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Simulation of entangled electronic states in semiconductor quantum wires
Physica B 314 (2002) 10–14
Simulation of entangled electronic states in semiconductor quantum 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), Unita" di Modena, via Campi 213/A, Modena, Italy b Dipartimento di Fisica, Universita" 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 quantum theory, entanglement seems the most astounding, revealing the non-locality of quantum states and having no correspondence in classical physics. Entanglement is also the key phenomenon on which the new field of quantum information theory (that covers, among others, quantum computing [1]) has its foundations. Thus, it becomes important to study and understand the conditions under which entangled states are produced. The aim of the present work is to propose a system based on coherent transport in
*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: [email protected] (A. Bertoni).
semiconductor quantum wires (QWRs), able to produce entangled states of two electrons. The proposed semiconductor devices are then used to construct an ideal setup able to test Bell inequality. The transformations that the two-electron wave function undergoes, obtained in previous works [2,4] by a formal approach, are studied by numerical simulations of the two-particle wave function dynamics.
2. The physical system The system under study was proposed by some of the authors as a possible realization of basic gates for quantum computation (see Ref. [2] for a detailed description). The qubit state (j0S or j1S) is defined as the localization in one of two parallel QWRs (left or right, respectively), of one injected
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
A. Bertoni et al. / Physica B 314 (2002) 10–14
11
Fig. 1. Set of universal gates.
electron. We stress the correspondence between the qubit space and the spin space for spin 12 particles. In the following, the usual notation for spin rotations will be used for qubits. A given transformation on the many-qubit state can be obtained by means of a network composed by three types of basic building blocks (universal gates): (a) electronic phase shifters, obtained with a small potential barrier (or well), able to delay the propagation of the wave function; (b) electronic beam splitters, obtained by a coupling window between the two wires of a qubit, able to split an incoming wave function in two parts; (c) Coulomb couplers, consisting of a region in which two electrons propagating along two different wires get close enough to each other to give rise to an effective interaction, able to delay both electrons. Fig. 1 shows a representation of the one-qubit gates a and b, and of the two-qubit gate c together with the matrix representation of the operations performed (on the basis fj0S; j1Sg for a and b, on the basis fj00S; j01S; j10S; j11Sg for c). 3. Numerical simulations and entanglement The proposed set of gates has been studied by solving numerically the 2D time-dependent . Schrodinger equation (in a Crank–Nicholson
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 Rx ðp=2ÞR0 ðpÞRx ðp=2Þj0S ¼ j0S:
A. Bertoni et al. / Physica B 314 (2002) 10–14
12
scheme [3]) for one or two single electrons injected in a number of QWR devices with different geometries [4] (see, as an example, Fig. 2). Once the results validated the transformations performed by the gates for non-entangled qubit states, a network has been studied, able to produce . entangled states. In this case, the Schrodinger equation for the full two-particle wave function must be solved. To reduce the computational effort, a semi-1D model is used. In fact, the variables x1 and x2 indicating the position orthogonal to the wires, are not discretized on a grid (as, instead, it is done for the variables y1 and y2 along the wires) but can assume only the values 0 or 1 to indicate simply one of the two wires, i.e. the state of the qubit. With this approach we pass . from a time-dependent Schrodinger equation for the five-variable wave function cðx1 ; y1 ; x2 ; y2 ; tÞ to . four Schrodinger equations: i_
contained in the two-particle potential Vx1 ;x2 ðy1 ; y2 Þ: It consists of three terms: the Coulomb interaction between the electrons, and the two structure potentials along the wires 0 and 1 of each qubit: Vx1 ;x2 ðy1 ; y2 Þ ¼
e2 þ Ux1 ðy1 Þ þ Vx2 ðy2 Þ; Dx1 ;x2 ðy1 ; y2 Þ
where Dx1 ;x2 ðy1 ; y2 Þ represents the distance between point y1 in x1 wire of the first qubit and point y2 in x2 wire of the second qubit. 3.1. Gate R0 ðfÞ The phase shift R0 ðfÞ (R1 ðfÞ) on the state j0S (j1S) of the first qubit is obtained with a delaying potential barrier inserted in the potential U0 ðy1 Þ (U1 ðy1 Þ). Similarly, for the second qubit the potential is inserted in V0 ðy2 Þ (V1 ðy2 Þ).
