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6

Quantum entanglement generation with surface acoustic waves

G. Giavaras,1,2 J. H. Jefferson,2 A. Ramsak,3 T. P. Spiller,4 and C. J. Lambert1

1Department of Physics, Lancaster University, Lancaster LA14YB, England2QinetiQ, St.Andrews Road, Malvern WR143PS, England

3Faculty of Mathematics and Physics, University of Ljubljana and J.Stefan Institute, 1000 Ljubljana, Slovenia4Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol, BS34 8QZ, UK

(Dated: February 6, 2008)

We propose a scheme to produce spin entangled states for two interacting electrons. One electronis bound in a well in a semiconductor quantum wire and the second electron is transported along thewire, trapped in a surface acoustic wave (SAW) potential minimum. We investigate the conditions forwhich the Coulomb interaction between the two electrons induces entanglement. Detailed numericalinvestigation reveals that the two electrons can be fully spin entangled depending on the confinementcharacteristics of the well and the SAW potential amplitude.

I. INTRODUCTION

In recent years it has become appreciated that entan-glement, one of the key fundamental features of quantumphysics, lies at the heart of numerous interesting researchareas. The ability to create entanglement between qubitsin a controlled manner is a necessary ingredient for anycandidate quantum information processing1 system. En-tanglement between quantum degrees of freedom of inter-est and those beyond our control—the environment—isresponsible for decoherence and the degradation of purequantum evolution. Entanglement can exist in solidseven at thermal equilibrium2,3 and it potentially givesa new perspective for critical phenomena4. In solid statesystems, whether from the perspective of fundamentalquantum phenomena or their assessment as candidatequantum processing devices, a real challenge is to es-tablish and control entanglement between chosen quan-tum degrees of freedom, whilst avoiding decoherence dueto entanglement with the relevant environment. In thiswork we study, from a theoretical and modelling perspec-tive, the generation of entanglement between electrons insemiconductor systems that are amenable to current fab-rication and experimental techniques.

Single electron transport (SET) in a GaAs/AlGaAssemiconductor heterostructure using a surface acousticwave (SAW) was demonstrated with a very high accuracyalmost a decade ago by Shilton et al.5. Originally, theSAW-based SET devices were investigated in the contextof metrological applications and specifically for defininga quantum standard for the current5,6,7. However, manyother novel applications based on this technology havebeen proposed aiming to manipulate the integer numberof electrons in various ways. For example, an extensionof a SAW-based SET device is a single photon source8 anecessary tool in quantum cryptography9,10.

Barnes et al.11,12 suggested how quantum computa-tions can be performed and quantum gates can be con-structed using the spins of single electrons, trapped inthe SAW potential minima, as qubits. The high SAWfrequency (∼ 2.7 GHz) allows a high computation rate,which is regarded as an advantage of the SAW-based

quantum computer. The electrons are carried by theSAW in a series of narrow parallel channels separated bytunnel barriers. At the entrance of the channels a strongmagnetic field is applied to produce a well defined initialstate for the electrons. As the electrons are driven alongthe channel they can interact with electrons in adjacentchannels. The degree of interaction may be controlledby altering the height and/or the thickness of the bar-riers between the channels using surface gates. Variousreadout schemes that use for example magnetic Ohmiccontacts or the Stern-Gerlach effect were proposed anddescribed11,12.

This novel proposal of flying qubits has attracteda lot of interest and theoretical work has supportedits efficiency, though quantum gates have yet to bedemonstrated experimentally. Specifically Gumbs andAbranyos13 calculated the entanglement of spins, via theexchange interaction, for two electrons driven by SAWsin two adjacent channels. More recently Furuta et al.14

performed detailed calculations of the qubit dynamicswhen the qubits pass through magnetic fields.

In recent theoretical work Rodriquez et al.15 and alsoBordone et al.16 proposed an experiment to observequantum interference of a single electron using SAWs.The proposed experiment may provide an estimation ofthe electron decoherence time which is an importantquantity if these devices are to be exploited in the fieldof quantum information and computation. Finally, theuse of single electrons trapped in SAW potential minimafor quantum computing was considered briefly in Ref. 11where a more general scheme to induce entanglement wasexamined in which ballistic electrons propagate along twoparallel quantum wires.

In this paper, motivated by recent work on conduc-tance anomalies18 and spin entanglement generation inquantum wires19, we propose a scheme to produce entan-gled states for two electrons utilizing SAWs. A schematicillustration of the SAW-based device is shown in Fig. 1.The SAW time-dependent potential is used to carry a sin-gle electron through the channel where the second elec-tron is bound in a quantum well. The two electrons in-teract via the Coulomb interaction and it has been shown

2

SAW

2DEG

G1

G2

G4

G3T

FIG. 1: Schematic illustration of the SAW-based device togenerate spin entanglement between a static and a flyingqubit. The gates G1, G2 define a pinched-off quasi-one di-mensional channel and the gate G3 is used to create the openquantum dot which binds the static qubit. A SAW is gener-ated by the transducer (T) above a 2DEG and the (negative)potential on G1 and G2 increased until a SAW minimum con-tains a single electron which interacts with a bound electronunder G3.

in various schemes11,13,19,20,21 that this interaction is ca-pable of inducing entanglement. We investigate the con-ditions for which the electron in the SAW, after passingthrough the region of the quantum well, will be entangledwith the electron remaining in the well. Considering thespins of the electrons as the qubits the proposed schemebelongs to the static-flying qubit category where specifi-cally the qubits interact in the same channel in contrastto11 which involves interaction between flying qubits indifferent channels.

This paper is organised as follows. In Sec. II a single-electron study is presented for the bound electron in thewell and the propagating electron in the SAW. SectionIII introduces the two-electron model and considers sometypical cases of entanglement generation. In Sec. IV aHartree approximation is employed to explain the resultsof the electron dynamics. Section V presents some gen-eral important features of the entanglement and the mainresults are summarised in Sec. VI.

II. SINGLE ELECTRON STUDY

A. Preliminaries

Before examining the dynamics of the two electronsand entanglement generation, we study the two electronsseparately. The spin of the bound electron in the wellconstitutes the static qubit for the proposed scheme andtherefore it is necessary to understand how this electron

behaves under the SAW propagation. In principle, thestatic qubit must remain localised in the quantum wellduring the computation cycle and this means that theSAW-induced time-dependent perturbation must be suchthat this condition is satisfied. It is also important thatthe electron in the SAW, whose spin constitutes the flyingqubit, remains bound in the same SAW potential mini-mum at least up to the region where Coulomb repulsionwith the bound electron becomes important. Althoughthis could be achieved simply by a large SAW amplitudethe degree of screening due to the applied gate bias usedto form the quantum wire is uncertain and it may benecessary to form the wire by an etching technique22.Finally, it is interesting to note that well-defined singleSAW pulses can be generated11 which can be employedin order to minimise the interaction between propagatingelectrons and to allow the read-out process.

The well in the wire could be formed by surface gates,whose geometric design and applied bias would controlthe confining characteristics of the well. A single elec-tron turnstile23 could then be used to launch an electrontowards the region of the quantum well. Whilst these as-pects of realization are experimentally feasible, details ofthe formation of the quantum well or the capture processare beyond the scope of this paper.

For all the calculations in the following sections wehave employed a one-dimensional model considering onlythe direction of SAW propagation, that is the positive x-direction. The quantum well potential is modelled by theexpression

V (x) = −Vw exp

(−x2

2l2w

)

, (1)

where the parameters Vw , lw control the depth and thewidth of the well respectively. The SAW time-dependentpotential is given by24

VSAW (x, t) = Vo{cos[2π(x/λ− ft)] + 1}, (2)

where the parameter Vo represents the SAW potentialamplitude and to be specific we have chosen the typicalvalues f=2.7 GHz for the SAW frequency and λ=1 µmfor the SAW wavelength24.

