Position-dependent dynamics of electronic spin-subband entanglement in a Rashba nanoloop

Superlattices and Microstructures 60 (2013) 540–547

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Position-dependent dynamics of electronic spin-subband entanglement in a Rashba nanoloop R. Safaiee a,b, M.M. Golshan a,⇑ a b

Physics Department, College of Sciences, Shiraz University, Shiraz 71454, Iran Nanotechnology Research Institute, Shiraz University, Shiraz, Iran

a r t i c l e

i n f o

Article history: Received 12 May 2013 Received in revised form 22 May 2013 Accepted 29 May 2013 Available online 6 June 2013 Keywords: Entanglement Semiconductors Nanoloops Rashba spin–orbit coupling

a b s t r a c t In this paper, as a complementation to our previous reports, we study the position dependency of the electronic spin-subband entanglement for different spin initial conditions in a quasi-onedimensional Rashba nanoloop acted upon by a strong perpendicular magnetic field. We compute the von Neumann entropy, a measure of entanglement, as a function of time by explicitly including the confining potential and the Rashba spin–orbit (SO) coupling into the Hamiltonian. An analysis of the von Neumann entropy demonstrates that, due to a fictitious magnetic field arising from Rashba SO coupling, the spin-subband entanglement strongly depends upon the location of electron within the loop. Moreover, it is shown that the position dependency of entanglement dynamics depends upon the spin initial state. When the initial state is a pure one formed by a subband excitation and the z-component of spin states, the entanglement exhibits periodic oscillations with just one local minimum in each oscillation. On the other hand, when the initial state is formed by the subband states and a coherent superposition of spin states, the entanglement still periodically oscillates either with two local minima in each oscillation at h – 0, p or without local minima at h = 0, p. The physical reasons behind such behavior are also discussed. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction As a fundamental concept in quantum communications, quantum computations, etc., quantum entanglement has attracted vast attention in recent years [1–6]. The protocols of super dense coding ⇑ Corresponding author. Tel.: +98 711 646 0831; fax: +98 711 646 0839. E-mail addresses: [email protected] (R. Safaiee), [email protected] (M.M. Golshan). 0749-6036/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.spmi.2013.05.037

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[7,8], quantum teleportation [9] and telecloning [10] cannot be performed without some forms of entanglement between the parties involved [11]. It is therefore a problem of great interest to find physical systems where the entanglement can be produced, manipulated and detected. Among the many proposals [12–14], combinations of electronic spin states and environmentally induced states in heterostructures are considered as outstanding candidates for realizing entanglement and, consequently, solid-state-based quantum computing [15–17]. It is due to the fact that heterostructural systems have the advantage of offering the perspective to integrate a large number of quantum gates into a quantum computers, once the single gate and qubits are established [18–22]. In particular, nanoloop heterostructures have provided new means for constructing quantum memories [23], quantum networks [24,25], spin interferometers [26] and so on. Accordingly, it is viable to investigate the dynamical behavior of spin-subband entanglement in a nanoloop at different electronic locations along the loop. In the present work we show that such is indeed position-dependent. Although the electronic spin-subband entanglement in nanoloops has been previously reported [27,28], in what follows, nonetheless, we specifically report how the entanglement behaves at different locations along the loop. In semiconductor heterostructures inversion asymmetry of the potential profile in the growth direction introduces spin–orbit (SO) interactions for the confined carriers, known as structure-asymmetry-induced or the Rashba coupling [29]. The Rashba SO interaction inherently (as apposed to external magnetic fields) provides a way to couple electronic spins with its orbital degrees of freedom [15,29]. This is, in turn, an important issue for quantum information processing applications based on spins of electrons confined in nanostructures (here, a nanoloop). The strength of this interaction not only depends on the characteristics of the material but also can be controlled by an external electric field applied on the top of the heterostructure [30,31]. Moreover, choices of heterostructural materials, along with laterally applied electric fields, reduce the electronic motional freedom, confining it to a quasi-one dimensional (1D) quantum loop [32,27]. The confinement may be shown to be parabolic, acting in the radial direction in a 1D loop [32,33]. The Kramers spin degeneracy (because of timereversal symmetry) may also be removed by the application of an external uniform magnetic field [34]. As we shall see in what follows, the Rashba SO interaction in a nanoloop gives rise to a fictitious radial magnetic field. Consequently, the torque that acts on the electronic spin depends on its location in the loop. This fact then leads to the position-dependency in the dynamical behavior of the electronic spin. As a result, the dynamical behavior of spin-subband entanglement in the loop is also expected to be position-dependent. In the present study, we investigate the space dependency of spin-subband entanglement dynamics for a model of electrons confined in a 1D quantum loop in a strong perpendicular magnetic field with Rashba SO interaction. In so doing, we first present the governing Hamiltonian in the bosonic second quantized form. For simplicity, we approximate the confining (gate) effects as a parabola and then the orbital motions of the electron can be reduced to collection of a harmonic oscillators. The von Neumann entropy [35], a measure of entanglement [36], is then computed as a function of time. Using the well established material parameters for InAs nanoloops [32,27], our numerical computation of von Neumann entropy shows that, in general, the spin-subband entanglement varies from one point to another along the nanoloop. Taking the initial condition as a pure disentangled one, formed by subband and a state belonging to the z-component of spin, the entanglement exhibits periodic oscillations with just one local minimum (dip) in each oscillation [27], independent of the electronic location in the nanoloop. On the other hand, when the initial state is formed by the subband states and a coherent superposition of spin states, the entanglement still periodically oscillates either with two local minima (dips) or without them, depending on the location of the electron in the loop. The material presented in this article thus supplements the results of Refs. [27,28], where the dynamical behavior of spin-subband entanglement is reported at a particular electronic position along the loop. This paper is organized as follows. In Section 2 we briefly describe the Hamiltonian of our model along with Rashba fictitious magnetic field. We summarize the numerical procedure of computing the von Neumann entropy, as a function of time, in Section 3. Section 4 is devoted to the numerical results of temporal behavior of spin-subband entanglement for two spin initial conditions. Interpretation of the results and concluding remarks are presented in the last section.

