A general formalism is presented to describe resonance line polarization for a two-level atom in an optically thick, three-dimensional medium embedded in an arbitrary varying magnetic field and irradiated by an arbitrary radiation field. The magnetic field is supposed sufficiently small to induce a Zeeman splitting much smaller than the typical line width. By neglecting atomic polarization in the lower level and stimulated emission, an integral equation is derived for the multipole moments of the density matrix of the upper level. This equation shows how the multipole moments at any assigned point of the medium are coupled to the multipole moments relative at a different point as a consequence of the propagation of polarized radiation between the two points. The equation also accounts for the effect of the magnetic field, described by a kernel locally connecting multipole moments of the same rank, and for the role of inelastic and elastic (or depolarizing) collisions. After having given its formal derivation for the general case, the integral equation is particularized to the one-dimensional and two-dimensional cases. For the one-dimensional case of a plane parallel atmosphere, neglecting both the magnetic field and depolarizing collisions, the equation here derived reduces to a previous one given by Rees (1978).
In a previous paper, I have presented a general theory of the optical pumping of atoms by a polarized multimode-laser beam, in the presence of a magnetic field. Equations of motion, valid at arbitrary laser intensities for the density matrix of the atomic system, have been obtained. In this paper, these equations are exactly solved for a linearly $\ensuremath{\sigma}$-polarized laser and levels having small angular momenta. More precisely, I consider the laser transitions ${J}_{b}=1\ensuremath{\rightarrow}{J}_{a}=0$, ${J}_{b}=1\ensuremath{\rightarrow}{J}_{a}=1$, and ${J}_{b}=1\ensuremath{\rightarrow}{J}_{a}=2$. Two particular results are emphasized: (i) For $J=1\ensuremath{-}J=0$ and $J=1\ensuremath{-}J=1$ transitions and for the
J=2 level of a $J=1\ensuremath{-}J=2$ transition, the laser-induced Hanle effects keep a Lorentzian shape, independent of the laser intensity. For the $J=1\ensuremath{-}J=1$ transition, it is shown that the depolarizing relaxation processes play a leading part in the coupling induced by the laser between the transverse alignments of the two levels. On the other hand, the power broadenings of the Hanle effects are shown to be similar. For the
J=1 level of a $J=1\ensuremath{-}J=2$ transition, the Hanle-effect shape presents a departure from the Lorentzian one, owing to a coupling with the laser-induced hexadecapole moment in the
J=2 level. (ii) The populations of the levels exhibit a "zero-field saturation resonance" which is due to the coupling with the Zeeman coherences. These resonances, which reproduce the Hanle effects of the transition, can be observed on the $\ensuremath{\pi}$-polarized fluorescence from the levels. In general, these resonances appear as a saturation effect, i.e., they reduce the laser-induced fluorescence of the levels in zero magnetic field. However, for the ${J}_{b}=1\ensuremath{\rightarrow}{J}_{a}=2$ transition, the ${J}_{a}=2\ensuremath{\rightarrow}{J}_{f}=2 \ensuremath{\pi}$ fluorescence presents an anomalous behavior; the saturation resonance increases the zero-field laser-induced fluorescence. The theoretical study shows the prominent part played by the value of the
Jb=1 level lifetime in the interpretation of this anomaly. Other results, such as the laser-induced depolarization of the fluorescence, are also obtained. All the theoretical predictions have been experimentally checked and will be presented in forthcoming papers.
A suitable theoretical scheme able to describe, in a wide variety of astrophysical situations, the phenomenon of atomic line polarization by electron impact is developed. Starting from the general principles of quantum mechanics and assuming the Born approximation, the rate equations for the density matrix elements of a multilevel atomic system, interacting with a nonrelativistic electron beam having any kind of angular distribution, are derived in full generality. The resulting theory generalizes the previous ones by accounting for the collisional rates and the cross sections concerning both inelastic and superelastic collisions (in any geometrical situation), and, moreover, by taking into account the coherences among Zeeman sublevels split by a magnetic field. As an example of particular relevance, the general formulas derived in the first sections of the paper are subsequently particularized to the case of the electric dipole interaction.