B. The bound electron in the well

In order to study the state of the electron in the well wesolved the time-dependent Schrodinger equation, using aCrank-Nicholson scheme25 for the Hamiltonian

Ho = − ~2

2m∗

∂2

∂x2+ Vt(x, t), (3)

where m∗ = 0.067mo is the effective mass of the electronin GaAs. The total time-dependent potential is givenby the combination of the SAW and the quantum wellpotential

Vt(x, t) = VSAW (x, t) + V (x). (4)

3

024

-303

-303

-303

-303

-303

-0.5 -0.25 0 0.25 0.5

(µm)-2024

V t

x

(meV

)

FIG. 2: (color online). Time evolution of the probabilitydistribution (dashed line in arbitrary units) of the bound stateof the quantum well and the total time-dependent potential(full line). The time sequence is from top to bottom andspecifically t/T=0, 0.2, 0.4, 0.5, 0.6, 0.8, 1.

The time evolution of the probability distribution isshown in Fig. 2 for one SAW period T = 1/f and for theparameters Vo = 2 meV, Vw = 6 meV and lw = 7.5 nm.The quantum well parameters have been chosen such thatthere is only a single bound state when Vo = 0. This be-comes a quasi-bound at specific times provided εw < 2Vo,where εw is the minimum required energy to delocalizethe electron from the well when Vo = 0. Although thisinequality is fulfilled for the chosen parameters, the elec-tron still remains very well-localised in the well for thewhole SAW period as we can see from Fig. 2. This issimply because the tunnelling time to escape from thewell is much greater than the SAW period.

The instantaneous eigenvalues versus time, obtainedby solving the time-independent Schrodinger equation ateach instant in time are shown in Fig. 3 and provideinsight into the dynamics. Only the first few eigenvaluesthat are relevant to the time evolution process are shownin order of increasing energy (En, n = 0, 1,...). Eigen-values corresponding to an odd (even) integer are shownwith a dashed (full) line. Note that none of the curvesactually cross, though the very small energy differencecannot be resolved in the figure. The characteristic sinefeature that develops from the left to the right of thegraph indicates that the state evolves via non-adiabaticLandau-Zener transitions26. The transition probabilityat an anti-crossing point, that is a point in the graphwhere the two curves have minimum separation, dependson this characteristic energy gap26,27. Specifically, if theenergy gap is large the state cannot undergo the tran-sition and thus it tunnels out of the well losing its ini-tial character, that is bound in the quantum well. Onthe other hand for small energy gaps, which is the casehere, the electron can successfully undergo Landau-Zener

0 0.2 0.4 0.6 0.8 1

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(m

eV

)E

n

/ Tt

FIG. 3: (color online). The first few bound instantaneouseigenenergies (En, n = 0, 1,...) as a function of time for thetotal time-dependent potential Vt(x, t) described in the text.Eigenenergies which correspond to an odd (even) integer areshown with a dashed (full) line.

transitions thus retaining the initial character of its stateas the time develops and the potential profile changes.Strictly speaking, after very many SAW cycles the elec-tron will be delocalised from its initial quantum well be-cause the Landau-Zener transitions do not occur withprobability of exactly one. In our study the sine featureis only shown for one SAW period and it describes howthe energy of the bound electron in the quantum wellchanges with time as the SAW propagates. An impor-tant characteristic is that via the Landau-Zener transi-tions the state retains its initial character by changingthe eigenvalue number at each anti-crossing point from nto n± 1. In particular, for t = 0 and after one SAW pe-riod t = T the SAW potential is maximum at x = 0 andtherefore the energy level that corresponds to the boundstate of the quantum well is maximum. For t = T/2the SAW potential is minimum at x = 0 and the en-ergy of the quantum well is the ground state energy ofthe system. For t < T/2 the state lowers its eigenvaluenumber at each anti-crossing point from n to n−1 in or-der to decrease its energy, whereas for t > T/2 the stateincreases its eigenvalue number from n to n + 1 in or-der to increase its energy, via successfully accomplishingLandau-Zener transitions. For t > T this pattern of tran-sitions is repeated. Increasing the SAW potential ampli-tude and keeping the characteristics of the well fixed theenergy gaps become larger and eventually the sine fea-ture will disappear. In this case the electron escapes fromthe quantum well tunnelling partly in the SAW potentialminimum and in the continuum. Decreasing the SAWamplitude there will be a value such that the state ofthe well will be a true-bound state at all times. In thiscase the state evolves adiabatically, its energy changes si-nusoidally and the electron remains localized in the wellwithout any effect from the SAW propagation.

4

To summarise for a particular quantum well there is aregime of a small SAW potential amplitude (that satis-fies εw < Vo) where the bound electron evolves adiabat-ically, followed by a regime of stronger SAW amplitudefor which the electron evolves via non-adiabatic Landau-Zener transitions. Finally, for an even stronger SAW am-plitude the electron escapes from the quantum well. TheSAW potential amplitude must be restricted to the firsttwo regimes for a particular well depth. Here we considerthe most interesting intermediate case and although weonly consider a quantum well with a single bound state,it is straightforward to generalise the results to cases withmore bound states in the well.

C. The propagating electron in the SAW

In this section we study how the electron in the SAWpotential minimum propagates along the quantum wirefar from the quantum well for which we may restrict thepotential of the Hamiltonian (3) to the SAW potentialonly. The set of coefficients Cm, m = 0, 1..., which satisfy

Cm = − Cm〈um|um〉

+∑

n6=m

Cn

~ ωmn

⟨

um

∣

∣

∣

∣

∂Ho

∂t

∣

∣

∣

∣

un

⟩

exp

[

i

∫ t

o

ωmn(t′)dt′]

,

(5)

determines the evolution of the wave function φ(x, t), viathe expansion28

φ(x, t) =∑

n

Cn(t)un(x, t) exp

[

− i

~

∫ t

o

En(t′)dt′]

, (6)

in the basis states of the instantaneous solutionsHo(x, t)un(x, t) = En(t)un(x, t), with ωmn = (Em −En)/~ the Bohr angular frequency. The wave functionφ(x, t) describes the electron in the SAW potential mini-mum and satisfies the time-dependent Schrodinger equa-tion. The system of Eqs. (5) is solved using a fourth-order Runge-Kutta method29, although for the calcula-tions we have dropped the first term of Eqs. (5), since itonly induces an unimportant phase difference in the finalcoefficients. Figure 4 shows the variation of the squaredmodulus of the coefficients, when the initial state is theground, the first and the second excited state of the SAWpotential minimum, (|Cj

n|2, j = n = 0, 1, 2 where the su-perscript j indicates the corresponding initial state) asa function of time for two SAW periods and for a SAWamplitude of Vo = 4 meV. As we can see, the electron re-mains to a very good approximation in the initially pop-ulated state of the SAW minimum throughout the timeevolution and furthermore the corresponding moduli ofthe expansion coefficients present an oscillating behavior.

