Enhancement of entanglement transfer in a spin chain by phase shift control Koji Maruyama1,2, Toshiaki Iitaka3, and Franco Nori1,4 1Frontier Research System, RIKEN (The Institute of Physical and Chemical Research), Wako-shi 351-0198, Japan 2Laboratoire d’Information Quantique and QUIC, CP 165/59, Universit´e Libre de Bruxelles, 1050 Bruxelles, Belgium 3Computational Astrophysics Laboratory, RIKEN (The Institute of Physical and Chemical Research), Wako-shi 351-0198, Japan 4 Center for Theoretical Physics, Physics Department, Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109-1040, USA (Dated: February 9, 2008) Westudytheeffectofaphaseshiftontheamountoftransferrabletwo-spinentanglementinaspin chain. We consider a ferromagnetic Heisenberg/XY spin chain, both numerically and analytically, 7 and two mechanisms togenerate a phase shift, theAharonov-Casher effect and theDzyaloshinskii- 0 Moriyainteraction. Inbothcases,themaximumattainableentanglementisshowntobesignificantly 0 enhanced,suggesting its potential usefulness in quantuminformation processing. 2 PACSnumbers: 03.67.Hk,03.67.Lx,75.10.Pq n a J I. INTRODUCTION A number of important and interesting results have 4 beenreportedinthisresearcharea: forexample,sending 2 quantuminformationthroughaspinchainwithoutmod- Transferring quantum information reliably and effi- ulation [1], entanglement transport with an anisotropic 2 cientlyisanimportanttaskinquantuminformationpro- v XY model [2], perfect transfer by manipulating the cou- cessing. For example, quantum communication proto- 3 pling strengths [3, 4], near perfect transfer with uni- cols,suchasquantumkeydistribution(orquantumcryp- 0 form couplings by a spatially varying magnetic field tography), usually require two (or more) distant parties 1 [5], Fourier analysis-based quantum information encod- 0 to share entanglement of high quality to achieve tasks ing [6], measurement-assisted transfer through two par- 1 that are impossible in the regime of classical mechanics. allel chains [7], specific realizations of the method in [1] 6 Also in a typical situation we encounter in the standard 0 quantumcomputationmodel,weneedtocoupletwospa- with superconducting qubit array [8, 9], perfect transfer / by local measurements on individual spins [10], and also h tiallyseparatedqubitsinordertoperformtwoqubituni- transfer through a chain of coupled harmonic oscillators p tary operations, e.g., a controlled-NOT gate. [11]. - Most common approaches to this task include meth- t n ods with an information bus, guided ions (atoms), fly- The geometry of the spin chain could be more gen- a ing photons, a sequence of swapping operations between eral in principle, but we will primarily consider a one- u neighboring qubits, etc. However,these methods require dimensionalchainorringofN spin-1/2particlesforsim- q : additional complexity in structures, manipulations and plicity. If we look at the chain as a whole, we can think v controls of the interaction between qubits, as well as re- of independent quantum states of collective modes, i.e., Xi peated conversions between the qubit state and another eigenstates of the Hamiltonian for the whole chain. The physical degree of freedom. Flying photons may be the propagation of quantum information encoded in a spin r a best information carrier over macroscopic distances, but can be thought of as the interference between all modes, may not be so for microscopic scales of the order of, say, which evolve independently. a few micrometers. This is why there has been intensive Hence, a naive strategy towards quantum information research activity in the past few years on quantum in- transfer of better quality would be to control the propa- formationtransfer via arrays/chainsof stationaryqubits gation of each mode, which will affect the (constructive) that are interacting with their neighboring qubits. interference at a certain target site. Here we take this Typically, in previous studies of this