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Long-Distance Entanglement of Soliton Spin Qubits in Gated Nanowires Pawel(cid:32) Szumniak,1,2 Jarosl(cid:32)aw Pawl(cid:32)owski,2 Stanisl(cid:32)aw Bednarek,2 and Daniel Loss1 1Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland 2AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, al. Mickiewicza 30, 30-059 Krako´w, Poland (Dated: January 9, 2015) We investigate numerically charge, spin, and entanglement dynamics of two electrons confined in a gated semiconductor nanowire. The electrostatic coupling between electrons in the nanowire and the charges in the metal gates leads to a self-trapping of the electrons which results in soliton- like properties. We show that the interplay of an all-electrically controlled coherent transport of 5 1 the electron√solitons and of the exchange interaction can be used to realize ultrafast SWAP and entangling SWAP gates for distant spin qubits. We demonstrate that the latter gate can be 0 used to generate a maximally entangled spin state of spatially separated electrons. The results 2 are obtained by quantum mechanical time-dependent calculations with exact inclusion of electron- n electron correlations. a J PACSnumbers: 73.21.La,03.67.Lx,73.63.Nm 8 ] Introduction. One of the most striking manifestations spin states of spatially separated electrons. Our scheme l l of the quantum laws of physics is entanglement [1, 2]. doesnotrequirecouplingwithanadditionalexternalsys- a The ability to entangle qubits is also an essential ingre- tem-‘quantumbus’-whichmaysimplifyitsimplementa- h - dient for quantum computation [3]. The spins of elec- tion. The proposed scheme is based on the interplay be- s trons confined in an array of electrostatically defined tween the exchange interaction and on-demand coherent e m quantum dots (QD) emerged as a promising candidate transport of self-trapped electron solitons [34, 35] con- for encoding quantum bits of information [4, 5]. Spin fined in gated semiconductor nanostructures. Exchange . t qubits weakly interacting with their environment can interaction,whichhasashort-rangecharacter,limitsthe a m be electrically controlled and show potential in scalabil- ability to couple spatially separated spin qubits confined ity [6]. Recent experiments have demonstrated fast and in stationary QDs. However, for mobile electron solitons - d precise manipulation, initialization, and measurement of this is not the case. Self-trapping allows for transport- n spin qubits confined in lateral QDs [7–14] and nanowire ing spatially separated initially not entangled electrons o QDs[15]. Furthermorelongspindecoherencetimeshave to the region where they can entangle their spins due c [ been reported, reaching ∼ 200ns [15] for InAs QDs and to the exchange interactionand finally be separated and ∼270µs [16] for GaAs QDs. transported back to distant regions as an entangled en- 1 tity. Tobespecificwepresenttheresultsforstructurally v However, theability to coupleand entangle solidstate defined InAs nanowires. However, one can expect quali- 2 spinqubitsoverlongdistances,whichisessentialforreal- 3 izations of scalable quantum computer architectures and tatively similar results for electrostatically defined quan- 9 tum wires in 2DEG/2DHG systems as proposed e.g. in for applications of fault-tolerant quantum error correc- 1 Refs. [36], and for different materials. One can integrate tion (QEC) schemes, still seems to be one of the key √ 0 SWAPand SWAPgateswithall-electricallycontrolled . challenges to overcome. First encouraging steps toward 1 singleultrafastquantumlogicgates[37,38]whichcanbe coupling remote spin qubits via microwave cavities [17– 0 arranged