Editing Ionization (section)

===Tunnel ionization===
[[File:Tunnel ionization 3.png|thumb|upright=1.5|Combined potential of an atom and a uniform laser field. At distances {{nowrap|''r'' < ''r''{{sub|0}}}}, the potential of the laser can be neglected, while at distances with {{math|''r'' > ''r''{{sub|0}}}} the Coulomb potential is negligible compared to the potential of the laser field. The electron emerges from under the barrier at {{math|''r'' {{=}} ''R''{{sub|c}}}}. {{math|''E''{{sub|i}}}} is the ionization potential of the atom.]]

[[Tunnel ionization]] is ionization due to [[quantum tunneling]]. In classical ionization, an electron must have enough energy to make it over the potential barrier, but quantum tunneling allows the electron simply to go through the potential barrier instead of going all the way over it because of the wave nature of the electron. The probability of an electron's tunneling through the barrier drops off exponentially with the width of the potential barrier. Therefore, an electron with a higher energy can make it further up the potential barrier, leaving a much thinner barrier to tunnel through and thus a greater chance to do so. In practice, tunnel ionization is observable when the atom or molecule is interacting with near-infrared strong laser pulses. This process can be understood as a process by which a bounded electron, through the absorption of more than one photon from the laser field, is ionized. This picture is generally known as multiphoton ionization (MPI).

Keldysh<ref>{{cite journal |last=Keldysh |first=L. V. |date=1965 |url=http://www.jetp.ac.ru/cgi-bin/e/index/e/20/5/p1307?a=list |title=Ionization in the Field of a Strong Electromagnetic Wave|journal=Soviet Phys. JETP |page=1307|volume=20|issue=5}}</ref> modeled the MPI process as a transition of the electron from the ground state of the atom to the Volkov states.<ref>Volkov D M 1934 Z. Phys. 94 250</ref> In this model the perturbation of the ground state by the laser field is neglected and the details of atomic structure in determining the ionization probability are not taken into account. The major difficulty with Keldysh's model was its neglect of the effects of Coulomb interaction on the final state of the electron. As it is observed from figure, the Coulomb field is not very small in magnitude compared to the potential of the laser at larger distances from the nucleus. This is in contrast to the approximation made by neglecting the potential of the laser at regions near the nucleus. Perelomov et al.<ref>{{cite journal |last1=Perelomov |first1=A. M. |last2=Popov |first2=V. S. |last3=Terent'ev |first3=M. V. |date=1966 |journal=Soviet Phys. JETP |volume=23 |issue=5 |page=924 |url=http://www.jetp.ac.ru/cgi-bin/e/index/e/23/5/p924?a=list |title=Ionization of Atoms in an Alternating Electric Field |bibcode=1966JETP...23..924P |access-date=2013-08-12 |archive-date=2021-03-18 |archive-url=https://web.archive.org/web/20210318094804/http://www.jetp.ac.ru/cgi-bin/e/index/e/23/5/p924?a=list |url-status=dead }}</ref><ref>{{cite journal |last1=Perelomov |first1=A. M. |last2=Popov |first2=V. S. |last3=Terent'ev |first3=M. V. |date=1967 |journal=Soviet Phys. JETP |volume=24 |issue=1 |page=207 |url=http://www.jetp.ac.ru/cgi-bin/e/index/e/24/1/p207?a=list |title=Ionization of Atoms in an Alternating Electric Field: II |bibcode=1967JETP...24..207P |access-date=2013-08-12 |archive-date=2021-03-03 |archive-url=https://web.archive.org/web/20210303205015/http://www.jetp.ac.ru/cgi-bin/e/index/e/24/1/p207?a=list |url-status=dead }}</ref> included the Coulomb interaction at larger internuclear distances. Their model (which we call the PPT model) was derived for short range potential and includes the effect of the long range Coulomb interaction through the first order correction in the quasi-classical action. Larochelle et al.<ref>{{cite journal |last1=Larochelle |first1=S. |last2=Talebpour |first2=A. |last3=Chin |first3=S. L. |doi=10.1088/0953-4075/31/6/009 |url=http://slchin-symposium.copl.ulaval.ca/MPublication/154_JPB_031_1215.pdf |title=Coulomb effect in multiphoton ionization of rare-gas atoms |date=1998 |journal=Journal of Physics B: Atomic, Molecular and Optical Physics |volume=31 |issue=6 |page=1215 |bibcode=1998JPhB...31.1215L |s2cid=250870476 |url-status=dead |archive-url=https://web.archive.org/web/20141121115047/http://slchin-symposium.copl.ulaval.ca/MPublication/154_JPB_031_1215.pdf |archive-date=November 21, 2014 }}</ref> have compared the theoretically predicted ion versus intensity curves of rare gas atoms interacting with a Ti:Sapphire laser with experimental measurement. They have shown that the total ionization rate predicted by the PPT model fit very well the experimental ion yields for all rare gases in the intermediate regime of the Keldysh parameter.

