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arXiv:1705.00860 · Physical Review Letters

Possibility to Probe Negative Values of a Wigner Function in Scattering of a Coherent Superposition of Electronic Wave Packets by Atoms

Dmitry V. Karlovets, Valeriy G. Serbo

Published Oct 24, 2017• 13 citations
arXivHTMLPDF

Abstract

Within a plane-wave approximation in scattering, an incoming wave packet's Wigner function stays positive everywhere, which obscures such purely quantum phenomena as nonlocality and entanglement. With the advent of the electron microscopes with subnanometer-sized beams, one can enter a genuinely quantum regime where the latter effects become only moderately attenuated. Here we show how to probe negative values of the Wigner function in scattering of a coherent superposition of two Gaussian packets with a nonvanishing impact parameter between them (a Schrödinger's cat state) by atomic targets. For hydrogen in the ground 1s state, a small parameter of the problem, a ratio a/σ_{⊥} of the Bohr radius a to the beam width σ_{⊥}, is no longer vanishing. We predict an azimuthal asymmetry of the scattered electrons, which is found to be up to 10%, and argue that it can be reliably detected. The production of beams with the not-everywhere-positive Wigner functions and the probing of such quantum effects can open new perspectives for noninvasive electron microscopy, quantum tomography, particle physics, and so forth.

