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Journal of Physics Condensed Matter

Scattering-induced entanglement between spin qubits at remote two-state structures

Matthew Habgood, J. H. Jefferson, G. Andrew D. Briggs

Published Jan 29, 2009• 8 citations
Publisher

Abstract

A theoretical scheme is presented for the entanglement of two-electron spin qubits bound in series within a quasi-one-dimensional mesoscopic structure at a distance beyond their normal range of interaction. A third electron is scattered from them, and full entanglement is achieved upon measurement of a transmitted electron in the correct spin state. Critically, each bound electron is trapped within an individual structure that has at least two spatial states. Two simple examples of such structures are discussed here. One is a 'stub', in which a quantum dot (for example) is coupled to one side of the quasi-one-dimensional structure. The other is a pair of degenerate, coupled quantum dots, with strong interdot Coulomb repulsion, placed within the one-dimensional superstructure. Both of these are shown to allow generation of entanglement with a significant probability of success. In contrast to the results of the authors' previous works, this allows for the generation of entanglement in a series, rather than in a parallel, configuration of the bound electrons with respect to the propagating electron.

DOI
10.1088/0953-8984/21/7/075503

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A theoretical scheme is presented for the entanglement of two-electron spin qubits bound in series within a quasi-one-dimensional mesoscopic structure at a distance beyond their normal range of interaction. A third electron is scattered from them, and full entanglement is achieved upon measurement of a transmitted electron in the correct spin state. Critically, each bound electron is trapped within an individual structure that has at least two spatial states. Two simple examples of such structures are discussed here. One is a 'stub', in which a quantum dot (for example) is coupled to one side of the quasi-one-dimensional structure. The other is a pair of degenerate, coupled quantum dots, with strong interdot Coulomb repulsion, placed within the one-dimensional superstructure. Both of these are shown to allow generation of entanglement with a significant probability of success. In contrast to the results of the authors' previous works, this allows for the generation of entanglement in a series, rather than in a parallel, configuration of the bound electrons with respect to the propagating electron.
The possibility of generating entanglement between two solid-state electron spin-qubits by spin-dependent scattering of a third, propagating electron is attractive. A process of this kind could work well beyond the range of the direct interaction between the two spins. In this study, the binding of electrons in one-dimensional structures is theoretically modeled using a real-space analog of the Anderson model. The respective binding sites are arranged in two different geometries. The first is a straightforward series arrangement, and the second a parallel arrangement in which the binding sites are embedded in either arm of an Aharonov-Bohm style ring. For the series arrangement, the entanglement between the bound electrons is calculated and is shown to be low (concurrence <0.2<0.2<0.2) in all regions of significant transmission. In contrast, the parallel arrangement allows full entanglement to be generated from a ${S}_{z}=1(\ensuremath{\mid}\ensuremath{\uparrow}\ensuremath{\uparrow}⟩)$ or ${S}_{z}=0(\ensuremath{\mid}\ensuremath{\uparrow}\ensuremath{\downarrow}⟩)$ starting configuration with an overall probability of success up to 0.5.
We study two simple mesoscopic models of interacting two electrons; first one consists of two quantum coherent parallel conductors with long-range Coulomb interaction in some localized region and the other is of an interacting quantum dot (QD) side-coupled to a noninteracting quantum wire. We evaluate exact two-particle scattering matrix as well as two-particle current which are relevant in a two-particle scattering experiment in these models. Finally we show that the on-site repulsive interaction in the QD filters out the spin-singlet two-electron state from the mixed two-electron input states in the side-coupled QD model.
The entanglement of flying qubits with static qubits in the solid state is important in the development of quantum computing. It has recently been shown theoretically that the spin-dependent scattering of a propagating electron from a bound electron is sufficient to give full entanglement between the qubits embodied in their respective spins [J. H. Jefferson et al., Europhys. Lett. 74, 764 (2006); D. Gunlycke et al., J. Phys.: Condens. Matter 18, S851 (2006)]. In this paper a generalized, real-space Anderson model is introduced for a quasi-one-dimensional structure consisting of a binding site coupled to ideal leads. Degeneracy of both the binding site and the leads is incorporated to represent, for example, conduction band degeneracy in carbon nanotubes. The model is used to calculate the spin-dependent scattering behavior and resultant static-flying qubit entanglement created by the scattering process. Degeneracy (and more generally, multiplicity) in the binding site gives rise to inelastic scattering processes. In the elastic scattering regime, a degenerate binding site gives rise to antiresonant structures in the transmission spectrum. Additional degeneracy in the leads restricts this effect to the second set of leads, raising the possibility of spin filtration, though this is eliminated in the inelastic scattering regime. Degeneracy in the binding site also gives rise to multiple resonances in transmission that will improve the probability of obtaining entangled pairs relative to the nondegenerate case. This effect is maximized when the components of the degenerate binding site are symmetrically coupled to the leads.
We propose schemes for generating spatially separated spin entanglement in systems of two quantum dots with onsite Coulomb repulsion weakly coupled to a joint electron reservoir. An enhanced probability for the formation of spin entanglement is found in nonequilibrium situations with one extra electron on each dot, either in the transient state after rapid changes of the gate voltage or in the steady state with applied bias voltage. In both cases, so-called Werner states with spin-singlet fidelity exceeding 1∕21∕21∕2 are generated, which indicates entanglement.
We analytically study entanglement generation through an open quantum-dot system described by the two-lead Anderson model. We exactly obtain the transition rate between the nonentangled incident state in one lead and the outgoing spin-singlet state in the other lead. We find that only the spin-singlet state can transmit in the cotunneling process. To discuss such an entanglement property in the open quantum system, we construct the exact two-electron scattering state of the Anderson model. It is striking that the scattering state contains spin-singlet bound states induced by the Coulomb interaction. The bound state describes a scattering process in which the set of momenta is not conserved and hence, it is not in the form of a Bethe eigenstate.
We analytically study entanglement generation through an open quantum dot system described by the two-lead Anderson model. We exactly obtain the transition rate between the non-entangled incident state in one lead and the outgoing spin-singlet state in the other lead. In the cotunneling process, only the spin-singlet state can transmit. To discuss such an entanglement property in the open quantum system, we construct the exact two-electron scattering state of the Anderson model. It is striking that the scattering state contains spin-singlet bound states induced by the Coulomb interaction. The bound state describes the scattering process in which the set of momenta is not conserved and hence it is not in the form of a Bethe eigenstate.