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Physical Review A

Classical description of angular-momentum motion due to optical pumping

K. A. Nasyrov

Published Mar 7, 2001• 8 citations
Publisher

Abstract

The Wigner representation of angular-momentum orientation in the classical limit on $J\ensuremath{\gg}1$ has been used to derive an equation of angular-momentum motion in the case of a closed two-level system interacting with an arbitrary polarized radiation. This equation enables us to carry out a theoretical analysis of angular-momentum motion and consider final states of the quantum system for all types of dipole transitions. We have found that in the case of a zero magnetic field in the final state of the system with $\stackrel{\ensuremath{\rightarrow}}{J}J$ or $\stackrel{\ensuremath{\rightarrow}}{J}J+1$ optical transition, the angular momentum is directed along or opposite to the wave vector of the radiation depending on the sign of the polarization ellipsity, while for a linear polarization the angular momentum is isotropically distributed on the plane orthogonal to the polarization vector. For the $\stackrel{\ensuremath{\rightarrow}}{J}J\ensuremath{-}1$ transition two directions of angular momentum are possible in the final state. These directions are determined by the ellipsity of polarization. The spatial size of the final distribution of the angular momentum has been defined by the quantum uncertainty, $1/\sqrt{J},$ of the angular-momentum orientation. In the case of the $\stackrel{\ensuremath{\rightarrow}}{J}J\ensuremath{-}1$ or $\stackrel{\ensuremath{\rightarrow}}{J}J,$ transition particles come into ``dark'' states while for the $\stackrel{\ensuremath{\rightarrow}}{J}J+1$ transition they occupy the ``brightest'' state. In the presence of a nonzero magnetic field, particles with $\stackrel{\ensuremath{\rightarrow}}{J}J$ transition have in the final state only one angular-momentum orientation, i.e., along or opposite to the direction of the magnetic field depending on the polarization ellipsity sign. For the $\stackrel{\ensuremath{\rightarrow}}{J}J+1$ transition both directions are possible, i.e., along and opposite to the magnetic field. In the case of the $\stackrel{\ensuremath{\rightarrow}}{J}J\ensuremath{-}1$ transition the directions of the angular momentum form a cone around the magnetic field.

DOI
10.1103/physreva.63.043406

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The dynamics of a two-level atom with optical transition J g → J e =J g + 1 under the action of elliptically polarized resonance radiation in the presence of a constant field directed along the radiation wave vector is studied in the approximation of a semiclassical description of the angular momentum orientation. It is shown that the atomic distribution over the angular momentum orientation exhibits a two-scale time evolution. At the first stage, after the beginning of irradiation, the angular momenta of atoms get oriented (over comparatively short time intervals) along the magnetic field, as well as in the opposite direction, depending on their initial orientation and the ellipticity of radiation. At the second (longer) stage, the redistribution of atoms takes place, as a result of which they are oriented predominantly in one of the above directions. The duration of the second stage is an exponential function of angular momentum J.
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We develop a theory that describes the interaction of radiation and multilevel particles on the basis of the classical description of the orientation of angular momentum. For examples of application of this theory we take the problems of stimulated Raman scattering and of parametric generation of light in four-wave mixing. We find that the efficiency of the radiation processes largely depends on the polarization of the pump waves and on the types of optical transitions (Δ J ) and to a much lesser extent on the value of the angular momentum J . We also explain some polarization phenomena observed in experiments on four-wave mixing.
In quantum mechanics, spatial wavefunctions describe distributions of a particle's position or momentum, but not of angular momentum jjj. In contrast, here we show that a spatial wavefunction, jm(ϕ,θ,χ)= eimϕδ(θ−θm) ei(j+1/2)χj_m (ϕ,θ,χ)=~e^{i m ϕ} δ(θ- θ_m) ~e^{i(j+1/2)χ}jm​(ϕ,θ,χ)= eimϕδ(θ−θm​) ei(j+1/2)χ, which treats jjj in the ∣jm>|jm>∣jm> state as a three-dimensional entity, is an asymptotic eigenfunction of angular-momentum operators; ϕϕϕ, θθθ, χχχ are the Euler angles, and cosθm=(m/∣j∣)cos θ_m=(m/|j|)cosθm​=(m/∣j∣) is the Vector-Model polar angle. The jm(ϕ,θ,χ)j_m (ϕ,θ,χ)jm​(ϕ,θ,χ) gives a computationally simple description of particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in jjj and mmm) which predicts the effective wavepacket angular uncertainty relations for ΔmΔϕΔm ΔϕΔmΔϕ, ΔjΔχΔj ΔχΔjΔχ, and ΔϕΔθΔϕΔθΔϕΔθ, and the position of the particle-wavepacket angular motion on the orbital plane. The particle-wavepacket rotation can be experimentally probed through continuous and non-destructive jjj-rotation measurements. We also use the jm(ϕ,θ,χ)j_m (ϕ,θ,χ)jm​(ϕ,θ,χ) to determine well-known asymptotic expressions for Clebsch-Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles, g=2g=2g=2, and the m-state-correlation matrix elements, $$. Interestingly, for low j, even down to j=1/2j=1/2j=1/2, these expressions are either exact (the last two) or excellent approximations (the first two), showing that jm(ϕ,θ,χ)j_m (ϕ,θ,χ)jm​(ϕ,θ,χ) gives a useful spatial description of quantum-mechanical angular momentum, and provides a smooth connection with classical angular momentum.