in a 2D scalable register (Ref. [39]) and be se- 5 20] have been recently reported [20–22]. Coherent long- lectively manipulated. Such an architecture may be par- 1 range spin qubit coupling based on cotunneling phenom- : ena[23]hasbeendemonstratedforanarrayconsistingof ticularly suitable for implementation of powerful QEC v surface codes [40]. i threequantumdots[24,25]. Therearealsoseveralinter- X estingproposalsforcouplingspatiallyseparatedQDspin r qubits, e.g. using ferromagnets [26], floating gates [27], a Majorana edge modes [28, 29] or superconductors [30]. Model. We consider two electrons confined in an InAs Another promising platform useful for coupling spin semiconductor nanowire covered by 7 electrodes e , e , L R qubits are mobile electrons shuttled by surface acoustic e , and e , to which the voltages V , V , V , and J 1−4 L R J waves [31, 32] or flying qubits [33], however, spin entan- V , are applied [41], respectively. The inter-electrode 1−4 glement of such moving electrons has not been reported distance is about 10nm. The radius of the nanowire is so far. l=5nm. The nanowire is separated from the metal by a In this paper we propose an all-electrically controlled dielectric material (InAlAs) [42], and the distance from and ultrafast method for realization of the SWAP and the center of the nanowire to the metal is d = 15nm. √ SWAP gates and for generating maximally entangled The presented system can be described by the quasi-1D 2 stead the image charge technique [35], e (cid:90) ρ(x(cid:48),t) V (x,t)= dx(cid:48) . (3) ind 4πεε0 (cid:112)(x−x(cid:48))2+4d2 This greatly simplifies and speeds up the numerical calculations. Quantum calculations [44] indicate that it is a good approximation of the actual re- sponse potential of the electron gas. Here, ε = 14.3ε is the dielectric constant for the InAs nanowire 0 [45]. The two-electron charge density is defined as ρ(x,t) = e(cid:82) dx(cid:48)(cid:0)|Ψ(x,x(cid:48),t)|2+|Ψ(x(cid:48),x,t)|2(cid:1), where FIG. 1. (color online) Schematic view of the nanostructure anditscross-section. Thenanowireiscoveredby7electrodes |Ψ(x1,x2,t)|2 = (cid:80)i,j=↑,↓|ψij(x1,x2,t)|2. The voltages which are labeled by e and e . Two electrons with applied to the electrodes generate an additional elec- 1−4 L,R,J opposite spins (blue arrows) are confined in the nanowire. trostatic potential in the nanowire region, φ (x,y ,z ), 0 0 0 Thepositivechargeinducedonthegatesurfaceismarkedby which we determine by solving the Laplace equation, red areas. ∇2φ (x,y,z) = 0, under the conditions φ (x,y ,z) = 0 0 0 V , where V is the voltage applied to the i-th elec- i i trode. Thus, according to the superposition princi- Hamiltonian, ple, the total confinement potential can be expressed as (cid:0) (cid:1) H = (cid:88) (cid:32)2pm2j +Vconf(cid:0)xj,ρ(x,t)(cid:1)(cid:33)+VC1D(x1,x2)+HBR, VthcoeTnhfinedxeu,lecρce(tdxr,iccth)fiaer=lgde−,sE|aey|rV,eignaedlns(exor,attth)ee−dsb|oeyu|φrt0ch(eexo,eyfle0ct,htzre0o)dR.eassahnbda j=1,2 (1) spin-orbit interaction in the nanowire, HRj = αEykxjσz wherep =−i(cid:126)∂/∂x isthex-componentofthemomen- [46]. However, for a chosen orientation of the electrodes, j j tumoperatorforthej-thelectron,j =1,2,m=0.023m wire, and initial electron spin (either up or down along 0 istheeffectivemassoftheelectronsconfinedintheInAs thezdirection)themotionoftheelectronsalongthewire nanowire, m isthefreeelectronmass. Thetwo-electron does not induce spin rotations. Such spin-orbit interac- 0 wave function is represented as [43] tiononlyslightlyincreasesspinswaptimes. Furthermore weassumethatthenanowireisgrownalong[111]crystal- (cid:0) (cid:1)T lographic direction which allows us to neglect the Dres- Ψ(x ,x ,t)= ψ ,ψ ,ψ ,ψ . (2) 1 2 ↑↑ ↑↓ ↓↑ ↓↓ selhausspin-orbit(DSO)interaction[45]whichcanaffect slightly gate fidelity [47]. It has to be antisymmetric with respect to simul- The effective 1D Coulomb interaction between charge taneous exchange of the space and spin coordinates: carriers in a nanowire with strong parabolic confinement ψ (x ,x ,t) = −ψ (x ,x ,t), where i,j =↑,↓ indicate ij 1 2 ji 2 1 in the y and z directions has the form [48] the spin projections of the first and the second electron on the z axis. (cid:18) (cid:19) 1 |x −x | Theelectronsinthenanowireinduceapositivecharge VC1D(x1,x2)= √ erfcx 1√ 2 . (4) 2π4ε εl 2l on the surface of the metal electrodes which in turn 0 leads to the self-confinement of the electron wave func- The time evolution of the system is described by the tion along the wire. This wave function has soliton-like time-dependent Schrödinger equation i(cid:126)∂ Ψ(x ,x ,t) = ∂t 1 2 properties [35]: It can be transported in the form of a HΨ(x ,x ,t) which we solve numerically using an ex- 1 2 stablewavepacketwhichmaintainsitsshapewhiletrav- plicit Askar-Cakmak scheme [49]. As initial condition eling. Moreover, it can reflect or pass through obstacles we take the ground state Ψ(x ,x ,t ) = Ψ (x ,x ) for 1 2 0 0 1 2 (potential barriers or wells) with 100% probability while the two self-confined electrons under the metal elec- preservingitsshape. Thiseffectcanbeexploitedtoreal- trodes which we obtain by solving the eigenvalue equa- izeon-demandtransportofself-confinedelectrons,whose tionHΨ (x ,x )=EΨ (x ,x )withtheimaginarytime 0 1 2 0 1 2 motions can be fully controlled by geometry and volt- propagation method [50]. In this approach electron- ages applied to the electrodes [36]. In the general case electron correlations are taken into account exactly. (e.g. arbitrary geometry of the electrodes) the induced To characterize properties of the system we evaluate self-confining potential is determined using the Poisson- thespindensity(i-thcomponent)ofthetwo-electronsys- Schrödinger self-consistent scheme which was described tem, in detail in Refs. [36, 37, 39]. However, since in the con- sidered structure electrodes form almost uniform wide ρ (x,t)= (cid:126)(cid:90) xRdx(cid:48)(cid:0)Ψ†(x,x(cid:48),t)σ ⊗IΨ(x,x(cid:48),t) Si 2 i plates (the inter-electrode distance is about 10nm), in xL order to determine the induced potential, we can use in- +Ψ†(x(cid:48),x,t)I⊗σ Ψ(x(cid:48),x,t)(cid:1), (5) i 3 √ where σ is i-th Pauli matrix, i = x,y,z. Consequently, In order to realize a quantum SWAP gate using i with this formula the expectation value of the electron the nanodevice from Fig. 1, we propose the follow- spin in the left (sL) or right (sR) part of the nanostruc- ing scheme. Initially (for t ) each of the electrons is z z 0 ture takes the form, localized under spatially separated electrodes e and L e , respectively. Since the electrons are significantly R (cid:90) x0(xR) away from each other (in our numerical simulation by sL(R)(t)= dxρ (x,t), (6) z Sz about 1.2µm) we assume that it is possible to prepare xL(x0) each of the electrons independently in a well-defined where x is the midpoint of the nanowire and x (x ) spin state, i.e., the electron confined under the elec- 0 L R is the left (right) end of the nanowire. The amount trode e (e ) is in spin up sL(t ) = (cid:126)/2 (spin down L R z 0 of entanglement between the spin in the left and the sR(t ) = −(cid:126)/2) state. Thus in the representation (2) z 0 right part of the nanowire can be quantified by calculat- the initial state has the following form [52]: Ψ (x,y) = 0 ing the√concurr√ence [5√1] whic√h is defined as C(ρSLR) = (0,ϕL(x1)ϕR(x2),−ϕR(x1)ϕL(x2),0)T, where ϕR(xi) max{0, λ − λ − λ − λ }. It varies from zero and ϕ (x ) are the single electron ground state orbitals 1 2 3 4 L i for completely separable (non-entangled) states to unity localized in the right (R) and in the left (L) dot. In for maximally entangled states. Here, λ are eigenvalues this situation there is no entanglement between electron i (in decreasing order) of ρ˜ = ρ (σ ⊗σ )ρ ∗(σ ⊗ spins, i.e., C(ρ(t ))=0. SLR y y SLR y 0 σ ), ρ is the reduced density matrix describing two- y SLR electron spin states in the left and right part of the nanowire. Results. First we investigate the charge dynamics and illustratedifferencesinpropagationbetweenself-trapped soliton-like electrons and ‘freely’ propagating electrons not interacting with the metal. Initially, electrons are confined in the nanowire under the electrodes e and L e (see Fig. 1), which is achieved by applying V = R 1,2,3,4 −1 meV and zero voltage to the other gates V =0. L,R,J The electrons are forced to move against each other by changing the voltage on electrodes e and e to V = 2 3 2 V =0 and by lowering the voltage on electrodes e and 3 1 e toV =V =−1.5mV.Thetimeevolutionofthetwo- 4 1 4 electron charge density ρ(x,t) along the wire is depicted in Fig. 2. It can be seen that the electrons being self- trapped under the metal maintain their charge density shape while moving, which is a characteristic feature of FIG.3. (a)Timeevolutionofthechargedensityρ(x,t)and schemeoftheelectrodescoveringthenanodevice(grey). Time soliton waves. Furthermore, the shape is not affected by evolution of (b) the z-component of spin density ρ , (c) the thecollisionwithotherelectrons[Fig.2(a)]. Howeverthis Sz z-component of expectation value of the electron spin in the isnotthecasefor‘free’electronsthatarenotinteracting lefts (t)(red)andtherights (t)(blue)partofthenanowire L R with the metal [Fig. 2(b)]. (referred to the left axis). The concurrence C (black) of the two-electronspinstateisshownwithrespecttotherightaxis in (c). In the inset (c’) of Fig. 3 (c) we plot the average distance|x −x |betweenelectronsforthefirstfewcollisions. 1 2 Then, by changing the voltage on the electrodes (in the same manner as in the previous example), the electrons start to move. When the electrons reach the region under the electrode e (for t = t ≈ 20ns) the J trap voltage on the neighboring electrodes e and e is set to 2 3 V = V = -1mV, which traps the electrons under e . 2 3 J Then the electrons collide periodically under this electrode. Thetimeevolutionofthecharge(z-thcomponent FIG.2. Timeevolutionofthechargedensityρ(x,t)illustrat- of spin) density for the two electrons ρ(x,t) (ρSz(x,t)) is ingthedifferenceinpropagationandcollisionoftwoincident plottedinFig.3(a)[Fig.3(b)]. Thetimeevolutionofthe electrons between the case with (a) and without (b) image expectation value of the spin in the left and right part charge potential. Note the self-trapping and the soliton-like of the nanodevice and the concurrence C is depicted in behavior in case (a). Fig.3(c). Duringeachcollision(duetoexchangeinterac- 4 tionwhichisintrinsicallypresentinourmodel)electrons exchange a fraction of the spin and consequently entanglement builds up between electron spins in the left and right part of the nanodevice. This is illustrated in Fig. 3 (c’)wherewealsoplottheaveragedistancebetweenelec- trons (orange line). Dips in the average value of the spin or concurrence is due to a local and temporary increase of double occupation probability [53] during the soliton collision. Aftermanycollisionsfort≈260pswehaveasituation where the spin density vanishes and the spin in the left FIG. 4. Probability densities of the two-electron wave