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American Journal of Physics

Monitoring Operators in Magnetic Resonance and Light Modulation

Thomas R. Carver, Robert B. Partridge

Published Apr 1, 1966• 34 citations
Publisher

Abstract

A density matrix description of magnetic resonance is constructed, with detailed application to spin J = 1 (or greater) particles. Different types of “monitoring” or “measurement” operators employed in atomic beam, induction, and optical experiments are constructed, and used as a compact introductory basis for the description of resonance line shapes and other characteristics of these experiments. Of particular interest are the similarities and unique differences between the optical experiments known by the names “optical pumping,” “double resonance,” and “light beats” on the one hand, and magnetic induction and atomic beam experiments on the other.

DOI
10.1119/1.1972965

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Physical Review A•Aug 14, 2009•29 citations
A density matrix description of magnetic resonance is constructed, with detailed application to spin J = 1 (or greater) particles. Different types of “monitoring” or “measurement” operators employed in atomic beam, induction, and optical experiments are constructed, and used as a compact introductory basis for the description of resonance line shapes and other characteristics of these experiments. Of particular interest are the similarities and unique differences between the optical experiments known by the names “optical pumping,” “double resonance,” and “light beats” on the one hand, and magnetic induction and atomic beam experiments on the other.
By expanding the atomic density matrix in powers of the light intensity, it is possible to derive simple formulas for the light shifts and optical broadening of magnetic-resonance lines in optical pumping experiments. Thus, we have obtained for the first time approximate analytic formulas for the light shifts in Zeeman multiplets with spins greater than 1/2. The predicted light shifts for a high-spin atom are similar to those for a spin-1/2 atom, and shifts due to real and virtual transitions can be identified. However, in contrast to the situation with a spin-1/2 atom, the light shifts for a high-spin atom depend on the magnitude of the rf field, and the shifts for the longitudinal and transverse components of orientation and alignment are not necessarily the same.
Cs and Rb atomic vapors have been optically pumped in their ground states. The optical/rf-field configuration used allowed for the optical detection of magnetic resonance spectra with subkilohertz resolution by a pump–probe beam technique, in which the free-precessional motion of spins is recorded in the time domain. A six-level system serves as a model to describe the dependence of the observed spectra on the optical frequencies involved. With an intense laser beam applied during the free-precession period, Zeeman light-shift effects are observed. The spatial electric-field distribution of the laser beams and particle diffusion have to be taken into account to let us explain the observed magnetic resonance line shapes adequately.
Recent experiments (Dodd, Fox, Series & Taylor, I959; Colegrove, Franken, Lewis & Sands, i959) have shown that coherence effects can be observed between the Zeeman components of an optical line provided some coherence exists in the excited state. This coherence may be due to a radio-frequency field inducing transitions between the Zeeman sublevels of this excited state, or to the excitation process itself or to both. These effects are actually more or less a generalization of the well-known Hanle's experiments on the polarization of resonance radiation.: In the case of 'double resonance' experiments (Dodd et al. I959), they appear as a modulation of the intensity of the fluorescence radiation at one of the harmonics of the r.f. field; they can be interpreted as interference between the radiations of different frequencies emitted by the atoms which are submitted to the 'double resonance' ('light beat'). These coherence effects are related to those observed in the trapping of resonance radiation. In this paper, the methods used in former papers (Barrat I959 a, b and c, hereafter referred to as I, II, III) for the problem of light trapping are applied to the light beats experiments, using a density matrix formalism. The expression for the density matrix representing the states of the atoms just after the excitation will first be derived; then the intensity of the re-emitted light will be calculated. It will be shown finally how this method can be applied to practical problems. Notation unexplained in this paper is the same as in papers on coherence in light trapping (papers I, II and III).
Optical pumping experiments have been performed on helium 4 atoms in the metastable level 2 3 S 1 with the object of studying the modulation of light in absorption in the longitudinal and transverse beams. Magnetic resonance curves were obtained similar to those found in emission in double-resonance experiments. With a particular geometrical arrangement employing circular polarizers, the Bloch magnetic resonance curves were obtained. Modulation at four times the applied frequency was obtained with a linear polarizer in the pumping beam. This modulation is ascribed to the circulation of coherence in the optical pumping cycle. From the behaviour of the resonance signals as a function of the frequency and amplitude of the stimulating field, it is inferred that a substantial part of the signal arises from coherence which has survived two complete cycles. The observations are satisfactorily explained by the theory developed in the previous paper.
The sensitivity of measurements in the different schemes of optical detection of magnetic resonance in an ensemble of optically orientated paramagnetic atoms has been studied. The authors have considered the standard schemes of transmission monitoring of the atomic angular momenta precession assuming a classical resonant and off-resonant probing light beam as well as interference probing scheme by squeezed radiation. The main attention has been given to the limit accuracy of Zeeman transition frequency determination in these experiments. The circumstances limiting the radiation intensity increase in these experiments have been discussed.
A theoretical study on double-resonance optical pumping for the transitions $5{S}_{1/2}\text{\ensuremath{-}}5{P}_{3/2}\text{\ensuremath{-}}4{D}_{3/2,5/2}$ of 87Rb^{87}\text{R}\text{b}87Rb atoms is presented. The full-density-matrix equations in the presence the probe and the coupling laser beams for all relevant magnetic sublevels of the hyperfine levels were solved numerically, and the absorption of a probe beam was calculated as a function of the detuning of the coupling beam. The calculated results were compared with experimental data, and they showed an excellent agreement. The results were also compared with the approximate results calculated when the coherence between the ground and the highest excited levels was neglected. We found that the dominant operating mechanism for double-resonance signals was the optical pumping rather than the two-photon coherence, although the latter affected the signals significantly.