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The Journal of Chemical Physics

Response to Double Irradiation of a Nuclear Spin System: General Treatment and New Multiple Quantum Transitions

Pietro Bucci, Massimo Martinelli, S. Santucci

Published Dec 15, 1970• 49 citations
Publisher

Abstract

A general theory has been developed to calculate the response, in NMR conditions, of a spin system irradiated by two nearly resonant waves having different frequencies and intense enough to produce an appreciable saturation. The absorption signal has been calculated employing the statistical operator technique, treating the electromagnetic field by the second quantization formalism. Two kinds of transitions are expected: The first one arises from processes in which n photons of one wave are emitted and n photons of the other wave are absorbed, the second one arises from processes involving the emission of n photons of one wave and the absorption of n ± 1 photons of the other. In the first case all the transitions coalesce in a single dispersionlike signal, which occurs when the frequencies of the two waves are nearly the same; in the second case a many-line spectrum occurs in which emission and absorption lines are present. The experimental results here reported strictly agree with theoretical previsions.

DOI
10.1063/1.1673983

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A general theory has been developed to calculate the response, in NMR conditions, of a spin system irradiated by two nearly resonant waves having different frequencies and intense enough to produce an appreciable saturation. The absorption signal has been calculated employing the statistical operator technique, treating the electromagnetic field by the second quantization formalism. Two kinds of transitions are expected: The first one arises from processes in which n photons of one wave are emitted and n photons of the other wave are absorbed, the second one arises from processes involving the emission of n photons of one wave and the absorption of n ± 1 photons of the other. In the first case all the transitions coalesce in a single dispersionlike signal, which occurs when the frequencies of the two waves are nearly the same; in the second case a many-line spectrum occurs in which emission and absorption lines are present. The experimental results here reported strictly agree with theoretical previsions.
The expression for the absorption coefficient in two-photon resonance transitions is deduced under general conditions. No particular restrictions are imposed on the level system or on the frequencies and intensities of the fields. The following cases are considered: (i) a two-level system (with and without singularities) irradiated by a single wave and (ii) a three-level system irradiated by two waves. Some theoretical and experimental disagreements reported by authors of previous papers are examined and clarified by the present theory.
Resonant two-photon absorption is discussed for the case in which the exciting field may be intense enough and near enough to resonance to cause appreciable saturation. A single pair of levels in an otherwise quite general atom is assumed to satisfy the appropriate resonance condition, and equations are derived governing the time evolution of the associated 2 \ifmmode\times\else\texttimes\fi{} 2 submatrix of the full atomic density matrix. Atomic relaxation is treated explicitly. The analysis, which treats both inversion-symmetric and inversion-nonsymmetric cases within the same general formalism, is developed systematically, the necessary approximations receiving careful discussion. The results are analogous to those for the corresponding one-photon process, but with a shift in the resonance frequency proportional to the intensity of the exciting field. This frequency shift is not identical to the one calculated by time-independent (second-order) perturbation theory, and becomes large when the energy of an intermediate state of the atom approaches the mean of the energies of the active states.
Two-photon transitions have provided the basis for numerous useful nonlinear interactions with radiation. In this paper we calculate the absorption and coupling coefficients for one-, two-, and three-wave mixing in two-photon homogeneously broadened media. The resulting field equations form the basis for understanding saturation spectroscopy, phase conjugation,1 and single-mode instabilities2 in lasers and optical bistability in two-photon media. We assume that the wave frequencies are not resonant with any intermediate state j and that the probe and conjugate waves cannot saturate the two-photon transition. This allows us to use the two-photon two-level model and to transfer much of our knowledge of the one-photon two-level systems to the two-photon case. Two major differences occur between the two models. First, dynamic Stark shifts of the active levels play an important role in generating asymmetric spectra and nonzero dispersion (≃ω s T1) at line center. Second, high pump intensities do not bleach the absorption coefficient at the Rabi sidebands.
For pt.I see Chem. Phys. Lett. vol.96, p.464 (1983). The non-linear multiphoton excitation dynamics of two-level systems under the influence of two strong linearly polarised monochromatic fields is studied exactly for the first time along the semiclassical many-mode Floquet theory. The present approach relaxes the restrictions of the conventional generalised rotating-wave approximation (GRWA) and allows correct treatment of various non-linear features such as power broadening, resonance shift, asymmetry in absorption lineshape, etc. By extending the nearly degenerate perturbative method of Salwen (1955), approximate analytical expressions for transition probabilities are obtained for multiphoton resonance processes for which (n+1) photons of one field are absorbed, whereas n photons of the other field are emitted, where n is an arbitrary integer. Detailed comparison of the analytical, the GRWA and the numerically exact results is given. Also presented is a case study of the time evolution of spin 1/2 systems in multi-quantum NMR transitions driven by pulsed bichromatic radiofrequency fields.
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.