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Physical Review A

Cooperative decay rate and cooperative Lamb shift from a nanospherical configuration of two adjoining atomic species

Jamal T. Manassah

Published Feb 22, 2013• 0 citations
Publisher

Abstract

The eigenmodes of a nanosphere configuration where the core is an ensemble of two-level atoms with resonance frequency ${\ensuremath{\omega}}_{C}$, and the shell is an ensemble of two-level atoms but with shifted resonant frequency ${\ensuremath{\omega}}_{B}$ are obtained for different values of the ratio of the core radius to the shell outer radius as function of $({\ensuremath{\omega}}_{C}\ensuremath{-}{\ensuremath{\omega}}_{B})$. I show that the eigenmodes belong to one of two branches. The dominant decay mode for superradiant emission may belong to either branch, depending on the system's parameters. In instances of configurations where the cooperative decay rate values for the two branches become equal, the cooperative exponential amplification (decay) is accompanied by temporal oscillations.

DOI
10.1103/physreva.87.025805

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The eigenmodes of a nanosphere configuration where the core is an ensemble of two-level atoms with resonance frequency \ensuremathωC{\ensuremath{\omega}}_{C}\ensuremathωC​, and the shell is an ensemble of two-level atoms but with shifted resonant frequency \ensuremathωB{\ensuremath{\omega}}_{B}\ensuremathωB​ are obtained for different values of the ratio of the core radius to the shell outer radius as function of (\ensuremathωC\ensuremath−\ensuremathωB)({\ensuremath{\omega}}_{C}\ensuremath{-}{\ensuremath{\omega}}_{B})(\ensuremathωC​\ensuremath−\ensuremathωB​). I show that the eigenmodes belong to one of two branches. The dominant decay mode for superradiant emission may belong to either branch, depending on the system's parameters. In instances of configurations where the cooperative decay rate values for the two branches become equal, the cooperative exponential amplification (decay) is accompanied by temporal oscillations.
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.
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Using the eigenmodes of the scalar-photon kernel of interacting NNN two-level atoms in a cylindrical configuration, I obtain the time dependence of the superradiant emission for this geometry in two cases of interest. In particular, I consider first the case when one or two eigenmodes dominate the system's dynamics for an inverted system which has been initially prepared in an isotropic state and show that for certain values of the radius of the cylinder ($r=m\ensuremath{\lambda},m \mathrm{integer}$) one obtains temporal modulation in the state function. In the second case of interest, I show that the angular distribution of the state function of a system initially prepared in a phased state appears to be chaotic; however, I show that this behavior is due to the high multiplicity of modes having close values for their amplification coefficients.