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Molecular Physics

Theoretical model for multiphoton transitions in rare earth ions

Humberto Siqueira Brandi, Oscar L. Malta

Published Nov 1, 1982• 0 citations
Publisher

Abstract

We propose a model to calculate multiphoton transition rates in rare earth ions based on the Green's function formalism. Analytical results are obtained by considering a truncated set of solutions of the time dependent Schrödinger equation which approximates the exact Green's function by one which we consider to be appropriated to describe multiphoton processes in rare earth ions for arbitrary field intensities. In the weak field limit the results of perturbation theory are recovered.

DOI
10.1080/00268978200100692

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We propose a model to calculate multiphoton transition rates in rare earth ions based on the Green's function formalism. Analytical results are obtained by considering a truncated set of solutions of the time dependent Schrödinger equation which approximates the exact Green's function by one which we consider to be appropriated to describe multiphoton processes in rare earth ions for arbitrary field intensities. In the weak field limit the results of perturbation theory are recovered.
The absorption of a single photon by the simultaneous excitation of two rare-earth ions was first reported by Varsanyi and Dieke in 1961.1 D. L. Dexter explained this unusual observation by including the electron-electron interactions (that also give rise to energy transfer) between the rare earth ions with perturbation theory.2 In the 70's Dexter became interested in electronic-vibrational coupling. A theory of two-center optical transitions was developed for a rare-earth ion weakly coupled to a high energy, internal vibrational mode of a near-lying ion.3 Using theoretical techniques that had been developed for rare-earth ion pairs, intensities and selection rules could be determined.
A general framework for description of quantum systems in strong external fields is established. The most important application of the theory is concerned with the multiphoton ionization of atoms and molecules. The adiabatic approximation is developed. The Hamiltonian of the system is diagonalized by a time-dependent unitary transformation, which generates an interaction energy vanishing for very slow fields. A general analysis of the photoionization rates is performed. A specific model is considered: one-dimensional motion in the δ-function potential and external harmonic electric field. This model allows the exact solution for time-dependent eigenvalues and eigenfunctions. These solutions are compared with the conventional free electron approach to the ionization states.
A theoretical procedure is presented for the evaluation of resonant multiphoton ionisation rates for atoms interacting with intense monochromatic lasers. The essence of the procedure is to treat first the strong coupling by solution of the truncated Schrodinger equation to obtain a time-dependent superposition state. This is then used as the lowest order approximation to iterate the full time-dependent Schrodinger equation to obtain the transition amplitude to the continuum. A model-atom study of the time evolution of the ionisation probability is carried out and the approach to rate behaviour in the long-time limit is demonstrated. Ionisation rates are evaluated for a variety of resonant processes in H, Na and Cs involving one or two lasers, as a function of intensities and detunings. Comparison with experiment is still in a fragmentary stage, and some large discrepancies are found to exist.
The two-photon transitions between the Stark levels of the 7F0^{7}\mathrm{F}_{0}7F0​ and 5D2^{5}\mathrm{D}_{2}5D2​ multiplets of the Sm2+{\mathrm{Sm}}^{2+}Sm2+ ion in BaClF are investigated both from an experimental and theoretical viewpoint. Direct two-photon excitation spectra of the 5D2^{5}\mathrm{D}_{2}5D2​ fluorescence resulting from an optical excitation with an infrared laser beam are presented and the polarization dependence of the observed transitions is analyzed. These new data are interpreted along the lines of a simple model which originates in the second-order theory developed by Axe [Phys. Rev. 136, A42 (1964)] for two-photon processes in complex atoms. The extension of this theory to the case of rare-earth ions embedded in crystalline fields is carried out by use of symmetry-adaptation techniques based on the use of an SU(2)\ensuremath{\supset}G∗{\mathrm{G}}^{\mathrm{*}}G∗ chain of groups. It is shown that the resulting model (limited up to second-order terms) describes correctly the relative intensities of the two-photon transitions between the Stark levels of the 7F0^{7}\mathrm{F}_{0}7F0​ and 5D2^{5}\mathrm{D}_{2}5D2​ multiplets of Sm2+{\mathrm{Sm}}^{2+}Sm2+ in BaClF.
A simple, "exactly solvable" atomic model will be presented that helps to discuss some of the interesting features of multiphoton ionization of atoms. The electromagnetic field is treated non-perturbatively, it can be strong. The atomic electrons are assumed to be non-interacting (Hartree model). They are bound in a harmonic oscillator potential (one- or three-dimensional). At the ionization energy, the potential is cut off and the free electron has no Coulomb attraction. The Schrodinger equation has exact analytic solutions valid in the inner or outer regions. If the field is turned on adiabatically, the exact solution can be found by matching these solutions at the boundary; both of their value and the normal derivative at all times. Thus the machine calculation reduces to repeated application of fast Fourier transforms and to the solution of linear equations (unfortunately a lot of them).