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Physical Review A

Rate equations versus Bloch equations in multiphoton ionization

Jay R. Ackerhalt, Bruce W. Shore

Published Jul 1, 1977• 108 citations
Publisher

Abstract

The Wilcox-Lamb method of deriving rate equations from Bloch equations is outlined using as an example a two-level atomic model. Computer-generated plots directly comparing solutions of these rate equations and of Bloch equations for a three-level atomic model are illustrated. Two cases of interest are discussed: copropagating and counterpropagating laser beams. Some comments concerning collisional accessing and ionizing of atoms in a very broad Doppler profile are made.

DOI
10.1103/physreva.16.277

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The Wilcox-Lamb method of deriving rate equations from Bloch equations is outlined using as an example a two-level atomic model. Computer-generated plots directly comparing solutions of these rate equations and of Bloch equations for a three-level atomic model are illustrated. Two cases of interest are discussed: copropagating and counterpropagating laser beams. Some comments concerning collisional accessing and ionizing of atoms in a very broad Doppler profile are made.
Laser excitation of a two-level atom is analytically studied for arbitrary pulse shape, area, and energy. The ionization mechanism is modeled as a homogeneous width $\ensuremath{\gamma}$ of the atomic excited state. A transformation of the optical Bloch equations allows the dynamics to be completely described by a single equation for the area of the pulse at time ttt, $\ensuremath{\Phi}(t)$. Analytic solutions for the total bound population for a wide variety of pulse shapes can be found in two regions: $\ensuremath{\Phi}(t)\ensuremath{\approx}sin\ensuremath{\Phi}(t)$ and $\ensuremath{\Omega}(t)>\ensuremath{\gamma}$, where $\ensuremath{\Omega}(t)$ is the resonance Rabi frequency. The generalization of the two-level atom results to the NNN-level atom is discussed.
We outline a formalism and develop a computational procedure to treat the process of multiphoton ionization (MPI) of atomic targets in strong laser fields. We treat the MPI process nonperturbatively as a decay phenomenon by solving a coupled set of the integral Lippmann-Schwinger equations. As basic building blocks of the theory, we use a complete set of field-free atomic states, discrete and continuous. This approach should enable us to provide both the total and differential cross sections of MPI of atoms with one and two electrons. As an illustration, we apply the proposed procedure to a simple model of MPI from a square-well potential and to the hydrogen atom.
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).
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 NNN-photon-resonant (N+1N+1N+1)-photon ionization to be obtained directly from the one-photon-resonant two-photon ionization rate. We treat explicitly the cases N=1N=1N=1 and N=2N=2N=2.