func- andtherightdotisequaltozero,sLz(t)≈sRz(t)≈0. Fur- tions|ψ↑↓(x1,x2,t)|2(upperpanel)and|ψ↓↑(x1,x2,t)|2(lower thermore,theconcurrencereachesC(ρ(t))≈1,whichin- panel with primes) for the instants (a) t , (b) t , (c) t 0 trap 1/2 dicatesthatthespinsaremaximallyentangled. However, [C(ρ(t ))=0.5],(d)t ,and(e)t duringtherealization 1√/2 rel stop thesystemisnotyetinthespatiallyseparatedentangled of the SWAP gate. state. In order to separate electrons from the area under theelectrodee ,thevoltageontheelectrodese ande is J 2 3 switched (for t ≈ 260ps) to V =V =0, and after the rel 2 3 last collision, the electrons start to move into opposite directions. When they reach the region under the electrodes e and e (initial position) for t ≈ 280 ps, L R stop respectively, they are trapped again by changing the voltage on electrodes e to V = 2mV. Finally, 1,2,3,4 1,2,3,4 the maximally spin-entangled state is obtained for spatially separated electrons characterized by the concurrence reaching C(ρ)>0.999. It is also instructive to analyze the probability densities of the components of the total two-electron wave functionforthecasewhentheelectronsspinsarenoten- tangled, partially entangled, and maximally entangled, respectively. For the chosen initial state with opposite spins, during the whole evolution components with par- allel spins are zero ψ = ψ = 0, while the other two FIG. 5. Same as Fig. 3 but for the SWAP gate. ↑↑ ↓↓ are nonzero. The corresponding values |ψ (x ,x ,t)|2 ↑↓ 1 2 and |ψ (x ,x ,t)|2 are depicted in Fig. 4 for the ini- ↓↑ 1 2 tial moment t - non-entangled spatially separated elec- more than 90%, C(ρ(t)) > 0.9. In our simulations the 0 trons,t -firstcollision,t whentheconcurrencebe- voltage switching time is about 4ps long. The operation trap 1/2 comes C(ρ(t1/2)) = 0.5, trel last collision when the elec- times τop of the proposed gates are on the order of hun- tron spins are maximally entangled, and finally for the dreds of picoseconds, which is three orders of magnitude maximally entangled and separated electron spins under shorter than the reported spin decoherence time in InAs electrodes e and e . QDs [15]. However, one can further tune (decrease) the L R A similar procedure can be used to realize the two- gate operation time by increasing the voltage applied to qubit SWAP gate, which fully exchanges the spin of the the gates e3 and e4. two electrons. In this case electrons have to be released Summary. We have shown that the interplay of fromundertheelectrodee fort ≈510psandtrapped soliton-like properties of self-trapped electrons in gated J rel againundereL andeR fortstop ≈530ps. Theresultsare semiconductor nanowires and the excha√nge interaction showninFig.5. Itisclearlyseenhowelectronsexchange can be exploited to realize SWAP and SWAP gates. theirspinsduringsolitoncollisionsunderelectrodee . It The latter gate can be used to realize maximally entan- J isquiteremarkablethatdespitemanycollisionstheshape gledspinstatesofspatiallyseparatedelectrons. Thepro- ofthefunction(chargedensity)isstillpreservedandwell posedgatesactinanultrafastmanner(subnanoseconds) localized. and are controlled only by small static voltages applied The proposed scheme is most sensitive to proper ad- to the electrodes which makes our proposal particularly justment of the t and t times. However, our suitable for addressing individual spin qubits in scalable trap stop method is quite robust against variations of t , the quantum registers. rel releasing time of electrons from under e . For t − We acknowledge support from the Swiss NSF, NCCR J rel 30%t <t<t +30%t theconcurrencereachesstill QSIT, SCIEX (P.S.), and IARPA. rel rel rel 5 [24] F. Braakman, P. Barthelemy, C. 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