q c ðy1 ; y2 ; tÞ qt x1 ;x2 _2 q2 q2 ¼ þ ðy1 ; y2 ; tÞ c 2m qy21 qy22 x1 ;x2
3.2. Gate Rx ðyÞ Within this semi-1D model it is not possible to simulate directly the dynamics of the wave function splitting by a coupling window leading to the one-qubit transformation Rx ðyÞ: To include the effect of an Rx gate, we apply the transformation matrix as obtained by the analytical development, validated through the 2D single particle simulations.
þ Vx1 ;x2 ðy1 ; y2 Þcx1 ;x2 ðy1 ; y2 ; tÞ with x1 ; x2 Af0; 1g: During the evolution, the four different components of the wave function are coupled by the transformations induced by the coupling windows. The geometry of the system is
0.9 angle performed (unit of pi)
0.8 0.7
CC length (nm)
0.6 0.5
0 50 100 150 200 250 300 350 400 450 500 550 600 650
0.4 0.3 0.2 0.1 0.0 0
10
20
30
40
50
60
distance between wires (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) 10–14
13
3.3. Gate TðgÞ To study the optimal geometry for the conditional phase shifter T realized by the Coulomb coupler, a number of simulations has been performed varying two geometric parameters: the length of the coupler and the distance between the coupled wires (Fig. 3). The maximum value obtained for the angle g is 0:79p that should be enough to produce a measurable effect with the network proposed in Section 4 for the Bell inequality test. The parameters for Si=SiO2 wires in a single parabolic band approximation have been used.
4. Bell states and Bell inequality With some straightforward calculations it is easy to see that the network (the superscript in a one-qubit transformation indicates the qubit on which it acts):
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 j01S þ j10S (see text).
ð2Þ ð1Þ Rð2Þ x ðp=2ÞTðpÞRx ðp=2ÞRx ðp=2Þ
depicted in left part of Fig. 4, is able to create the four maximally entangled Bells states. The simulations of this network for the two initial conditions j10S and j01S are shown in Figs. 5 and 6. They lead to the states Cþ ¼ j01S þ j10S and
Fig. 6. Same as Fig. 5 but with initial and final conditions, respectively, j01S and j00S þ j11S (see text).
Fig. 4. Network for Bell states preparation (left) and Bell’s inequality test (right).
Fþ ¼ j00S þ j11S; respectively. The reason for the residues of the wave function in other states is that the TðgÞ gate used is not able to perform a complete p rotation, as mentioned in Section 3.3.
A. Bertoni et al. / Physica B 314 (2002) 10–14
14
As suggested in Ref. [5], entangled states produced by the proposed devices can be used to test Bell inequality [6]. The network needed is ð2Þ ð1Þ ð1Þ Rð2Þ x ðp=2ÞRx ðp=2ÞR1 ðf2 ÞR0 ðf1 Þ ð2Þ ð1Þ Rð1Þ x ðp=2ÞTðpÞRx ðp=2ÞRx ðp=2Þ
and is represented in right part of Fig. 4 (see Ref. [5] for details). The realization of the proposed devices should be on the borderline of the present semiconductor technology. The main difficulty to face in an experimental realization of a physical structure is the onset of interactions between the system and the environment that produce decoherence. An analysis of the effects of phonon scattering on the functioning of the proposed QWR devices, is presented in Ref. [4] and indicates that the coherent component of the current should be experimentally detectable.
Acknowledgements Work partially supported by MURST (40% project on Quantum Computing) and by the US Office of Naval Research (Contract No. N0001498-1-0777).
