The problem of tunneling ionization and the associated questions of how long it takes for an electron to tunnel through the barrier, and what the tunneling rate has fascinated scientists for almost a century. In strong field physics, tunnel ionization plays an important role, and accurate knowledge of the time-dependent tunnel rate is of paramount importance. The Keldysh theory and other more advanced related theories are often used, but their accuracy is still controversial. In previous work, we suggested using a curved waveguide as a quantum simulator to simulate the tunnel ionization process. Here we implemented for the first time such a curved waveguide and observed the simulated tunneling ionization process. We compare our results with the theory.
The general theory of multiphoton ionization of hydrogen atoms by simultaneous absorption of several photons is discussed. In the nonrelativistic dipole approximation, the calculations lead to a recursion relationship concerning angular functions and involve radial matrix elements containing many summations over intermediate states. A general method for exactly evaluating several sums is described in detail and applied to the photo-ionization of atomic hydrogen in the ground state. Numerical results are presented as dispersion curves of the transition rate $\frac{{\ensuremath{\sigma}}_{n}}{{I}^{N\ensuremath{-}1}}$ for a range of wavelengths from 1000 up to 7200 \AA{}.
A perturbation theory of the ionization of atoms by simultaneous absorption of several photons, each of whose energy is less than the ionization potential, is developed from the evolution-operator formalism. A precise computation is made for the hydrogen atom, giving transition rates as a function of photon energy for two- through twelve-photon photoionization. The eighth-order ionization rate (in cgs units) at the 1.78-eV ruby-laser line is found to be \ensuremath{\sim}${10}^{\ensuremath{-}244}$\ifmmode\times\else\texttimes\fi{}
(photonflux)8 and should be observable using available techniques. Good agreement is obtained with Zernik's exact calculation of the two-photon ionization rate of metastable
2S hydrogen. Approximate calculations are made for the rare gases. Assuming "typical" experimental conditions of a gas density of \ensuremath{\sim}
1020 atoms ${\mathrm{cm}}^{\ensuremath{-}3}$ and a ruby laser focused into a volume of \ensuremath{\sim}${10}^{\ensuremath{-}8}$
cm3, we find that the flux required to liberate one electron during a 10-nsec pulse is \ensuremath{\sim}
1029 ${\mathrm{cm}}^{\ensuremath{-}2}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ for Xe, Kr, and Ar and \ensuremath{\sim}5\ifmmode\times\else\texttimes\fi{}
1030 photons ${\mathrm{cm}}^{\ensuremath{-}2}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ for Ne and He. These gases ionize with the simultaneous absorption of 7, 8, 9, 13, and 14 photons, respectively. The predicted rate for Xe is found to be in excellent agreement with the recent direct measurements of Voronov and Delone. We conclude that multiphoton ionization provides the initial electrons required for the optical breakdown of gases, though it does not account for the over-all growth of the discharge except possibly at very low pressures. Impurity atoms (particularly heavy rare gases) may be the source of "initiating" electrons in Ne and He.
Detailed descriptions of one- and two-photon
ionization are presented. We go beyond conventional perturbation
theory to investigate the dynamics of both processes for a broad
range of field intensities.
The general
theory of unstable decaying states is applied to the problem of
one-photon ionization (the photoeffect). It is found that a
non-trivial dependence of the transition dipole moment on the
continuum state energy can lead to corrections to exponential
decay, and even oscillations in the bound state population. As
examples we consider the photoionization of hydrogen and atoms with
autoionizing states. Only in the absence of autoionizing states
corrections to exponential decay remain inconsequential for any
experimentally relevant intensity.
Our
description of two-photon ionization is formulated in the
Heisenberg picture. Besides the usual two-level atom projection
operators we introduce explicitly the operators referring to the
atomic continuum. By eliminating from the equations of motion the
terms related to the electromagnetic self field as well as the
terms referring to the continuum, transverse and longitudinal
damping corresponding to spontaneous emission and photoionization
is obtained. The use of field correlation functions allows us to
account for the partial coherence of the applied field.
We develop a technique to extract from the
equations of motion a unique two-photon ionization rate. Such a
characterization of the ionization process is not only instructive
because of its simplicity, it is also in many cases completely
satisfactory. The resulting general expression for the two-photon
ionization rate is the central result of our treatment. From it we
recover several familiar results in different limiting cases. In
addition, our general rate expression provides the first
description of the complete hierarchy of saturation effects that
one should expect.
We present a number of
examples where we calculate the two-photon ionization rate as a
function of the detuning, and where we emphasize the saturation and
field correlation effects.