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Proceedings of the Royal Society of London A Mathematical and Physical Sciences

Density matrix formalism applied to light beat experiments

J.P. Barrat

Published Sep 19, 1961• 44 citations
Publisher

Abstract

Abstract A density matrix formalism is developed to account for various coherence effects observed in double resonance experiments. It is shown in particular that the intensity of the resonance light can be modulated at frequencies pw|2r (p = 1, 2, 3, 4) where wl2r is the radio frequency. Explicit expressions are derived in the case of the 63P1 state of the even isotopes of mercury.

DOI
10.1098/rspa.1961.0166

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Density matrix formalism applied to light beat 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).

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Abstract This work is an experimental study of the modulation which has been found in resonance radiation when the fluorescing vapour is subjected to static and radio-frequency magnetic fields, as in the double resonance experiment of Brossel & Bitter. The particular example chosen for study in this work also was the resonance line λ 2537 Å of mercury. The experimental observations are compared with the predictions of a theoretical treatment which has already been published. The phenomenon was studied under a variety of geometrical configurations. When the polarization of the exciting light allowed excitation to only one excited state (a component of a Zeeman multiplet), the depth of modulation was found to depend on the closeness to resonance of the frequency, ω0, of the radio-frequency field to the Larmor precessional frequency, ω = γH , in the static field H. When the polarization of the exciting light allowed excitation to more than one component of the multiplet, resonance effects were found in the depth of modulation and in the mean intensity of the fluorescent light at fields where the applied frequency was equal, not to the Larmor frequency alone, but to combinations of the Larmor frequency with the nutational frequency. From a quantum-mechanical point of view, these new phenomena are related to the interference effects which are found in resonance fluorescence when there is degeneracy between excited states: in the present case the degeneracy is induced by the radio-frequency field. The ease with which the geometrical conditions could be altered allowed the theory to be tested in considerable detail without quantitative assessment of the resonance line contours: nevertheless, a quantitative study was made of one particular feature. The predictions of the theory were confirmed at all points where they were tested.

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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.

Proceedings of the Physical Society•Aug 1, 1966•17 citations
Abstract A density matrix formalism is developed to account for various coherence effects observed in double resonance experiments. It is shown in particular that the intensity of the resonance light can be modulated at frequencies pw|2r (p = 1, 2, 3, 4) where wl2r is the radio frequency. Explicit expressions are derived in the case of the 63P1 state of the even isotopes of mercury.
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).
Abstract A theory is formulated to describe the modulation which has been observed in fluorescent light from atoms subjected simultaneously to optical and radio-frequency radiation. The optical field stimulates one or more of a set of excited states of the atom, between which the radio-frequency field establishes coherence. This coherence is manifest in the fluorescent radiation. Interference between radiations of different frequency leads to modulation. General expressions are given for the intensity of the fluorescent light as a function of time. The Zeeman structure of the transition (63P1 — 6 1S0), A2537 A, in mercury is studied in detail. Modulation at frequencies, 1, 2, 3 and 4 times that of the radio-frequency field, w0, is predicted, and resonant effects at static magnetic fields, 0, 1/2, 1, 3/2, 2 and 3 times H0, the field for which w0 is the Larm or frequency. Resonances at fields other than H0 are due to excitation with light of mixed polarization. Most of the predicted effects have been found experimentally. A ‘frequency diagram’ is introduced and discussed, to represent the combined effects of static and radio-frequency magnetic fields. To each excited state belong a set of r frequencies, where r is the number of states linked by the radio-frequency perturbation. The 9 levels are drawn, as functions of H, for the states mJ = 0, ± 1, of 3P1. The resonances at fields other than H0 may be associated with intersections of frequency levels belonging to different mJ.
In radio frequency (RF) experiments on aligned atoms (e.g. optical double resonance) new phenomena can be found in the case of longitudinal excitation if a strong oscillating RF field is used: both the positions of the RF resonances (RF shift) and the widths (power broadening) depend on the direction of the light observed and on the state of its (linear) polarisation. This applies to the main (single-quantum) transitions as well as the (odd) multi-quanta transitions. These effects have been observed in an experiment on aligned mercury atoms in the metastable 6 3 P 2 state and have been confirmed by numerical calculations based on the semiclassical approach.
Abstract This work is an experimental study of the modulation which has been found in resonance radiation when the fluorescing vapour is subjected to static and radio-frequency magnetic fields, as in the double resonance experiment of Brossel & Bitter. The particular example chosen for study in this work also was the resonance line λ 2537 Å of mercury. The experimental observations are compared with the predictions of a theoretical treatment which has already been published. The phenomenon was studied under a variety of geometrical configurations. When the polarization of the exciting light allowed excitation to only one excited state (a component of a Zeeman multiplet), the depth of modulation was found to depend on the closeness to resonance of the frequency, ω0, of the radio-frequency field to the Larmor precessional frequency, ω = γH , in the static field H. When the polarization of the exciting light allowed excitation to more than one component of the multiplet, resonance effects were found in the depth of modulation and in the mean intensity of the fluorescent light at fields where the applied frequency was equal, not to the Larmor frequency alone, but to combinations of the Larmor frequency with the nutational frequency. From a quantum-mechanical point of view, these new phenomena are related to the interference effects which are found in resonance fluorescence when there is degeneracy between excited states: in the present case the degeneracy is induced by the radio-frequency field. The ease with which the geometrical conditions could be altered allowed the theory to be tested in considerable detail without quantitative assessment of the resonance line contours: nevertheless, a quantitative study was made of one particular feature. The predictions of the theory were confirmed at all points where they were tested.
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.