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Physical Review A

Generation of tripartite states of flying electronic qubits and their characterization by energy measurements

S. Parida, Naresh Chandra

Published Jun 1, 2009• 5 citations
Publisher

Abstract

A scheme for generating states of three free electrons entangled with respect to their spins is suggested. It consists of sequential ejection of two Auger electrons (${e}_{2}$ and ${e}_{3}$) in the nonradiative decay of an inner-shell vacancy created due to the emission of photoelectron ${e}_{1}$ from an atom, say $A$. In the absence of spin-orbit interaction, the entanglement among the spin angular momenta of the flying $({e}_{1},{e}_{2},{e}_{3})$ is generated simply by the Coulomb interaction experienced by them inside $A$. Their states are classified according to the hierarchic structure suggested by D\"ur, Cirac, and Tarrach [Phys. Rev. Lett. 83, 3562 (1999); D\"ur and Cirac, Phys. Rev. A 61, 042314 (2000)]. The generation of fully separable, 1-electron biseparable, fully inseparable, or ``$1\ensuremath{\rightarrow}2$ entangled'' tripartite (in addition to various kinds of bipartite) states is shown to be completely determined only by the spin multiplicities of the electronic states of $A$ and of its ionic species $({A}^{{+}^{\ensuremath{\ast}}},{A}^{2{+}^{\ensuremath{\ast}}},{A}^{3+})$ participating in the suggested scheme. The entanglement of three electrons is Greenberger-Horne-Zeilinger type. The experimental characterization of these states is fully achieved merely by the measurements of the energies of $({e}_{1},{e}_{2},{e}_{3})$, without requiring any entanglement witness or other similar protocols hitherto developed in quantum information.

