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arXiv:2010.11845 · Optics Express

Quantum theory of Rayleigh scattering

A. P. Vinogradov, Vladislav Yu. Shishkov, I. V. Doronin, E. S. Andrianov, A. A. Pukhov, A. A. Lisyansky

Published Dec 29, 2020• 29 citations
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Abstract

We suggest a quantum description of Rayleigh light scattering on atoms. We show that an entangled state of the excited atom and the incident photon is formed during the scattering. Due to entanglement, a photon is never completely absorbed by the atom. The formation of the scattering spectrum is considered as a relaxation of incident photons to the reservoir of free space modes that are in thermal equilibrium. Additional excitations of the reservoir modes occurring during scattering are treated as scattered light. We show that even if the frequency of incident photons is incommensurate with an atomic transition frequency, the scattered light spectrum has a maximum at the frequency of incident photons. In addition, the linewidth of the scattered light is much smaller than that of the spontaneous emission of a single atom. Therefore, the process can be considered as elastic.

DOI
10.1364/oe.412852

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We suggest a quantum description of Rayleigh light scattering on atoms. We show that an entangled state of the excited atom and the incident photon is formed during the scattering. Due to entanglement, a photon is never completely absorbed by the atom. The formation of the scattering spectrum is considered as a relaxation of incident photons to the reservoir of free space modes that are in thermal equilibrium. Additional excitations of the reservoir modes occurring during scattering are treated as scattered light. We show that even if the frequency of incident photons is incommensurate with an atomic transition frequency, the scattered light spectrum has a maximum at the frequency of incident photons. In addition, the linewidth of the scattered light is much smaller than that of the spontaneous emission of a single atom. Therefore, the process can be considered as elastic.
Semi-classical calculation of an oscillating dipole induced in a two-level atom indicates that spherical radiation from the dipole under coherent interaction, i.e., Rayleigh scattering, has a power level comparable to that of spontaneous emission resulting from an incoherent process. Whereas spontaneous emission is nearly isotropic and has random polarization generally, Rayleigh scattering is strongly anisotropic and polarized in association with incident light. In the case where Rabi frequency is much larger than frequency detuning and spontaneous emission rate, while the power of spontaneous emission varies with time at the Rabi frequency, the power of Rayleigh scattering oscillates at twice the Rabi frequency. Moreover, the radiation pressure force acting on an atom due to Rayleigh scattering exceeds that caused by spontaneous emission when frequency detuning is large.
Since in quantum optics light is represented in terms of photons, light propagation through a linear medium is discussed quantum mechanically in this paper by following the multiple-scattering process of one incident photon from the medium. To treat the photon and the medium on the same quantum footing, the medium is assumed to be an ensemble of uniformly distributed identical two-level atoms. It is found that inside the medium the incident photon follows the same propagation rules as a plane wave does in the classical domain, and has a possibility to become entangled with the atoms. It is also found that when interacting with a two-level test atom outside the medium, the output photon appears to be formally in a single mode identical to that of the incident photon.
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.
We investigate the resonant scattering of photons from an atom in the thermal equilibrium with a phonon bath. A two-level model of the atom is discussed in detail. In the model problem, the atom is coupled directly to the lattice through a diagonal interaction which is bilinear in the lattice quanta. Relaxation effects are introduced indirectly via correlation functions. In the limit in which the relaxation time is much shorter than the radiative lifetime, the frequency spectrum of the scattered radiation consists of three parts: (1) a coherent, elastic term weighted by the factor ${tanh}^{2}(\frac{\ensuremath{\hbar}{\ensuremath{\omega}}_{0}}{2KT})$, (2) an incoherent quasielastic term weighted by the factor [$1\ensuremath{-}{tanh}^{2}(\frac{\ensuremath{\hbar}{\ensuremath{\omega}}_{0}}{2KT})$], (3) a fluorescent term characterizing emission at the natural frequency ${\ensuremath{\omega}}_{0}$, which arises from an effective two-photon-two-phonon interaction. The extension of the analysis to the problem of resonant scattering from paramagnetic ions in crystals is outlined with particular emphasis on the influence of neighboring ionic levels and the relative importance of the one- and two-phonon processes.
A fully quantum-mechanical treatment of resonant light scattering is presented. The incident field is assumed to be described by a coherent state, and is allowed to be intense enough to cause saturation. Complete solutions are obtained for the correlated atom-field pure state vector, including multiphoton contributions of arbitrary order. The frequency spectrum of the scattered field is evaluated and is found to agree exactly with the result previously obtained by means of the quantum fluctuation-regression theorem. A derivation of the fluctuation-regression theorem and of the optical Bloch equations is given which is fully quantum mechanical and which relies upon no assumption of statistical factorization of atom and field states. The accuracy of the result found for the scattered - field spectrum is thus shown to be limited only by the assumption of the smallness of the saturated linewidth compared to the (optical) atomic resonance frequency. The one-photon approximation is analyzed in some detail. The method of adding an imaginary term to the upper-atomic-state energy is clarified, and it is shown how the vacuum and one-photon amplitudes thereby obtained may be used, within a simple and plausible iteration scheme, to construct the complete multiphoton spectrum. A variety of commonly used injection schemes and methods of representing atomic relaxation are discussed, and comparisons are made with results found by other authors. The entire analysis is performed with the aid of a canonical transformation which replaces the applied field by a ccc number. It is thus proved quite rigorously and generally that the use of a ccc-number applied field is a fully quantum-mechanical procedure, provided only that radiation-reaction terms are retained.