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Propagation of coherent states through conical intersections

Clotilde Fermanian Kammerer, Stephanie Gamble and Lysianne Hari

Vol. 5 (2023), No. 2, 323–376
Abstract

We analyze the propagation of a wave packet through a conical intersection. This question was addressed for Gaussian wave packets in the 90s by George Hagedorn and we consider here a more general setting. We focus on the case of the Schrödinger equation but our methods are general enough to be adapted to systems presenting codimension-2 crossings and to codimension-3 ones with specific geometric conditions. Our main theorem gives explicit transition formulas for the profiles when passing through a conical crossing point, including precise computation of the transformation of the phase. Its proof is based on a normal form approach combined with the use of superadiabatic projectors and the analysis of their degeneracy close to the crossing.

Dedicated to the memory of George Hagedorn (1953–2023)

Keywords
mathematical physics, semiclassical analysis, Schrödinger equation, wave packets, conical intersections
Mathematical Subject Classification
Primary: 35B40, 35G35, 35Q40, 35Q41
Secondary: 81Q05, 81Q20, 81R30
Milestones
Received: 6 January 2022
Revised: 14 July 2022
Accepted: 15 November 2022
Published: 26 June 2023
Authors
Clotilde Fermanian Kammerer
Université Paris Est Creteil
CNRS, LAMA
Créteil
France
Université Gustave Eiffel
LAMA
Marne-la-Vallée
France
Stephanie Gamble
Department of Mathematics
Virginia Tech
Blacksburg, VA
United States
Savannah River National Laboratory
Aiken, South Carolina
United States
Lysianne Hari
Laboratoire de Mathématiques de Besançon (LMB)
Université Bourgogne Franche-Comté
CNRS, UMR 6623
Besançon
France