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Proceedings of the Physical Society

Theory of the modulation of light in optical pumping experiments

George William Series

Published Aug 1, 1966• 17 citations
Publisher

Abstract

The absorption of light by optically pumped atoms undergoing magnetic resonance is calculated from expressions based on a simplification of Barrat and Cohen-Tannoudji's theory of optical pumping. The density matrix for the system is first derived in orders of approximation which correspond to successive cycles of pumping. The off-diagonal elements of the matrix, and consequently the absorption of light, are modulated. The amplitudes of modulation are resonance functions which are, in general, more complicated than solutions of the Bloch equations. The selection rules for electric dipole radiation limit the possible frequencies of modulation to ω0 and 2ω0 for primary interactions (ω0 is the applied frequency), but modulation at higher frequencies can arise through the circulation of coherence in optical pumping cycles. The functions which represent the amplitude of modulation at 4ω0 are derived in first and second order, for comparison with experiment.

DOI
10.1088/0370-1328/88/4/318

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The modulated absorption of light in an optical pumping experiment on4He

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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.

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We derive quantum-mechanical kinetic equations for the matter density $\ensuremath{\rho}$ and radiation density matrix $R$, which describe optical pumping phenomena. The resultant kinetic equations are a set of coupled nonlinear equations for $\ensuremath{\rho}$ and $R$. With appropriate linearizations, we can obtain the present theories of optical pumping. The nonlinear equations describe multiple scattering and line narrowing due to imprisonment of resonant radiation. We show that the coupled equations for $\ensuremath{\rho}$ and $R$ are equivalent to coupled equations for $\ensuremath{\rho}$ and a generalized polarization matrix II, whose matrix elements are the second moments of $R$. The polarization matrix II constitutes a complete description of linear phenomena in the same manner as present theories describe optical pumping phenomena by using the matter density matrix $\ensuremath{\rho}$ alone. As a consequence of our nonphenomenological treatment of radiation, we can provide a completely microscopic treatment of the externally modulated light-beam experiment of Bell and Bloom. We show that the atom absorbs the modulation envelope directly from the external thermal light beam in the same way that the atom absorbs transverse polarization directly from the light beam in optical pumping experiments.

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ABS>A detailed physical interpretation of the results of the Barrat- Cohen-Tannoudji theory of the optical pumping cycle is presented. The experimental verification of the essentially new effects that the calculation of the pumping cycle predicts is described. The odd isotopes of mercury were chosen for the experimental verification. The two most important effects predicted by the theory and verified by the experimental work are the radiative effects associated with the absorption process and the partial conservation of coherence'' during the optical pumping cycle. (J.S.R.)

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This is a brief account of recent results obtained in Paris by the development of optical-pumping techniques, and especially on the thesis of Cagnac and the thesis of Cohen-Tannoudji. Cagnac has used the optical-pumping technique to achieve nuclear orientation in the ground state of the odd mercury isotopes and has made an extensive study of the longitudinal relaxation time. Barrat and Cohen-Tannoudji have developed the quantum-mechanical theory of the optical-pumping cycle of an atom and Cohen-Tannoudji in his thesis has confirmed the theoretical predictions experimentally by applying Dehmelt’s cross-beam technique to Hg199: If a collection of atoms is irradiated permanently by light which can be absorbed and re-emitted, this pumping cycle has different effects on the ground-state properties of the atoms; these effects are experimentally detectable by observing the magnetic resonance of the atomic ground state. A broadening of the magnetic-resonance line proportional to the light intensity occurs. It is due to the shortening of the lifetime of the ground-state Zeeman levels by light absorptions. Other effects are displacements of the Zeeman levels of the ground state caused by irradiation resulting in a change of the magnetic-resonance frequency. Theory predicts displacements of 2 different kinds: (1) Displacements caused by real transitions—During an up-and-down transition of an atom, coherence is partly conserved. As a result, a certain amount of the g factor of the excited state is mixed with the g factor of the ground state. (2) Displacements caused by virtual transitions—These displacements are related to the dispersion of light. All these effects, predicted by the theory, have been qualitatively and quantitatively confirmed by Cohen’s experiments.

