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Communications in Computational Physics

A Numerical Study of Quantum Decoherence

Riccardo Adami, Claudia Negulescu

Published May 22, 2012• 7 citations
Publisher

Abstract

The present paper provides a numerical investigation of the decoherence effect induced on a quantum heavy particle by the scattering with a light one. The time dependent two-particle Schrödinger equation is solved by means of a time-splitting method. The damping undergone by the non-diagonal terms of the heavy particle density matrix is estimated numerically, as well as the error in the Joos-Zeh approximation formula.

DOI
10.4208/cicp.011010.010611a

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The present paper provides a numerical investigation of the decoherence effect induced on a quantum heavy particle by the scattering with a light one. The time dependent two-particle Schrödinger equation is solved by means of a time-splitting method. The damping undergone by the non-diagonal terms of the heavy particle density matrix is estimated numerically, as well as the error in the Joos-Zeh approximation formula.
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.
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.
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.
A logarithmic Schr\"odinger equation with time-dependent coupling to the nonlinearity is presented as a model of collisional decoherence of the wave function of a quantum particle in position space. The particular mathematical form of the logarithmic Schr\"odinger equation has been shown to follow from conditional wave theory, but the validity of the logarithmic Schr\"odinger equation has not yet been investigated numerically for general initial conditions. Using an operator-splitting approach, we solve the nonlinear equation of motion for the wave function numerically and compare it to the solution of the standard Joos-Zeh master equation for the density matrix. We find good agreement for the time-dependent behavior of the ensemble widths between the two approaches, but note curious zero-pinning behavior of the logarithmic Schr\"odinger equation, whereby the zeros of the wave function are not erased by continued propagation. By examining the derivation of the logarithmic Schr\"odinger equation from conditional wave theory, we indicate possible avenues of resolution to this zero-pinning problem.
Abstract We study quantum decoherence numerically in a system consisting of a relativistic quantum field theory coupled to a measuring device that is itself coupled to an environment. The measuring device and environment are treated as quantum, non-relativistic particles. We solve the Schrödinger equation for the wave function of this tripartite system using exact diagonalization. Although computational limitations on the size of the Hilbert space prevent us from exploring the regime where the device and environment consist of a truly macroscopic number of degrees of freedom, we nevertheless see clear evidence of decoherence: after tracing out the environment, the density matrix describing the system and measuring device evolves quickly towards a matrix that is close to diagonal in a subspace of pointer states. We measure the speed with which decoherence spreads in the relativistic quantum field theory for a range of parameters. We find that it is less than the speed of light but faster than the speed of the massive charges in the initial state.