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Physical Review A

Absorption and fluorescence in frequency-modulated fields under conditions of strong modulation and saturation

N. Nayak, G. S. Agarwal

Published May 1, 1985• 50 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.31.3175

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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.
Optical non-linearities and spectroscopy in fields that are strongly modulated and also intense are examined by obtaining the solution of the optical Bloch equations in modulated fields as a matrix continued fraction. Numerical results for emission and absorption in such fields are presented. The saturation optical susceptibilities and the different types of spectra show resonances at the Rabi frequency if the modulation index is small. With increase in this index, additional resonant features appear at submultiples of the Rabi frequency. These additional features become more and more prominent as the modulation index increases further. For modulated fields on resonance, a dressed-atom formulation is developed to explain the results obtained from the numerical evaluation of the continued fraction. The results are also applicable to non-linear spectroscopic measurements such as Raman gain when the probe field is modulated.
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.
Based on the vanishing-damping continued-fraction method which I developed previously for the analysis of the resonance behavior of a two-level atom in a two-frequency field with equal intensities in the two field modes [W. M. Ruyten, Phys. Rev. A 40, 1447 (1989)], I analyze here the related problem in which the two field intensities are unequal. Several analytical results are derived, namely, expressions for the lowest-order Bloch-Siegert shifts of all resonances, and an expression for the positions of the resonances in the limit of one strong and one weak field. It is shown that, in this limit, the resonances exhibit level crossings of the field-atom system.
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.
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.