topic, the quan- approach and consider the effect of changes in the en- tum information channel consists of one or more chains ergy spectrum and dispersion relation of the mode (or of spin-1/2 particles, each of which interacts with their ‘spin wave’), that induce a spin current in the chain. As nearest neighbors. The interaction between spins can be a result of the induced change in the energy spectrum, described by either the Heisenberg or the XY (or varia- there will also be a change in the time evolution of each tions of these, e.g., the XXZ) model with some relevant modeandthustheinterferencebetweenthesemodes. An parametersforcouplingstrengths,anisotropy,etc. These advantage of this approach is that the pairwise coupling types of models attract muchattention because they are strengths J between i-th and j-th spins and the exter- ij inprinciplesufficientforimplementingquantuminforma- nalelectromagneticfield canbe takenas a constantover tionprocessing: examplesofproposedmethodsarethose the whole chain, unlike some schemes proposed before. with quantum dots and particles trapped in an optical Thatis, they do nothave to be manipulatedsite by site, lattice. regardlessofthestartingandtargetsitesforthetransfer. 2 In this paper, we study the effect of a phase shift on whereS+ =Sx+iSy istheraisingoperatordefinedwith j j j theamountoftransferrableentanglementthroughaspin the spin-1/2 operators Sα(α = x,y,z) for the j-th spin. j chain. We shall primarily focus on the transfer of entan- The lowering operator is defined as its Hermite conju- glement, rather than the state itself, since entanglement gate, S− = Sx iSy. Throughout this paper, we will j j − j is the key to achieving highly non-classical information let 0 denote the spin-down state, while 1 is the spin- processing. Besides, keeping the fidelity of the (entan- ups|taite. Asinstandardentanglementtra|nsiferscenarios gled) two-spin system is harder than keeping the fidelity withtheone-magnonassumption,themagnonisinitially ofasinglespin. Thiscanbestatedmorepreciselyas“the localizedatasinglesite. Withsuchinitialconditions,we entanglement fidelity associated with a trace-preserving can identify the amplitudes α (t) as the propagators, or j map is lower than or equal to the corresponding input- the Green functions, from the point of view of wave me- output fidelity of a subsystem” [12]. In short, attaining chanics. They have all information to describe the time an entanglement transfer of high quality guarantees a evolution of a magnon wave packet. state transfer of high quality. In order to evaluate the pairwise entanglement in Further, we assume that the whole system is in the Ψ(t) , we employ the concurrence [19] as its measure. | i ‘one-magnon’state,inwhichthetotalnumberofupspins The concurrence C in a bipartite state ρ is defined as in the chain is one. This situation is simple enough to start our analysis with and is indeed reasonable when C :=max 0,λ1 λ2 λ3 λ4 , (2) { − − − } considering information transfer. Thus, all spins in the initial state, except the one that is the subject of the where λi are the square roots of the eigenvalues of ma- transfer operation, are initially down. As for the in- trix R in descending order. The matrix R is given as a teraction between spins, we will consider the standard product of ρ and its time-reversed state, namely isotropicHeisenbergmodel. TheHeisenbergmodelisac- R=ρ(σy σy)ρ∗(σy σy), (3) tuallyequivalenttotheXYmodelundertheone-magnon ⊗ ⊗ assumption,thereforeouranalysiswillbeapplicabletoa where σy is one of the standard Pauli matrices and the widerangeofphysicalsystems. Also,asassumed(mostly star denotes the complex conjugate. The concurrence tacitly) in the literature listed above, we make another takesitsmaximumvalue1forthemaximalentanglement, assumption not to let useful entanglement pass by the while it is 0 for all disentangled qubits. target site: the entanglement at the target site(s) can WewillcomputetheconcurrenceC (t),betweenthe be extracted later on at will or can be retained for fur- l1,l2 l -th and l -th spins, at time t by tracing out all spins ther operations, including entanglement distillation [13], 1 2 except those two. Then, C (t) can be written as to maximize the efficiency of subsequent processes. l1,l2 Wewillconsidertwodifferentwaysofgeneratingphase C (t)=2α (t) α (t). (4) shifts. One is due to the Aharonov-Casher effect [14], l1,l2 | l1 |·| l2 | while the other is induced by the Dzyaloshinskii-Moriya HenceEq. (4)statesthattheconcurrencebetweenthe interaction [15, 16]. There are other means to generate two sites in a chain can be expressed as a product of the a phase shift, or equivalently a (persistent) spin current, absolute values of propagators. That the entanglement in a chain/ring, such as those reported in [17, 18]. How- is determined by the propagatorssupports our approach ever, we focus on the two above because these seem to towards enhancing the entanglement by phase shift, be- be sufficient to demonstrate a significant entanglement- causethepropagatorsarenaturallyaffectedbychangein enhancing effect due to phase shifts. We discuss only dispersion relation caused by phase shifts. chains of ring geometry because the effect is absent in open ended chains as shown in the Appendix. III. ENTANGLEMENT TRANSFER WITH THE AHARONOV-CASHER EFFECT II. PAIRWISE ENTANGLEMENT IN SPIN First, we consider the Aharonov-Casher effect [14] as CHAINS a physical mechanism that causes a phase shift in the collective modes. When a neutral particle with mag- Let us start with a description of the entanglement netic moment µ~ travels from ~r to (~r+∆~r) in the pres- between an arbitrary pair of spins in an N-spin chain ence of electric field E~, the wave function of the particle within the one-magnon condition. The properties of the acquires an extra phase, which is the Aharonov-Casher spin chain, such as its geometry, the nature of the inter- (AC) phase, action, etc. could be any at this point. As there is only one up spin intotal, the state of the whole chainattime 1 ~r+∆~r t can be given by ∆θ = ~µ E~(~x) d~x, (5) ~c2 × · Z~r Ψ(t) = α (t)S+ 0 ⊗N, (1) in addition to the ordinary dynamical phase. The phys- | i j j | i ical origin of the AC effect is that a particle moving in j X 3 shapedconfigurationisrepresentedbyperiodicboundary conditions, i.e., N +1 = 1. The Hamiltonian H can be diagonalized with the help of the Jordan-Wigner trans- formation [22, 23] that maps spins under H to spinless fermions. Theannihilationandcreationoperatorsforthe fermion at site j are j−1 c = exp πi S+S− S− j l l j FIG. 1: Examplesof configurations for theAharonov-Casher l=1 ! X effect in a ring-shaped spin chain. In (a), the z-axis is taken j−1 tobeparalleltothedirectionperpendiculartotheplanethat and c† = S+exp πi S+S− . (7) contains the spin chain. With an electric field E~, which is j j − l l ! directed to the radial direction, the term ~µ E~(~x) d~x in Xl=1 Eq. (5) takes its largest value. Alternatively,×as in (b·), the Under the one-magnon condition, the Hamiltonian H is directions for the spin and the electric field can be swapped now diagonalized as to have the same ~µ E~(~x). The magnetic moment of spin eigenstate, |↑i or |↓×i, is parallel to theradial direction. H = − N 21 eiθc†jcj+1+e−iθc†j+1cj Xj=1(cid:20) (cid:16) (cid:17) 1 1 1 c†c +c† c +h c†c + an electric field feels a magnetic field as well due to rela- −2 j j j+1 j+1 j j − 2 4 tivistic effects: the AC effect is essentially equivalent to (cid:16) (cid:17) (cid:18) (cid:19) (cid:21) = E η†η (8) spin-orbitcoupling. Ifthereisnoexternalfieldappliedto k k k the ring of spins, the dispersion relation should be sym- Xk metric with respect to the zero wave number (k = 0), withafurtherlineartransformationη = φ∗ c . The k j kj j i.e. E = E , due to the rotational symmetry of the energy eigenvalues E can be computed as k −k k P system. However, if the accumulated AC phase along a N N