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arXiv:1706.06772 · Journal of Physics B Atomic Molecular and Optical Physics

Cooperative scattering of scalar waves by optimized configurations of point scatterers

Frank Schäfer, Felix Eckert, Thomas Wellens

Published Oct 12, 2017• 3 citations
arXivHTMLPDF

Abstract

Abstract We investigate multiple scattering of scalar waves by an ensemble of N resonant point scatterers in three dimensions. For up to N = 21 scatterers, we numerically optimize the positions of the individual scatterers, to maximize the total scattering cross section for an incoming plane wave, on the one hand, and to minimize the decay rate associated to a long-lived scattering resonance, on the other. In both cases, the optimum is achieved by configurations where all scatterers are placed on a line parallel to the direction of the incoming plane wave. The associated maximal scattering cross section increases quadratically with the number of scatterers for large N , whereas the minimal decay rate—which is realized by configurations that are not the same as those that maximize the scattering cross section—decreases exponentially as a function of N . Finally, we also analyze the stability of our optimized configurations with respect to small random displacements of the scatterers. These results demonstrate that optimized configurations of scatterers bear a considerable potential for applications such as quantum memories or mirrors consisting of only a few atoms.

DOI
10.1088/1361-6455/aa92d8

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Abstract We investigate multiple scattering of scalar waves by an ensemble of N resonant point scatterers in three dimensions. For up to N = 21 scatterers, we numerically optimize the positions of the individual scatterers, to maximize the total scattering cross section for an incoming plane wave, on the one hand, and to minimize the decay rate associated to a long-lived scattering resonance, on the other. In both cases, the optimum is achieved by configurations where all scatterers are placed on a line parallel to the direction of the incoming plane wave. The associated maximal scattering cross section increases quadratically with the number of scatterers for large N , whereas the minimal decay rate—which is realized by configurations that are not the same as those that maximize the scattering cross section—decreases exponentially as a function of N . Finally, we also analyze the stability of our optimized configurations with respect to small random displacements of the scatterers. These results demonstrate that optimized configurations of scatterers bear a considerable potential for applications such as quantum memories or mirrors consisting of only a few atoms.
We study the influence of cooperative effects such as superradiance and subradiance on the scattering properties of dilute atomic gases. We show that cooperative effects lead to an effective potential between two atoms that decays as 1∕r1∕r1∕r. In the case of superradiance, this potential is attractive for close enough atoms and can be interpreted as a coherent mesoscopic effect. We consider a model of multiple scattering of a photon among superradiant pairs and calculate the elastic mean free path and the group velocity. We study first the case of a scalar wave which allows us to obtain and to understand basic features of cooperative effects and multiple scattering. We then turn to the general problem of a vector wave. In both cases, we obtain qualitatively similar results and derive, for the case of a scalar wave, analytic expressions for the elastic mean free path and for the group velocity for an arbitrary detuning (near resonance).
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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.
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Subwavelength atomic arrays are a leading platform for engineering light-matter interactions, enabling near-perfect single-photon mirrors and robust quantum memories based on long-lived dark spin waves. However, a comprehensive theory of their nonlinear, multi-excitation dynamics has remained a significant challenge. We present a unified quantum scattering theory that treats both photons and collective atomic spin waves as distinct propagating excitations interacting across different spatial dimensions. Our central result is a powerful analytical reduction: we demonstrate that the complete multi-channel S-matrix and the associated scattering cross sections are exactly determined by the effective scattering dynamics solely within the atomic spin wave subspace. This maps the complex physical problem of photon-atom interactions to a conceptually simpler one involving only atomic modes. We apply this formalism to the two-excitation case, deriving the complete analytical S-matrix and scattering cross sections for systems with two-level nonlinearities. Our work provides a versatile analytic tool for analyzing and engineering complex quantum nonlinear phenomena, including multi-excitation subradiance, in large-scale atomic systems.