Investigation of two photon emission in strong field QED using channeling in a crystal

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 electrons 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.