This behavior may be explained within the adiabaticapproximation28. Starting with Cj

n(t = 0) = δnj , (initialstate uj) and assuming that all the coefficients in (5)

0.988

0.992

0.996

1

|C |

0.96

0.98

1

|C |

0 0.5 1 1.5 2

0.94

0.96

0.98

1

|C |

0 01

22

2

2

12

t / T

FIG. 4: (color online). From top to bottom the initial electronstate (for t = 0) is the ground, the first, and the second excitedstate of a SAW potential minimum respectively. The plotsshow how the corresponding probability for each initial stateevolves with time for two SAW periods.

remain constant with time Cjn(t > 0) ≈ δnj , we obtain

the approximate formula for all m 6= j

Cjm ≈ ωΛmj

~ ωmjexp

[

i

∫ t

o

ωmj(t′)dt′

]

, (7)

where we have set Λmj = 〈um|∂Ho

∂t |uj〉/ω, with ω = 2πfthe SAW cyclic frequency. The matrix elements Λmj andthe frequencies ωmj are time-independent and hence thefinal expression for the squared modulus of the coeffi-cients for m 6= j becomes

|Cjm(t)|2 ≈ ω2|Λmj|2

~2 ω4mj

4 sin2

(

ωmjt

2

)

. (8)

For the SAW potential given by (2) the matrix elementsare real for bound states and therefore Λmj = Λjm. Also,the transitions are allowed when Λmj 6= 0 which occurswhen m+ j is odd. If ω|Λmj| ≪ ω2

mj~ then |Cjm| ∼ 0 and

the electron remains at all times in the initial state uj.This limit corresponds to the adiabatic approximationand is satisfied when the system changes very slowly com-pared to the transition frequency ωmj associated with thestates. In general, the higher the states the smaller thetransition frequency between them and as a result the lessvalid the adiabatic approximation. This can be seen di-rectly from Fig. 4 by observing the minimum magnitudeof the oscillations which gives the maximum deviationfrom the initial state and hence the deviation from theadiabatic approximation. On the other hand, the lowerthe SAW frequency the better the adiabatic approxima-tion. In the extreme limit of a ’frozen’ wave, ω = 0, thestates become stationary acquiring only a phase.

For the ground state, in the time interval of interest,the adiabatic approximation is excellent. During the time

5

evolution there is a very small contribution from excitedstates and |C0

0 |2 ≈ 1 − |C01 |2 with |C0

1 |2 given by Eq. (8)with m = 1 and j = 0. Hence, the frequency of theoscillations is equal to ω10 and the amplitude dependson the quantity ω2|Λ10|2/(~2 ω4

10). In the evolution ofthe first excited state there is some contribution not onlyfrom the ground but also from the second excited state.We may approximate the numerical results by |C1

1 |2 ≈1 − |C1

0 |2 − |C12 |2 and using Eq. (8)

|C11 (t)|2 ≈ 1−D01 sin2

(

ω01t

2

)

−D21 sin2

(

ω21t

2

)

, (9)

where the auxiliary constant is Dm,j =4ω2|Λmj |2/(~2ω4

mj). Defining the quantitiesδ = (ω10 − ω21)/2,δo = (ω10 + ω21)/2 andD = (D21 − D10)/2, Do = (D21 + D10)/2 we mayfurther write

|C11 (t)|2 ≈1 −Do +Do cos(δot) cos(δt)

+D sin(δot) sin(δt),(10)

which explains the sinusoidal variation of the amplitudein the oscillations with a period equal to 2π/δ. This pe-culiar form is due to the very small difference betweenthe Bohr frequencies which define the frequency of theoscillations and the small difference between the matrixelements which control the amplitude of the oscillations.A similar situation occurs for the evolution of the secondexcited state as shown in the bottom frame of Fig. 4. Allthe relevant quantities can be defined similarly, consid-ering that the small deviation from this state is mainlydue to transitions to the first and the third excited state.

To summarise, the electron in the SAW propagates adi-abatically along the quantum wire even for a relativelylow SAW amplitude. In other words, it remains well-localised in the particular SAW potential minimum as itis driven towards the bound electron in the well, whichis the main requirement for the entanglement generationdescribed in the next section.

III. ELECTRON DYNAMICS AND

ENTANGLEMENT

A. The two-electron model

The dynamics of the two-electron system is governedby the time-dependent Schrodinger equation, with theHamiltonian

H =∑

i=1,2

[

− ~2

2m∗

∂2

∂x2i

+ Vt(xi, t)

]

+ Vc(x1, x2). (11)

The single electron term Vt(x, t) was described in SectionIIB and the Coulomb term Vc(x1, x2) is modelled by thequasi-one-dimensional form

Vc(x1, x2) =q2

4πǫrǫo√

(x1 − x2)2 + γ2c

, (12)

where ǫr = 13 is the relative permittivity of GaAs.This simplified form of the Coulomb interaction as-sumes that all excitations take place in the x-direction,whereas in the other two directions the electrons occupyat all times the corresponding ground states (transversemodes). This is a good approximation provided that theparameter γc that models the confinement lengths in they and z directions is relatively smaller than the confine-ment length scales in the x-direction. For all the calcula-tions we choose γc = 20 nm, for which the restriction tolowest transverse modes is an excellent approximation.

The resulting two-electron time-dependentSchrodinger equation is solved numerically with anexplicit scheme based on a finite difference method,which is described in detail for the case of a singleelectron by Visscher30. The extension for two electronsis straightforward.

For the initial state at t = 0 we choose one electronto have spin up in the ground state of the SAW poten-tial minimum, ψ(x), and the other electron to have spindown in the ground state of the quantum well, ϕ(x). Itis important to note that ψ(x) and ϕ(x) are exactly or-thogonal, with no spatial region of overlap, i.e. ψ peaksaround the region where the particular SAW potentialminimum is located (far from the quantum well) and ϕpeaks around x ∼ 0 where the quantum well is located.To study the dynamics of the electrons, at time t > 0 it isnecessary to take into account the fact that the electronsare fermions and thus indistinguishable. This is impor-tant when the electron carried by the SAW interacts withthe electron in the quantum well, giving rise to a spin ex-change interaction. The initial state is thus representedby the Slater determinant

Ψ↑↓(x1, x2, 0) =1√2

∣

∣

∣

∣

ψ(x1)χ↑(1) ϕ(x1)χ↓(1)ψ(x2)χ↑(2) ϕ(x2)χ↓(2)

∣

∣

∣

∣

. (13)

This state is unentangled according to the criteria of Ref.31. Note that Ψ↑↓(x1, x2, 0) can also be expressed as acombination of a singlet and a Sz=0 triplet state

Ψ↑↓(x1, x2, t) =1√2[ΨS

↑↓(x1, x2, t)+ΨT↑↓(x1, x2, t)], (14)

which is also the general form of the total two-electronwave function at all times, due to the fact that theHamiltonian Eq. (11) contains no spin-dependent terms.Furthermore, for the case of two electrons, the or-bital and spin parts factorize, i.e. ΨS

↑↓(x1, x2, t) =

ΦS(x1, x2, t)χS↑↓(1, 2) and similarly ΨT

↑↓(x1, x2, t) =

ΦT (x1, x2, t)χT↑↓(1, 2). With this notation the spin

components are given by χS/T↑↓ (1, 2) = [χ↑(1)χ↓(2) ∓

χ↓(1)χ↑(2)]/√

2 (with the negative sign for the singlet)and the corresponding orbital components at t = 0, byΦS/T (x1, x2, 0) = [ψ(x1)ϕ(x2) ± ϕ(x1)ψ(x2)]/

√2 (with

the positive sign for the singlet). For t > 0, the spineigenstates are unchanged whereas the orbital states aregiven directly by the solution of the time-dependent

6

Schrodinger equation. The form of the orbital compo-nents at t = 0 implies that the two electrons do notinteract, i.e. they are well separated with negligibleCoulomb interaction, and therefore are written as sym-metric and antisymmetric products of non-interactingsingle-electron states. Finally, in this work we consideronly cases where the electron in the quantum well is well-localized before (t = 0) and after the scattering event(t = tf ), when the electrons are well separated. Theenergy parameters (SAW amplitude and quantum wellcharacteristics) are thus chosen such that the final elec-tron probability distribution in the well is to a good ap-proximation the same as before interaction. However,this restriction is not imposed on the propagating elec-tron in the SAW, which can gain energy due to a combi-nation of the effect of the time-dependence of the SAWpotential and Coulomb repulsion.

