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Journal of Physics B Atomic and Molecular Physics

On the classical model for resonance fluorescence

C. Gerald Carrington

Published Sep 1, 1971• 14 citations
Publisher

Abstract

The results of a quantum mechanical formulation of resonance fluorescence in the white light approximation are expressed in a cartesian form equivalent to the classical expressions used in a recent publication. Explicit formulae are derived which allow the cartesian equations to be applied to light, both plane and elliptically polarized and to atomic states with arbitrary angular momenta.

DOI
10.1088/0022-3700/4/9/011

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Journal of Physics B Atomic and Molecular Physics•Jan 1, 1970•10 citations
The results of a quantum mechanical formulation of resonance fluorescence in the white light approximation are expressed in a cartesian form equivalent to the classical expressions used in a recent publication. Explicit formulae are derived which allow the cartesian equations to be applied to light, both plane and elliptically polarized and to atomic states with arbitrary angular momenta.
The angular distribution and polarization of the resonance radiation are deduced for elliptically polar- ized light incident on an atomic system of arbitrary initial and intermediate total angular momenta. The incident and re-radiated light are characterized by Stokes's representation, which allows an immediate generalization to partially polarized incident radiation. The results follow closely in form those deduced for free electron (Rayleigh) scattering but give, in general, a rather smaller degree of anisotropy for the resonance radiation. The results also determine the complete correlation, in polarization and direction of propagation, of two successive quanta emitted by a radiating system
The polarization of the resonance fluorescence of interacting atoms and the transfer of coherence, due to the exchange of excitation, are studied theoretically. The treatment is for the general case where both the excited and the ground states can have arbitrary angular momenta, except that they can be connected to each other by an allowed transition. Electrostatic dipole–dipole interaction and electromagnetic radiative interaction are considered simultaneously as the perturbations on the system. Heitler and Ma's generalized time-dependent perturbation method is used. The system is quantized with respect to the space-fixed z axis and a static magnetic field along the z axis may be present, so that the magnetic sublevels (m, m′, etc., for the excited state J, and α, β, etc., for the ground state J′) are not necessarily degenerate. The intensity of the scattered light of a given polarization, in a given direction, is expressed as a function of kR (where ℏck is the energy of excitation and R is the internuclear distance), and of the orientation of R with respect to z axis. The matrix elements (ℏ / 2)γmβ,αm′ and Vmβ,αm′, which cause resonant transfer of excitation between two atoms through radiative and electrostatic interaction, respectively, are also expressed as a function of kR, and of the orientation of R. For randomly oriented atoms, the polarization after being averaged over the random orientation of R becomes a function of kR. Numerical calculations have been carried out for S01 → 1P1 transition. The coherence is shown to be conserved upon resonant transfer of excitation when R is parallel to the quantization axis.
The cooperative resonance fluorescence steady state is discussed within the context of an operator master equation which conserves total pseudospin. Emphasis throughout is on quantum fluctuations and their significance in relation to a background of factorised dynamics. Atom-atom correlations are shown to play a fundamental role for systems driven beyond the linear regime. Use of the atomic coherent state representation yields a Fokker-Planck description closely allied to the dynamics for a classical angular momentum oscillator. For intense incident fields the quantum-mechanical steady state is understood in terms of diffusion both around and between classical trajectories on the Bloch sphere. In the limit of infinite systems simple closed-form expressions for steady-state features are derived. Coherent and incoherent fluorescent intensities are obtained together with the second-order correlation function for fluorescent light. Specific features are illustrated by numerical results for systems of from two to fifty atoms.
A treatment of resonance fluorescence is given in which the properties of the incident and scattered light are expressed in terms of coherency matrices. Calculations are made for two successive scattering processes corresponding to the experiments described in an accompanying paper. For the first process the scattering atoms are subjected to a magnetic field consisting of parallel static and oscillating components. The fluorescent light is frequency-modulated and may also be polarization-modulated. This light excites fluorescence in a second cell in which the atoms are subjected to a static magnetic field. The equations show under what conditions the second fluorescence is intensity-modulated and describe the resonances in the amplitude of modulation which occur when the modulation frequency matches a Zeeman interval of the excited atoms.