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arXiv:quant-ph/0702197

Particle Localization by Decoherence and Classical Lensing

I. Knittel

Published Feb 20, 2007• 0 citations
arXivHTMLPDF

Abstract

A phenomenological model of the time evolution of a particle wavepacket is presented that is subject to scattering event with small momentum transfer. It is suited for three dimensions and allows for an additional potential. For a random value of a phase parameter, it is equivalent to the decoherence by scattering off plane wave field states. Imposing some condition on the same phase parameter, the model delivers definite outcomes in a local pointer base. Common sense notions about the quantum-classical transition are readily obtained: Decoherent free particle solutions are localized constant-width wavepackets, quantum barrier scattering results in random reflection and transmission. The scattering process assumed in the phenomenological model is realized by scattering of a particle pair interacting by a force-distance law with small momentum transfer. Particles are localized not by decoherence but by a classical lensing effect. Schmidt paths in the Schmidt representation of the multiparticle state are products of single-particle wavepackets. Therefore the model has a natural 'Many Worlds' interpretation, identifying each Schmidt path with a classical branch. Different runs of the phenomenological model map the state of the same particular particle within different branches.

DOI
10.48550/arxiv.quant-ph/0702197

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A phenomenological model of the time evolution of a particle wavepacket is presented that is subject to scattering event with small momentum transfer. It is suited for three dimensions and allows for an additional potential. For a random value of a phase parameter, it is equivalent to the decoherence by scattering off plane wave field states. Imposing some condition on the same phase parameter, the model delivers definite outcomes in a local pointer base. Common sense notions about the quantum-classical transition are readily obtained: Decoherent free particle solutions are localized constant-width wavepackets, quantum barrier scattering results in random reflection and transmission. The scattering process assumed in the phenomenological model is realized by scattering of a particle pair interacting by a force-distance law with small momentum transfer. Particles are localized not by decoherence but by a classical lensing effect. Schmidt paths in the Schmidt representation of the multiparticle state are products of single-particle wavepackets. Therefore the model has a natural 'Many Worlds' interpretation, identifying each Schmidt path with a classical branch. Different runs of the phenomenological model map the state of the same particular particle within different branches.
A particle initially in a pure state but interacting with some environment evolves into a discrete ensemble of pure states, the eigenstates of its reduced density operator, with ensemble probabilities given by the corresponding eigenvalues. In this work, we use numerics to present explicit results for the time-dependence of these eigenvalues and eigenstates for simple scattering experiments in one and two dimensions. This provides a time-resolved picture of the scattering process, showing in detail how an initial state described entirely in terms of continuous parameters evolves into a discrete set of possible outcomes, each with an associated probability and time-evolving wavefunction. We find that for scattering of Gaussian wavepackets in one dimension, the late time spectrum is dominated by two large eigenvalues nearly equal to the transmission and reflection probabilities associated with the central value of momentum. The corresponding eigenstates appear as single-peaked reflected or transmitted wavepackets. The remaining smaller eigenvalues, which increase to a maximum during scattering and then decrease to small values, correspond to reflected or transmitted wavepackets with multiple spatially separated parts. In this case and also for two-dimensional scattering, we find that successively smaller eigenvalues correspond to probability distributions with successively more peaks. These multi-peaked states correspond to outcomes of the scattering experiment where a particle initially in a single wavepacket ends up in a superposition of separated wavepackets after scattering.
I will present some of the results obtained regarding the emergence of decoherence in neutrino oscillations. In our model all the particles, including the source and detector, are treated dynamically and evolved consistently with Quantum Field Theory; decoherence can emerge naturally given the time evolution of the initial state and the final state considered. We have shown that some of the assumptions commonly used in the literature, such as the covariance of the wavepackets, are inconsistent. We found that a crucial ingredient for decoherence is the localization in space-time of the neutrino creation and detection: in Nature, such a measurement is usually carried out by environmental interactions, however it could also be approximated by considering localized wavefunctions in the final state. On the other hand, if the environmental interactions are not present (for example, if the decay happens in vacuum), the final position of the daughter particles will not be measured, i.e. they will be described by plane waves instead: in this case the neutrino is not localized either, and we don't have decoherence. A consequence of the time-evolution is that a Gaussian wavepacket will gradually spread: I will show that such an effect could in principle affect decoherence; moreover it would depend on the absolute mass scale of the neutrino, not on the Δm2\Delta m^2Δm2, which could offer a possible way to probe such a parameter by studying the neutrino oscillations.
In the context of nonrelativistic quantum mechanics, Gaussian wavepacket solutions of the time-dependent Schrödinger equation provide useful physical insight. This is not the case for relativistic quantum mechanics, however, for which both the Klein-Gordon and Dirac wave equations result in strange and counterintuitive wavepacket behaviors, even for free-particle Gaussians. These behaviors include zitterbewegung and other interference effects. As a potential remedy, this paper explores a new trajectory-based formulation of quantum mechanics, in which the wavefunction plays no role [ Phys. Rev. X , 4 , 040002 (2014)]. Quantum states are represented as ensembles of trajectories, whose mutual interaction is the source of all quantum effects observed in nature—suggesting a "many interacting worlds" interpretation. It is shown that the relativistic generalization of the trajectory-based formulation results in well-behaved free-particle Gaussian wavepacket solutions. In particular, probability density is positive and well-localized everywhere, and its spatial integral is conserved over time—in any inertial frame. Finally, the ensemble-averaged wavepacket motion is along a straight line path through spacetime. In this manner, the pathologies of the wave-based relativistic quantum theory, as applied to wavepacket propagation, are avoided.
Experiments involving single or few elementary particles are completely described by Quantum Mechanics. Notwithstanding the success of that quantitative description, various aspects of observations, as nonlocality and the statistical randomness of results, remain as mysterious properties apart from the quantum theory, and they are attributed to the strangeness of the microscopic world. Here we restart from the fundamental relations of uncertainty to reformulate the probability law of Born including the temporal variable. Considering that both the spatial and the temporal variables play a symmetric role in the wave-function Ψ(x,t) , a temporal wavepacket is built and analysed. The probability density is written as p(x,t) = | Ψ(x,t) |^2, where the probabilistic interpretation for the temporal wavepacket is equivalent to Born's law for the spatial variable, x. For the convenience of the discussion of the role of the temporal variable, we write p(x_0,t) = | Ψ(x_0,t) |^2 for a free particle, expressing only the temporal wavepacket, then we discuss its spread. In the light of the evolution of this temporal wavepacket we analyse basic processes of matter-wave interaction, involving single and entangled entities. Nonlocality appears then as a consequence of the spread of the temporal wavepacket; and the position of each detected event in two-slits interferometry as due to the independent phases of the spatial and temporal wavepackets.
Quantum state diffusion provides a dynamics for the localization or reduction of a quantum-mechanical wavepacket during a measurement or similar physical process. The essence of this localization dynamics is captured in the classical theory presented here, which applies to the common situation where the localization takes place over macroscopic distances and the wave properties of the system are no longer relevant. It provides a picture of the localization process in classical phase space, and a practical aid for computations on open quantum systems. The theory is developed from classical Hamiltonian dynamics in phase space and the known properties of localization. As an example it is used to illustrate how absorption by a screen leads to quantum jumps. A derivation from quantum state diffusion using the Wigner function is given in the companion paper.