Math @ Duke

Publications [#338113] of Alexander Watson
Papers Published
 Watson, A; Weinstein, MI, Wavepackets in Inhomogeneous Periodic Media: Propagation Through a OneDimensional Band Crossing,
Communications in Mathematical Physics, vol. 363 no. 2
(October, 2018),
pp. 655698, Springer Nature America, Inc [doi]
(last updated on 2019/10/16)
Abstract: © 2018, SpringerVerlag GmbH Germany, part of Springer Nature. We consider a model of an electron in a crystal moving under the influence of an external electric field:Schrödinger’s equation in one spatial dimension with a potential which is the sum of a periodic function V and a smooth function W. We assume that the period of V is much shorter than the scale of variation of W and denote the ratio of these scales by ϵ. We consider the dynamics of semiclassical wavepacket asymptotic (in the limit ϵ↓ 0) solutions which are spectrally localized near to a crossing of two Bloch band dispersion functions of the periodic operator 12∂z2+V(z). We show that the dynamics is qualitatively different from the case where bands are wellseparated: at the time the wavepacket is incident on the band crossing, a second wavepacket is ‘excited’ which has opposite group velocity to the incident wavepacket. We then show that our result is consistent with the solution of a ‘Landau–Zener’type model.


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