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arXiv:1306.3902

Scattering of an exponential pulse by a single atom

Markus Sondermann, Gerd Leuchs

Published Jun 17, 2013• 1 citation
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Abstract

We discuss the scattering of a light pulse by a single atom in free space using a purely semi-classical framework. The atom is treated as a linear elastic scatterer allowing to treat each spectral component of the incident pulse separately. For an increasing exponential pulse with a dipole radiation pattern incident from full solid angle the spectrum resulting from interference of incident and scattered components is a decreasing exponential pulse.

DOI
10.48550/arxiv.1306.3902

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We discuss the scattering of a light pulse by a single atom in free space using a purely semi-classical framework. The atom is treated as a linear elastic scatterer allowing to treat each spectral component of the incident pulse separately. For an increasing exponential pulse with a dipole radiation pattern incident from full solid angle the spectrum resulting from interference of incident and scattered components is a decreasing exponential pulse.
In this article we theoretically study the phase shift a single atom imprints onto a coherent state light beam in free space. The calculations are performed in a semiclassical framework. The key parameters governing the interaction and thus the measurable phase shift are the solid angle from which the light is focused onto the atom and the overlap of the incident radiation with the atomic dipole radiation pattern. The analysis includes saturation effects and discusses the associated Kerr-type non-linearity of a single atom.
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.
The absorbed/emitted radiation will be treated classically, as a monochromatic plane wave, whereas the atomic system that can absorb/emit such radiation will be described quantum-mechanically. However, no incident radiation beam is needed, and the atom “falls down” after a certain time interval to the ground state without any external reason. In the spontaneous emission processes the emitting atom can be thought of as a classical oscillating dipole, whose amplitude decreases due to the energy loss through the radiation. If an ensemble of identical atoms is considered and if, for simplicity, each atom emits radiation with the same spectral shape and full width at half maximum, it is impossible to ascribe a definite spectral component with intensity to a definite atom. In this case, such a broadening is identical for all atoms in the ensemble. On the microscopic scale, absorption of radiation is reduced to the transitions between pairs of energy levels of a discrete atomic spectrum.
In this work, scattering of an incident electric field from a moving atom is reexamined classically in two steps: the time-dependent current density created by the field inside the atom is first calculated under the electric-dipole approximation, and is then used to calculate the field scattered from the atom. Unlike the conventional frame-hopping method, the present method does not need to treat the Doppler effect as an effect separated from the scattering process, and it derives instead of simply uses the Doppler effect.
Variations in the spatial intensity distribution of light caused by coherent interaction with two-level atoms are determined by semi-classically calculating a term for interference between incident light and spherical radiation from the atom. The interference term is divided into two components: one oscillates at the Rabi frequency corresponding to absorption and stimulated emission, and the other contributes to dispersion. The spatial distribution of both components involves alternation of constructive and destructive parts in the form of parabolic surfaces where light energy flows. Although the time average over the Rabi oscillation of the former component vanishes at every position, that of the latter remains. The light-related reaction of a dipole force acting on an atom is considered to appear in the non-vanishing latter component.
The semi-classical method for calculating the observable properties of light scattered from an optically thin atomic vapour is placed in the context of the well known input-output relations for random signals transmitted through a linear system. The autocorrelation of the scattered field amplitude, the statistic from which the mean intensity spectral density and intensity fluctuations may be deduced, is expressed in terms of the autocorrelation of the incident optical field and the autocorrelation of the scattering system; these latter are quantities that can be readily measured or calculated. An appendix contains a classical and a semi-classical calculation of the scattering system autocorrelation function for a vapour of atoms optically excited from a J=0 level under the conditions of a level crossing experiment. In the course of this calculation the relationship between the matrix element of the electric dipole moment operator and the corresponding classical quantity is carefully examined.