We study how two magnetic impurities embedded in a solid can be entangled by
an injected electron scattering between them and by subsequent measurement of
the electron's state. We start by investigating an ideal case where only the
electronic spin interacts successively through the same unitary operation with
the spins of the two impurities. In this case, high (but not maximal)
entanglement can be generated with a significant success probability. We then
consider a more realistic description which includes both the forward and back
scattering amplitudes. In this scenario, we obtain the entanglement between the
impurities as a function of the interaction strength of the electron-impurity
coupling. We find that our scheme allows us to entangle the impurities
maximally with a significant probability

A. T. Costa

A. C. Hewson

S. Bose

Y. Omar

English

arXiv

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Entanglement of Two Impurities through Electron Scattering

We study how two magnetic impurities embedded in a solid can be entangled by an injected electron scattering between them and by subsequent measurement of the electron's state. We start by investigating an ideal case where only the electronic spin interacts successively through the same unitary operation with the spins of the two impurities. We find conditions for the impurity spins to be maximally entangled with a significant success probability. We then consider a more realistic description which includes both the forward and backscattering amplitudes. In this scenario, we obtain the entanglement between the impurities as a function of the interaction strength of the electron-impurity coupling. We find that our scheme allows us to entangle the impurities maximally with a significant probability

Costa, AT

Bose, S

Omar, Y

UCL Discovery

arXiv:quant-ph/0503183v1  23 Mar 2005Entanglement of Two Impurities through Electron ScatteringA. T. Costa Jr.Departamento de Ciˆ encias Exatas, Universidade Federal de Lavras, 37200-000 Lavras, BrazilS. BoseDepartment of Physics and Astronomy, University College London,Gower Street, London WC1E 6BT, United KingdomY. OmarCentro de F´ isica de Plasmas, Instituto Superior T´ ecnico, P-1049-001 Lisbon, Portugal(Dated: 22 March 2005)We study how two magnetic impurities embedded in a solid can be entangled by an injectedelectron scattering between them and by subsequent measurement of the electron’s state. We startby investigating an ideal case where only the electronic spin interacts successively through the sameunitary operation with the spins of the two impurities. In this case, high (but not maximal) entan-glement can be generated with a signiﬁcant success probability. We then consider a more realisticdescription which includes both the forward and back scattering amplitudes. In this scenario, weobtain the entanglement between the impurities as a function of the interaction strength of theelectron-impurity coupling. We ﬁnd that our scheme allows us to entangle the impurities maximallywith a signiﬁcant probability.PACS numbers: 03.67.Lx, 05.50.+q, 73.23.Ad, 85.35.DsRecently there has been an increasing interest onthe generation of entanglement among spins in meso-scopic solid state structures [1, 2, 3, 4, 5, 6, 7, 8, 9].While there are several schemes for entangling adja-cent stationary spins through a direct quantum gate be-tween them [1, 2, 3], there is an unfortunate dearth ofschemes for entangling well separated stationary spins insuch structures. An overwhelming majority of the pro-posed schemes in which one obtains a reasonable separa-tion between the entangled spins are for mobile entities[4, 5, 6, 7, 8, 9]. Entangling well separated stationaryspins is practically important because they can be con-stituents (qubits) of distinct quantum computers. Es-tablishing entanglement between them is equivalent tolinking these computers. Even if they themselves arenot parts of quantum computers, they can each have aswitchable interaction with static spin qubits of well sep-arated quantum computers. In absence of a method ofentangling well separated stationary spins, one has to de-sign a scheme to either stop mobile spins after they havetraversed a distance, or ﬁnd a scheme for mapping theirstate on to stationary spins. In addition to the abovepragmatic application, such entangled stationary spinswill also enable one to test Bell’s inequalities with mas-sive particles, which is yet to be done for a