Communications in Mathematical Sciences

Volume 14 (2016)

Number 5

Decoherence for a heavy particle interacting with a light particle: New analysis and numerics

Pages: 1373 – 1415

DOI: http://dx.doi.org/10.4310/CMS.2016.v14.n5.a8

Authors

Riccardo Adami (Dipartimento di Scienze Matematiche G.L. Lagrange, Politecnico di Torino, Italy)

Maxime Hauray (Université d’Aix-Marseille, CNRS, École Centrale Marseille, Marseille, France)

Claudia Negulescu (Université de Toulouse, CNRS, UPS, Institut de Mathématiques de Toulouse, France)

Abstract

We study the dynamics of a quantum heavy particle undergoing a repulsive interaction with a light particle. The main motivation is the detailed description of the loss of coherence induced on a quantum system (in our model, the heavy particle) by the interaction with the environment (the light particle).

The content of the paper is analytical and numerical.

Concerning the analytical contribution, we show that an approximate description of the dynamics of the heavy particle can be carried out in two steps: first comes the interaction, then the free evolution. In particular, all effects of the interaction can be embodied in the action of a collision operator that acts on the initial state of the heavy particle. With respect to previous analytical results on the same topics, we turn our focus from the Møller wave operator to the full scattering operator, whose analysis proves to be simpler.

Concerning the numerical contribution, we exploit the previous analysis to construct an efficient numerical scheme that turns the original, multi-scale, two-body problem into two one-body problems which can be solved separately. This leads to a considerable gain in simulation time. We present and interpret some simulations carried out on specific one-dimensional systems by using the new scheme.

According to simulations, decoherence is produced by an interference-free bump which arises from the initial state of the heavy particle immediately after the collision. We support such a picture by numerical evidence as well as by an approximation theorem.

Keywords

quantum mechanics, Schrödinger equation, heavy-light particle scattering, interference, decoherence, asymptotic analysis, numerical discretization

2010 Mathematics Subject Classification

35Q41, 65M06, 81S22

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