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Journal of Physics B Atomic and Molecular Physics

Operator treatment of RF multiple-quantum transitions

E. Arimondo, G. Moruzzi

Published Apr 1, 1976• 6 citations
Publisher

Abstract

A set of operators, which act on the eigenstates of the atomic Hamiltonian in the presence of a partial I-J decoupling in a way analogous to the way the usual irreducible tensor operators act on the eigenstates of the total angular momentum, is introduced. Starting from these operators a set of operators, analogous to Haroche's pseudotensor operators, is obtained for the dressed-atom Hilbert space. They are thus particularly suitable to the theoretical treatment of the Delta m F >1 multiple quantum transitions within the Zeeman levels of a hyperfine state of an alkali atom. As an example, this treatment is applied to a system having a 'fictitious' spin F=1, and it is shown that the contribution of the single and multiple quantum transitions to the multipole components of the dressed-atom density matrix is separated.

DOI
10.1088/0022-3700/9/5/016

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Journal of Physics Condensed Matter•Oct 26, 1998•1 citation
A set of operators, which act on the eigenstates of the atomic Hamiltonian in the presence of a partial I-J decoupling in a way analogous to the way the usual irreducible tensor operators act on the eigenstates of the total angular momentum, is introduced. Starting from these operators a set of operators, analogous to Haroche's pseudotensor operators, is obtained for the dressed-atom Hilbert space. They are thus particularly suitable to the theoretical treatment of the Delta m F >1 multiple quantum transitions within the Zeeman levels of a hyperfine state of an alkali atom. As an example, this treatment is applied to a system having a 'fictitious' spin F=1, and it is shown that the contribution of the single and multiple quantum transitions to the multipole components of the dressed-atom density matrix is separated.
The theoretical behaviour of radiofrequency transitions with Delta m>or=1 in an atomic vapour under the simultaneous action of a longitudinal optical pumping, a static field introducing small decoupling between I and J and a relaxation mechanism is investigated. It is shown that the information contained in the eigenvalues and eigenvectors of the Hamiltonian of the atom 'dressed' by the RF field can be used to simplify significantly the equations of the time evolution of the atomic density matrix. Although an analytical solution is obtained only in a few simple cases a numerical solution is possible for any RF polarization. Several previous treatments are shown to be limiting cases of the general equations presented here. From the analysis of the recent experimental results of Arimondo and Corbalan (1974) on the ground state of 87 Rb, it turns out that the observable quantities are strongly dependent on all the parameters describing the simultaneous action of the pumping cycle, relaxation and the RF interaction. Thus accurate descriptions of the pumping cycle and the relaxation mechanism are required for any comparison with theory of experimental results for the widths and shifts of multiple-quantum transitions.
A formalism is presented that describes the time behavior of the spin density matrix of a nuclear spin system with arbitrary spin in terms of fictitious spin −(1/2) operators. This formalism is an extension of that used earlier for nuclei with spin I=1. For a spin system with n eigenstates we define for every pair of eigenstates ‖i〉 and ‖j〉 three operators Ii−jp, with p=x, y, and z, according to the three 2×2 Pauli matrices σx, σy, and σz. These operators together constitute a complete set of n2−1 independent Hermitian operators, and we can write the n×n density matrix and the spin Hamiltonian of the system in terms of the Ii−jp operators. The commutation relations among the operators make it possible in many cases to solve the equation of motion of the density matrix analytically. Three examples of the use of the Ii−jp operators are presented. Firstly a system of noninteracting spins with I=1 is considered. The Ii−jp operators for this case are compared with the Iq,k operator defined earlier. The cw signal intensities of the single and double quantum transitions for the I=1 spin system are derived. Secondly a spin system with I=3/2 is described as a simple extension of the use of the Ii−jp operators. The excitation and the detection of the triple quantum coherence are discussed for noninteracting nuclei in solids. Finally the double quantum effects in a system of two interacting spins with I=1/2 are discussed.
The solution of the time-dependent wave equation describing the interaction of two coupled spins with a rotating magnetic field in the presence of a steady field, as in the usual magnetic resonance situation, has been reexamined. By extending the method previously developed by Salwen, an exact expression is derived for all components of the wave function describing a three-state system under magnetic-resonance conditions. This solution describes in analytical form all the single and multiple resonances of which the system is capable for fields of any intensity. It is shown that when zzz is the axis of quantization, the projection of the wave function on an eigenstate of Fx{F}_{x}Fx​ oscillates at twice the resonance frequency at exact double quantum resonance. An approximate form is derived for the transition probability near double quantum resonance, showing that our solution is consistent with Salwen's in the limit of well-separated single and double quantum resonances. The theory is applied to the uppermost three substates of the upper hyperfine multiplet of an alkali metal of arbitrary nuclear spin, a case of interest to the analysis of magnetic resonance in optically pumped vapors.
As part of our general programme to develop the field of quantum nucleonics, we have studied Mössbauer spectroscopy when there is Zeeman splitting of the nuclear levels and a further interaction due to an applied rf-radiation field. We have applied the `dressed' state concept, developed in quantum electronics, to this situation. In particular, we have studied the case when the rf frequency is in the neighbourhood of the ground state (spin =1/2) splitting. The dressed-state approach in this case treats the coupling of nuclear Zeeman levels, due to an rf field, by considering the total system made up of nucleus, static magnetic field and rf field as one global quantum system. This allows the time evolution of the system to be handled straightforwardly. The energy levels and corresponding eigenstates of the system are calculated as a function of the rf frequency and the magnitude of the rf magnetic flux density. Mössbauer spectra are calculated for the case in which the source is subjected to both the static and radiation fields while the absorber nuclear levels are unsplit. Spectra are given as a function of the rf frequency, the magnitude of rf-field magnetic flux density, as well as the photon direction.