B. Entanglement measure

In this paper, concurrence will be used as a measureof spin entanglement. An expression for concurrencemay be obtained using the form suggested by Wooters32,starting with the total density matrix for the pure scat-tering state and integrating over the orbital degrees offreedom. For the axially symmetric problems consideredhere, for which total spin projection along the quantisa-tion axis is conserved, concurrence is physically relatedto spin-spin correlation functions for the two domains Aand B and takes the form33 C = 2|〈S+

AS−B 〉|, where S±

are the usual spin flip operators. Equivalently in termsof the symmetric (singlet) and antisymmetric (triplet)orbital states concurrence is given by the formula33

C(t) =1

N(t)|∫

A,B

dx1dx2Φ∗−(x1, x2, t)Φ+(x1, x2, t)|

(15)where Φ±(x1, x2, t) = ΦS(x1, x2, t) ± ΦT (x1, x2, t) andthe normalization constant N equals

N(t) =

∫

A,B

dx1dx2(|ΦS(x1, x2, t)|2 + |ΦT (x1, x2, t)|2).

(16)In these expressions the regions of integration A, B arechosen to be regions which the electrons are expectedto occupy before and after scattering with A being thedomain of one electron and B the domain of the other.Physically, the regions A and B can be viewed as (posi-tion) measurement domains. For example, sensing of thepresence of an electron charge34,35,36 with sufficient posi-tional information only to identify it as being located insome region could correspond to such a “fuzzy” positionmeasurement. Since the quantum well always containsat least one electron, we choose A to be this region, i.e.A=[xl, xr], where xl and xr denote the points where thebound state of the electron in the quantum well has de-cayed to zero at the left and right respectively. For the

numerical calculations, these points were chosen to corre-spond to a value of approximately 10−4 of the probabilitydensity at the peak. The regionB is chosen to correspondto the region of occupation of the propagating electron.For the two-electron scattering problem under study, wemay choose this to be the total domain excluding thewell, i.e. B=[-L, xl]∪[xr , L], where [-L, L] defines thetotal region of space within which the electron dynamicsis studied. Note that the corresponding concurrence isreally only meaningful when the electrons are well sepa-rated, before and after scattering, i.e. at sufficiently smallor large t, though it may be calculated at any time. Werefer to this concurrence as the total concurrence, C(t),. We may also define (the potentially more useful) re-flected or transmitted concurrence, for which the mea-surement domains are restricted to either the left or theright of the quantum well respectively, i.e. B=[-L, xl],or B=[xr, L], with corresponding concurrences Cr andCt. A “fuzzy” position measurement (charge sensing)which gives sufficient information to resolve the outgoingelectron as reflected or transmitted could project the twoelectrons into a state with the associated concurrence Cr

or Ct. It is also useful to define the quantities PS/Tt and

PS/Tr as the transmitted and reflected probabilities for

singlet and triplet states. The maximum probability inthe whole space of either state equals 0.5 due to the gen-eral form (14) of the wave function. The transmitted andreflected concurrence are considered only when the cor-responding singlet or triplet probabilities are not negli-gible. For the numerical calculations the minimum limitwas taken to be approximately 10−2. We should alsomention that by definition33 0 ≤ C ≤ 1, where the limitC = 0 corresponds to an unentangled state and C = 1 toa fully entangled state. For the time-dependent problemunder study, the concurrence is also time-dependent andit is easily verified that for the initial state, C(t = 0) = 0.

C. Entanglement generation

In this section we present some typical scattering re-sults that take place when the two electrons interact viathe Coulomb interaction and demonstrate how the en-tanglement develops with time due to this interaction.

Figure 5 shows the initial and final electron densityρS/T (x, t) = 2

∫

dx′ |ΦS/T (x, x

′

, t)|2, and Fig. 6 showshow the concurrence and the relative probabilities de-velop with time for the parameters Vw = 6 meV, lw = 7.5nm and Vo = 2 meV. For these parameters the quantumwell can accommodate only a single bound electron, asecond electron being delocalised due to the Coulombinteraction. From the figure we see that there is a veryhigh transmission for the singlet state and high reflectionfor the triplet state after scattering. The concurrence(C,Ct, Cr) varies with time, due to the interactions of thewave packets mediated by the Coulomb interaction, andeventually saturate to a constant value when the over-lap is once again negligible. The transient time interval

7

-1.2 -0.8 -0.4 0 0.4 0.8(µm)x

ρS

(arb

. uni

ts)

ρT

(arb

. uni

ts)

FIG. 5: (color online). Initial (dashed) and final electrondistribution (full) in arbitrary units when the singlet state(top) is mostly transmitted and the triplet state (bottom) ismostly reflected.

0

0.1

0.2

0.3

0.4

C

0

0.2

0.4

0.6

0.8

1

Ct ,

P

t

0.28 0.32 0.36 0.4 0.44 0.48 (ns)

0

0.2

0.4

0.6

0.8

1

Cr

,

Pr

t

Ct

Cr

Pt

S

Pt

T

Pr

T

Pr

S

FIG. 6: (color online). Concurrence and relative probabil-ities as a function of time when the singlet state is mostlytransmitted and the triplet state is mostly reflected.

is not of main interest since the degree of entanglementis important after the scattering process when the twoelectrons are well separated. Note that the reflected con-currence is close to unity since at the left hand side the re-flection probability of the singlet state is very small com-pared with that of the triplet, and a pure Sz = 0 tripletis fully spin entangled. In this case, the reflection process

-1 -0.8 -0.6 -0.4 -0.2 0 0.2(µm)x

ρS

(arb

. uni

ts)

ρT

(arb

. uni

ts)

FIG. 7: (color online). Initial (dashed) and final electrondistribution (full) in arbitrary units when both singlet (top)and triplet (bottom) are almost totally reflected.

may be regarded as a filtering process in which the singletpart of the initial wave function is essentially removed bytransmission to the right. One the other hand, if we lookin transmission, although the singlet part is almost fullytransmitted, the transmission of the triplet is not negligi-ble and interference results in an asymptotic concurrencewhich is somewhat less than unity. We also see that themaximum concurrence is also significantly reduced whenthe measurement domain includes both transmitted andreflected parts after scattering.

In Fig. 7 we present results for a smaller SAW am-plitude (Vo = 0.5 meV) for which the initial and finalelectron density of both singlet and triplet states are al-most totally reflected, the electron carried by the SAWbeing almost completely reflected by the Coulomb re-pulsion with the bound electron. Fig. 8 illustrates howthe concurrence develops in time, the concurrence of thereflected part and that over the whole domain being ap-proximately the same due to the high reflection. We cansee again that the concurrence builds up with time dueto the Coulomb interaction and saturates to a constantvalue after reflection. Note however, that this asymptoticvalue is much smaller, due essentially to the Coulomb re-pulsion inhibiting significant overlap of the wave packets.

A third regime of interest is when both singlet andtriplet are almost fully transmitted. Figure 9 shows theinitial and final electron density for such a case with pa-rameters Vw = 70.5 meV, lw = 10 nm and Vo = 10 meV.Although for this choice of parameters the quantum wellcan bind two electrons in the absence of the SAW, thesecond electron does not in fact become bound when itis carried by the SAW and after scattering the proba-bility of finding both electrons in the well is negligible.This is because the SAW period is too short for the sec-ond electron to become trapped in the well. We thussee that both singlet and triplet states are almost per-fectly transmitted, whilst the electron in the quantum

8

0.21 0.24 0.27 0.3 0.33 0.36 0.39 (ns)

0

0.001

0.002

0.003

0.004

C

t

FIG. 8: (color online). Concurrence as a function of timewhen both singlet and triplet are almost totally reflected.

well remains very well-localised. However, the electronin the SAW potential minimum, after passing throughthe region of the bound electron in the well, gains energyfrom the interaction, resulting in a superposition statewhich includes excited states of the SAW. It is easilyverified that for this special case the concurrence takesthe form33 C = |Im〈T |S〉|, where |T 〉 and |S〉 are the sin-gle electron states of the transmitted electron in a SAWminimum which result from triplet and and singlet statesrespectively. We see from this formula that C = 1 onlywhen |T 〉 and |S〉 differ by a phase factor of π/2. This isnot the case in general, for which not only the phases butalso the amplitudes of |T 〉 and |S〉 are different. Limitswhich give zero concurrence are when |S〉 = |T 〉 (such asthe trivial case of no interaction between the electrons)and when |S〉 and |T 〉 are orthogonal. As explained inthe next section the latter can occur, or approximatelyso, when an electron in the well resonantly tunnels outof the well into an excited state of a SAW minima forsinglet but not triplet or visa versa. More generally, theoverlap (and hence C) may be small but not preciselyzero due again to different tunnelling rates for singletand triplet. Figure 10 shows how the concurrence andthe relative probabilities develop in time. Similarly withthe previous cases the concurrence increases and satu-rates to a constant value, while for intermediate timesit oscillates. Since the reflected part is very small, weget the expected result that the asymptotic value of thetransmitted concurrence approximately equals the totalconcurrence C ≈ Ct ∼ 0.53.

