Separation of scales: a quantum-classical approach for complex systems and a system-bath ansatz

Irene Burghardt (1) , Rémi Carles (2) , Clotilde Fermanian Kammerer (3) , Benjamin Lasorne (4) , Caroline Lasser (5)

1 Institute of Physical and Theoretical Chemistry [Frankfurt am Main]
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 LAMA - Laboratoire Analyse et de Mathématiques Appliquées
4 ICGM - Institut Charles Gerhardt Montpellier - Institut de Chimie Moléculaire et des Matériaux de Montpellier
5 Zentrum Mathematik, Technische Universität München

We consider quantum-dynamical systems that consist of two different parts as in system-bath Hamiltonians, and assume that each of these parts is described by a set of parameters. We investigate régimes of intersystem couplings that are partially flat because they vary slowly with respect to one set of variables (those of the bath, for example). We also study the situation where one of the sets of variables is semiclassically scaled; this case effectively leads to a quantum-classical formulation, due to the classical parameter evolution. In both situations, we propose two kinds of schemes of dimension reduction : one based on the Taylor expansion and the other one based on partial averaging. We analyze the error estimates in terms of appropriate norms of the coupling both for the wave function and for the action of observables, showing that the choice between an approach by Taylor expansion or by average is more a question of numerical convenience than a matter of errors, since both approaches lead to comparable estimates.

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Chimie-Physique [physics.chem-ph]; Chimie théorique et/ou physique

Irene Burghardt, Rémi Carles, Clotilde Fermanian Kammerer, Benjamin Lasorne, Caroline Lasser. Separation of scales: a quantum-classical approach for complex systems and a system-bath ansatz. 2020. ⟨hal-03064636v1⟩