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Path Integral Technique for Inter-Landau Level Tunneling

We write a coherent state Feynman path integral for tunneling between Landau levels by developing a continuum representation for the Landau level index and also by showing that a gradient expansion of a sufficiently smooth potential can be written in terms of creation and annihilation operators for the cyclotron motion. We restrict our consideration to only the lowest two Landau levels, n=0 and n=1. We employ a steepest descent method for determining the coupled equations of motion for the path of the guiding center drift coordinate R(τ) and the single Landau index continuous coordinate θ(τ) in a smooth but otherwise arbitrary 2D potential.