Separation of scales: Dynamical approximations for composite quantum systems

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 composite quantum-dynamical systems that can be partitioned into weakly interacting subsystems, similar to system-bath type situations. Using a factorized wave function ansatz, we mathematically characterize dynamical scale separation.Specifically, we investigate a coupling régime that is partially flat, i.e., slowly varying with respect to one set of variables, for example, those of the bath. Further, we study the situation where one of the sets of variables is semiclassically scaled and derive a quantum-classical formulation. In both situations, we propose two schemes of dimension reduction: one based on Taylor expansion (collocation) and the other one based on partial averaging (mean-field). We analyze the error for the wave function and for the action of observables, obtaining comparable estimates for both approaches. The present study is the first step towards a general analysis of scale separation in the context of tensorized wavefunction representations.

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: Dynamical approximations for composite quantum systems. Journal of Physics A: Mathematical and Theoretical, 2021, 54 (41), pp.article n°414002. ⟨10.1088/1751-8121/ac219d⟩. ⟨hal-03064636v3⟩