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 \(x\)--\(y\) 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 \(\ell =(eaB/h)^{-1}\) 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 \(\propto\hbar\).