Attenuation of energy relaxation in chiral one-dimensional quantum channels

In order to probe energy relaxation in a one-dimensional chiral channel, we consider the injection of a dilute beam of electrons at a sharply-defined energy above the channel's Fermi sea. The injected electrons undergo finite-range interactions with channel electrons. In order to obtain the energy distribution in the channel as a function of the injected electrons' propagation time, we perform a non-perturbative calculation using bosonization. In this approach, low energy excitations in the channel (plasmons) propagate at a constant velocity that is higher than the injected electrons' velocity. We find that injected electrons lose only about one quantum of energy, measured in units of the ratio of plasmon velocity and interaction range. To investigate the resulting energy distribution also in the presence of a non-linear plasmon dispersion relation, we present a solution of the problem that is exact in the semiclassical limit, applicable when the injection energy is much higher than the typical plasmon energy, and when the energy loss is not too large.