Multiple-scattering approach to interatomic interactions and superradiance in inhomogeneous dielectrics

Abstract

The dynamics of a collection of resonant atoms embedded inside an inhomogeneous nondispersive and lossless dielectric is described with a dipole Hamiltonian that is based on a canonical quantization theory. The dielectric is described macroscopically by a position-dependent dielectric function and the atoms as microscopic harmonic oscillators. We identify and discuss the role of several types of Green tensors that describe the spatio-temporal propagation of field operators. After integrating out the atomic degrees of freedom, a multiple-scattering formalism emerges in which an exact Lippmann-Schwinger equation for the electric field operator plays a central role. The equation describes atoms as point sources and point scatterers for light. First, single-atom properties are calculated such as position-dependent spontaneous-emission rates as well as differential cross sections for elastic scattering and for resonance fluorescence. Secondly, multi-atom processes are studied. It is shown that the medium modifies both the resonant and the static parts of the dipoledipole interactions. These interatomic interactions may cause the atoms to scatter and emit light cooperatively. Unlike in free space, differences in position-dependent emission rates and radiative line shifts influence cooperative decay in the dielectric. As a generic example, it is shown that near a partially reflecting plane there is a sharp transition from two-atom superradiance to single-atom emission as the atomic positions are varied.

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“…We stress, however, that the general formalism above rigorously covers both weak and strong coupling regimes in a self-consistent way, naturally recovering phenomena such as vacuum Rabi oscillations and reverse spontaneous emission [40][41][42]. Our expression for the spontaneous emission is also in agreement with derivations from Dung et al [43,44] and Wubs et al [38], who have pioneered many of the general theories of quantized light emission within inhomogeneous materials. In a similar manner, one can also obtain a frequency shift of the QD exciton, de-…”

“…The following relationships can also be derived: K = G+1 (r r H )a4(r), and K = G T + T (rY r H )a4(r), where T (rY r H ) = f (r)f £ (r H )4(r) is a generalized transverse delta function [38]. Next, we introduce some of the basic quantum mechanics.…”

“…The subtle difference between the quantum and classical fields, is that in the quantum case, the dipole source above stems from quantum noise or vacuum fluctuations, and the Green function K appears, rather than G [38]. We can rearrange the quantum field operator to better connect to the more familiar self-energy term, ¦(3):…”

“…These modes are easily computed for most cavities and PC media using standard electromagnetic techniques, and one can also frequently compute the numerically-exact Green function. Another useful Green function is defined through [38] K(rYr H ; 3) = 3 2 f (r)[f (r H )] £ a(3 2 3 2 ), which, like G T , only depends on the transverse modes of the system. Note that we have exploited the relation,…”

“…Next, we introduce some of the basic quantum mechanics. To describe quantized light emission in an arbitrary inhomogeneous dielectric, we employ a canonical Hamiltonian of the form [38,39] (the time dependence is implicit):…”