Physica B 314 (2002) 10–14

Simulation of entangled electronic states in semiconductorquantum wires

Andrea Bertonia,b,*, Radu Ionicioiua,c, Paolo Zanardia,c, Fausto Rossia,c,Carlo Jacobonia,b

a Istituto Nazionale per la Fisica della Materia (INFM), Unit "a di Modena, via Campi 213/A, Modena, Italyb Dipartimento di Fisica, Universit "a di Modena e Reggio Emilia, via Campi 213/A, I-41100 Modena, Italy

c Institute for Scientific Interchange (ISI), Torino, Italy

Abstract

A system able to produce entangled two-electron states is proposed and studied by means of numerical simulations.

The basic device consists of a couple of semiconductor quantum wires in which single electrons are injected and

propagated coherently. Coulomb coupling between two electrons in two different wires arises in a region where the

wires get close to each other. The strength of this interaction can be tuned with a proper design of the system geometry.

It is shown that it is possible to create the four entangled Bell states for the two-particle wave function. r 2002 Elsevier

Science B.V. All rights reserved.

Keywords: Entanglement; Quantum computation; Bell inequality; Quantum wires; Coherent transport

1. Introduction

Among the phenomena predicted by quantumtheory, entanglement seems the most astounding,revealing the non-locality of quantum states andhaving no correspondence in classical physics.Entanglement is also the key phenomenon onwhich the new field of quantum information theory(that covers, among others, quantum computing[1]) has its foundations. Thus, it becomes impor-tant to study and understand the conditions underwhich entangled states are produced.

The aim of the present work is to pro-pose a system based on coherent transport in

semiconductor quantum wires (QWRs), able toproduce entangled states of two electrons. Theproposed semiconductor devices are then used toconstruct an ideal setup able to test Bell inequality.The transformations that the two-electron wavefunction undergoes, obtained in previous works[2,4] by a formal approach, are studied bynumerical simulations of the two-particle wavefunction dynamics.

2. The physical system

The system under study was proposed by someof the authors as a possible realization of basicgates for quantum computation (see Ref. [2] for adetailed description). The qubit state (j0S or j1S)is defined as the localization in one of two parallelQWRs (left or right, respectively), of one injected

*Corresponding author. Dipartimento di Fisica, Universit"a

di Modena e Reggio Emilia, via Campi 213/A, I-41100

Modena, Italy. Fax: +39-05-9367488.

E-mail address: bertoni.andrea@unimo.it (A. Bertoni).

0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 9 2 1 - 4 5 2 6 ( 0 1 ) 0 1 4 5 5 - 7

electron. We stress the correspondence betweenthe qubit space and the spin space for spin 1

2

particles. In the following, the usual notation forspin rotations will be used for qubits.

A given transformation on the many-qubit statecan be obtained by means of a network composedby three types of basic building blocks (universalgates): (a) electronic phase shifters, obtained witha small potential barrier (or well), able to delay thepropagation of the wave function; (b) electronicbeam splitters, obtained by a coupling windowbetween the two wires of a qubit, able to split anincoming wave function in two parts; (c) Coulombcouplers, consisting of a region in which twoelectrons propagating along two different wires getclose enough to each other to give rise to aneffective interaction, able to delay both electrons.Fig. 1 shows a representation of the one-qubitgates a and b, and of the two-qubit gate c togetherwith the matrix representation of the operationsperformed (on the basis fj0S; j1Sg for a and b, onthe basis fj00S; j01S; j10S; j11Sg for c).

3. Numerical simulations and entanglement

The proposed set of gates has been studiedby solving numerically the 2D time-dependentSchr .odinger equation (in a Crank–Nicholson

Fig. 1. Set of universal gates.

Fig. 2. Square modulus of the electron wave function, at

different time steps, in a device composed of two beam splitters

and one phase shifter. The transformation simulated is

Rxðp=2ÞR0ðpÞRxðp=2Þj0S ¼ �j0S:

A. Bertoni et al. / Physica B 314 (2002) 10–14 11

scheme [3]) for one or two single electrons injectedin a number of QWR devices with differentgeometries [4] (see, as an example, Fig. 2).

Once the results validated the transformationsperformed by the gates for non-entangled qubitstates, a network has been studied, able to produceentangled states. In this case, the Schr .odingerequation for the full two-particle wave functionmust be solved. To reduce the computationaleffort, a semi-1D model is used. In fact, thevariables x1 and x2 indicating the positionorthogonal to the wires, are not discretized on agrid (as, instead, it is done for the variables y1 andy2 along the wires) but can assume only the values0 or 1 to indicate simply one of the two wires, i.e.the state of the qubit. With this approach we passfrom a time-dependent Schr .odinger equation forthe five-variable wave function cðx1; y1; x2; y2; tÞ tofour Schr .odinger equations:

i_qqt

cx1;x2ðy1; y2; tÞ

¼ �_2

2m

q2

qy21

þq2

qy22

� �cx1;x2

ðy1; y2; tÞ

þ Vx1;x2ðy1; y2Þcx1;x2

ðy1; y2; tÞ

with x1; x2Af0; 1g: During the evolution, the fourdifferent components of the wave function arecoupled by the transformations induced by thecoupling windows. The geometry of the system is

contained in the two-particle potentialVx1;x2

ðy1; y2Þ: It consists of three terms: theCoulomb interaction between the electrons, andthe two structure potentials along the wires 0 and 1of each qubit:

Vx1;x2ðy1; y2Þ ¼

e2

Dx1;x2ðy1; y2Þ

þ Ux1ðy1Þ þ Vx2

ðy2Þ;

where Dx1;x2y1; y2ð Þ represents the distance between

point y1 in x1 wire of the first qubit and point y2 inx2 wire of the second qubit.