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. Flannery, 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.
Simulation of entangled electronic states in semiconductor quantum 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), Unita" di Modena, via Campi 213/A, Modena, Italy b Dipartimento di Fisica, Universita" 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 quantum theory, entanglement seems the most astounding, revealing the non-locality of quantum states and having no correspondence in classical physics. Entanglement is also the key phenomenon on which the new field of quantum information theory (that covers, among others, quantum computing [1]) has its foundations. Thus, it becomes important to study and understand the conditions under which entangled states are produced. The aim of the present work is to propose a system based on coherent transport in
*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: [email protected] (A. Bertoni).
semiconductor quantum wires (QWRs), able to produce entangled states of two electrons. The proposed semiconductor devices are then used to construct an ideal setup able to test Bell inequality. The transformations that the two-electron wave function undergoes, obtained in previous works [2,4] by a formal approach, are studied by numerical simulations of the two-particle wave function dynamics.
2. The physical system The system under study was proposed by some of the authors as a possible realization of basic gates for quantum computation (see Ref. [2] for a detailed description). The qubit state (j0S or j1S) is defined as the localization in one of two parallel QWRs (left or right, respectively), of one injected
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
A. Bertoni et al. / Physica B 314 (2002) 10–14
11
Fig. 1. Set of universal gates.
electron. We stress the correspondence between the qubit space and the spin space for spin 12 particles. In the following, the usual notation for spin rotations will be used for qubits. A given transformation on the many-qubit state can be obtained by means of a network composed by three types of basic building blocks (universal gates): (a) electronic phase shifters, obtained with a small potential barrier (or well), able to delay the propagation of the wave function; (b) electronic beam splitters, obtained by a coupling window between the two wires of a qubit, able to split an incoming wave function in two parts; (c) Coulomb couplers, consisting of a region in which two electrons propagating along two different wires get close enough to each other to give rise to an effective interaction, able to delay both electrons. Fig. 1 shows a representation of the one-qubit gates a and b, and of the two-qubit gate c together with the matrix representation of the operations performed (on the basis fj0S; j1Sg for a and b, on the basis fj00S; j01S; j10S; j11Sg for c). 3. Numerical simulations and entanglement The proposed set of gates has been studied by solving numerically the 2D time-dependent . Schrodinger equation (in a Crank–Nicholson
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 Rx ðp=2ÞR0 ðpÞRx ðp=2Þj0S ¼ j0S:
A. Bertoni et al. / Physica B 314 (2002) 10–14
12
scheme [3]) for one or two single electrons injected in a number of QWR devices with different geometries [4] (see, as an example, Fig. 2). Once the results validated the transformations performed by the gates for non-entangled qubit states, a network has been studied, able to produce . entangled states. In this case, the Schrodinger equation for the full two-particle wave function must be solved. To reduce the computational effort, a semi-1D model is used. In fact, the variables x1 and x2 indicating the position orthogonal to the wires, are not discretized on a grid (as, instead, it is done for the variables y1 and y2 along the wires) but can assume only the values 0 or 1 to indicate simply one of the two wires, i.e. the state of the qubit. With this approach we pass . from a time-dependent Schrodinger equation for the five-variable wave function cðx1 ; y1 ; x2 ; y2 ; tÞ to . four Schrodinger equations: i_
contained in the two-particle potential Vx1 ;x2 ðy1 ; y2 Þ: It consists of three terms: the Coulomb interaction between the electrons, and the two structure potentials along the wires 0 and 1 of each qubit: Vx1 ;x2 ðy1 ; y2 Þ ¼
e2 þ Ux1 ðy1 Þ þ Vx2 ðy2 Þ; Dx1 ;x2 ðy1 ; y2 Þ
where Dx1 ;x2 ðy1 ; y2 Þ represents the distance between point y1 in x1 wire of the first qubit and point y2 in x2 wire of the second qubit. 3.1. Gate R0 ðfÞ The phase shift R0 ðfÞ (R1 ðfÞ) on the state j0S (j1S) of the first qubit is obtained with a delaying potential barrier inserted in the potential U0 ðy1 Þ (U1 ðy1 Þ). Similarly, for the second qubit the potential is inserted in V0 ðy2 Þ (V1 ðy2 Þ).
