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Physical Review A

Conditions of availability of the Markoff approximation for the description of cooperative spontaneous emission

Elisabeth Ressayre, Andree A. Tallet

Published Oct 1, 1975• 2 citations
Publisher

Abstract

We show that the conditions of validity of the Markoff approximation for the description of cooperative spontaneous emission depend on the Fresnel number associated with the radiating system. Let $\frac{S}{\ensuremath{\lambda}L}$ be the Fresnel number, where $S$ and $L$ are, respectively, the section and the length of the radiating system, and $\ensuremath{\lambda}$ is the radiation wavelength. In the case of large Fresnel numbers, we find that the characteristic time ${\ensuremath{\tau}}_{R}$ of the cooperative emission of radiation must be much greater than $\frac{L}{c}$; but for small Fresnel numbers, we show that ${\ensuremath{\tau}}_{R}$ must be much greater than $\frac{S}{\ensuremath{\lambda}c}$.

DOI
10.1103/physreva.12.1725

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We show that the conditions of validity of the Markoff approximation for the description of cooperative spontaneous emission depend on the Fresnel number associated with the radiating system. Let S\ensuremathλL\frac{S}{\ensuremath{\lambda}L}\ensuremathλLS​ be the Fresnel number, where SSS and LLL are, respectively, the section and the length of the radiating system, and \ensuremathλ\ensuremath{\lambda}\ensuremathλ is the radiation wavelength. In the case of large Fresnel numbers, we find that the characteristic time \ensuremathτR{\ensuremath{\tau}}_{R}\ensuremathτR​ of the cooperative emission of radiation must be much greater than Lc\frac{L}{c}cL​; but for small Fresnel numbers, we show that \ensuremathτR{\ensuremath{\tau}}_{R}\ensuremathτR​ must be much greater than S\ensuremathλc\frac{S}{\ensuremath{\lambda}c}\ensuremathλcS​.
Basic properties for the emission of radiation are derived from the diagonalization of the matrix (${\ensuremath{\Gamma}}_{\mathrm{ij}}$), where an element ${\ensuremath{\Gamma}}_{\mathrm{ij}}$ gives the coupling between two atoms iii and jjj via the electromagnetic field. This method gives naturally an estimate of the solid angles in which the radiation can take place. By using these estimates, we discuss whether the emission process can be described by a one- (two-) mode model. As expected, this model fails in the two limiting cases of large and small Fresnel numbers. Also for Fresnel numbers of the order of a of few units, it is not obvious whether such a model is valid.
In order to study the cooperative emission of radiation from an extended system of NNN two-level atoms, we use the master-equation approach within the Born and Markov approximations. All the properties of the emission of radiation can be deduced from the diagonalization of the coupling matrix $\ensuremath{\Gamma}$, where an element ${\ensuremath{\Gamma}}_{\mathrm{jk}}$ gives the coupling between the atoms labeled jjj and kkk via the electromagnetic field. We choose a pencil-shaped volume and study analytically the coupling matrix in the limit for small and large Fresnel numbers. For Fresnel numbers of order unity numerical methods are used. Thus we prove that, for the pencil-shaped geometry, any elaborated model must take into account the interaction of the atoms with the infinity of electromagnetic field modes lying inside two small solid angles about the axis of the largest linear dimension, which we label the zzz axis. In other words, the intensity of radiation emitted by any atom of the system depends on its position with respect to the others and this "propagation effect" cannot be neglected when interpreting superradiance experiments. For small Fresnel numbers the zzz dependence predominates, in the opposite case the (x,yx, yx,y) dependence does, while for Fresnel numbers of order unity the dependence of all three components must be taken into account. Preliminary calculations are given for which only the zzz dependence is considered; for Fresnel numbers of magnitude of order unity or smaller, the effect of propagation along the zzz axis displays large delays for the appearance of superradiance with respect to the time at which the superradiant pulse is maximum when the "propagation effects" are neglected (one-mode model).
The cooperative spontaneous emission by a fully inverted array of N atoms, interacting with and via the radiation field is studied. The arrays to which this treatment is applicable are symmetric under the exchange of any pair of atoms, including the dipole-dipole interaction in the energy and are confined to a volume with linear dimensions that are small relative to the radiation wavelength. Analytic expressions are obtained for the populations of the levels and the intensity of emission as functions of time, and for the spectrum of the radiation emitted in the cascade to the ground state.
Coherent spontaneous emission from a system of NNN two-level atoms interacting with a quantized radiation field is treated for the case in which Dicke's "cooperation number" rrr is macroscopically large. It is shown how to modify the quasiclassical approach to this problem to incorporate quantum effects that are lost in the self-consistent-field approximation. The statistics of the emitted radiation are found to vary markedly with the initial state of the system of atoms. The photon statistics tend to that typical of blackbody radiation when the initial state of the atomic system is that which would result from incoherent pumping ($m\ensuremath{\sim}r$). When, on the other hand, the atoms are initially in a superradiant state ($m\ensuremath{\ll}r$), the emitted radiation may be represented approximately by a coherent state.
An analysis is made of the deexcitation kinetics of strongly excited extended systems of initially uncorrelated two-level radiators. The possibility of partially cooperative states of the system is predicted when the radiation intensity acquires values intermediate between the spontaneous decay intensity and the Dicke superradiance intensity. The conditions for the onset of cooperative processes are obtained. The threshold for the oscillatory emission conditions is determined and the oscillation frequency is found. Modulation of the emission intensity is predicted with different modulation frequencies before and after the superradiant pulse. The modulation frequencies are determined. Formulas are derived for the superradiant pulse delay times and intensities. The possibility of weakly anisotropic cooperative emission is predicted.