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Physical Review

Polarization of the Resonance Radiation from Degenerate Systems

Frank C. Hoyt

Published Sep 1, 1928• 1 citation
Publisher

Abstract

The problem of resonance between light and an atom is treated by solving the wave equation through the method of variation of constants. It is found that an initial degeneracy is removed by the action of the light, and that there occur fluctuations between the two states involved. Calculation of the electric moment of the perturbed atom on the Schr\"odinger density hypothesis shows that when the incident light is linearly polarized the radiation has the same degree of polarization as though the atom were in a weak magnetic field parallel to the electric vector of the light. In the case of circular polarization this fictious field is perpendicular to the plane of polarization. These results confirm the Heisenberg rule deduced from the correspondence principle.

DOI
10.1103/physrev.32.377

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The problem of resonance between light and an atom is treated by solving the wave equation through the method of variation of constants. It is found that an initial degeneracy is removed by the action of the light, and that there occur fluctuations between the two states involved. Calculation of the electric moment of the perturbed atom on the Schr"odinger density hypothesis shows that when the incident light is linearly polarized the radiation has the same degree of polarization as though the atom were in a weak magnetic field parallel to the electric vector of the light. In the case of circular polarization this fictious field is perpendicular to the plane of polarization. These results confirm the Heisenberg rule deduced from the correspondence principle.
We study the behaviour of an atomic wave packet in a circularly polarized light, and especially give the calculation of the radiative force exerted by the circularly polarized light on the atomic wave packet under the resonance condition. A general method of the calculation is presented and the result is interesting. For example, under the condition that the wave packet is very narrow or/and the interaction is very strong, no matter whether the atom is initially in its ground state or excited state, as time approaches to infinity, the resonance-radiation force exerted by the light on the atom approaches to zero. If the atom is initially in its ground state and excited state with the probability 1/2 respectively, and if the momentum density is a even function, then the resonance-radiation force exerted by the light on the atom is equal to zero.
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
Bohr’s correspondence principle is that associated with each observable in classical mechanics is an observable in quantum mechanics that follows certain quantization axioms which can be used to explain the energy spectra of atoms and quantum oscillators. In Schrodinger’s wave mechanics, the state of the system at any time is described by a density operator in a Hilbert space. The chapter describes how the Pauli equation can be used to calculate the average electric dipole moment and average magnetic dipole moment of an atom in an external electromagnetic field. This calculation enables to calculate the electric polarization field and the magnetization field of a material.
An equation has been developed for the polarization, and the angle of maximum polarization of resonance radiation as observed along a magnetic field perpendicular to the electric vector and the direction of propagation of the incident beam. Explicit application has been made to the polarization and the angle for the line $^{1}S_{0}\ensuremath{-}^{1}P_{1}$ for various values of the ratio of the hyperfine separation constant to the natural breadth of the levels, for a nuclear moment of \textonehalf{}. It was noted here that the form of the curves might differ widely from that to be expected from classical considerations. The polarization, and the angle of maximum polarization for the sodium D lines, for several values of the same ratio, were computed and compared with experimental observations. Thus a range of possible values of AAA was determined, which is not in disagreement with previous figures, and a value for $\ensuremath{\tau}$ established which agrees well with that calculated from optical dispersion data.
The optical-pumping theory of the passage of polarised radiation through a resonant medium was developed in the irreducible tensorial formalism. Transitions with arbitrary momenta J 1.2 are considered in the cubic approximation. For transitions with small values of momenta (J 1 =1 to J 2 =0; J 1 = 1 / 2 to J 2 = 1 / 2 ) the vector form of the equation system in density-matrix notation allows the problem to be solved exactly for light intensity. In the general case of an elliptically polarised wave, the equation systems for the interaction between circular components of polarisation of light in a resonant medium are obtained. Numerical calculations for deformation and rotation of the polarisation ellipse in a medium subject to saturation effects are also carried out. Irrespective of the initial state the polarisation tends either to a linear or to a circular form as it propagates, depending on times of collisional relaxation of orientation and alignment. The polarisation of linearly and circularly polarised waves does not change as the waves propagate, but the values of the saturation power for these polarisations differ considerably.
We study the stimulated emission of a two-level atom in an electromagnetic wave of circular polarization. The correlation function G(r1t,r2t)=(E^-E^+) of atom radiation fields at dipole approximation are computed. Under the resonance condition, the atom stimulated emission is imfluenced by the circularly polarized electromagnetic wave discussed. We have found that the time-averaged value of energy density does not depend on the initial conditions. We have also deduced the relation between the emission power of an atom and the Rabi frequency Ω.