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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).

•Jan 1, 2016•0 citations
The absorption of light by optically pumped atoms undergoing magnetic resonance is calculated from expressions based on a simplification of Barrat and Cohen-Tannoudji's theory of optical pumping. The density matrix for the system is first derived in orders of approximation which correspond to successive cycles of pumping. The off-diagonal elements of the matrix, and consequently the absorption of light, are modulated. The amplitudes of modulation are resonance functions which are, in general, more complicated than solutions of the Bloch equations. The selection rules for electric dipole radiation limit the possible frequencies of modulation to ω0 and 2ω0 for primary interactions (ω0 is the applied frequency), but modulation at higher frequencies can arise through the circulation of coherence in optical pumping cycles. The functions which represent the amplitude of modulation at 4ω0 are derived in first and second order, for comparison with experiment.
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.
We derive quantum-mechanical kinetic equations for the matter density $\ensuremath{\rho}$ and radiation density matrix RRR, which describe optical pumping phenomena. The resultant kinetic equations are a set of coupled nonlinear equations for $\ensuremath{\rho}$ and RRR. With appropriate linearizations, we can obtain the present theories of optical pumping. The nonlinear equations describe multiple scattering and line narrowing due to imprisonment of resonant radiation. We show that the coupled equations for $\ensuremath{\rho}$ and RRR are equivalent to coupled equations for $\ensuremath{\rho}$ and a generalized polarization matrix II, whose matrix elements are the second moments of RRR. The polarization matrix II constitutes a complete description of linear phenomena in the same manner as present theories describe optical pumping phenomena by using the matter density matrix $\ensuremath{\rho}$ alone. As a consequence of our nonphenomenological treatment of radiation, we can provide a completely microscopic treatment of the externally modulated light-beam experiment of Bell and Bloom. We show that the atom absorbs the modulation envelope directly from the external thermal light beam in the same way that the atom absorbs transverse polarization directly from the light beam in optical pumping experiments.
ABS>A detailed physical interpretation of the results of the Barrat- Cohen-Tannoudji theory of the optical pumping cycle is presented. The experimental verification of the essentially new effects that the calculation of the pumping cycle predicts is described. The odd isotopes of mercury were chosen for the experimental verification. The two most important effects predicted by the theory and verified by the experimental work are the radiative effects associated with the absorption process and the partial conservation of coherence'' during the optical pumping cycle. (J.S.R.)
This is a brief account of recent results obtained in Paris by the development of optical-pumping techniques, and especially on the thesis of Cagnac and the thesis of Cohen-Tannoudji. Cagnac has used the optical-pumping technique to achieve nuclear orientation in the ground state of the odd mercury isotopes and has made an extensive study of the longitudinal relaxation time. Barrat and Cohen-Tannoudji have developed the quantum-mechanical theory of the optical-pumping cycle of an atom and Cohen-Tannoudji in his thesis has confirmed the theoretical predictions experimentally by applying Dehmelt’s cross-beam technique to Hg199: If a collection of atoms is irradiated permanently by light which can be absorbed and re-emitted, this pumping cycle has different effects on the ground-state properties of the atoms; these effects are experimentally detectable by observing the magnetic resonance of the atomic ground state. A broadening of the magnetic-resonance line proportional to the light intensity occurs. It is due to the shortening of the lifetime of the ground-state Zeeman levels by light absorptions. Other effects are displacements of the Zeeman levels of the ground state caused by irradiation resulting in a change of the magnetic-resonance frequency. Theory predicts displacements of 2 different kinds: (1) Displacements caused by real transitions—During an up-and-down transition of an atom, coherence is partly conserved. As a result, a certain amount of the g factor of the excited state is mixed with the g factor of the ground state. (2) Displacements caused by virtual transitions—These displacements are related to the dispersion of light. All these effects, predicted by the theory, have been qualitatively and quantitatively confirmed by Cohen’s experiments.
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.
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).