signiﬁcantseparation [10]. Of course, mobile entangled electronscan also enable such tests with the individual spins be-ing measured by spin ﬁlters or spin selective detectors[11, 12]. There are also proposals for Bell’s inequalitymeasurements using the orbital or path, as opposed tothe spin, degree of freedom of mobile entities [9, 13, 14].However, quite a few of the proposals for the measure-ment of a single spin, such as those based on scanningtunnelling microscopy or magnetic resonant force mi-croscopy are speciﬁc to stationary spins [15]. These spinmeasurement methods could be used in a Bell inequalityexperiment if one were able to entangle well separatedstationary spins.With the above motivations is mind, in this article wepropose a scheme to entangle two magnetic impurities(stationary spins 1/2) embedded in a solid state system.The main idea is to use a ballistic electron as an agentwhich scatters oﬀ the two impurities in succession andentangles them. Being a scattering based scheme, it re-quires no control over the ability to switch interactions onand oﬀ between entities in a solid, as is required by manyexisting entangling proposals [4]. Moreover, even in com-parison to other reduced control proposals, such as thosebased on scattering or two particle interference [5, 6], ourcurrent scheme has the simplicity that it involves onlyone mobile entity, namely the ballistic electron, and doesaway with the diﬃculty of having to make two electronscoincide at the same place at the same time.We comment ﬁrst on the geometry of the system. Sinceentanglement generation depends on a conduction elec-tron interacting with both impurities, it is most conve-nient to make the system’s cross section as small as pos-sible. In this spirit, and for the sake of simplicity, weconsider a one-dimensional metallic atomic chain (of non-magnetic atoms), with two embedded (substitutional)spin-1/2 magnetic impurities. This is shown in Fig. 1.We know that in an ideal case, where a mediating agentis allowed to interact with two systems through distinctunitary operations, it can then perfectly entangle them.The ﬁrst of these unitaries perfectly entangles the ﬁrstsystem with the agent, and then the second operation2Impurity 1 Impurity 2ElectronFIG. 1: Setup for our scheme to entangle two magnetic im-purities of spin 1/2 in a solid through electron scattering. Weconsider the simple case of a one-dimensional metallic atomicchain (of non-magnetic atoms) where the two impurities areembedded. The electron ﬂies along the chain and its spin in-teracts with the spins of the impurities, scattering the electronoﬀ and entangling the two impurities.swaps the state of the agent with that of the secondsystem. This technique has, for example, been used inproposals for entangling the state of cavities using ﬂy-ing atoms [16]. The diﬀerent unitaries are implementedby diﬀerent interaction times or strengths between theagent and each of the systems. Such a technique obvi-ously requires either a great control over the motion ofthe agent, or the non-trivial engineering of diﬀerent in-teraction strengths of the agent with the systems. Underthese circumstances, it becomes interesting to investigatethe reduced control situation where an agent interactswith both systems through the same unitary operation.How well can the systems be entangled under these cir-cumstances? In the context of quantum optics, one canthink of this as an atom having the same ﬂight time andinteraction strength with two cavities through which itﬂies in succession. An analog of this scenario in solidstate systems would be to have an electron ﬂying pasttwo identical impurities with the same velocity, inter-acting magnetically with them successively without anyscattering. We ﬁrst consider this simpliﬁed case, just inorder to investigate how much entanglement can be es-tablished between two impurities, even when the electroninteracts with them through the same unitary. We ﬁndthat a signiﬁcant entanglement can be established with asigniﬁcant probability as long as we can make an appro-priate measurement of the state of the electron after itssuccessive interaction with the two impurities. This casemay not be realistic from the solid state scattering sce-nario, but it is an interesting precursor to the case whenspatial scattering is involved. Moreover, it can be real-ized in a quantum optics setting where the electronic spinstates are replaced by the atomic internal states and