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Physical Review A

Generation of tunable entangled states of two electrons and their characterization without entanglement witness

Naresh Chandra, R. Ghosh

Published Dec 30, 2004• 12 citations
Publisher

Abstract

In this Rapid Communication, we show that a simple process of two-step double ionization taking place in an atom $A$, 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., $A,{A}^{+},{A}^{2+}$) 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.

DOI
10.1103/physreva.70.060306

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Journal of Physics B Atomic Molecular and Optical Physics•May 9, 2002•19 citations
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.
Quantum entanglement is a unique phenomenon of quantum mechanics that explains how two subatomic particles are correlated even if they are separated by a vast distance. The phenomena of quantum entanglement are useful resources for quantum information. In this chapter, we will study the entanglement properties of bipartite states of two electronic qubits, without observing spin-orbit interaction (SOI), produced by single-step double photoionization in helium atom following the absorption of a single photon. In absence of SOI, Russell-Saunders coupling (L-S coupling) is applicable. We observe that the entanglement depends significantly on the direction of the ejection, as well as the spin quantization of photoelectrons.
This paper shows an electron and a photon, emitted in two consecutive steps from an atom following the absorption of a single photon, may be entangled in the presence of the spin-orbit interaction only. This entanglement strongly depends upon the polarization of the absorbed and of the radiated photons, kinematics of two emitted particles, and dynamics of photoionization; however, the photoemission dynamics plays no role in this entanglement. This hybrid entanglement can be used in teleporting a quantum state encoded in a flying/stationary material particle onto a light pulse, or vice versa. Such an electron-photon entanglement, in addition, will make it possible to learn about the polarization of a single photon or spin-polarization of a free electron without making any measurements on the corresponding particle itself.
We propose a scheme for detecting entanglement between two-electron spin qubits in a double quantum dot using an entanglement witness operator. We first calculate the optimal configuration of the two electron spins, defined as the position in the energy level spectrum where, averaged over the nuclear spin distribution, (1) the probability to have two separated electrons and (2) the degree of entanglement of the quantum state quantified by the concurrence are both large. Using a density matrix approach, we then calculate the evolution of the expectation value of the witness operator for the two-spin singlet state, taking into account the effect of decoherence due to quantum charge fluctuations modeled as a boson bath. We find that, for large interdot coupling, it is possible to obtain a highly entangled and robust ground state.
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 ``$1\ensuremath{\rightarrow}2$ 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 AAA 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 (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.
In this paper, we study the entanglement properties of a pair of electrons ejected simultaneously from an atom following the absorption of a single photon. We show that, in the absence of spin-orbit interaction (SOI), the non-separability of the state of two photoelectrons in a pair depends upon the spins of the atom and of its residual dication. The two electrons produced in a double photoionization experiment may always be entangled, on the other hand, if the SOI is taken into account. In the latter case, entanglement will be affected by, among other things, the photoionization dynamics and the state of polarization of the absorbed radiation.