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Physical Review A

Magnetic and optical resonance of two-level quantum systems in modulated fields. II. Floquet Hamiltonian approach

Wilhelmus M. Ruyten

Published Oct 1, 1990• 13 citations
Publisher

Abstract

It is shown that, in modulated magnetic resonance on spin-1/2 systems, the time-averaged absorption signal of the optical pump beam can be obtained directly from the dressed-atom energy eigenvalues of the Floquet Hamiltonian for arbitrary orientations of the optical pump beam, namely, by a straightforward generalization of Shirley's well-known result [Phys. Rev. 138, B979 (1965)]. Analytical and numerical results derived from this generalized diagonalization technique are compared with the vanishing damping solution of the Bloch equations from the preceding paper [W. M. Ruyten, Phys. Rev. A 42, 4226 (1990)] and are shown to be in complete agreement with the latter.

DOI
10.1103/physreva.42.4246

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Magnetic and optical resonance of two-level quantum systems in modulated fields. I. Bloch equation approach

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It is shown that, in modulated magnetic resonance on spin-1/2 systems, the time-averaged absorption signal of the optical pump beam can be obtained directly from the dressed-atom energy eigenvalues of the Floquet Hamiltonian for arbitrary orientations of the optical pump beam, namely, by a straightforward generalization of Shirley's well-known result [Phys. Rev. 138, B979 (1965)]. Analytical and numerical results derived from this generalized diagonalization technique are compared with the vanishing damping solution of the Bloch equations from the preceding paper [W. M. Ruyten, Phys. Rev. A 42, 4226 (1990)] and are shown to be in complete agreement with the latter.
The resonance behavior of magnetic and optical two-level systems under the influence of strong modulated fields is studied in detail by solving the Bloch equations in the limit of vanishing damping, namely, in terms of matrix continued fractions. Thus both analytical and numerical results are obtained for the positions of the various multiple quantum resonances in the time-averaged as well as modulated interaction components. Specifically, the role of the orientation of the optical pump beam in magnetic-resonance configurations is investigated, and it is shown that there are several classes of resonances for which the behavior of the system is independent of the orientation of the pump beam. These are the geometries in which the static and oscillating magnetic fields are either orthogonal or parallel, the geometries leading to Haroche-like resonances (which, however, play a special role in modulated interaction components also), and a previously unknown class, labeled saddle-point resonances. However, other than for these special classes, the behavior of the system depends generally on the orientation of the optical pump beam, unlike as implied in some previous work on this problem. As in the recently studied case of a fully amplitude-modulated optical interaction [W. M. Ruyten, Phys. Rev. A 40, 1447 (1989)], it is found that the resonances in modulated interaction components are generally doubly branched, with one branch duplicating the resonances found in the time-averaged component. A new feature found for not-fully-amplitude-modulated and phase-modulated, off-resonance optical excitation (or magnetic resonance with nonparallel or nonorthogonal static and oscillating magnetic fields) is the occurrence of an extra type of resonance in modulated components of the interaction that cannot, like all other resonances of the system, be classified as power-shifted multiple quantum transitions.
We present a detailed theoretical treatment of the response of a two-level atom to a fully-amplitude-modulated, narrow-band optical field. Specifically, we calculate the resonance behavior of modulated fluorescence and susceptibility components from the scalar continued-fraction solution of the optical Bloch equations. In the limit of vanishing damping the resonances are generally doubly branched: One set of branches reproduces the resonances of the time-averaged fluorescence, while the second set is different for each harmonic component. However, each branch is uniquely associated with an odd multiple quantum resonance. In analogy with magnetic resonance work, we derive analytical expressions for the generalized Bloch-Siegert shifts of the additional resonances. We discuss also the possibility of experimental verification of the theory.
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.
Using a novel semiclassical approach, the authors analyse the resonance behaviour of spin-1/2 atoms subjected to static and oscillating magnetic fields. Unlike a number of previous calculations, the orientation of the pump beam is taken into account explicitly. Thus it is shown that much previous work is valid only if the pump beam is along the static magnetic field. They also show how one such previous calculation can be generalised, namely that based on diagonalisation of the Floquet Hamiltonian.