sample with radius much smaller than the radiation wavelength. Inside the dense small sample the polarization depends on position, and this space dependence is expressed in terms of spherical Bessel functions and vector spherical harmonics. It is shown that outside the dense small sample, a broadening of the original resonant line and a shift of the resonance frequency occur in the radiation spectrum of the atoms.
We investigate the collective spontaneous emission from an ensemble of two-level atoms evenly distributed inside a sphere with a low density. An initial symmetric single-excitation state is considered and polarizations for all atoms are assumed to be aligned in the same direction. We find that the superradiant decay rate of the ensemble exhibits oscillatory features dependent on the radius of the sphere and the atomic radiation wavelength, indicating that the collective emission rate can be less than the single-atom decay rate for certain parameters. Moreover, the system exhibits a hidden equivalence where the collective emission from the ensemble gives the same radiation rate as the case of a two-atom model where one atom is at the center of the sphere and the other effective large atom, composed of all other atoms, is localized at the edge of the sphere along the polarization direction. Our result hence provides a potential method towards exploring complex many-body physics by simplifying the model into a two-body problem.
We present a general formalism for studying the cavity quantum electrodynamics for an atom or a radiating dipole inside a hollow spherical cavity. The atomic-energy-level shifts and the modified dipolar decay rates are calculated in specific cases. In particular, the energy shifts for an atom located at the center of an empty sphere (radius a) vary as 1/a4{\mathit{a}}^{4}a4 for large ka and 1/a2{\mathit{a}}^{2}a2 for small ka, where k is the wave number. We attribute these variations to the retarded Casimir-Polder interaction and the state-independent vacuum-field contribution, respectively. \textcopyright{} 1996 The American Physical Society.