q c ðy1 ; y2 ; tÞ qt x1 ;x2 _2 q2 q2 ¼ þ ðy1 ; y2 ; tÞ c 2m qy21 qy22 x1 ;x2
3.2. Gate Rx ðyÞ Within this semi-1D model it is not possible to simulate directly the dynamics of the wave function splitting by a coupling window leading to the one-qubit transformation Rx ðyÞ: To include the effect of an Rx gate, we apply the transformation matrix as obtained by the analytical development, validated through the 2D single particle simulations.
þ Vx1 ;x2 ðy1 ; y2 Þcx1 ;x2 ðy1 ; y2 ; tÞ with x1 ; x2 Af0; 1g: During the evolution, the four different components of the wave function are coupled by the transformations induced by the coupling windows. The geometry of the system is
0.9 angle performed (unit of pi)
0.8 0.7
CC length (nm)
0.6 0.5
0 50 100 150 200 250 300 350 400 450 500 550 600 650
0.4 0.3 0.2 0.1 0.0 0
10
20
30
40
50
60
distance between wires (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) 10–14
13
3.3. Gate TðgÞ To study the optimal geometry for the conditional phase shifter T realized by the Coulomb coupler, a number of simulations has been performed varying two geometric parameters: the length of the coupler and the distance between the coupled wires (Fig. 3). The maximum value obtained for the angle g is 0:79p that should be enough to produce a measurable effect with the network proposed in Section 4 for the Bell inequality test. The parameters for Si=SiO2 wires in a single parabolic band approximation have been used.
4. Bell states and Bell inequality With some straightforward calculations it is easy to see that the network (the superscript in a one-qubit transformation indicates the qubit on which it acts):
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 j01S þ j10S (see text).
ð2Þ ð1Þ Rð2Þ x ðp=2ÞTðpÞRx ðp=2ÞRx ðp=2Þ
depicted in left part of Fig. 4, is able to create the four maximally entangled Bells states. The simulations of this network for the two initial conditions j10S and j01S are shown in Figs. 5 and 6. They lead to the states Cþ ¼ j01S þ j10S and
Fig. 6. Same as Fig. 5 but with initial and final conditions, respectively, j01S and j00S þ j11S (see text).
Fig. 4. Network for Bell states preparation (left) and Bell’s inequality test (right).
Fþ ¼ j00S þ j11S; respectively. The reason for the residues of the wave function in other states is that the TðgÞ gate used is not able to perform a complete p rotation, as mentioned in Section 3.3.
A. Bertoni et al. / Physica B 314 (2002) 10–14
14
As suggested in Ref. [5], entangled states produced by the proposed devices can be used to test Bell inequality [6]. The network needed is ð2Þ ð1Þ ð1Þ Rð2Þ x ðp=2ÞRx ðp=2ÞR1 ðf2 ÞR0 ðf1 Þ ð2Þ ð1Þ Rð1Þ x ðp=2ÞTðpÞRx ðp=2ÞRx ðp=2Þ
and is represented in right part of Fig. 4 (see Ref. [5] for details). The realization of the proposed devices should be on the borderline of the present semiconductor technology. The main difficulty to face in an experimental realization of a physical structure is the onset of interactions between the system and the environment that produce decoherence. An analysis of the effects of phonon scattering on the functioning of the proposed QWR devices, is presented in Ref. [4] and indicates that the coherent component of the current should be experimentally detectable.
Acknowledgements Work partially supported by MURST (40% project on Quantum Computing) and by the US Office of Naval Research (Contract No. N0001498-1-0777).
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. Flannery, 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.