Self consistent nonlinear transverse quantum dynamics of a cold relativistic electron beam in a magnetized plasma

A relativistic electron/positron beam travelling in a cold plasma in the overdense regime (n0>> nb) is the driver of large amplitude plasma oscillations that are generated by the Plasma Wake Field excitation. The beam experiences the 3D effects of the wake field that it has produced by itself (self interaction). In the long beam limit, the transverse effects due to the self interaction (f.i., self focusing/defocusing) are dominant compared to the longitudinal ones. Here, ignoring the longitudinal beam dynamics, a theoretical investigation of the quantum transverse beam motion is carried out when a relativistic electron/positron beam is travelling along an external magnetic field. This is done by adopting a fluid model of a magnetized plasma describing the Plasma Wake Field excitation driven by the beam density and current. On the other hand, taking into account the quantum nature of the single particle of the beam, it is shown that the transverse electron/positron dynamics is governed by a 2D Schrödinger equation in the form of the Gross-Pitaevskii equation. The latter accounts for the collective effects due to both the plasma wake field and the external magnetic field. The above set of equations governing the beam-plasma system (i.e., fluid plus 2D Schrödinger quations), is then reduced to a pair of coupled equations that can be thought as a quantum Zakharov system, leading in general to a 2D nonlocal and nonlinear Schrödinger equation. In the weakly focussed regime, the analysis of this equation is carried out, both analytically and numerically. Remarkably, the existence of quantum beam vortices is shown and the conditions for the self focusing and collapse that include the quantum nature of the beam particles are discussed.