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Physical Review A

Matrix continued-fraction solution for saturation effects in spin-½ radio-frequency spectroscopy

M. Allegrini, E. Arimondo, Arturo Bambini

Published Feb 1, 1977• 18 citations
Publisher

Abstract

We present a semiclassical solution for a Zeeman-split, optically pumped, spin-\textonehalf{} system interacting with a strong radio-frequency field. The method of solution presented here is based upon a matrix continued-fraction expansion, and can explain any resonance phenomena which occur for arbitrary geometrical configurations of the pumping beam or the rf field polarization axis. Theoretical predictions of line shapes for some cases are reported and discussed. The solution can also explain some deviations of the observed Bloch-Siegert shift at high rf power from the behavior predicted by a previous semiclassical theory, which can account for simple geometrical configurations only.

DOI
10.1103/physreva.15.718

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The semiclassical interaction of a two-level system with a multimode electromagnetic field is discussed in detail. The equations of motion for the macroscopic density matrix are transformed into a single integral equation for the population density, with a complex kernel which can take into account both running- and standing-wave configurations. The iterative solution to this integral equation, which corresponds to an expansion in terms of powers of the electromagnetic field amplitudes, turns out to be divergent when the field amplitudes are greater than some critical value. In the case in which the kernel is a periodic or quasiperiodic function of time, a general solution to the integral equation is found, and this solution turns out to be unique and convergent for every value of the electromagnetic (e.m.) field amplitude. This solution is expressed through a continued-fraction expansion, whose terms are matrices, which can be readily obtained from the kernel. This solution is a generalization of the continued-fraction expansion usually met in the strong-signal semiclassical theory of lasers. Some particular cases are then treated in detail; among them, the solution to the multimode operation in laser devices is given in which the excited level lifetime is long compared with the period of the beating term between two adjacent modes. Although the theory presented is directed towards the steady-state regime in multimode lasers, it may also treat the transient regime of laser operations, or it may be extended to include phenomena where a strong e.m. field, made up of several modes equally spaced in frequency, interacts with a two-level system.

Physical Review A•Oct 1, 1976•15 citations
We present a semiclassical solution for a Zeeman-split, optically pumped, spin-\textonehalf{} system interacting with a strong radio-frequency field. The method of solution presented here is based upon a matrix continued-fraction expansion, and can explain any resonance phenomena which occur for arbitrary geometrical configurations of the pumping beam or the rf field polarization axis. Theoretical predictions of line shapes for some cases are reported and discussed. The solution can also explain some deviations of the observed Bloch-Siegert shift at high rf power from the behavior predicted by a previous semiclassical theory, which can account for simple geometrical configurations only.
For pt.III, see abstr. A46187 of 1973. The authors solve the problem of a doublet excited state under the influence of an arbitrarily strong RF field and optically pumped by a transverse beam. The solution is obtained as a continued fraction, which is easily evaluated on a computer. Haroche's resonances at even multiples of the radio frequency are investigated and their Bloch-Siegert shifts are determined. The lines are less power broadened than the multi-photon resonances earlier investigated, and they are hence more exactly measurable. In the perturbation limit the authors regain results derived by Haroche and Cohen-Tannoudji from a quantum theory. Detection at either the DC component or any of the oscillating components is discussed. For the Hanle effect the authors find a generalization of the Cohen-Tannoudji dressed g-factor. The experimental results for a rotating RF field are discussed, and experiments using a linearly oscillating field performed by Tsukada, Yabuzaki and Ogawa are compared with the theory. Good agreement is found.
The semiclassical description of an RF resonance measurement on a doublet excited state is formulated. The solution of the density matrix equation of motion is obtained as a continued fraction. The single quantum and multiple quantum transitions are seen as resonances in the measured quantities. An analytic solution is obtained for the case of degenerate doublet (the Hanle phenomena) and the RF modification of the g-value is obtained. It is shown that the continued fraction solution gives simple computations of strong field saturation effects.
The purpose of this paper is to develop a theory of saturation effects of the two-level system in two rotating rf fields with different frequencies and arbitrary intensities. We analyze this problem with the density matrix equation of motion including the relaxation processes, and the solution is obtained in the form of continued fraction. We give numerical calculations of the positions and shapes of the single and multiple quantum resonances, caused by large intensity rf fields using this continued fraction solution. It is also shown that the irradiation of an atomic system with a saturating field quasiresonant to a given transition drastically changes the absorption line shape of a probe field.
The dynamical behavior of a two-level system in an intense frequency-modulated field is obtained without recourse to the weak-modulation assumption. Solutions to the optical Bloch equations are expressed in terms of continued fractions involving 2\ifmmode\times\else\texttimes\fi{}2 matrices. Such matrix-continued fractions are numerically evaluated to obtain a wide variety of absorption and fluorescence spectra in frequency-modulated fields. Results in various special cases involving combinations of weak fields, weak modulation, strong modulation, and strong fields are given. Analytical results for weak fields but with an arbitrary degree of modulation are also presented. The spectra exhibit additional resonances for strong modulation. The main resonance shifts and broadens considerably due to strong modulation and saturation effects.
The semiclassical interaction of a two-level system with a multimode electromagnetic field is discussed in detail. The equations of motion for the macroscopic density matrix are transformed into a single integral equation for the population density, with a complex kernel which can take into account both running- and standing-wave configurations. The iterative solution to this integral equation, which corresponds to an expansion in terms of powers of the electromagnetic field amplitudes, turns out to be divergent when the field amplitudes are greater than some critical value. In the case in which the kernel is a periodic or quasiperiodic function of time, a general solution to the integral equation is found, and this solution turns out to be unique and convergent for every value of the electromagnetic (e.m.) field amplitude. This solution is expressed through a continued-fraction expansion, whose terms are matrices, which can be readily obtained from the kernel. This solution is a generalization of the continued-fraction expansion usually met in the strong-signal semiclassical theory of lasers. Some particular cases are then treated in detail; among them, the solution to the multimode operation in laser devices is given in which the excited level lifetime is long compared with the period of the beating term between two adjacent modes. Although the theory presented is directed towards the steady-state regime in multimode lasers, it may also treat the transient regime of laser operations, or it may be extended to include phenomena where a strong e.m. field, made up of several modes equally spaced in frequency, interacts with a two-level system.