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Theory of multiphoton processes for atoms and ions

Bo Gao

Published Jan 1, 1989• 0 citations
OpenAlex

Abstract

In the first part of this thesis, a new variational principle is presented for the calculation of perturbative transition amplitudes of any order but free from any singularities. Applications are made to the multiphoton ionization and harmonic generation for atomic hydrogen. Its formulations in hyperspherical coordinates as well as in Hartree-Fock representation are also presented. In the second part of this thesis, theories for multiphoton detachment in a static electric or magnetic field are developed. They are based on the analytic solutions of an electron in the combined field of a laser and a static electric or magnetic field, and, therefore, are nonperturbative in both the laser intensity and the static electric or magnetic field strength. Heretofore unexpected cyclotron resonances, which are due to the infinite range character of the electron-photon coupling induced by the external magnetic field, are shown to exist under certain conditions.

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Short Wavelength Coherent Radiation: Generation and Applications•Jan 1, 1986•0 citations
In the first part of this thesis, a new variational principle is presented for the calculation of perturbative transition amplitudes of any order but free from any singularities. Applications are made to the multiphoton ionization and harmonic generation for atomic hydrogen. Its formulations in hyperspherical coordinates as well as in Hartree-Fock representation are also presented. In the second part of this thesis, theories for multiphoton detachment in a static electric or magnetic field are developed. They are based on the analytic solutions of an electron in the combined field of a laser and a static electric or magnetic field, and, therefore, are nonperturbative in both the laser intensity and the static electric or magnetic field strength. Heretofore unexpected cyclotron resonances, which are due to the infinite range character of the electron-photon coupling induced by the external magnetic field, are shown to exist under certain conditions.
The variationally stable procedure for Nth-order perturbative-transition-matrix elements introduced by the authors [Phys. Rev. Lett. 61, 404 (1988)] is presented here in detail. Its key features are that it is noniterative, involves only two unknown functions regardless of the value of N, is stable even near intermediate-state resonances (due to the presence of energy numerators rather than energy denominators), and uses the inverse of the perturbation operator for N\ensuremath{\ge}3. Explicit formulas are presented for application to high-order multiphoton processes in atomic hydrogen. Numerical results for multiphoton-ionization cross sections (for two, three, and seven photons), for the frequency dependence of the nonlinear susceptibilities for harmonic generation (for the third, fifth, and seventh harmonics), and for harmonic-generation transition rates up to 11th order for \ensuremath{\lambda}=1064 nm are presented for the atomic hydrogen ground state and compared with results of others by more standard procedures. We also present detailed analyses of the Z scaling of our results for hydrogenic systems, of the use of the imaginary part of the appropriate 2N-photon nonlinear susceptibility to obtain N-photon-ionization cross sections, and of the relation of the variational principle presented here to variational principles for scattering processes of Nuttall and Cohen [Phys. Rev. 188, 1542 (1969)] and of Schwinger [Phys. Rev. 72, 742 (1947)].
A general program has been developed to calculate multiphoton processes in atomic hydrogen to arbitrarily high order in perturbation theory. This program can be used to calculate quantities in intense field multiphoton processes, such as the generalized cross section of multiphoton absorption, the a.c. Stark effect (shift and the broadening of the level due to the field interaction), and the parameters in nonlinear optics such as the nonlinear index of refraction and the nonlinear susceptibility. Effective parametrization of high-order susceptibilities has been achieved over a wide range of frequencies, utilizing concepts from quantum defect theory to express the results of lengthy calculations in a compact form that also permits extrapolation across thresholds and resonance poles. Significant progress has been made in direct numerical solution of the time-dependent Schroedinger equation. In addition, contributions have been made in: understanding the role of spatial dimensionality in the solution of model systems (e.G., the one-dimensional delta-function potential); the development of a general R-matrix code for perturbative computations of multiphoton processes in many-electron atoms; and multiphoton ionization with two commensurate laser frequencies.
The infinite summations over the complete set of unperturbed atomic states appearing in the NNNth-order perturbation theory for multiphoton ionization are performed using the Dalgarno-Lewis method. The relevant transition matrix elements are written in a closed integral form exhibiting all the analytic properties of the amplitude as a function of incident photon energy. The cross sections for two- and three-photon ionization for atomic hydrogen are calculated for both linearly and circularly polarized light with a wide range of photon energy spectrum including the near resonance, and numerical comparison is made with the values obtained by different methods.
The present paper applies the method of high-intensity quantum electrodynamics developed earlier to several problems of current experimental interest. In this paper the intensitydependent shifts and widths of atomic levels are calculated and applied to a given transition near resonance. The technique is demonstrated with three examples: the multiphoton ionization of atomic cesium and hydrogen in atomic beams and third-harmonic production in an atomic vapor. It is found that the multiphoton-ionization rate in a relatively weak laser beam (of order 108{10}^{8}108 W/cm2{\mathrm{cm}}^{2}cm2) is determined by the intensity-dependent level width (6f6f6f configuration) as well as the interference of the fine-structure states, and that the rates for hydrogen, in a much more intense beam (of order 1012{10}^{12}1012 W/cm2{\mathrm{cm}}^{2}cm2) are determined by both the intensity-dependent widths and shifts as‘\stackrel{\mathrm{`}}{\mathrm{s}}s‘ well as the wave-function renormalization constants. The agreement of these two examples of calculations with the experiments done recently is satisfactory. For third-harmonic production the role of two-photon resonance is emphasized, and the dependence of the generated third-harmonic intensity on the intensity and frequency of the fundamental is discussed. Near resonance, the ratio of third-harmonic intensity to fundamental intensity saturates.
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 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).