# Calculating the ionization rate

here is a kind of silly question: in quantumn physics, we assume that the charge is reserved. So if we use a laser, which is strong enough to make ionization happen, to shot at the system, how can we calculate the decreasing rate of the electrons? Or the ionization rate can be calculated by other ways?

I will very appreciate for helping me understand this question!

Your question is never silly, but many physicists are struggling to obtain an accurate expression for the ionization rate you are asking.

When you drive an atom with a laser field, electrons bound to the atom start to oscillate. To make our discussion simple and short, let us deal with a single electron bound to a parent ion, rather than many electrons and nucleus, to describe laser-atom interaction. If the laser field is too weak and the frequency of the laser field is not high enough, which is the case for lasers in our daily life, then effectively nothing happens to anything.

If the frequency of the laser field is high enough, then the atom can either be excited or ionize to emit an electron by a single photon of the laser field, and this ionization process is called single photon ionization.

In this case, the ionization rate can be calculated by obtaining bound-free transition rate in the presence of a weak high frequency laser field. In this case, you can either use quantum electrodynamics or use quantum mechanics to calculate accurate transition rate.

Even when the frequency of the laser field is not high enough to ionize the atom with a single photon, an atom can be ionized when the intensity of the laser field is high enough. This process can be classified into three, depending on the intensity of the laser field.

1. If the intensity of the laser field is high enough but rather weak, then it is called multiphoton ionization.
2. 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)
3. If the intensity of the laser field is too high, then it is called over-the-barrier ionization.

In the case 1 multiphoton ionization, an electron can absorb two or more photons to be ionized, rather than one at a time. In the case 2 tunneling ionization, the laser field is strong enough to "bend" the Coulomb(-like) potential near the parent ion and form a slope toward the opposite direction of the laser field. Then "the potential slope due to the laser field + the Coulomb(-like) potential" form a barrier and high wall. The bound electron is forced toward the barrier at one moment and be freed tunneling through the potential barrier. This process flips the direction every half-cycle of the laser field and repeats over and over until the laser field is turned off. In the case 3 over-the-barrier ionization, the laser field is even stronger than the case 2, so that the potential barrier is low enough, lower than the bound energy of the electron and the bound electron can be freed. Yeah, electrons are spilled over the barrier. It is like breaking Coulomb potential using the laser field. In this case, our assumption from the beginning is not valid anymore because many electrons can be spilled out freely.

The ionization rate in the cases 1 and 2 can be estimated by the Keldysh theory proposed in 1964(Link to a review paper).

One of rough but intuitive expression in the case of DC (a laser field is oscillating over time) tunneling ionization is in a textbook written by L. D. Landau and E. M. Lifshitz. It can be obtained using WKB approximation over the potential barrier of a hydrogen atom, and the final expression in atomic units can be written as

$$W = \frac{1}{|E|}\exp(-\frac{2}{3|E|}),$$

where $$W$$ is the static tunneling ionization rate, and $$E$$ is the static electric field. Rougly speaking, the static tunneling ionization rate is exponentially increasing as the amplitude of the laser fied increases, and is zero when the laser field is absent. There are more accurate expressions for more general cases but those are rather complicated and I refer one paper. You can search more of them starting from this paper. Nonadiabatic tunnel ionization: Looking inside a laser cycle, published in 2001, Physical Review A.

You can obtain tunneling ionization rate exactly in the case of a hydrogen by solving Schrodinger equation for the Coulomb potential numerically, including laser interaction term.

More precise number can be obtained by solving Dirac equation in the case when the electron can move with relativistic speed but if you need to solve the Dirac equation, then it is most likely the case of over-the-barrier ionization.