ring does not vanish after a 2π rotation, the dispersion E = cos(k+θ)+ 1 h 1 , (9) k − − 4 − − 2 relation will change as the applied field breaks the (spa- (cid:18) (cid:19) (cid:18) (cid:19) tial) symmetry. Consequently, the propagation speed of where k = 2πn/N with N/2 < n N/2, and φ = kj − ≤ each mode will be affected and the concurrence between 1/√Neikj. As the secondand third terms of Eq. (9) are any two sites can be expected to change accordingly. constant, we will omit them hereafter. A one-magnon Figure 1 sketches two possible configurations to have eigenstate can be obtained accordingly with the form of the AC phase effectively. The geometry in Fig. 1(a) is η† as verysimilar to thatfor electrons in anatom. An electric k field diverges radially, and the z-axis is taken to be per- k :=η† 0 ⊗N = 1 eikjS+ 0 ⊗N. (10) pendicular to the plane containing the ring. The term | i k| i √N j | i j ~µ E~(~x) d~x in Eq. (5) takes its largest value when X × · The presence ofthe extra phase θ is reflectedonly in the a (quasi-) magnetic moment (magnon) travels along the energy spectrum, while the expression for eigenstates is chain. The electric field could be generated by, for ex- unchanged. The change in the dispersion relation by a ample, a charge on a wire at the center of the ring. Al- phase shift is illustrated in Fig. 2(a). ternatively, the directions for the spin and the electric Note thatapplying the Jordan-Wignertransformation field can be swapped as in Fig. 1(b) to have the same ~µ E~(~x). to the Hamiltonian (6) gives an additional term to Eq. × (8), 1/2(eiθc†c + e−iθc† c )(exp(iπ N c†c ) + 1), WeconsideraHeisenbergchainofN spin-1/2particles − 1 N N 1 l=1 l l whichis aresultofthe periodic boundarycondition. We interacting ferromagnetically with their nearest neigh- P have alreadyomitted this term in Eq. (8) since it equals bors. ThephaseacquisitionduetotheACeffectmodifies zero as long as we consider the one-magnon state. This the standard Heisenberg model Hamiltonian to [20, 21] is because exp(iπ N c†c )+1 = 0 for any N. Also, l=1 l l N the one-magnon condition makes the Heisenberg model 1 H = eiθS+S− +e−iθS−S+ +SzSz equivalent to the XPY model, as the interaction between − 2 j j+1 j j+1 j j+1 j=1(cid:20) (virtual) fermions after the Jordan-Wigner transforma- X (cid:0) (cid:1) +hSz , (6) tion is absent under this condition in both models. j where the inte(cid:3)raction strength is taken as J = 1 for all neighboring pairs and h is the magnetic field,−which A. Entanglement with an isolated spin is taken to be uniform and parallel to the z-direction. The phase change θ between neighboring spins is given Letusanalyzetheentanglementpropagationalongthe by Eq. (5) with ~r = ~r and ∆~r = r~ r~. The ring- spin chain. Suppose that at t = 0 a physically isolated j j+1 j − 4 FIG. 3: Schematic picture of entanglement transfer in a spin chain. (a) The configuration of the Heisenberg ring and the entanglement it has at t=0, with a spatially separated sys- tem that is here represented as the 0th spin. (b) The ideal goaloftheentanglementtransferoperation: wewishtotrans- fer as much entanglement with the 0th spin as possible to a specific (target) spin in thechain. obtain C0,1(t) = √2α(t), where the superscripts on C 0,l | l | denote the initially entangled pair. In the limit of large N, this takes a simple analytical form C00,,l1(t)= e−i(l−1)(θ−π2)Jl−1(t) =|Jl−1(t)|, (14) FIG.2: EnergyspectrumEk =−cos(k+θ)(intheunitsofJ). with the Bess(cid:12)(cid:12)el function of the fi(cid:12)(cid:12)rst kind J (x). The (a) A non-zero θ changes the dispersion relation represented (cid:12) (cid:12) ν withopencircles(nophaseshift)totheonewithfilledcircles effect of the AC phase θ disappears in this limit, since (θ = 0). The number of sites N is taken to be 8. (b) The the energy spectrum becomes continuous and displacing 6 energy spectrum when N is large. all modes by θ does not change the overall dispersion relation. This can be clearly seen