DOI
10.1103/physrevlett.119.173601

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Within a plane-wave approximation in scattering, an incoming wave packet's Wigner function stays positive everywhere, which obscures such purely quantum phenomena as nonlocality and entanglement. With the advent of the electron microscopes with subnanometer-sized beams, one can enter a genuinely quantum regime where the latter effects become only moderately attenuated. Here we show how to probe negative values of the Wigner function in scattering of a coherent superposition of two Gaussian packets with a nonvanishing impact parameter between them (a Schrödinger's cat state) by atomic targets. For hydrogen in the ground 1s state, a small parameter of the problem, a ratio a/σ_{⊥} of the Bohr radius a to the beam width σ_{⊥}, is no longer vanishing. We predict an azimuthal asymmetry of the scattered electrons, which is found to be up to 10%, and argue that it can be reliably detected. The production of beams with the not-everywhere-positive Wigner functions and the probing of such quantum effects can open new perspectives for noninvasive electron microscopy, quantum tomography, particle physics, and so forth.
Within a plane-wave approximation in scattering, an incoming wave packet's Wigner function stays everywhere positive, which obscures such quantum phenomena as non-locality and entanglement. With the advent of the modern electron microscopes with subnanometer-sized beams one can enter the genuinely quantum regime where the latter effects become only moderately attenuated. Here we show how to probe negative values of the Wigner function in scattering of a coherent superposition of two Gaussian packets with a non-vanishing impact-parameter between them (a Schr\odinger's cat state) by atomic targets. For hydrogen in the ground 1s1s1s state, a small parameter of the problem, a ratio a/σ⊥a/\sigma_{\perp}a/σ⊥​ of the Bohr radius aaa to the beam width σ⊥\sigma_{\perp}σ⊥​, is no longer vanishing. We predict an azimuthal asymmetry of the scattered electrons, which is found to be up to 10%10 \%10% when the impact-parameter is comparable to the beam width, and argue that it can be reliably detected. Production of subnanometer-sized Schr\odinger's cat states and probing such purely quantum effects can open new perspectives in studying the interiors of atoms.
While a plane-wave approximation in high-energy physics works well in a majority of practical cases, it becomes inapplicable for scattering of the vortex particles carrying orbital angular momentum, of Airy beams, of the so-called Schrödinger cat states, and their generalizations. Such quantum states of photons, electrons and neutrons have been generated experimentally in recent years, opening up new perspectives in quantum optics, electron microscopy, particle physics, and so forth. Here we discuss the non-plane-wave effects in scattering brought about by the novel quantum numbers of these wave packets. For the well-focused electrons of intermediate energies, already available at electron microscopes, the corresponding contribution can surpass that of the radiative corrections. Moreover, collisions of the cat-like superpositions of such focused beams with atoms allow one to probe effects of the quantum interference, which have never played any role in particle scattering.
Abstract The generation of quantum entanglement through scattering processes is a fundamental mechanism in quantum information science. In this work, we investigate how coherent-state superpositions, represented as spatial Gaussian wave-packet superpositions, modify nonclassical phase-space features and enhance entanglement production during the scattering of a single particle from a localized spin impurity governed by a Heisenberg-type interaction. We construct even and odd superpositions of coherent states and analyze their time evolution using Fourier-space propagation. The corresponding Wigner distributions are computed analytically, revealing os-cillatory interference fringes and negative-valued regions that quantify nonclassicality. Using the nonclassical volume as a metric, we show that increasing the spatial separation between the Gaussian components enhances nonclassicality, eventually saturating at a stable asymptotic maximum. By incorporating these superposed states as incident wave packets, we compute the post-scattering spin entanglement between the reflected particle and the impurity using the negativity measure. Our results demonstrate that the symmetry of the initial coherent-state superposition fundamentally governs the stability of the generated entanglement: while the even superposition exhibits parameter-sensitive, oscillatory entanglement as the wave-packet separation increases, the odd superpo-sition yields a robust, decay-resistant entanglement plateau across all separation scales. These findings highlight the critical role of engineered nonclassical initial states in optimizing scattering-based quantum technologies
Quantum information and quantum communication are both strongly based on concepts of quantum superposition and entanglement. Entanglement allows distinct bodies, that share a common origin or that have interacted in the past, to continue to be described by the same wave function until evolution is coherent. So, there is an equivalence between coherence and entanglement. In this paper, we show the relation between quantum coherence and quantum interference and the negative parts of the Wigner quasi-distribution, using the Wigner signed-particle formulation. A simple physical problem consisting of electrons in a nanowire interacting with the potential of a repulsive dopant placed in the center of it creates a quasi two-slit electron system that separates the wave function into two entangled branches. The analysis of the Wigner quasi-distribution of this problem establishes that its negative part is principally concentrated in the region after the dopant between the two entangled branches, maintaining the coherence between them. Moreover, quantum interference is shown in this region both in the positive and in the negative part of the Wigner function and is produced by the superposition of Wigner functions evaluated at points of the momentum space that are symmetric with respect to the initial momentum of the injected electrons.
We introduce an operational criterion to identify Wigner function (WF) negativity for an arbitrary quantum state within the framework of quantum non-demolition measurements. This criterion corresponds to experimentally accessible schemes that enable a direct measurement of the WF, and establishes the coherent-state basis as a privileged basis for determining when the WF exhibits negative regions. We show that the absence (presence) of coherent superpositions in the coherent-state basis provides direct information about the positivity (negativity) of the WF. In particular, the absence of such superpositions constitutes a sufficient condition for WF positivity. Although a general proof of necessity remains elusive, we demonstrate that this condition is also necessary in two relevant cases: Schrödinger-cat states and higher-order cat states on a circle. More precisely, for Schrödinger-cat states we establish a necessary and sufficient condition for the positivity of the WF in full generality, whereas for high-order cat states on a circle we derive an analogous condition in the limit of a large number of densely packed coherent states.
The problem of elastic scattering of a nonrelativistic electron Airy beam on a hydrogen atom in the ground 1s1s1s state is considered in the Born approximation. It is demonstrated that the angular dependence of the scattering probability density is, in general, azimuthally asymmetric. When the position of the atom happens to coincide with a minimum of the probability density of the Airy beam, the asymmetric pattern is represented by four separated peaks. We show that this behavior is very sensitive to the precision with which the relative position of the atom and the minima is defined and study how uncertainty in the position can affect observation of the azimuthal asymmetry. Finally, we consider a spatially localized target and discuss the difficulties of observing the asymmetry for targets with sizes exceeding a critical value determined by the beam parameters and by the position of the target center.