Chaotic Landau Level Mixing in Classical and Quantum Wells

We analyze the 3D motion of a charged particle between parallel planar potential barriers with a magnetic field applied at an angle $\theta$ to the barriers and an electric field normal to the barriers. A nonlinear map is derived for the classical system which gives analytic conditions for the occurrence of a chaotic energy exchange between the cyclotron and longitudinal motion as a function of $\theta$ and other system parameters. This energy exchange can lead to a population of very high Landau levels. Quantizing this problem suppresses energy exchange up to a critical angle determined by the localization transition.