in the plot in Fig. 2(b). In other words,θ appears only as a common phase factor for all modes, e−i(l−1)(θ−π2), thus there is no θ- (the 0th)spinandthefirstspinaremaximallyentangled dependence in α . j as (01 + 10 )/√2 and the restof the spins in the chain An example|of t|he plots of concurrence as a function | i | i are all in 0 . Thus, the initial state of the whole system of t andθ is shownin Fig. 4, whichis the plot of C0,1(t) | i 0,3 can be expressed as for N = 5. The phase θ is the same as that in Eq. (6), that is, the phase a magnon acquires when hopping 1 1 Ψ(0) = 0 eik k + 1 0 ⊗N . (11) from the j-th to the (j+1)-th site. Some improvement 0 0 | i √2"| i √N | i | i | i # in the amount of transferred entanglement due to the k X nonzerophaseshiftisevident: themaximumconcurrence Hence, we find the state at time t, taking ~=1, as when θ = 0 is Cθ=0 = 0.647 (at t = 59.05), and when max θ = 0, C can reach as high as 0.996 (at t = 23.71 max Ψ(t) = 1 1 exp(ik(j 1) iE t) 0 S+ 0 ⊗N an6d tanθ =1.376) in the region we have calculated, i.e., | i √2N − − k | i0 j | i t [0,200] and θ [ π,π]. k,j ∈ ∈ − X Figure 5 showsthe comparisonbetween the maximum +eit|1i0|0i⊗N . (12) valuesofconcurrenceCmax with andwithoutphase shift for various (N,l) from (3,2) to (13,13), where N is the Figure3depictstheproc(cid:3)essweconsider: Fig. 3(a)shows total number of sites and l is the site where the concur- the initial correlation in Ψ(0) , and Fig. 3(b) is the | i rence is evaluated. Plotted are the highest values found desired goal of our entanglement transfer operation. numerically in the range of 0 t 200. The horizon- Now we can evaluate the entanglement between the ≤ ≤ tal axis represents (N,l). The blue lines with diamond 0th and l-th spins. Equation (12) can be written in the markers are for the geometry with an isolated spin, and form of Eq. (1) with amplitudes the red lines are for the entanglement transfer when the 1 initial entanglement is held by a pair in the chain. The α (t)= exp[ik(j 1) iE t]. (13) j k latter case will be discussed in the following subsection. √2N − − Xk For both cases, the open markers show the maximum Because the 0th spin is not interacting with other spins, concurrence Cm1,m2 when θ = 0, while the filled ones l1,l2 we can take α (t) = 1/√2 for all t. From Eq. (4) we mark max Cm1,m2 in the range of π θ π. | 0 | { l1,l2 } − ≤ ≤ 5 AsN tends toinfinity, Cm1,m2(t)approachesthe form l,l+1 Cm1,m2(t) = J (t)+e−i(m1−m2)(θ−π/2)J (t) l1,l2 l1−m1 l1−m2 (cid:12) (cid:12) J (cid:12)(t)+e−i(m1−m2)(θ−π/2)J (t) . (16(cid:12)) × l2−m1(cid:12) l2−m2 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) Interesti(cid:12)ngly, unlike Eq. (14), there is still a dep(cid:12)endence on θ, regardless of l’s and m’s. Nonzero θ can indeed alwaysincrease the maximumattainable pairwise entan- glement in a long chain, no matter which pair is initially entangled,andnomatterwhichpairweevaluatethecon- currence for. We can see in Fig. 5 that the degree of enhancement is larger when the two initially entangled spins are in the chain, compared with the case of entanglement with an isolated one. This difference can be understood intu- FIG.4: AnexampleofplotsoftheconcurrenceC transferred itively when N is large: the physical reasoning for small inafive-spinchain(ring)inthepresenceofaphaseshift. The initial entanglement (of the form (01 + 10 )/√2) is in the N’s is essentially the same. If we look at a propagator first spin of the ring and an isolate|d i(0th|) sipin. The phase froma single site, the phase shift θ gives analmostcom- shift θ is an extra phase a magnon acquires when travelling mondisplacementtothephaseofallmodesatanysingle toaneighboringsite. TheconcurrenceC isevaluatedforthe site as we have seen in Eq. (14). When the entangled pairofthe0thandthirdspins. Unitsfortandθare~seconds pair,m andm ,isembeddedinthechaininitially,there 1 2 and radian, respectively,where ~ is thePlanck constant. aretwoindependentpropagatorsstemmingfromthetwo sites. Since these propagators have different phase dis- placementsatthesamesite,saythel-th,theinterference due to non-zero θ still occurs. As a result, there remains The enhancement of entanglement by the phase shift a dependence of the concurrence on the phase shift. isclearlyseeninFig. 