im-purity states are replaced by zero and one photon statesin the cavities. We then proceed to the realistic caseof the electron being spatially scattered by the interac-tion with the impurities. Interestingly, in this case, weﬁnd that the electron can entangle the two impuritiesnear perfectly (conditional on a favorable outcome of ameasurement of the electron’s spin). Moreover, the prob-ability of this favorable outcome is signiﬁcant (above 40percent), which means that on average one should be ableto perfectly entangle the impurities with three attempts.We begin by considering the ideal scenario where theelectron’s spin interacts in succession with each of theimpurity spins through the HamiltonianH = J− → S .− → σ , (1)where − → σ refers to the Pauli operators of the electronicspin, − → S refers to Pauli operators for the impurity spinsand J is the coupling constant between the spins. Wenow assume that the electron interacts with the two im-purities in succession for equal intervals of time, so thatwith both impurities the same unitary operation is im-plemented. The joint unitary operation between the elec-tron and impurity ι (with ι = 1,2) as a function of theinteraction time t is given by (expressing the unitary op-eration in terms of its eigenstates):Ueι(t) = e−itH= ei3Jt|Ψ−  Ψ−|eι+ e−iJt (|↑↑  ↑↑| + |↓↓  ↓↓|)eι+ e−iJt|Ψ+  Ψ+|eι, (2)where|Ψ± eι =1√2(|↑↓ eι ± |↓↑ eι). (3)Let us consider the following initial state where the im-purity spins are aligned, and the electron’s spin is anti-aligned with them|ψ0  = |↑ e|↓ 1|↓ 2. (4)The ﬁnal state is then given by|ψf  = Ue2(t)Ue1(t)|ψ0  = (5)e−i2Jt2(α|↑ e|↓ 1|↓ 2 + β|↓ e|↑ 1|↓ 2 + γ|↓ e|↓ 1|↑ 2),where α = (1 + ei4Jt)2/2, β = 1 − ei4Jt and γ =(1 − ei8Jt)/2. Each interaction either leaves the direc-tion of the spins unchanged, either ﬂips the interactingpair. Note that if we now measure the spin of the elec-tron and observe the state |↓ e, the impurities will be leftin the entangled state β|↑ 1|↓ 2 + γ|↓ 1|↑ 2. In Fig. 2we present the probability of this outcome (dashed line),as well as the resulting amount of entanglement quan-tiﬁed by the entanglement of formation [17] (solid line)between the two impurities, both as a function of Jt, theproduct of the interaction strength and the interactiontime. We study the probability and the entanglementas a function of Jt in the interval [0,π/2] as they areperiodic functions, and observe that in this ideal modelmaximal entanglement is generated only with zero prob-ability. This, however, does not rule out the possibility of3obtaining a high amount of entanglement with a signif-icant probability: for example, an entanglement of 0.99with a probability of 0.41, or an entanglement of 0.84with a probability of 0.86 as seen from Fig. 2.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.600.10.20.30.40.50.60.70.80.91JtP,EFIG. 2: Ideal case — Plots of the probability of success P(dashed line) of detecting the transmitted electron in the spindown state, and the entanglement E (solid line) obtained be-tween the impurity spins in that case, as a function of Jt, forthe ideal scenario where the electron successively interactswith the two impurities through the same unitary operation.It is clear that one can obtain a high amount of entanglementwith a signiﬁcant probability: for example, an entanglementof 0.99 with a probability of 0.41, or an entanglement of 0.84with a probability of 0.86.Let us now move to a more realistic scattering scenario.Magnetic impurities embedded in a conduction electronsea are traditionally modelled by a s-d Hamiltonian [18].In this model the magnetic impurities are localized spinsinteracting with the conduction electrons via an exchangeterm. The full hamiltonian of a system with one impurityreadsH =Xk,σεka†kσakσ +Xkk′Jkk′   S.  skk′ , (6)where   S is the impurity spin operator, a†kσ creates anelectron with wavevector k and spin σ and  skk′ = ˆ a†  σˆ a, (7)with ˆ a =￿ak↑ak↓￿. The s-d Hamiltonian is actually de-rived from the more fundamental Anderson Hamiltonianthrough the Schrieﬀer-Wolﬀ transformation. As a con-sequence, the interaction strength J is related to thestrength of the Coulomb interaction between electronsand the hybridization of narrow and conduction bands[18]. In our calculation we will adopt the usual assump-tion that J is independent of k,k′.We want to ﬁnd out how much entanglement may begenerated by a conduction electron that is injected inthe system and interacts with both