Electron dynamics in slowly modulated crystals: theorems of power and adiabatic drifting

We discuss electron dynamics in crystals which are modulated slowly in space and time. The equations of motion for the center-of-mass position and crystal momentum of a wave packet involve Berry phase terms between every pair of these variables plus time. We derive general theorems about the time evolution of energy and adiabatic drifting of position and crystal momentum. Together with a semiclassical quantization rule, these results enable us to derive the pattern of band splitting in one-dimensional quasiperiodic systems. The effect of external electromagnetic fields on electron dynamics in a modulated crystal is also discussed.