Finally, we consider the possibility of choosing param-eters such that by changing the confining characteris-tics of the well the singlet and triplet orbital compo-nents, after the scattering event, may be chosen to dif-fer only by a phase factor eiδϕ, with δϕ = ϕS − ϕT .This may be done, at least approximately, for param-eters which give almost perfect transmission for bothsinglet and triplet states. This occurs, for example,when the SAW amplitude is sufficiently large. For thisregime the concurrence takes the form33 C = | sin δϕ|,as can be seen directly from Eq. (15), using the form

ΦS(x1, x2, tf ) = eiϕS [ψf (x1)ϕ(x2) + ϕ(x1)ψf (x2)]/√

2,

-0.6 -0.4 -0.2 0 0.2 0.4 0.6(µm)x

ρρ

S(a

rb. u

nits

)T

(arb

. uni

ts)

FIG. 9: (color online). Initial (dashed) and final electron dis-tribution (full) in arbitrary units for a typical case when bothsinglet (top) and triplet (bottom) are almost fully transmit-ted.

0

0.2

0.4

0.6

C

0

0.2

0.4

0.6

Ct ,

Pt

0.15 0.175 0.2 0.225 0.25 0.275

(ns)

0

0.2

0.4

0.6

0.8

Cr ,

Pr

t

Ct

Cr

Pt

S

Pt

T

Pr

S

Pr

T

FIG. 10: (color online). Concurrence and relative probabili-ties as a function of time for a typical case when both singletand triplet are almost fully transmitted.

ΦT (x1, x2, tf ) = eiϕT [ψf (x1)ϕ(x2) − ϕ(x1)ψf (x2)]/√

2,for the singlet and triplet orbital components respec-tively. Note that ψf describes the electron in the SAWpotential after scattering and ϕ describes the electron inthe well. We see immediately from this form that theconcurrence may be controlled by changing the relativephase of singlet and triplet whilst maintaining approxi-mately full transmission, giving full entanglement whenthe magnitude of this phase difference is π/2. Howeverthe value of the phase difference cannot be easily con-trolled and, indeed, the phase difference picture is itself

9

-0.5 -0.25 0 0.25 0.5(µm)x

ρS

(arb

. uni

ts)

ρ(a

rb. u

nits

)T

FIG. 11: (color online). Initial (dashed) and final electron dis-tribution (full) in arbitrary units when both singlet (top) andtriplet (bottom) are fully transmitted with a phase differenceas it is defined in the text.

only an approximate since the singlet and triplet prob-abilities distributions in the SAW minimum after scat-tering are never precisely identical. However for somespecial cases this model is an excellent approximation asshown for example in Fig. 11 for the parameters Vw = 66meV, lw = 10 nm for which the well can bind at leasttwo electrons and Vo = 20 meV. We see that the electronin the SAW after the scattering event is well-bound inthe SAW potential minimum occupying the characteris-tic ground state both for singlet and triplet. In Fig. 12we show the concurrence as a function of time. In thiscase the concurrence increases relatively smoothly com-pared to the previous cases because we are in a regimewhere the one electron orbital states in the scatteringprocess are the same for singlet and triplet, apart froma phase factor. Note that in order to even have the pos-sibility of achieving high concurrence the electrons musthave sufficient time to interact as the SAW propagates.The timescale to give spin entanglement is of order ~/|J |,with J = ET −ES the exchange energy and ET , ES thetriplet and singlet energies when we fix the SAW withboth electrons in the proximity of the well. Hence theSAW period has to be at least as long as this for highconcurrence to be possible and this is indeed the casefor typical SAWs which used to give high accuracy sin-gle electron quantisation5,6,7, as we demonstrate in nextsection. Finally, it is worth mentioning that to a good ap-proximation a phase difference may be present even whenboth states are reflected backwards, as described earlier.However in this case the phase difference is expected verysmall due to the weak effect of the Coulomb interaction.Furthermore, the regime of near perfect transmission alsohas the advantage that the electron in the SAW, after thescattering event is very well-localised in a particular SAWminimum driven along the wire.

For all the cases we have described so far the induced

0

0.1

0.2

0.3

0.4

0.5

C

0

0.1

0.2

0.3

0.4

0.5

Ct ,

Pt

0.16 0.18 0.2 0.22

(ns)

0

0.2

0.4

0.6

Cr ,

Pr

Ct

Cr

Pt

S

Pt

T

Pr

SPr

T

t

FIG. 12: (color online). Concurrence and relative probabil-ities as a function of time when both singlet and triplet arefully transmitted with a phase difference as it is defined inthe text.

entanglement between the two-electron spins is subjectedto quantum decoherence which is an undesirable factorpresent to all solid state systems. The spin lifetime inGaAs, within which the process of generation-detectionneeds to take place, is estimated to be ∼100 ns37 arisingprimarily from phonon scattering. Typical times to gen-erate entanglement in the SAW-based system are almosttwo orders of magnitude shorter while methods to readthe final spin states have been described theoreticallyin Refs 11, 14 for the SAW-qubit and already demon-strated experimentally for the static qubit38. Other im-portant sources of decoherence that can affect the en-tanglement generation are coupling of electron spins tonuclear spins39,40, noise on surface gates and tempera-ture effects. These are discussed in the original proposalfor SAW-based quantum computation11,14.

IV. A HARTREE APPROXIMATION

The mechanism that controls the different scatteringof singlet and triplet may be understood with an ap-proximate treatment which also gives insight into theorigin of the differences in transmission and reflectionprobabilities and concurrence. In a mean-field approx-imation, the electron which is carried by the SAW po-tential feels an effective time-dependent potential of theform V e

SAW (x, t) ≈ Vt(x, t) + VH(x, t), where the sec-ond term represents the Hartree potential due to theCoulomb repulsion of the trapped electron in the well:VH(x, t) =

∫

|ϕ(x′

, t)|2Vc(x, x′

)dx′

. This assumes that

10

the trapped electron in the quantum well remains well-localised and is described at a specific time t by the stateϕ(x, t). Below we explain the different scattering resultsof Sec. III.C by employing the effective potential formV e

SAW for the propagating electron.

A. The single bound energy level regime

First we consider the cases for which the quantum wellhas a single bound energy level, i.e. the first two cases ofSec. III.C. Figure 13(a) illustrates the effective potentialV e

SAW (x, t) when t is such that VSAW (x, t) is minimumat x ∼ 0 and for parameters that result in high transmis-sion for the singlet state and high reflection for the triplet(that is the first case that we described in Sec. III.C). Theeffective potential may be described as a triple well struc-ture that changes with time due to the time-dependentnature of the SAW potential. Specifically, due to theSAW propagation, the right well becomes deeper thanthe left well with increasing time, with the middle wellshifting upwards and downwards in energy at x ∼ 0. Ini-tially, the electron that is carried by the SAW resides inthe left well and has a tendency to tunnel through themiddle well into the right well, in order to remain boundin the SAW potential. The tunnelling mechanism is moreefficient when there are resonance conditions for the elec-tron to first tunnel from the left well into the middle welland then from the middle well to the right-hand well, i.e.in the time interval when resonant bound state energywidths of the left, middle and right-hand wells overlap.Of course the SAW potential amplitude, the width of thewire and the characteristic width and depth of the wellshould be chosen in such a way that the resulting effectivepotential guarantees at least one resonance energy levelfor the middle well, that will lie above the bottom of theright and left wells. A sufficiently large SAW amplitudeis also necessary to ensure that the barrier between theresonance condition between left and middle wells is sat-isfied, otherwise the electron in the SAW will be reflected.Finally, a necessary criterion for high transmission is thatthere must be sufficient time for the whole process to takeplace, i.e. the tunnelling time into and out of the middlewell must be much smaller than the period of the SAW,a condition that is fulfilled in the simulations.

