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arXiv:quant-ph/0210199 · Journal of Mathematical Physics

Decoherence in a two-particle model

Detlef Dürr, Rodolfo Figari, Alessandro Teta

Published Mar 17, 2004• 24 citations
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Abstract

We consider a simple one-dimensional quantum system consisting of a heavy and a light particle interacting via a point interaction. The initial state is chosen to be a product state, with the heavy particle described by a coherent superposition of two spatially separated wave packets with opposite momentum and the light particle localized in the region between the two wave packets. We characterize the asymptotic dynamics of the system in the limit of small mass ratio, with an explicit control of the error. We derive the corresponding reduced density matrix for the heavy particle and explicitly compute the (partial) decoherence effect for the heavy particle induced by the presence of the light one for a particular set up of the parameters.

DOI
10.1063/1.1647692

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Abstract. We consider a non relativistic quantum system consisting of K heavy and N light particles in dimension three, where each heavy particle interacts with the light ones via a two-body potential αV. No interaction is assumed among particles of the same kind. Choosing an initial state in a product form and assuming α sufficiently small we characterize the asymptotic dynamics of the system in the limit of small mass ratio, with an explicit control of the error. In the case K = 1 the result is extended to arbitrary α. The proof relies on a perturbative analysis and exploits a generalized version of the standard dispersive estimates for the Schrödinger group. Exploiting the asymptotic formula, it is also outlined an application to the problem of the decoherence effect produced on a heavy particle by the interaction with the light ones. 1.
We study the dynamics of a quantum heavy particle undergoing a repulsive interaction with a light one. 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 M{\o}ller 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, multiscale, two-body problem in 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. Finally, we propose a new description of the mechanism of decoherence induced by scattering: 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.
The dynamical process of generation of entanglement between subsystems of an evolving quantum system is analysed in this paper. We consider one of the simplest possible continuous systems in which such a process can be rigorously studied: a quantum ‘heavy’ particle (the system) interacting via zero range forces with a lighter particle (the environment). In a three-dimensional model we study the asymptotic dynamics when the ratio between the masses of the light and the heavy particle is very small. The effect of decoherence on the heavy particle induced by the interaction is explicitly computed.
We recall the classical picture about the decoherence effects induced on a quantum sub-system by the interaction with a macroscopic environment. We discuss the need of a rigorous treatment, in mathematically reasonable models, of the dynamical process bringing the system in an entangled state with the environment. We consider a simple system consisting of two particles interacting via a two-body potential and we discuss a rigorous derivation of the asymptotic wave function of the system in the limit of small mass ratio. We apply the result to the explicit computation of the decoherence effect induced on the heavy particle, in concrete examples of quantum evolution.