5. Thetransferredentanglementis significantly increased by nonzero phase shifts for many values of (N,l). Yet, the degree of enhancement varies IV. SPIN SHIFT INDUCED BY THE because quite an effective constructive interference can DZYALOSHINSKII-MORIYA INTERACTION occur even when θ =0 for some (N,l). Phase shifts can be generated by a different type of B. Entanglement in a pair of spins in the chain interaction, that is, the antisymmetric exchange interac- tion, a.k.a. the Dzyaloshinskii-Moriya (DM) interaction [15, 16], in solids. The DM interaction could be quite Letus nowlookatanalternativescenarioofentangle- significant in some solid-state-based qubit systems, such ment transfer. Instead of being entangled with an iso- as quantum dots [24]. lated (0th) spin, both spins of an entangled pair can be The Hamiltonian for the DM interaction between two in the same chain. Considering the effect of the phase spins, say 1 and 2, can be written as shiftmentionedabove,wecannaturallyexpectsomeim- provementin the efficiency of transferin this scenarioas H =d~ (S~ S~ ), (17) well. WealreadyhavetheexpressionEq. (4)forthecon- DM · 1× 2 currence at time t between the l-th and (l+1)-th spins Cm1,m2(t) = 2αm1,m2(t) αm1,m2(t), where m and whered~isthecouplingvectorthatreflectstheanisotropy l,l+1 | l | · | l+1 | 1 of the system. Assuming that only the z-component of m denote the initial sites that are entangled. Factors αm21,m2(t) are given explicitly by d~ has a nonzero value, i.e., dz = 0,dx = dy = 0, and j 6 thatallcomponentsareconstantalongthechain,wecan 1 write the totalinteractionHamiltonianandits spectrum αm1,m2(t) = (exp[ik(j m ) iE t] j √2N − 1 − k as (omitting the terms that give only a constant bias) k X +exp[ik(j m ) iE t]). (15) 1 − 2 − k H = S+S− +S−S+ +id (S+S− S−S+ ) −2 j j+1 j j+1 z j j+1− j j+1 Thecomparisonofthemaximumconcurrencebetween j X(cid:2) (cid:3) the cases with and without the phase shift is shown in 1 = (eiφS+S− +e−iφS−S+ ) Fig. 5 with red triangular markers. For the plot in Fig. −2cosφ j j+1 j j+1 5,theinitialentanglementoftheformof(01 + 10 )/√2 Xj | i | i 1 isassumedtobeinthefirstandsecondspinsandtherest = cos(k+φ)η†η , (18) are in 0 . −cosφ k k | i Xk 6 FIG. 5: Comparison of the maximum concurrence attainable Cmax between the cases with/without phase shift. The blue plots (diamonds) show the concurrence between the 0th and l-th spins when the 0th and first spins have the entanglement (01 + 10 )/√2att=0. Thefilledandopendiamondscorrespondtononzeroandzerophaseshift,respectively. Theredplots | i | i (triangles)arefortheconcurrencebetweenthel-thand(l+1)-thspinswhenthefirstandthesecondspinsareinitiallyentangled in the same form. The horizontal axis represents the total number N of sites in the chain and the location l (2 l N) at ≤ ≤ which theconcurrence is evaluated. whereφ=tan−1d . Thus,thedifferencefromtheenergy enough. Aroughcalculationgivesanestimateofthenec- z spectrumunder the AC effect, Eq. (9), is onlythe factor essarystrengthoftheelectricfieldofatleast107 V/mto 1/cosφfortheenergyeigenvalues. Asthisfactorisinde- havea meaningfulphaseshift, if the systemsize isofthe pendentofk,itonlyrescalestheenergyspectrumlinearly order of a µm. Nevertheless, such a strong electric field (for a givenφ), hence increases the speed of propagation canberealizedbytwodimensionalelectrongasesformed of all modes by 1/cosφ. Consequently, the maximum in heterostructured SiGe, GaAs, or other types of III-V concurrence attainable stays the same as that in the AC materials. Furthermore,theso-calledband-enhancement effect case, though the time at which the maximum is of spin-orbit coupling in crystals [25] could be useful to