Propagation of Coherent States through Conical Intersections

In this paper, we analyze the propagation of a wave packet through a conical intersection. This question has been 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 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.

[1]  K. Kay,et al.  The Herman–Kluk approximation: Derivation and semiclassical corrections , 2006 .

[2]  C. Zener.  Non-Adiabatic Crossing of Energy Levels , 1932 .

[3]  Semiclassical analysis , 2019, Graduate Studies in Mathematics.

[4]  C. Lasser,et al.  An Egorov Theorem for Avoided Crossings of Eigenvalue Surfaces , 2016, 1605.03520.

[5]  Jianfeng Lu,et al.  Frozen Gaussian approximation with surface hopping for mixed quantum-classical dynamics: A mathematical justification of fewest switches surface hopping algorithms , 2016, Math. Comput..

[6]  P. Gérard,et al.  Mesures semi-classiques et ondes de Bloch , 1991 .

[7]  Caroline Lasser,et al.  Single switch surface hopping for molecular dynamics with transitions. , 2008, The Journal of chemical physics.

[8]  Caroline Lasser,et al.  Propagation through Generic Level Crossings: A Surface Hopping Semigroup , 2008, SIAM J. Math. Anal..

[9]  R. Carles,et al.  Nonlinear Coherent States and Ehrenfest Time for Schrödinger Equation , 2009, 0912.1939.

[10]  Alexander B. Watson,et al.  Wavepackets in Inhomogeneous Periodic Media: Propagation Through a One-Dimensional Band Crossing , 2017, Communications in Mathematical Physics.

[11]  D. Robert,et al.  Coherent States and Applications in Mathematical Physics , 2012, Theoretical and Mathematical Physics.

[12]  André Martinez,et al.  Twisted Pseudodifferential Calculus and Application to the Quantum Evolution of Molecules , 2008, 0809.3663.

[13]  George A. Hagedorn,et al.  Landau-Zener transitions through small electronic eigenvalue gaps in the Born-Oppenheimer approximation , 1998 .

[14]  D. Robert,et al.  On time dependent Schrödinger equations: Global well-posedness and growth of Sobolev norms , 2016, 1610.03359.

[15]  Patrick Gérard,et al.  A Landau-Zener Formula for Non-Degenerated Involutive Codimension 3 Crossings , 2003 .

[16]  Didier Robert,et al.  ON THE HERMAN-KLUK SEMICLASSICAL APPROXIMATION , 2009, 0908.0847.

[17]  P. Markowich,et al.  Homogenization limits and Wigner transforms , 1997 .

[18]  Y. Meyer.  Nouvelles estimations pour les solutions d'équations aux dérivées partielles non linéaires , 1982 .

[19]  R. Carles,et al.  A nonlinear adiabatic theorem for coherent states , 2010, 1010.5977.

[20]  E. Kluk,et al.  A semiclasical justification for the use of non-spreading wavepackets in dynamics calculations , 1984 .

[21]  Alexander Elgart,et al.  The Adiabatic Theorem of Quantum Mechanics , 1998 .

[22]  Torben Swart,et al.  A Mathematical Justification for the Herman-Kluk Propagator , 2007, 0712.0752.

[23]  A. G. Childs.  On the ²-boundedness of pseudo-differential operators , 1976 .

[24]  C. Lubich.  From Quantum to Classical Molecular Dynamics: Reduced Models and Numerical Analysis , 2008 .

[25]  Stefan Teufel,et al.  Adiabatic perturbation theory in quantum dynamics , 2003 .

[26]  C. Kammerer.  A non‐commutative Landau‐Zener formula , 2004 .

[27]  David Sattlegger,et al.  Discretising the Herman–Kluk propagator , 2016, Numerische Mathematik.

[28]  P. Gérard,et al.  Mesures semi-classiques et croisement de modes , 2000 .

[29]  Y. D. Verdière.  The level crossing problem in semi-classical analysis I. The symmetric case , 2003 .

[30]  Clotilde Fermanian Kammerer,et al.  Semiclassical analysis of generic codimension 3 crossings , 2004 .

[31]  K. Kay,et al.  Integral expressions for the semiclassical time‐dependent propagator , 1994 .

[32]  Mouez Dimassi,et al.  Spectral asymptotics in the semi-classical limit , 1999 .

[33]  George A. Hagedorn,et al.  Molecular propagation through electron energy level crossings , 1994 .

[34]  D. Robert.  PROPAGATION OF COHERENT STATES IN QUANTUM MECHANICS AND APPLICATIONS , 2006 .

[35]  Y. D. Verdière.  The level crossing problem in semi-classical analysis , 2003 .

[36]  D. Robert,et al.  Adiabatic and non-adiabatic evolution of wave packets and applications to initial value representations , 2020, 2011.01618.

[37]  Caroline Lasser,et al.  Propagation of Wave Packets for Systems Presenting Codimension One Crossings , 2020, Communications in Mathematical Physics.

[38]  G. Nenciu,et al.  Linear adiabatic theory. Exponential estimates , 1993 .

[39]  Lysianne Hari.  PROPAGATION OF SEMICLASSICAL WAVE PACKETS THROUGH AVOIDED EIGENVALUE CROSSINGS IN NONLINEAR SCHRÖDINGER EQUATIONS , 2014, Journal of the Institute of Mathematics of Jussieu.