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Journal of Physics B Atomic Molecular and Optical Physics

Multiphoton ionization of an atom with a finite-range binding potential

I. N. Kosarev, Gennady L. Yudin

Published Oct 28, 1992• 4 citations

Abstract

The authors have formulated a new non-perturbative gauge-invariant analytical approach for solving the multiphoton ionization problem for a one-dimensional atomic system with a finite-range binding potential. Their method is based on the quasistationary state decay picture of ionization. They have applied the general solution to the model potential with the oscillator-like well and carried out the detailed analytical analysis of the ionization process. In the limit of moderately strong low-frequency laser fields and shallow potential wells, the authors' results reduce to those obtained for the case of an atom with the delta -well binding potential. For a finite-range binding potential one can distinguish the following peculiarities: (1) even for the case of non-resonant external field a sufficiently large number of binding states can lead to considerable changes in the multiphoton ionization; (2) in the case when the one-photon channel is open and the laser field is strong enough an atom becomes more stable as the field intensity is increased.

DOI: 10.1088/0953-4075/25/20/014

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The authors have formulated a new non-perturbative gauge-invariant analytical approach for solving the multiphoton ionization problem for a one-dimensional atomic system with a finite-range binding potential. Their method is based on the quasistationary state decay picture of ionization. They have applied the general solution to the model potential with the oscillator-like well and carried out the detailed analytical analysis of the ionization process. In the limit of moderately strong low-frequency laser fields and shallow potential wells, the authors' results reduce to those obtained for the case of an atom with the delta -well binding potential. For a finite-range binding potential one can distinguish the following peculiarities: (1) even for the case of non-resonant external field a sufficiently large number of binding states can lead to considerable changes in the multiphoton ionization; (2) in the case when the one-photon channel is open and the laser field is strong enough an atom becomes more stable as the field intensity is increased.

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).

The photoionization of atoms in a strong electromagnetic field leads to the effect of "above-threshold" ionization, displaying a number of peaks in the energy spectrum of the photoelectrons. In the case of a one-electron model atom in a circularly polarized field, we derive expressions that describe the position as well as the intensity of those peaks as a function of the field parameters by means of a nonperturbative method. It is immediately clear from our formulas that the lowest-energy peaks in general disappear with increasing field intensity, for Coulomb-like as well as shortrange potentials. For a zero-range potential, a numerical evaluation is easily performed, leading to results that are in every respect qualitatively similar to the experimental results obtained thus far on these kinds of processes.

We formulate a computationally feasible method for calculating multiphoton ionization rates for atoms exposed to intense fields in the intensity regime where perturbation theory ceases to apply. The method is based on the time-independent picture of ionization, which starts with the Floquet ansatz. The question of the gauge of the radiation field is discussed in some detail. Various expressions for the ionization amplitude are derived from a variational principle, with the radiation field expressed in the velocity gauge. We consider in particular an approximation in which the wave vector developing from the initial state is replaced by a trial vector that is the Floquet expansion truncated just below the threshold for ionization, and in which the wave vector developing into the final state is replaced by a trial vector that is just the wave vector appearing in the Kroll-Watson low-frequency approximation in scattering theory. We have applied this approximation to hydrogen and we present some results for both nonresonant and resonant multiphoton ionization. We argue that the experimentally observed resonance structure in the above-threshold peaks of the ionization signal occurs through the electron jumping from one dressed-state energy-eigenvalue curve to another.

A time-independent single-rate theory for multiphoton ionization processes is developed to cover cases with a partially coherent (finite-bandwidth) driving laser. We use the Ornstein-Uhlenbeck fluctuating-phase model to describe the driving field, which has therefore a non-Lorentzian spectrum, and the effective two-level model to represent the atomic system. We present analytic and numerical methods that permit discussion of the effects of non-Lorentzian band-shape laser light, and we give analytically the bandwidth dependence of the rate in the wings of the ionization curve. Our model contains a simple scaling relationship that permits the rate for

N

-photon-resonant (

N+1

)-photon ionization to be obtained directly from the one-photon-resonant two-photon ionization rate. We treat explicitly the cases

N=1

and

N=2

.