To explain qualitatively the difference in the evolutionbetween the singlet and the triplet states, we also needto consider explicitly the symmetry of the orbital statesand take into account the fact that only for the singletstate can both electrons occupy the same one electronorbital state. More specifically, if ϕo(x, t) is the low-est resonant bound state of the combined well and SAWpotential that peaks in the region of the well x ∼ 0,then the instantaneous energy of the two-electron sin-glet state on resonance is ES(t) ≈ εo(t) + Uo(t) whereUo(t) =

∫

|ϕo(x1, t)|2Vc(x1, x2)|ϕo(x2, t)|2dx1dx2 is theCoulomb energy when both electrons occupy the singleelectron state ϕo(x, t). This is analogous to resonant

-6

-4

-2

0

2

4

(me

V)

-0.6 -0.4 -0.2 0 0.2 0.4 0.6(µm)

0

1

2

3

4

(m

eV

)

=2.0 meV

=0.5 meV

E s

εο

E s

(a)

(b)

VH

Vt

V e

x

V e

SA

W

SAW

Vo

Vo

FIG. 13: (color online). (a) The effective potential, and itsconstituent parts, that a SAW electron feels at a time forwhich the SAW potential is minimum at x = 0 and for aSAW amplitude Vo = 2 meV. (b) The effective potential closeto the resonant tunnelling regime for the singlet state for aSAW potential amplitude Vo=2 meV (solid line). For Vo=0.5meV (dotted line) the resonance condition can not be fulfilled(see text).

tunnelling in the Anderson impurity model. A similarapproach could be applied to the triplet state but inthis case the two electrons must occupy different oneelectron resonance levels, ϕo(x, t) and ϕ1(x, t) due tothe Pauli principle. If the quantum well had a secondresonant state then the two-electron resonance wouldoccur at the generally higher, triplet resonance energyET (t) ≈ ε1(t) + U1,0(t) − J1,0(t) > ES(t), where U1,0(t)and J1,0(t) are the Coulomb and the exchange integralsrespectively.

From the above description it is clear that the electronwhich is carried by the SAW feels an effective potentialwhich is independent of the character of the two-electronorbital state (symmetric or antisymmetric), however thisis not the case for the energy levels of the tunnelling pro-cess. For the first regime described in Sec. III.C thesinglet resonance level gives high resonance transmissiononly for the singlet state, as expected. The transmissionof the triplet state is much weaker and is in fact due tonon-resonant tunnelling, since with the chosen param-eters the energy of the electron in the effective poten-tial V e

SAW is always below the barriers which define themiddle well. Note that for this case the SAW potentialamplitude is strong enough to drive the propagating elec-tron to the resonance level. On the other hand, in thesecond regime described in Sec. III.C the SAW poten-

11

tial well is so shallow that, for both singlet and triplet,the propagating electron is reflected before it reaches theresonance energy for tunneling into the middle well. Fig-ure 13(b) shows the effective potential profile close to theresonant tunnelling regime for the singlet state for bothSAW potential amplitudes. Note that the resonance levellies above the bottom of the left and right wells, ensur-ing that tunnelling may occur. For these two regimes aneffective antiferromagnetic exchange interaction controlsthe scattering process since the singlet scattering involveslower energy levels than the triplet.

B. Beyond the single bound energy level regime

For the last two cases of Sec. III.C the quantum wellhas more than a single bound energy level and can bindat least two electrons. One effect of making the quan-tum well deeper is to reduce the barriers to the SAWwells to the left and right, as can be seen by comparingFigs. 13(a) and 14(c) for the effective one electron po-tential. This results in almost perfect transmission forboth singlet and triplet states provided the SAW ampli-tude is much larger than the small residual barriers whenthe quantum well is at a SAW potential minimum (Fig.14(c)). However, the different positions of the singletand triplet resonances still affect the final orbital statesof the transmitted electron in the SAW, depending onthe magnitude of the tunnel barrier when the SAW po-tential minimum energy is close to the resonance level inthe quantum well (e.g. Fig. 14(b)).

Specifically, when this barrier is large the propagatingelectron emerges in the lowest state of the SAW potentialminimum. This is illustrated in Fig. 14, where we plotsome of the instantaneous eigenenergies of the effectiveone electron potential V e

SAW for the parameters that re-sult in a phase difference between singlet and triplet (thisis the last case that we considered in Sec. III.C). Notethat the quasi-bound state levels within the well includethe effect of Coulomb repulsion due to the bound elec-tron which shifts the potential well up in energy by VH

and also gives rise to the very small peaks in the effectivepotential. At t = 0 (Fig. 14(a)) the propagating electronis in the lowest energy of the V e

SAW potential minimumto the left of the quantum well. This is actually the firstexcited state of the system since the lowest state is in thewell. Between t = 0 and t = 0.3T (Figs. 14(a),(b)) theenergy levels corresponding to the electron in the V e

SAW

minimum and in the second state of well are almost thesame (anti-crossing region) but there is insufficient timefor the electron to tunnel into the well and therefore itremains in the V e

SAW potential minimum. It thereforemakes a (non-adiabatic Landau-Zener) transition fromthe first to the second excited state of the system. Att ≃ 0.3T there is a further anti-crossing region and tran-sition to the third excited state of the system with theelectron remaining in the SAW. Between t = 0.5T andt = T (Figs. 14(d),(e)) there are further transitions back

to the initial state. This is the reason that the propagat-ing electron emerges in the lowest state of the V e

SAW po-tential minimum which actually coincides with the origi-nal SAW potential minimum when the electrons are wellseparated at t = T (Fig. 14(e)) when the SAW cycle iscompleted. It is clear from Fig. 14(c) that the highestresonance level of the well, in this case is the fourth level,gives rise to the largest interaction with the propagatingelectron as long as tunnelling to lower excited states isblocked due to the large barrier. Of course lower excitedresonances are involved for shallower quantum wells.

Although the scattering process does not excite theelectron in the SAW, it does affect its wave function byinducing a phase shift and this phase shift is different forsinglet and triplet cases due to Coulomb interaction. Inparticular, the evolution of singlet and triplet states willbe different but, unlike the lowest singlet-triplet pair, thehigher-lying levels will generally have the triplet lowerin energy than the singlet, due essentially to Hund’srule41,42. This results in a ferromagnetic exchange in-teraction, rather than the antiferromagnetic exchange ofthe lowest singlet-triplet pair. It is the small energy dif-ference between the relevant singlet and triplet energylevels which induces the relative phase between singletand triplet states as a consequence of the interaction timefor the two electrons which is set by the SAW period. Weshow in the next section how the phase difference may bedirectly related to a ferromagnetic exchange interactionbetween the spins of the two electrons as they interact,changing their entanglement.