The rate of MPI on atom with an ionization potential <math> E_i </math> in a linearly polarized laser with frequency <math> \omega </math> is given by
:<math>W_{PPT} =
  \left|C_{n^* l^*}\right|^2 \sqrt{\frac{6}{\pi}} f_{lm}
  E_i \left(\frac{2}{F} \left(2E_i\right)^{\frac{3}{2}}\right)^{2n^* - |m|- \frac{3}{2}}
  \left(1 + \gamma^2\right)^{\left|\frac{m}{2}\right|+ \frac{3}{4}}
  A_m (\omega, \gamma) e^{-\frac{2}{F}\left(2E_i\right)^{\frac{3}{2}} g\left(\gamma\right)}
</math>

where 
* <math> \gamma=\frac{\omega \sqrt{2E_i}}{F} </math> is the Keldysh parameter,
* <math> n^*=\frac{\sqrt{2E_i}}{Z^2} </math>,
* <math> F </math> is the peak electric field of the laser and
* <math> l^*=n^* - 1 </math>.
The coefficients <math> f_{lm} </math>, <math> g(\gamma) </math> and <math> C_{n^* l^*} </math> are given by
:<math>\begin{align}
           f_{lm} &= \frac{(2l + 1)(l + |m|)!}{2^m |m|!(l - |m|)!} \\
        g(\gamma) &= \frac{3}{2\gamma} \left(1 + \frac{1}{2\gamma^2} \sinh^{-1}(\gamma) - \frac{\sqrt{1 + \gamma^2}}{2\gamma}\right) \\
  |C_{n^* l^*}|^2 &= \frac{2^{2n^*}}{n^* \Gamma(n^* + l^* + 1) \Gamma(n^* - l^*)}
\end{align}</math>

The coefficient <math> A_m (\omega, \gamma)</math> is given by
:<math>
  A_m (\omega, \gamma) =
    \frac{4}{3\pi} \frac{1}{|m|!}
    \frac{\gamma^2}{1 + \gamma^2}
    \sum_{n>v}^\infty e^{-(n - v) \alpha(\gamma)}
    w_m \left(\sqrt{\frac{2\gamma}{\sqrt{1 + \gamma^2}} (n - v)}\right)
</math>

where
:<math>\begin{align}
          w_m(x) &= e^{-x^2} \int_0^x (x^2 - y^2)^m e^{y^2}\,dy \\
  \alpha(\gamma) &= 2\left(\sinh^{-1}(\gamma) - \frac{\gamma}{\sqrt{1 + \gamma^2}}\right) \\
               v &= \frac{E_i}{\omega} \left(1 + \frac{2}{\gamma^2}\right)
\end{align}</math>

====Quasi-static tunnel ionization====
The quasi-static tunneling (QST) is the ionization whose rate can be satisfactorily predicted by the ADK model,<ref>{{cite journal |last1=Ammosov |first1=M. V. |last2=Delone |first2=N. B. |last3=Krainov |first3=V. P. |date=1986 |journal=Soviet Phys. JETP |volume=64 |issue=6 |page=1191 |url=http://www.jetp.ac.ru/cgi-bin/e/index/e/64/6/p1191?a=list |title=Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field |bibcode=1986JETP...64.1191A |access-date=2013-08-12 |archive-date=2021-03-01 |archive-url=https://web.archive.org/web/20210301174150/http://www.jetp.ac.ru/cgi-bin/e/index/e/64/6/p1191?a=list |url-status=dead }}</ref> i.e. the limit of the PPT model when <math> \gamma </math> approaches zero.<ref name="SharifiTalebpour2010">{{cite journal |last1=Sharifi |first1=S. M. |last2=Talebpour |first2=A |last3=Yang |first3=J. |last4=Chin |first4=S. L. |title=Quasi-static tunnelling and multiphoton processes in the ionization of Ar and Xe using intense femtosecond laser pulses |journal=Journal of Physics B: Atomic, Molecular and Optical Physics |volume=43 |issue=15 |date=2010 |page=155601 |issn=0953-4075 |doi=10.1088/0953-4075/43/15/155601|bibcode=2010JPhB...43o5601S |s2cid=121014268 }}</ref> The rate of QST is given by
:<math>W_{ADK} =
  \left|C_{n^* l^*}\right|^2 \sqrt{\frac{6}{\pi}}
  f_{lm} E_i \left(\frac{2}{F} \left(2E_i\right)^{\frac{3}{2}}\right)^{2n^* - |m|- \frac{3}{2}}
  e^{-\frac{2}{3F} \left(2E_i\right)^{\frac{3}{2}}}
</math>

As compared to <math>W_{PPT}</math> the absence of summation over n, which represent different [[above threshold ionization]] (ATI) peaks, is remarkable.