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Journal of the Physical Society of Japan

Resonant Scattering of Intense Beam of Radiation by N-Atoms Contained in a Small Volume

Hari Prakash

Published Mar 1, 1977• 2 citations
Publisher

Abstract

Resonant scattering of am intense beam of radiation by an arbitrary number of atoms contained in a volume whose dimensions are small as compared to the radiation wavelength is studied. It is seen that the scattered radiation is an equally spaced triplet whose middle frequency is equal to the incident frequency. Counting rates exact to all orders in incident intensity are obtained. Special cases of (i) atoms in ground state, (ii) atoms in excited state, (iii) assembly of atoms in super-radiant state, and (iv) assembly of atoms in thermal equlibrium are studied.

DOI
10.1143/jpsj.42.947

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Resonant scattering of am intense beam of radiation by an arbitrary number of atoms contained in a volume whose dimensions are small as compared to the radiation wavelength is studied. It is seen that the scattered radiation is an equally spaced triplet whose middle frequency is equal to the incident frequency. Counting rates exact to all orders in incident intensity are obtained. Special cases of (i) atoms in ground state, (ii) atoms in excited state, (iii) assembly of atoms in super-radiant state, and (iv) assembly of atoms in thermal equlibrium are studied.
A simple theoretical model is formulated which describes dynamical features of the interaction of two-level atoms with a continuum of radiation oscillators. In particular, this paper is concerned with resonance scattering by single atoms in the presence of an external classical field, and spontaneous emission by a system of identical atoms. In the case of resonance scattering, the response of an atom to rapid changes of the amplitude of the external field is studied. The model for spontaneous emission by a system of atoms is based on the assumptions that all atoms have the same resonance frequency, and that the atoms are contained in a volume whose extension is small compared with the wavelength of the emitted radiation. The complete evolution of a system of mmm atoms is determined by a set of coupled first-order linear differential equations, and explicit solutions are presented for the values m=1,2,3,4,and8m=1,2,3,4, \mathrm{and} 8m=1,2,3,4,and8. Pair correlations of the atoms and photon-number fluctuations are discussed. An approximation for resonance fluorescence by a system of atoms with a broad distribution of resonance frequencies is included in an Appendix.
Interaction with a multimode quantized radiation field of an assembly of NNN two-level atoms contained in a small volume is considered for the special cases of (i) highly excited, (ii) weakly excited, and (iii) superradiant atomic assemblies, and solutions of equations of motion are obtained in closed form. For (i) and (ii) we show that, if the atomic assembly is in the Dicke state ∣r,m>|r,m>∣r,m>, the spontaneously emitted radiation is Lorentzian, and its spectral width and the numerical value of the Lamb shift are r+∣m∣r+|m|r+∣m∣ times that for a single excited atom. For (i) the sign of Lamb shift is opposite to that for (ii) and for a single excited atom. Change in the frequency shift gives chirping in the pulse radiation emitted spontaneously by an assembly of atoms. We obtain the reduced density operator for a single mode and for the complete spontaneous emitted radiation, and discuss its characteristic function, weight function in diagonal representation, coherence functions, and normalized variance. We find that the emitted radiation is chaotic if all atoms are excited or if the assembly is weakly excited and is in thermal equilibrium. For a superradiant assembly, the radiation is coherent; it is only amplitude coherent if the initial atomic density operator is nondiagonal. We also study stimulated emission and scattering for the cases (i) and (ii). For case (iii) these phenomena do not take place.
It is well known, that for large Rabi frequencies the spectrum of resonance fluorescence of a single two-level atom contains three separate lines (Mollow triplet). Question naturally arise as to what effect sidebands of this triplet might have on the spectrum of the scattered radiation in the multiatom case. The purpose of the present paper, therefore, is to obtain a space-frequency dependent spectrum of the scattered radiation in backward direction (reflection) by a system of N noninteracting atoms. We neglect the translational motion of the atoms and multiple scattering and concentrate on the effects of the collective scattering. It is shown that spectral energy of the scattered radiation consists of two different parts. One is the incoherent and isotropic emission of fluorescence. The energy of radiation in this case is the sum of the atomic energies and then is proportional to N . The other is the collective emission (interference effect). The energy of radiation in this case is spatially focused in a small solid angles about three propagation directions and then is proportional to N2. Thus the frequency Mollow triplet is transformed into the space-frequency dependent triplet.
The power spectrum of the radiation emitted by a driven collision-damped two-level system is evaluated. The driving field, which is treated classically, is assumed to oscillate harmonically near the atomic resonance frequency, and its intensity is allowed to assume arbitrary values. The collisions are assumed to be strong, i.e., to instantaneously thermalize the state of the atom. The limiting forms of the power spectrum of the radiated field are discussed for the cases of low and high excitation of the atomic system.
We study theoretically the scattering of light by an ensemble of N resonant atoms in a subwavelength volume. We consider the low intensity regime so that each atom responds linearly to the field. While N noninteracting atoms would scatter N^{2} more than a single atom, we find that N interacting atoms scatter less than a single atom near resonance. In addition, the scattered power presents strong fluctuations, either from one realization to another or when varying the excitation frequency. We analyze this counterintuitive behavior in terms of collective modes resulting from the light-induced dipole-dipole interactions. We find that for small samples and sufficiently large atom number, their properties are governed only by their volume.
The problem of resonance Rayleigh scattering by an atom in an intense laser beam, whose bare spectrum consists of one discrete fundamental state and an excited continuum of states, is studied with the use of the density-matrix formalism. Evolution equations for the density-matrix elements are solved, leading to the determination of the populations of the atom in the discrete state ∣g〉|g〉∣g〉 in the presence of nnn laser photons, and to the calculation of the dipole-moment correlation function. The importance of cascade effects is discussed. The emitted spectrum is calculated and its dependence on the intensity of the laser beam and on the frequency of the incident photons is studied.