Abstract:
We study the dynamics of wavepackets in crystals whose structure is spatially inhomogeneous. We make the assumption that inhomogeneities occur over a length scale which is long compared to the lattice period so that we may treat the two scales as approximately independent. We work mainly in the setting of Schrodinger's equation, where the crystal structure is modeled by a `two-scale' potential which varies periodically on the `fast' scale and smoothly on the `slow' scale, but our methods can be applied also to Maxwell's equations where the crystal structure is modeled by a `two-scale' matrix of constitutive relations. Phenomena which result from spatial variation of the crystal structure are: the anomalous velocity of wavepackets due to Berry curvature of the Bloch spectral band (responsible for the spin Hall effect of light), and Landau-Zener-type inter-band transitions, in the presence of spectral band crossings. This is joint work with Michael Weinstein and Jianfeng Lu. |