Finally, when the parameters are such that the proba-bility to tunnel from the SAW potential minimum to thewell is not negligible in the anti-crossing region, (e.g. thethird case considered in Sec. III.C) then the electron willemerge in a superposition state of the low-lying states ofthe SAW. This can be understood qualitatively again byreferring to Fig. 14. Since the tunnelling probability is nolonger negligible at the point where the V e

SAW minimumcrosses a resonance level, then the electron in the regionof the well (e.g. Fig. 14(c)) will emerge in a superposi-tion state of the third and fourth energy levels. Similarly,at later times when the energy level corresponding to theelectron in the well sweeps through higher excited levelsin the V e

SAW potential minimum the electron will even-tually leave the well and emerge in an asymptotic statethat is a superposition of the low-lying states of the SAW.In this regime the different orbital states for singlet andtriplet are due to different tunnel barriers for the highest-lying singlet-triplet pair due to different positions of theresonances.

To conclude, an effective antiferromagnetic exchangeinteraction controls the scattering events when the quan-tum well has only one bound state, due to the singletresonance channel. However, by increasing the depth ofthe well the ground singlet resonance level becomes in-active simply because it lies much lower than the energyof the propagating electron at all times. In this regimethe scattering is controlled by an effective ferromagnetic

12

-0.5 -0.25 0-30

0

30

-0.5 -0.25 0-60

-30

0

30

-0.25 0 0.25-60

-30

0

30

0 0.25 0.5-60

-30

0

30

0 0.25 0.5 (µm)

-30

0

30

(a)

(b)

(c)

(d)

(e)

V e SA

W(m

eV

)

x

FIG. 14: Propagation of an electron in the V e

SAW minimumpassing through the well region. The dashed line indicates theenergy of the propagating electron at each particular time, ifafter the SAW cycle it exits the well region in the lowest stateof the SAW minimum. The full lines indicate the energy lev-els of the quantum well. In the anti-crossing regions (b),(d)the electron makes a non-adiabatic Landau-Zener transitionand always remains in the V e

SAW minimum when the tun-nel barrier is large and as a result after the SAW cycle theelectron emerges in the lowest state of the SAW potentialminimum (e). When the tunnelling probability into and outof the quantum well is not negligible, the electron emerges ina superposition state of the low-lying states of the SAW. Thetime sequence is from (a) to (e) and specifically t/T=0, 0.3,0.5, 0.7, 1.

exchange interaction involving excited states for singletand triplet in which the triplet is lower. It is interestingto note that in the flying qubit scheme11,13 it is alwaysan antiferromagnetic exchange interaction that generatesthe entanglement, whereas in the scheme that we proposeboth ferromagnetic and antiferromagnetic type interac-tions can generate entanglement depending on SAW andwell parameters.

V. SOME GENERAL FEATURES OF THE

ENTANGLEMENT

In this section we generalise some of the above resultsand demonstrate quantitatively the sensitivity of the sys-

0

0.2

0.4

0.6

0.8

1

C ,

Cr ,

Ct

0.6 0.8 1 1.2 1.4 1.6 1.8 2

(meV)

0

0.1

0.2

0.3

0.4

0.5

Pr ,

Pt

Ct

C

Cr

Pr

S

Pt

T

Pr

T

Pt

S

(a)

(b)

Vo

FIG. 15: (color online). Variation of asymptotic (final time)concurrence (a) and relative probabilities (b), as defined in thetext, versus the SAW potential amplitude when the quantumwell is such that only a single electron can be bound.

tem to changes in the well and the SAW characteristics.

Figure 15(a) illustrates the variation of the concur-rence versus the SAW potential amplitude for a quan-tum well with Vw = 6 meV and lw = 7.5 nm, which canaccommodate only a single bound electron. This plotshows the total, transmitted and reflected concurrenceat the final time for which the overlap of propagatingand bound electron wave packets is negligible. Figure15(b) presents the corresponding probabilities. Note thatthe SAW potential amplitude is restricted to the specificregime for which the electron in the quantum well re-mains well-localised, as described in Sec. II B. We seethat the very small transmission of singlet and triplet(which we include here for completeness), correspondingto the minimum value of the SAW amplitude, gives riseto a concurrence, Ct ∼ 0.5. With increasing SAW am-plitude there is then a regime in which the transmittedconcurrence increases to Ct ∼ 1, for which the singletstate is on resonance and simultaneously there is mini-mum transmission for the triplet state. We may regardthis as a two-electron spin filter for which the initial un-entangled state, that is an equal superposition of singletand Sz = 0 triplet states, has its triplet component fil-tered out (reflected) with resonant transmission of thefully entangled singlet component. For this SAW ampli-tude, a “fuzzy” position measurement applied to the out-going electron (say through charge sensing34,35,36), whichmerely resolves whether it is transmitted or reflected,could be used to probabilistically prepare a highly en-tangled state. The form of the state prepared is her-alded by the measurement outcome. Further increase of

13

30 40 50 60 70 80 90

(meV)

0

0.2

0.4

0.6

0.8

1C =10 meV

=20 meV

42 44 46 48 50 52 54 56

(meV)

0

0.2

0.4

C

Vw

Vw

Vo

Vo

FIG. 16: (color online). Variation of asymptotic (final time)concurrence for a fixed SAW potential amplitude (Vo = 10meV, Vo = 20 meV) as a function of the well depth.

the SAW potential amplitude from this point causes thetransmitted concurrence to gradually decrease due sim-ply to the higher transmission of the triplet state. Thereflected concurrence is very small for low SAW poten-tial amplitude and approximately equals the total con-currence due to the very high reflection for both states.It then increases smoothly as the singlet transmitted partincreases and remains almost constant and close to unitywith further increase of the SAW amplitude, since thereflected component is mainly triplet. Finally, the to-tal concurrence has a relatively more complicated behav-ior, although it is clear that it has a maximum valueC ∼ 0.4 when PS

r = PSt ∼ 0.25, namely when the singlet

is equally transmitted and reflected. Also it is alwayslower than the transmitted and reflected concurrence ex-cept when Pt ∼ 0 and then C ≈ Cr. We may concludefrom Fig. 15 that the degree of entanglement for the twoelectrons can be changed significantly with SAW ampli-tude in the regime that scans through the singlet reso-nance and that there exists a point where in principle,through “fuzzy” position measurement, highly entangledstates could be prepared.

A more relevant case for simpler experiments, whichdon’t require position measurement to project into thetransmitted or reflected outcomes for the outgoing elec-tron, is when the SAW potential amplitude and the quan-tum well are such that the electron in the SAW is alwaysfully transmitted, or approximately so, leaving the part-ner electron bound in the quantum well. The backwardreflection, which is more likely to occur for a low poten-tial amplitude, may cause undesirable effects, since thereflected electron will occupy multiple wells and involveshighly excited components. This case of a reflected non-bound electron is more efficiently studied using kineticinjection without the presence of the SAW potential, asit is described for example in Refs. 19, 21. In addition,

a strong SAW potential amplitude has the advantage ofpreventing the trapped electron from leaking into adja-cent minima, thus minimizing possible errors. We havecalculated the concurrence as a function of the well depthfor two different, though relatively strong, SAW potentialamplitudes of Vo = 20 meV and Vo = 10 meV, and forfixed lw = 10 nm. The SAW potential amplitude that isused in the experiments for SAW-based SET applicationscan be even stronger than this (Vo ∼ 40 meV)24, thoughalong the channel there is likely to be some screeningdue to the gate bias. The chosen parameters guaranteethat there is very high transmission both for singlet and

triplet states (PS/Tt ∼ 0.5). In this study the range of

the well depth ensures high localisation of the trappedelectron resulting in a truly bound singlet ground state,as calculated within a Hartree approximation and there-fore, a ferromagnetic type exchange interaction generatesthe entanglement as described in the previous section.