DOI
10.1103/physreva.79.062501

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A scheme for generating states of three free electrons entangled with respect to their spins is suggested. It consists of sequential ejection of two Auger electrons (e2{e}_{2}e2​ and e3{e}_{3}e3​) in the nonradiative decay of an inner-shell vacancy created due to the emission of photoelectron e1{e}_{1}e1​ from an atom, say AAA. In the absence of spin-orbit interaction, the entanglement among the spin angular momenta of the flying (e1,e2,e3)({e}_{1},{e}_{2},{e}_{3})(e1​,e2​,e3​) is generated simply by the Coulomb interaction experienced by them inside AAA. Their states are classified according to the hierarchic structure suggested by D"ur, Cirac, and Tarrach [Phys. Rev. Lett. 83, 3562 (1999); D"ur and Cirac, Phys. Rev. A 61, 042314 (2000)]. The generation of fully separable, 1-electron biseparable, fully inseparable, or ``1\ensuremath→21\ensuremath{\rightarrow}21\ensuremath→2 entangled'' tripartite (in addition to various kinds of bipartite) states is shown to be completely determined only by the spin multiplicities of the electronic states of AAA and of its ionic species (A+\ensuremath∗,A2+\ensuremath∗,A3+)({A}^{{+}^{\ensuremath{\ast}}},{A}^{2{+}^{\ensuremath{\ast}}},{A}^{3+})(A+\ensuremath∗,A2+\ensuremath∗,A3+) participating in the suggested scheme. The entanglement of three electrons is Greenberger-Horne-Zeilinger type. The experimental characterization of these states is fully achieved merely by the measurements of the energies of (e1,e2,e3)({e}_{1},{e}_{2},{e}_{3})(e1​,e2​,e3​), without requiring any entanglement witness or other similar protocols hitherto developed in quantum information.
In this paper, we show that bipartite states of an electronic qubit and ionic qudit, produced in a laboratory in as simple an experiment as photoionization in electric dipole approximation, can be separable or can possess either free or bound entanglement. The dimension of the Hilbert space of the qudit depends upon the electronic state $|{1}^{+}\ensuremath{\rangle}$ of the residual photoion A1+{A}^{1+}A1+: In the absence of spin-orbit interaction (SOI), when Russell-Saunders coupling is applicable, it is equal to the spin multiplicity of $|{1}^{+}\ensuremath{\rangle}$, but becomes (2JJJ+1) in $j\ensuremath{-}j$ coupling, where JJJ is the total angular momentum of $|{1}^{+}\ensuremath{\rangle}$ when SOI is taken into account. In the case of $L\ensuremath{-}S$ coupling, all the properties (relevant to a study in quantum information) of a qubit-qudit state can readily be predicted merely with a knowledge of the spins of the atomic target and of the residual photoion which can experimentally be verified by measuring them using an entanglement witness, or any other such protocols, or by doing a full quantum-state tomography.
We present a theoretical analysis of spin-entanglement between a photoelectron and an Auger electron sequentially ejected from an atom (say, AAA) following the absorption of a single photon. Entanglement, both without and with spin-orbit interaction has been analyzed. In the former case, entanglement in a photo-Auger electron pair is generated only by the electrostatic Coulomb forces inside an atom and completely characterized merely by the multiplicities of the electronic states of AAA, A+∗{A}^{{+}^{*}}A+∗, A2+{A}^{2+}A2+ participating in the process, without using any protocols already suggested in the literature for this purpose. The presence of both the Coulomb and spin-orbit interactions, on the other hand, couples the entanglement of this bipartite state with the dynamics and kinematics of the whole process, as well as with the properties of the absorbed photon. In such cases, therefore, it is not possible to predict, a priori, the photo-Auger electron entanglement. These discussions have quantitatively been illustrated by several examples.
In this Rapid Communication, we show that a simple process of two-step double ionization taking place in an atom AAA, following the absorption of a single photon, produces a spin-entangled state of two electrons. The degree of entanglement of this state can be tuned to the desired value by selecting appropriate total spin quantum numbers of the electronic states of each of the three atomic species (i.e., A,A+,A2+A,{A}^{+},{A}^{2+}A,A+,A2+) participating in the process in Russell-Saunders coupling. These entangled states are readily characterized by measuring only energies of two emitted electrons, without requiring the entanglement witness, or any other such protocol.
Objectives: To study of entanglement between two electronic qubits,without observing spin orbit interaction (SOI), produced by double photo ionization from Xeon atom following the absorption of a photon. Methods: In absence of SOI, Russel-Salunders coupling (L-S coupling) is applicable. As the estimations of entanglement we have considered PeresHorodecki condition and negativity. In case of L-Scoupling, all the properties of electronic qubits are predicted merely with the knowledge of the spins of the target atom, the residual photoion, emitted electrons and the state of polarization of the absorbed photon. Findings: We have found that the electronic qubits may be totally entangled, partially entangled or separable depending on spin states of target Xe, residualdication Xe2+,emitted electrons and the state of polarization of incident photon as well as of the direction sspin quantization and ejection of the photoelectrons. Applications/ Improvements: Studies of quantumen tanglement and its paradoxical properties hold the key to various Quantum information (QI) tasks such as quantum teleportation, quantum cryptography and quantum computer technology. Keywords: Density Operator PACS No. 32.30, L-S Coupling, Negativity, Peres-Horodecki Condition
Quantum entanglement holds the key to a revolution in information processing.This paper investigates the entanglement characteristics of tripartite states of two electronic qubits and an ionic qubit produced by single-step double photoionization (DPI) from a Xeon atom after absorption of a single photon, without observing spin orbit interaction (SOI). DPI is the most natural processes for simultaneously producing two electrons in continuum in a single step. The dimension of the Hilbert space of the qudit depends upon the electronic state of the residual photoion Xe2+. Russel-Salunders coupling (L-S coupling) is applicable in absence of SOI. We consider Peres-Horodecki condition and negativity as an estimation of entanglement. With L-S coupling, all of the properties of a qubit-qudit system can be predicted simply by knowing the spins of the target atom, the residual photoion, the emitted electrons, and the polarisation state of incident photons.
Tripartite entangled states, such as Greenberger-Horne-Zeilinger (GHZ) and W states, are appealing for tests of fundamental aspects of quantum mechanics, as well as for implementations of quantum communication protocols. While the information is typically encoded in the polarization or path of the photons, we propose a feasible scheme to generate W states that are entangled in the energy degree of freedom. This is achieved by propagating photon pairs generated by parametric fluorescence in a linear optical circuit and post-selecting on the output channels. Such source can be easily integrated, profiting from high stability and a compact footprint. We also present a simple protocol to characterize the generated tripartite state, which is based on the reduced density matrix approach. Unexpectedly, although we rely on the energy degree of freedom, the full state tomography does not require any frequency conversion.