magnetic impurities.One may determine the system’s ﬁnal state by calculat-ing the scattering matrix associated with each impurityand combining them together. The result is a sequence of(inﬁnitely many) scattering processes, in which the out-put of a scattering event is the input of the subsequentone. The result of each individual scattering process isdetermined by use of Fermi’s golden rule. The relevantT-matrix is calculated to ﬁrst order in the interaction.If we consider that the conduction electron is beinginjected under low bias, its energy and wavevector willbe the Fermi energy and Fermi wavevector of the system,respectively. We thus assume a initial state of the form|ψin  = |kF,↑ e|↓↓ 12 , (8)that represents a conduction electron with positive Fermiwavevector kF and spin ↑ along the quantization axis(z, for instance) propagating towards the two impurities,whose spins are both ↓ along the z-axis.As a result of the multiple scatterings of the conductionelectron by the two impurities, a ﬁnal state is generatedwhich is a superposition of states in which the conductionelectron has been reﬂected (r) or transmitted (t),|ψout  = |ψrout  + |ψtout , (9)and the transmitted component reads|ψtout  = (10)A|kF,↑ e|↓↓ 12 + B|kF,↓ e|↑↓ 12 + C|kF,↓ e|↓↑ 12 .The coeﬃcients A, B and C may be expressed as aninﬁnite sum of powers of the product Jρ(εF), which, ac-cording to our estimates [5], is of the order of unit. Weveriﬁed numerically that the series converges rapidly forJρ(εF) ∈ [0,2]. Below we present the series up to sixthorder in Jρ(εF) (corresponding to three iterations of thescattering matrix),A(3) =1N(t2 + t2λ2 − 8tλ3 + 16λ6 − 7t2λ4)B(3) =1N(−2λt + 2tλ3 − 2t2λ2 + 6tλ5 + 8t2λ4)C(3) =1N(−2λt + 8λ4 − 2tλ3 + 2t2λ2 + 6tλ5), (11)where λ = πiJρ(εF)/2, t = 1 − λ and 1N = p|A(3)|2 + |B(3)|2 + |C(3)|2.We now proceed to calculate the amount of entangle-ment (as quantiﬁed by the entanglement of formation)generated conditional on an electron being transmitted,which is the entanglement contained in the state |ψtout .Notice that if the transmitted electron has spin up theﬁnal state has zero entanglement. Thus we will only eval-uate the entanglement of the state in which the transmit-ted electron has spin down. Fig. 3 shows the entangle-ment in this state (solid line) and the probability P of4observing a transmitted electron with spin up (dashedline). One may notice that there is some entanglementfor most of the range 0 < Jρ(εF) < 2, and the probabil-ity P is also considerable. Moreover, there are values ofJρ(εF) for which the entanglement is maximum, and P issigniﬁcant (0.41). It is somewhat interesting to note thatalthough in the ideal case one cannot prepare the impuri-ties in a maximally entangled state with a non-vanishingprobability, one can do so in the realistic case.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 200.10.20.30.40.50.60.70.80.91Jρ(εF)P,EFIG. 3: Realistic case — Plots of the probability P (dashedline) of detecting the transmitted electron in the spin downstate, and the amount of entanglement E (solid line) obtainedbetween the impurity spins in that case, as a function of theinteraction strength Jρ(ǫF), for the realistic scenario of theelectron successively scattering oﬀ the two impurities. It isclear that at certain values of Jρ(ǫF), as shown by the dottedvertical line, the impurities can be projected on to a maxi-mally entangled state with a signiﬁcant (0.41) probability ofsuccess.In this article, we have presented a scheme for entan-gling two magnetic impurities in a solid through the scat-tering of a single ballistic electron. While much work hasbeen done on entangling spins in mesoscopic solid statesystems, this is the ﬁrst proposal for entangling distantstationary spins without the aid of an array of interveningspins. An immediate consequence will be in testing Bell’sinequalities with well separated stationary spins in a solid(of course, we still have to measure the spin of a mobileelectron, but not when testing Bell’s inequalities). Amore far reaching and more signiﬁcant consequence willbe in interfacing distant spin quantum computers. Ourwork shows that maximal entanglement can be obtainedbetween the distant stationary spins with a signiﬁcantprobability. 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Entanglement of two impurities through electron scattering

Entanglement of Two Impurities through Electron Scattering

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