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Physical Review A

Comparison of the cooperative emission profile from a spherical distribution of two-level atoms resulting from the choice of the interaction kernel

Jamal T. Manassah

Published Jan 4, 2012• 3 citations
Publisher

Abstract

The choice of the wrong kernel for describing the cooperative effects in an ensemble of identical atoms, in most instances, leads to wrong predictions for the system's dynamics whether the system initially was excited weakly or nearly was inverted. In this Brief Report, using previously derived theoretical expressions, I compare the numerical values for the emission amplitude as a function of time for the complex exponential and the sine kernels for different-size spheres. The deviations found in the respective results show that, in general, the two kernels are not, as often claimed, up to a frequency shift in their spectra, interchangeable.

DOI
10.1103/physreva.85.015801

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The choice of the wrong kernel for describing the cooperative effects in an ensemble of identical atoms, in most instances, leads to wrong predictions for the system's dynamics whether the system initially was excited weakly or nearly was inverted. In this Brief Report, using previously derived theoretical expressions, I compare the numerical values for the emission amplitude as a function of time for the complex exponential and the sine kernels for different-size spheres. The deviations found in the respective results show that, in general, the two kernels are not, as often claimed, up to a frequency shift in their spectra, interchangeable.
Using the formalism developed earlier, we treat spontaneous emission from a pair of identical two-level atoms A1{A}_{1}A1​, A2{A}_{2}A2​, whose separation r21{r}_{21}r21​ can be comparable to the wavelength $\ensuremath{\lambda}$. We obtain expressions for time-dependent intensities and damping rates with the initial conditions (a) both atoms inverted, (b) prior excitation by a short $\frac{1}{2}\ensuremath{\pi}$ pulse, and (c) only A1{A}_{1}A1​ inverted. The results in (a) are compared with those obtained for a model consisting of two initially excited harmonic oscillators O1{O}_{1}O1​, O2{O}_{2}O2​. The atoms exhibit superradiant behavior, whereas O1{O}_{1}O1​, O2{O}_{2}O2​ tend to trap radiation. In (a), the intensity pattern $\mathcal{R}(\ensuremath{\theta},t)$ develops lobes in different directions at different times, so that the spatial distribution of photons at time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ is the same as in the independent-atom case ${r}_{21}\ensuremath{\gg}\ensuremath{\lambda}$. For the oscillators, the lobes of $\mathcal{R}(\ensuremath{\theta},t)$ do not change direction, but only become more pronounced as time increases. In (b) and (c), the lobes oscillate back and forth at frequency $2{\ensuremath{\Omega}}_{12}$ corresponding to the shifts $\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$ of the triplet and singlet states due to the ${A}_{1}\ensuremath{-}{A}_{2}$ interaction. The intensity can therefore have a sinusoidal component. Field correlation functions calculated for (a) and (c) show that A2{A}_{2}A2​ and A2{A}_{2}A2​ radiate simultaneously around the frequencies $\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{\ensuremath{\Omega}}_{12}$, where $\ensuremath{\epsilon}$ is the single-atom resonant frequency. The spectrum is calculated for case (c), and shows the effects of coherent linewidth enhancement in addition to the frequency shifts.
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.
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.
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.