Breathing Mode in Open-Orbit Magnetotransport: A Magnetic Lens with a Quantum Mechanical Focal Length

We consider the propagation of electrons in a lattice with an anisotropic dispersion in the $x\ensuremath{-}y$ plane (lattice constant $a$), such that it supports open orbits along the $x$ axis in an out-of-plane magnetic field $B$. We show that a point source excites a ``breathing mode,'' a state that periodically spreads out and refocuses after having propagated over a distance $\ensuremath{\ell}={(eaB/h)}^{\ensuremath{-}1}$ in the $x$ direction. Unlike known magnetic focusing effects, governed by the classical cyclotron radius, this is an intrinsically quantum mechanical effect with a focal length $\ensuremath{\propto}\ensuremath{\hbar}$.