3.1. Gate R0ðfÞ

The phase shift R0ðfÞ (R1ðfÞ) on the state j0S(j1S) of the first qubit is obtained with a delayingpotential barrier inserted in the potential U0ðy1Þ(U1ðy1Þ). Similarly, for the second qubit thepotential is inserted in V0ðy2Þ (V1ðy2Þ).

3.2. Gate RxðyÞ

Within this semi-1D model it is not possible tosimulate directly the dynamics of the wavefunction splitting by a coupling window leadingto the one-qubit transformation RxðyÞ: To includethe effect of an Rx gate, we apply the transforma-tion matrix as obtained by the analytical develop-ment, validated through the 2D single particlesimulations.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 10 20 30 40 50 6 0

distance between wires (nm)

ang

lep

erfo

rmed

(un

ito

fp

i)

050100150200250300350400450500550600650

CC length(nm)

Fig. 3. Rotation angle performed by a Coulomb coupler TðgÞ with variable length and distance between wires.

A. Bertoni et al. / Physica B 314 (2002) 10–1412

3.3. Gate TðgÞ

To study the optimal geometry for the condi-tional phase shifter T realized by the Coulombcoupler, a number of simulations has beenperformed varying two geometric parameters: thelength of the coupler and the distance between thecoupled wires (Fig. 3). The maximum valueobtained for the angle g is 0:79p that should beenough to produce a measurable effect with thenetwork proposed in Section 4 for the Bellinequality test. The parameters for Si=SiO2 wiresin a single parabolic band approximation havebeen used.

4. Bell states and Bell inequality

With some straightforward calculations it iseasy to see that the network (the superscript in aone-qubit transformation indicates the qubit onwhich it acts):

Rð2Þx ðp=2ÞTðpÞRð2Þ

x ðp=2ÞRð1Þx ðp=2Þ

depicted in left part of Fig. 4, is able to create thefour maximally entangled Bells states. The simula-tions of this network for the two initial conditionsj10S and j01S are shown in Figs. 5 and 6.They lead to the states Cþ ¼ j01Sþ j10S and

Fþ ¼ j00Sþ j11S; respectively. The reason for theresidues of the wave function in other states is thatthe TðgÞ gate used is not able to perform acomplete p rotation, as mentioned in Section 3.3.

Fig. 4. Network for Bell states preparation (left) and Bell’s

inequality test (right).

Fig. 5. Square modulus, at four different time steps, of the two

particle wave function injected in the network of Fig. 4. White

region represents the two-particle potential. Initial and final

conditions, are, respectively, j10S and j01Sþ j10S (see text).

Fig. 6. Same as Fig. 5 but with initial and final conditions,

respectively, j01S and j00Sþ j11S (see text).

A. Bertoni et al. / Physica B 314 (2002) 10–14 13

As suggested in Ref. [5], entangled statesproduced by the proposed devices can be used totest Bell inequality [6]. The network needed is

Rð2Þx ðp=2ÞRð1Þ

x ðp=2ÞRð2Þ1 ðf2ÞR

ð1Þ0 ðf1Þ

Rð1Þx ðp=2ÞTðpÞRð2Þ

x ðp=2ÞRð1Þx ðp=2Þ

and is represented in right part of Fig. 4 (see Ref.[5] for details).

The realization of the proposed devices shouldbe on the borderline of the present semiconductortechnology. The main difficulty to face in anexperimental realization of a physical structure isthe onset of interactions between the system andthe environment that produce decoherence. Ananalysis of the effects of phonon scattering on thefunctioning of the proposed QWR devices, ispresented in Ref. [4] and indicates that thecoherent component of the current should beexperimentally detectable.

Acknowledgements

Work partially supported by MURST (40%project on Quantum Computing) and by the USOffice of Naval Research (Contract No. N00014-98-1-0777).

References

[1] D.P. DiVincenzo, C. Bennett, Nature 404 (2000) 247.

[2] A. Bertoni, et al., Phys. Rev. Lett. 84 (2000) 5912;

R. Ionicioiu, et al., Int. J. Mod. Phys. B 15 (2001) 125,

quant-ph/9907043;

S. Reggiani, et al., IEEE 00TH8502, Proceedings of the

SISPAD 2000, p. 184.

[3] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flan-

nery, Numerical Recipes, Cambridge University Press, UK,

1992.

[4] A. Bertoni, et al., J. Mod. Opt., 2001, accepted for

publication.

[5] R. Ionicioiu, et al., Phys. Rev. A 63 (2001) 50101.

[6] J.S. Bell, Physics 1 (1964) 195.

A. Bertoni et al. / Physica B 314 (2002) 10–1414