achieved should be rescaled as well. All quantitative re- have a substantial AC effect. sultsintheprevioussectionarevalidforthissystemwith Despite a number of technical difficulties, some qubit H if t is replaced with t/cosφ. DM arrays, in which the effect of the phase shift can be observed, could be fabricated with present-day technol- ogy. For example, consider an array formed with charge V. SUMMARY AND OUTLOOK qubits, one type of superconducting qubits [26]. Quan- tum information in a charge qubit is represented by the We have investigated the effect of externally induced number of excess Cooper pairs in the superconducting phase shifts on the amount of entanglement that can be Cooper-pair box. When neighboring qubits are coupled transportedinaspinchain. Aswehavequalitativelyan- via a Josephson junction, the effective interaction can ticipated in the Introduction, these phase shifts can sig- be described by the XY model [27]. As the informa- nificantly enhance the efficiency of entanglement trans- tion carrier in this case is a pair of electrons, a phase fer. Although we have only studied two shift generating shift can be induced to the wave function of the pair by mechanisms, we believe that phase shifts are useful in the Aharonov-Bohmeffect. A magnetic flux Φ thread- B quantum information processing, particularly for short- ing through a ring formed by charge qubits with the distance transfers, regardless of the mechanism. Also, Josephson-junction-mediated coupling would generate a we have found that there is an interesting clear differ- phaseshift(e/~c)Φ forthewavefunctionoftheCooper B ence in the response to nonzero shift when the chain is pair. Then we could expect the same effect discussed in sufficiently long. this paper. Nonetheless, observing this effect is by no In the AC-effect-related experiment, there could be meansstraightforward: alltechnicalproblemsfromnano- a difficulty in providing an electric field that is intense structure fabrication to measurement method should be 7 addressed. Weshallleavethesechallengingexperimental same as Eq. (6), but instead of the periodic boundary problems for future investigation. condition we have the open boundary condition (OBC), α = α = 0. The one-magnon eigenstates are given 0 N+1 by Acknowledgments N 2 KM acknowledges Charlie Tahan for stimulating dis- k OBC = e−ijθsin(kj)S+ 0 ⊗N, (19) | i N +1 j | i cussions, and Sahel Ashhab for a careful reading of the r j=1 X manuscript and helpful suggestions. This work was sup- ported in part by the National Security Agency (NSA), where k = πn/(N +1) with N/2 < n N/2. Corre- theLaboratoryforPhysicalSciences(LPS)andtheArmy − ≤ sponding energy eigenvalues are ResearchOffice(ARO);andalsobytheNationalScience Foundation (NSF) grant No. EIA-0130383. EOBC = cosk. (20) k − Appendix: Note on open ended chains Clearly, the phase shift θ has no effect on the energy spectrum and thus the propagationspeed of each mode. As a geometry for information transfer, open ended Thereforeθ causesnochangeintheconcurrencebetween linear chains may look more natural. If the dispersion any two sites. relation can be affected by the phase shift in the case This result can also be paraphrased in the following of open ended chain, then the amount of entanglement way. The phase shifts at all sites can be cancelled by a transferred can be expected to change as well. However, product of local gauge transformations, Π exp[ijθ(Sz + j j this is not the case. Let us briefly look at this. 1/2)], in the case of open ended chains. The same can- Suppose thatachainofN spins isplacedinauniform cellation cannot be made for ring-shaped chains because electric field as in Section III. The Hamiltonian is the of the accumulated phase along the chain. [1] S.Bose, Phys. Rev.Lett. 91, 207901 (2003). [15] I. Dzyaloshinskii, Phys.Chem. Solids 4, 241 (1958). [2] V.Subrahmanyam,Phys.Rev. A 69, 034304 (2004). [16] T. 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