Editing Tunnel ionization (section)

=== Analytical formula for the rate of MPI ===
The dynamics of the MPI can be described by finding the time evolution of the state of the atom which is described by the Schrödinger equation. The form of this equation in the electric field gauge, assuming the single active electron (SAE) approximation and using dipole approximation, is the following
: <math>i\frac{\partial}{\partial t}\Psi(\mathbf{r},\,t)=-\frac{1}{2m}\nabla^2\Psi(\mathbf{r},\,t) + (\mathbf{E}(t)\cdot\mathbf{r}+V(\mathbf{r}))\Psi(\mathbf{r},\,t) ,</math>  
where <math> \mathbf{E}(t) </math> is the electric field of the laser and <math> V(r) </math> is the static Coulomb potential of the atomic core at the position of the active electron. By finding the exact solution of equation (1) for a potential <math> \sqrt{2 E_\text{i}}.\delta(\mathbf{r}) </math> ({{tmath| E_\text{i} }} the magnitude of the ionization potential of the atom), the probability current <math> \mathbf{J}(\mathbf{r}, t) </math> is calculated. Then, the total MPI rate from short-range potential for linear polarization, {{tmath| W(\mathbf{E}, \omega) }}, is found from
: <math> W(\mathbf{E}, \omega)=\lim_{x\to\infty}\int_0^\frac{2\pi}{\omega} \int_{-\infty}^\infty \int_{-\infty}^\infty \mathbf{J}(\mathbf{r}, t)\,dz\,dy\,dt </math>
where <math> \omega </math> is the frequency of the laser, which is assumed to be polarized in the direction of the <math> x </math> axis. The effect of the ionic potential, which behaves like <math> {Z} / {r} </math> (<math> Z </math> is the charge of atomic or ionic core) at a long distance from the nucleus, is calculated through first order correction on the semi-classical action. The result is that the effect of ionic potential is to increase the rate of MPI by a factor of
: <math> I_\text{PPT}=(2(E_\text{i})^{\frac{3}{2}}/F)^{n^{*}} </math>
Where <math> n^{*}=Z/\sqrt{2 E_\text{i} } </math> and <math> F </math> is the peak electric field of laser. Thus, the total rate of MPI from a state with quantum numbers <math> l </math> and <math> m </math> in a laser field for linear polarization is calculated to be
: <math> W_\text{PPT}=I_\text{PPT}W(\mathbf{E}, \omega)=|C_{n^{*}l^{*}}|^{2}\sqrt{\frac{6}{\pi}}f_{lm}E_{i}(2(2 E_\text{i})^{\frac{3}{2}}/F)^{2n^{*}-|m|-3/2}(1+\gamma^{2})^{|m/2|+3/4}A_{m}(\omega, \gamma)e^{-\frac{2}{3}g(\gamma)(2 E_\text{i})^{\frac{3}{2}}/F} </math>
where  <math> \gamma= \frac{\omega \sqrt {2 E_\text{i}}}{F} </math> is the Keldysh's adiabaticity parameter and {{tmath|1= l^{*} = n^{*} - 1 }}. The coefficients <math> f_{lm} </math>, <math> g(\gamma) </math> and   <math> C_{n^{*}l^{*}} </math> are given by
: <math> f_{lm}= \frac{(2l+1)(l+|m|){!}}{2^{|m|}|m|{!}(l-|m|){!}} </math>
: <math> g(\gamma)=\frac{3}{2\gamma} ((1+\frac{1}{2\gamma^{2}})\sinh^{-1}(\gamma)-\frac{\sqrt{1+\gamma^{2}}}{2\gamma})</math>
: <math>|C_{n^{*}l^{*}}|^{2}= \frac{2^{2n^{*}}}{n^{*}\Gamma(n^{*}+l^{*}+1)\Gamma(n^{*}-l^{*})}</math>   
The coefficient <math> A_{m}(\omega, \gamma)</math> is given by
: <math> A_{m}(\omega, \gamma)=\frac{4}{\sqrt{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> w_{m}(x)=e^{-x^{2}}\int_0^x (x^2-y^2)^m e^{y^2}\,dy </math>
: <math> \alpha(\gamma)= 2(\sinh^{-1}(\gamma)-\frac{\gamma}{\sqrt{1+\gamma^{2}}})</math>
: <math> v= \frac{E_\text{i}}{\omega}(1+\frac{1}{2\gamma^{2}}) </math>
The ADK model is the limit of the PPT model when <math> \gamma </math> approaches zero (quasi-static limit). In this case, which is known as quasi-static tunnelling (QST), the ionization rate is given by
: <math> W_\text{ADK}=|C_{n^{*}l^{*}}|^{2}\sqrt{\frac{6}{\pi}}f_{lm}E_{i}(2(2 E_\text{i})^{\frac{3}{2}}/F)^{2n^{*}-|m|-3/2}e^{-(2(2 E_\text{i})^{\frac{3}{2}}/3F)} </math>.
In practice, the limit for the QST regime is {{tmath|1= \gamma < 1/2 }}. This is justified by the following consideration.<ref>{{cite book | last=CHIN | first=S. L. | title=Advances in Multi-Photon Processes and Spectroscopy | chapter=From multiphoton to tunnel ionization | publisher=WORLD SCIENTIFIC | year=2004 |volume=16| isbn=978-981-256-031-5 | issn=0218-0227 | doi=10.1142/9789812796585_0003 | pages=249–271}}</ref> Referring to the figure, the ease or difficulty of tunneling can be expressed as the ratio between the equivalent classical time it takes for the electron to tunnel out the potential barrier while the potential is bent down. This ratio is indeed {{tmath|1= \gamma }}, since the potential is bent down during half a cycle of the field oscillation and the ratio can be expressed as 
: <math> \gamma =\frac {\tau_\text{T}} {\frac{1}{2}\tau_\text{L}}</math>,
where <math> \tau_\text{T} </math> is the tunneling time (classical time of flight of an electron through a potential barrier, and <math> \tau_\text{L} </math> is the period of laser field oscillation.