Modified atomic decay rate near absorptive scatterers at finite temperature

Abstract

The change in the decay rate of an excited atom that is brought about by extinction and thermal-radiation effects in a nearby dielectric medium is determined from a quantum-mechanical model. The medium is a collection of randomly distributed thermally excited spherical scatterers with absorptive properties. The modification of the decay rate is described by a set of correction functions for which analytical expressions are obtained as sums over contributions from the multipole moments of the scatterers. The results for the modified decay rate as a function of the distance between the excited atom and the dielectric medium show the influence of absorption, scattering and thermal-radiation processes. Some of these processes are found to be mutually counteractive. The changes in the decay rate are compared to those following from an effective-medium theory in which the discrete scatterers are replaced by a continuum.

Hierarchies of sum rules for squares of spherical Bessel functions

“…For small p > 0 the z-dependent polynomials occurring at the right-hand side of (3.9) have got a small degree, as in (3.3). Sum rules closely related to (3.9), with spherical Hankel functions as before, have been used in the quantum optics problem in [6]. Comparing the two hierarchies (3.3) and (3.9) we see that the coefficients in the weighted sums differ considerably.…”

“…Comparing for instance the evaluation times of both sides of (3.20) for p = 0, = 50 and z = 50 with the help of the numerical software contained in Mathematica, one finds that calculating the right-hand side is more than 10 times faster than calculating the left-hand side. Such an increase in the efficiency of the numerical evaluation proved to be advantageous in producing the plots in [6] for a range of values of z.…”

“…In [6] the sum rules (3.3) and (3.9), with spherical Hankel functions instead of j (z), have been used to determine indefinite integrals over squares of these functions. The general form of these indefinite integrals is z duu −n h The identities (3.3), (3.9) and (3.20) yield an efficient way to evaluate the sums at their left-hand sides numerically, in particular for small p and large .…”

“…For small p > 0 and arbitrary the coefficients of the squared spherical Bessel functions at the right-hand side of (3.3) are small-degree polynomials in 1/z that are easily evaluated. For general p ≥ 0 the sum rules (3.3), with spherical Hankel functions instead of spherical Bessel functions, were of crucial importance in the analysis of the modified atomic decay rates in [6]. A rather different hierarchy follows by starting from the sum rule (2.7) and choosing g k = 2k + 1 in (2.2).…”

“…In contrast, information on finite sums of this type, with an arbitrary large but finite number of terms, is less well available. Such finite sums occur in various branches of mathematical physics, for instance in atomic orbital theory [4], in acoustic diffraction problems [5] and more recently in quantum optics [6].…”