The results for the total concurrence, which almostequals the transmitted concurrence, are shown in Fig.16, while Fig. 17 illustrates the singlet and triplet com-ponents of the electron in the SAW (after scattering)for various well depths and for the SAW amplitude ofVo = 10 meV. Figure 16 presents two distinct maximafor each of the two amplitudes considered with an in-termediate region of relatively low concurrence (whichis shown in detail in the inset of Fig. 16). Analysis ofthe data shows that in the rise up to the first maximum,the asymptotic state is approximated well by the simplephase difference picture described in section III.C, i.e.with the electron in the SAW potential minimum beingin its ground state, to a good approximation, but with aphase difference between singlet and triplet components.This concurrence of almost unity at the maximum thencorresponds to a phase difference of δϕ ∼ π/2. As thewell depth is increased from this point the electron inthe SAW occupies additional excited states which aredifferent for singlet and triplet (the phase difference pic-ture is no longer valid) and this is why the concurrencedecreases. Figure 17 helps us understand how the sin-glet and triplet components of the electron in the SAWchange with well depth and specifically how we pass froma region of different probability distribution to a regionwhere the phase difference picture is valid. This behavioris clear for example by considering Figs. 17(a),(b) and(c). Similar behavior is valid for a SAW amplitude ofVo = 20 meV. Within the intermediate region for bothSAW amplitudes the concurrence fluctuates due to spin-dependent scattering events which involve excited statesof the SAW potential minimum. Figure 17(c) shows anexample within this region. Note that zero concurrencecorresponds to cases where the orbital states in the SAWfor singlet and triplet components are exactly orthogo-nal. Further increase of the well depth gives rise to asecond concurrence maximum, due to the fact that thefinal states of the SAW electron for singlet and tripletcomponents are approximated by the ground state of theSAW potential minimum. Similar to the first concurrence

14

ρρ

0.4 0.5 0.6 0.7 0.8

(µm)x

ρ ,

ρ(a

rb. u

nits

)S

TS

T(a)

(b)

(c)

(d)

(e)

(f)

FIG. 17: (color online). Final electron distribution for singletand triplet states in the SAW potential minimum, for a SAWamplitude of Vo = 10 meV and a well depth from (a) to (f)of Vw=26, 36, 47, 62, 75, 85 meV.

maximum, the phase difference picture is again valid asshown for example in Fig. 17(d). Excited states withhigh probability however are involved in the scatteringprocess, different for singlet and triplet as we demon-strate in Fig. 17(e) (reflecting spin-dependent scatter-ing) lowering the concurrence. This occurs up to theregime of the very deep quantum well (at the right-handside of Fig. 16 and for each SAW amplitude) wherethe phase difference picture becomes valid as Fig. 17(f)demonstrates. This is because the Coulomb interactionis effectively reduced due to the high confinement of thetrapped electron. Singlet and triplet components arescattered mainly due to the presence of the potential wellin the same final states with only a small effect from theCoulomb interaction, which will become negligible for ex-tremely deep wells. When this extreme limit is reachedthe two-electron states will be approximated at all timesby single electron states.

As we have said in previous sections, the asymptoticvalue of the concurrence (at the final time) which emergeswhen a relative phase difference is present between sin-glet and triplet states, depends on the magnitude ofthe so-called exchange energy J(t) = ET (t) − ES(t)and the SAW period which sets the interaction time.In the phase difference regime an approximate Heisen-berg Hamiltonian43 H(t) = J(t)S1 · S2, with Sı thespin operator of the ıth electron can provide insight intothe spin entanglement generation. This is because inthis regime the two electrons at all times occupy dif-ferent and well-defined orbital states and as we havedescribed in Sec. IV.B these states are the same for

0.4 0.5 0.6-0.04

-0.03

-0.02

-0.01

0

(m

eV)

Vw =

Vw =

J

t

62 meV

36 meV

/ T

FIG. 18: Exchange energy as a function of time, for two dif-ferent well depths (Vw=36 meV, Vw=62 meV) and a SAWpotential amplitude of Vo=10 meV.

singlet and triplet apart from a phase factor. Theexchange energy as a function of time can be deter-mined by solving the instantaneous (time-independent)two-electron Schrodinger equation for singlet and triplet

states treating time as a parameter i.e. H(t)ΦS/Tn (t) =

ES/Tn (t)Φ

S/Tn (t), with the Hamiltonian given by Eq. (11).

A common diagonalisation procedure is described in Refs.13, 41, 42. The instantaneous solutions provide the setsES

n (t) and ETn (t) with n the eigenvalue index. By fol-

lowing the non-adiabatic Landau-Zener transitions whichsuccessfully take place in the phase difference regime, wecan extract the energy of the two electrons during thescattering event i.e. ET (t), ES(t) and from these theJ(t) = ET (t) − ES(t) curve. In Fig. 18 we show the ex-change energy as a function of time for two different welldepths Vw=36 meV, Vw=62 meV and a SAW potentialamplitude Vo=10 meV which result in a phase differenceas shown in Figs. 17(b),(d). As we have analysed in Sec.IV.B and we see in Fig. 18, the exchange energy J is neg-ative in the phase difference regime. The lower J for thecase of the Vw =62 meV well depth is because a higherexcited energy level for the singlet-triplet pair is involvedin the scattering process, compared to the Vw =36 meVcase and in general higher excited energy levels have asmaller separation41,42. For the phase difference regimeand within the Heisenberg model we can calculate theasymptotic concurrence C = | sin δϕ| by extracting therelative phase difference δϕ directly from the J(t) curve

as δϕ =∫ T

0J(t)/~dt. Note that the time interval of the

integration is set by the SAW period T which is fixed inthe experiments. The values that we take by this approx-imate treatment are in excellent agreement with the val-ues that we take by solving the two-electron Schrodingerequation and by calculating the concurrence by the orig-inal formula (15).

VI. SUMMARY

In summary, we have presented and investigated ascheme to produce entangled states for two electrons

15

utilizing a SAW. One electron is carried by the time-dependent SAW potential along a semiconductor quan-tum wire where a second electron is bound in a quan-tum well. The Coulomb interaction induces entangle-ment between the two electrons that can cover the fullrange from zero to full entanglement, depending on SAWpotential amplitude and the shape of the confining po-tential. There are two regimes of interest, depending onthe SAW and well parameters.

The first is when there is a significant difference be-tween transmission probabilities for singlet and tripletstates. In this regime, entanglement generation may beinterpreted as a spin-filtering effect in which the sin-glet component of an initially unentangled state has ahigher transmission probability than the triplet due tospin-dependent scattering. This gives maximal entangle-ment (C ∼ 1) for resonant singlet tunnelling with fulltransmission for the singlet case and almost full reflec-tion for the triplet case. “Fuzzy” position measurement(possibly through charge sensing34,35,36), just resolvingwhether the outgoing electron is transmitted or reflected,would be needed to make this useful entanglement. Themeasurement result would identify and herald the formof entangled state produced in each run of an experiment.

The second regime occurs for the parameters chosensuch that there is approximately full transmission forboth singlet and triplet cases. Within this regime thetransmitted electron in a SAW minimum can be left inan excited state which in some cases is different for singletand triplet and in some cases the same (or approximately

so). In the latter cases, concurrence is given by a simpleexpression involving the relative phase difference betweenthe transmitted SAW potential minimum wave functionsarising from singlet and triplet. This demonstrates max-imal entanglement when the phase difference is π/2 andwe have identified a physically reasonable set of param-eters for which this occurs. For other cases, the con-currence cannot reach the unitary limit and can fluctu-ate significantly due to spin-dependent resonance effectswhen the electrons interact. In this regime of near fulltransmission, the concurrence is low when the transmit-ted SAW minima wave functions are significantly differ-ent from each other for singlet and triplet cases, becomingzero in the limiting cases when these wave functions areorthogonal.

The physical system we have considered and the pa-rameter ranges we have investigated suggest that itshould be experimentally possible to produce useful en-tanglement between a travelling and a trapped electron,using a SAW. This could be achieved either by sensingwhether the outgoing electron is transmitted or reflected,or by working in a regime where there is essentially com-plete transmission.

VII. ACKNOWLEDGMENTS

GG acknowledges funding from EPSRC and AR ac-knowledges support from the Slovenian Research Agencyunder contract PI-0044. This work was supported by theUK Ministry of Defence.

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