What is the real difference between tunneling ionization and multiphoton ionization (MPI)? Ionization is when an EM neutral atom gaing Em charge by losing or gaining an electron.


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).


https://en.wikipedia.org/wiki/Ionization
This wiki article says that tunneling ionization and MPI is essentially the same thing. 


The progress of strong-field laser-atom physics has been caused and significantly
    motivated by the first experimental observations of above threshold ionization (ATI) [1-3] of
    atoms more than thirty years ago. ATI can be usually understood as a process of multiphoton
    absorption, when the atom (or ion or molecule) absorbs more photons from an
    electromagnetic (laser) field than the minimum number required to overcome the ionization
    threshold. Alternatively, for sufficiently intense fields and sufficiently low frequencies of the
    laser, the process may be understood as a tunneling through a suppressed barrier of the
    Coulomb potential [4,5].
    Usually, the Keldysh parameter γ [7] is used to distinguish between multiphoton and
    tunneling ionization.


Now this says that the two are different things and the Keldysh parameter is used to distinguish between them.


Multiphoton ionization
    dominates when γ >1, and tunneling ionization prevails when γ <1.


Now what I do not understand is that one of them says that the two are the same thing, the other one says they are different.
Question:


*

*Is tunneling ionization the same as MPI or what is the real difference?

 A: The Keldysh parameter $\gamma$ is defined as, in atomic units,
$$
\gamma = \sqrt{\frac{I_p}{2U_p}},
$$
where $I_p$ is the ionization energy needed to ionize a target gas, and $U_p=E/(4\omega_0^2)$ is the ponderomotive energy of an electron in a laser field you illuminate to the target, and $\omega_0$ is the center frequency of the laser field. Since the ponderomotive energy is the average kinetic energy of a free electron driven by the laser field, the Keldysh parameter can be interpreted as a ratio between the binding energy of the parent ion vs laser driven oscillatory energy of the electron. If the laser driven energy (i.e. ponderomotive energy) is greater than the ionization energy of the atom $(\gamma < 1)$ then the laser field can drive electron outward from the parent ion without any help of resonant processes, like multiphoton process. If the ponderomotive energy is less than the ionization energy of the atom, then the laser field drive the electron rather weak, and there can be a room for multiphoton process to take place, or even dominate.
As you have quoted, the Keldysh parameter is used to distinguish the two regimes of ionization. The Keldysh parameter can give you an idea which process would be dominant, but it cannot clearly distinguish the two regimes in some cases. For example, when the Keldysh parameter $\gamma$ is close to $1$, then essentially two processes are mixed up so that both multiphoton ionization and tunneling ionization can take place at the same time. The Keldysh parameter can clearly distinguish the two regime only in its limiting values. In your last quotation, the inequality should be modified, $\gamma \ll 1$ for tunneling dominant, $\gamma \gg 1$ for multiphoton dominant. I would like to leave a link to a paper related to the Keldysh parameter. The following paper is a review paper on strong field approximation. Click this link to the paper(Kasra Amini et al, Symphony on strong field approximation, Reports on Progress in Physics, Volume 82, Number 11, 2019.)
I would like to quote my answer for a question about calculating ionization rate.

  
*
  
*If the intensity of the laser field is high enough but rather weak,
  then it is called multiphoton ionization.
  
*If the intensity of the laser field is high enough, but rather
  moderate compared to the last case, then it is called tunneling
  ionization. (In the extension of the case 1, it is often called above-threshold ionization)
  
*If the intensity of the laser field is too high, then it is called over-the-barrier
  ionization.
  

So, for the case when the laser field is overwhelmingly stronger than the Coulomb potential of the parent ion, the electron bound to the atom is spilling out through over-the-barrier ionization process. There are some range of parameters that these processes can overlap. They can be mixed up at certain intensity, frequency, and ionization energy etc..
