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European Journal of Physics

Free evolution of a mixture of quantum states with classical space and momentum uncertainties

R O Barrachina, Francisco Navarrete, Marcelo F. Ciappina

Published Aug 15, 2019• 3 citations
Publisher

Abstract

Abstract This article continues and completes a recent one regarding a quantum version of van Cittert–Zernike’s theorem (Fabre et al 2018 Eur. J. Phys. 39 015401). In that paper, we considered a mixture of quantum states that only differed in their initial positions, but not on their momenta. Here, we relax this nonphysical assumption and analyse how the inclusion of both classical uncertainties in the initial position and momentum of a mixture of—otherwise identical—free quantum states which do not interact with an environment, affects its quantum coherence length. We show that the mathematical development and the final result are simple enough to make possible the incorporation of this subject into a basic course of quantum physics. Furthermore, we demonstrate that these results provide a comprehensive solution to a recent controversy in atomic collision physics.

DOI
10.1088/1361-6404/ab3b6f

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Abstract This article continues and completes a recent one regarding a quantum version of van Cittert–Zernike’s theorem (Fabre et al 2018 Eur. J. Phys. 39 015401). In that paper, we considered a mixture of quantum states that only differed in their initial positions, but not on their momenta. Here, we relax this nonphysical assumption and analyse how the inclusion of both classical uncertainties in the initial position and momentum of a mixture of—otherwise identical—free quantum states which do not interact with an environment, affects its quantum coherence length. We show that the mathematical development and the final result are simple enough to make possible the incorporation of this subject into a basic course of quantum physics. Furthermore, we demonstrate that these results provide a comprehensive solution to a recent controversy in atomic collision physics.
The fundamental indication of the departure of quantum mechanics from the classical world is the so-called quantum coherence.Typically, we define it as the characteristic of systems which are in a superposition of states yielding interference patterns in certain kinds of experiments.In addition to its captivating philosophical implications, quantum coherence turned out to be a valuable tool in different areas, ranging from quantum information to biology, where it was used to describe several fundamental processes.Here, we go one step further to study how classical uncertainties in a mixture of similar states reduce quantum coherence in quantum scattering theory.To this end, we deal with different examples, all of them with roots in the widely studied Feynman's two-slit thought experiment.We finally propose an operational and intuitive definition of the concept of coherence length whose implications largely transcend the simplicity of the corresponding mathematical development, as it is demonstrated when applied to the analysis of some recent atomic and molecular processes.
The mechanism of the transition of a dynamical system from quantum to classical mechanics is one of the remaining challenges of quantum theory. Currently, it is considered to occur via decoherence caused by entanglement and/or stochastic interaction with a quantum environment. Here we prove that, in the absence of de-phasing environmental interaction, the asymptotic spatial wave function of any quantum system, propagating over distances and times large on an atomic scale but still microscopic, and subject to arbitrary deterministic fields, becomes proportional to the initial momentum wave function, \emph{where the particle positions and momenta are related by classical mechanics.} This implies that detection of particle positions and momenta at different times will yield results in accordance with classical mechanics, without the need to postulate decoherence of the wave function.
Recent progress on quantum state engineering has enabled the preparation of quantum photonic systems comprising multiple interacting particles. Interestingly, multiphoton quantum systems can host many complex forms of interference and scattering processes that are essential to perform operations that are intractable on classical systems. Unfortunately, the quantum coherence properties of multiphoton systems degrade upon propagation leading to undesired quantum-to-classical transitions. Furthermore, the manipulation of multiphoton quantum systems requires of nonlinear interactions at the few-photon level. Here, we introduce the quantum van Cittert-Zernike theorem to describe the scattering and interference effects of propagating multiphoton systems. This fundamental theorem demonstrates that the quantum statistical fluctuations, which define the nature of diverse light sources, can be modified upon propagation in the absence of light-matter interactions. The generality of our formalism unveils the conditions under which the evolution of multiphoton systems can lead to surprising classical-to-quantum transitions. Specifically, we show that the implementation of conditional measurements may enable the all-optical preparation of multiphoton systems with attenuated quantum statistics below the shot-noise limit. Remarkably, this effect had not been discussed before and cannot be explained through the classical theory of optical coherence. As such, our work opens new paradigms within the established field of quantum coherence.
INTRODUCTION The purpose of this article is to bring out in as conceptually clear terms as possible what seems to be a major incompleteness in the probability theory of particles offered by classical quantum mechanics. The exact nature of this incompleteness is illustrated by consideration of some simple quantum-mechanical examples. In addition, these examples are contrasted with the fundamental assumptions of Brownian motion in classical physics on the one hand, and with a controversy of a decade ago in mathematical psychology. (The psychological example is described in detail in the appendix). Our central claim is that classical quantum mechanics is radically incomplete in its probabilistic account of the motion of particles. In the last part of the article we derive the time-dependent joint distribution of position and momentum of the linear harmonic oscillator, and show how the apparently physically paradoxical statistical independence of position and momentum has a natural explanation. The explanation is given within the framework of the non-quantum-mechanical stochastic theory we construct for such oscillators. Before entering into any technical details, we shall attempt to spell out in intuitive terms our incompleteness claim. To begin with, there are two senses of completeness that we want to distinguish. One sense of completeness is that demanded by any hidden variable theories of quantum mechanics. States should be found, or be theoretically characterizable at least, that lead to precise values for all observables; that is, no observable is to have a variance greater than zero.