Breathing oscillations accompanying Bloch oscillations of wavepackets in periodic potentials

Using a 1D tight-binding model, we study the evolution of a well-localized wavepacket of Bloch states under an applied electric field. We apply a novel algorithm~\footnote{ along the lines of I. Souza {\it et. al.}, Phys. Rev. B {\bf 69}, 085106 (2004)} for solving numerically the equations of motion which does not rely on the single-band approximation and can thus be used to explore interband Zener tunneling effects. In addition to the well-known Bloch oscillations of the center of the packet, we show that as the waveform moves in k-space, its real-space width varies in response to the change in the local quantum metric,~\footnote{N. Marzari and D. Vanderbilt, Phys. Rev. B {\bf 56}, 12\,847 (1997).} $g(k)$, of the underlying Bloch states. A generalized uncertainty relation is obtained between the spread in position and in {\it crystal} momentum of a wavepacket. It differs from the usual position/momentum uncertainty relation because of the interband matrix elements of the position operator in the crystal-momentum representation, which introduce a correction in terms of $g(k)$.