The ultra narrow resonances, obtained by means of polarization spectroscopy of alkali atoms, originate from the destruction of the laser-induced coherence in the ground state. In the case of single-frequency, near resonance excitation and suitable geometry the well-known ground state Hanle effect is observed. Coherent population trapping in Hanle configuration is a nonlinear modification of this effect, observed in the fluorescence from the upper level. The ground state coherence is transmitted from the laser field and higher rank polarization moments are created. Using the irreducible tensor operator formal- ism programs for numerical modelling of this effect has been developed and presented here. The velocity distribution of the atoms, the Gaussian distribu- tion of the laser beam, stray magnetic field, and the experimental geometry are taken into account. The scheme of conversion of the octupole (� = 3) and hex- adecapole (� = 4) polarization moments (PM) through the laser and magnetic fields into the upper level quadropoles f 2 q is proposed. The conversion of the high-rank polarization moments (PM) into observable components is also dis- cussed. The results of the modelling are compared with the measured ones at different experimental conditions.
The history of the discovery, study, and explanation of the effect discovered fifty years ago by the German physicist Hanle and its development and generalization by the French school of physicists led by Kastler are described. The effect is observed optically in an increase in the degree of depolarization of the resonance fluorescence. It is studied in the range of magnetic fields in which the Zeeman splitting of the atomic magnetic sublevels does not exceed the level widths determined by thermal relaxation mechanisms and electromagnetic interaction. It is a direct optical manifestation of interference of degenerate atomic states and is intimately connected with the conservation law for the angular momentum of the photon-atom system. In the review, different applications of the effect to the measurement of mean lifetimes, linewidths, and Stark splitting constants of atoms in excited states are given. The resonance scattering of light by atoms in the ground state in a weak magnetic field is considered in detail—in particular, in the presence of radio-frequency magnetic fields oriented in different ways. The different possible types of parametric resonances in one or two radio-frequency fields for arbitrary orientation of the constant magnetic field are interpreted as the effect of crossing of the levels of an atom that is dressed by the radio-frequency field. Applications of the effect in the quantum magnetometry of ultraweak fields (down to 10-10 Oe) are examined.
For atomic media with resonant Doppler-broadened transitions of J=0 to J'=1 or J=1 to J'=0 type and taking into account the relaxation due to depolarizing collisions a theory of nonlinear magneto-optic polarization phenomena is developed. The theory is valid in the case of both arbitrary resonance linearly-polarized radiation intensity and arbitrary external magnetic field spatial orientation. The Faraday and Voigt geometries are analysed in detail, and in the case of the Faraday geometry the effect is considered of a noncompensated magnetic field transverse component on the shape of a dependence relating the radiation polarization plane rotation angle to a magnetic field longitudinal component. In the case of the Voigt geometry an essential effect is found of the angle between the radiation initial polarization and transverse magnetic field vectors on the character of the dependence relating the radiation polarization plane rotation angle to the transverse magnetic field magnitude.
Abstract A theoretical and experimental study has been made of the intensity of forward scattered light in conditions of radio-frequency double resonance and level-crossing. The main conclusions are (i) that the magnetic resonance and level-crossing effects which are familiar in the laterally scattered light are to be found also in the forward scattered light, but (ii) that the resonance curves are Doppler-broadened, and the strength of the oscillatory field necessary to generate them is correspondingly increased, and (iii) that coherence narrowing is much more pronounced in forward than in lateral scattering. The differences derive from the coherence between the radiation from different atoms which is present in forward, but not in lateral scattering, and which is implicitly recognized in the classical theory of the propagation of light in polarizable media. The equations of electromagnetism form the basis of the present analysis. In the presence of the oscillatory magnetic field the polarization tensor is complex and time-dependent. The time dependence leads to the result that the number of eigenwaves necessary to describe propagated waves in the medium is four, as against the usual two. An arbitrary wave is resolved into a superposition of the four eigenwaves, all of which have different propagation constants. The analysis is very general, and is applicable to experiments with lasers. The components of the polarization tensor are derived by finding first the polarizability of individual atoms from a semi-classical theory of their interaction with light and magnetic fields. The macroscopic polarization, obtained by summing over individual atoms, is then given by folding the individual polarizabilities with the Doppler distribution of resonance frequencies. In the particular case studied in detail (the inter-combination resonance line, 2537 A, in mercury), the Doppler width is much greater than the radiation width, and the imaginary (absorptive) part of the polarizability is Gaussian. Expressions are derived in a ‘weak scattering’ approximation for the intensity of the forward scattered light. They are magnetic resonance functions which are found to be the Doppler-broadened equivalents of the familiar functions, based on Lorentzians, characteristic of scattering by a single atom. In the case of double resonance the forward scattered light is strongly modulated, as is the laterally scattered light. To investigate coherence narrowing, the zero-field level-crossing has been studied with the restriction of ‘weak scattering’ removed. The result is obtained that the width of levelcrossing curves is inversely proportional to the vapour density for magnetic fields such that the Zeeman splitting is small in relation to the Doppler width. This analytical result has been verified experimentally.