  We investigate the 2nd order process of two photons being emitted by a
high-energy electron dressed in the strong background electric field found
between the planes in a crystal. The strong crystalline field combined with
ultra relativistic electrons is one of very few cases where the Schwinger field
can be experimentally achieved in the electron's rest frame. The radiation
being emitted, the so-called channeling radiation, is a well studied
phenomenon. However only the first order diagram corresponding to emission of a
single photon has been studied so far. We elaborate on how the 2 photon
emission process should be understood in terms of a two-step versus a one-step
process, i.e., if one can consider one photon being emitted after the other, or
if there is also a contribution where the two photons are emitted
'simultaneously'. From the calculated full probability we see that the two-step
contribution is simply the product of probabilities for single photon emission
while the additional one-step terms are, mainly, interferences due to several
possible intermediate virtual states. These terms can contribute significantly
when the crystal is thin. Therefore, in addition, we see how one can, for a
thick crystal, calculate multiple photon emissions quickly by neglecting the
one-step terms, which represents a solution of the problem of quantum radiation
reaction in a crystal beyond the usually applied constant field approximation.
We explicitly calculate an example of 180 GeV electrons in a thin Silicon
crystal and argue why it is, for experimental reasons, more feasible to see the
one-step contribution in a crystal experiment than in a laser experiment.

High Energy Physics - Phenomenology