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2. Rashba induced fictitious magnetic field in nanoloops We consider a 2D electron gas in the x–y plane confined in a quantum loop of mean radius R, acted ! upon by a perpendicular magnetic field, B (along the z-axis). The one-particle Hamiltonian describing an electron in the foregoing system is of the form,

H ¼ Horbit þ Hspin þ Hspin—orbit :

ð1Þ

The orbital part of the Hamiltonian is,

Horbit ¼

P2 2m

þ V con ðrÞ:

ð2Þ

! ! ! where P ¼ P þ ec A is electronic kinetic momentum and A is the vector potential associated with the ! magnetic field, B ¼ B^z. In the first part of Eq. (1) (the electronic kinetic energy), we assume that the electron gas in the loop has vanishing thickness, i.e. the spatial averages obey the relation D E D E D E P2z  P2x ; P2y [37]. In this case, the size quantization in z direction is so strong, that only one sub!

band is occupied by electrons. Therefore, the Pz-dependent term in the Hamiltonian of Eq. (2) giving just a constant has been omitted. The confining potential, Vcon, that localizes the electronic states within the loop is approximated by [38],

V con ðrÞ ¼

1  2 m xc ðr  RÞ2 : 2

ð3Þ

This potential arises from externally applied electric fields (gate potentials) and the materials building the nanoloop. The second part of the Hamiltonian,

Hspin ¼

1 g l Brz ; 2 B

ð4Þ

is the Zeeman term and the third term,

Hspin—orbit ¼

a^ ~ ! z  ½r  P ;

ð5Þ

h

2

e h Ez ~ is the Rashba SO interaction [35] in which a ¼ 4m 2 c2 is the Rashba constant and r ¼ ðrx ; ry ; rz Þ is the vector of pauli matrices. The coefficient a is tunable in strength by the external gates perpendicular to ! the plane of the loop. Using the symmetric-gauged vector potential, A ¼ 2B ð0; r; 0Þ (¼ B2 ðy; x; 0Þ in cartesian coordinates), the Hamiltonian of Eq. (1), along with Eqs. (2)–(5), reads,

"  2 # 1 eB a 2 þ V con ðrÞ þ Hspin þ ðcos hrx þ sin hry ÞPh P þ Ph þ r H¼ 2m r 2c h     a eB eB þ rx sin hPr þ r cos h  ry ðcos hPr  r sin hÞ ; 2c 2c h

ð6Þ

in the cylindrical coordinates, r and h. From the structure of the Hamiltonian of Eq. (6), it is evident that eikhU(r)g(s) form a solution to the unperturbed part (the part without Rashba term). In addition, m @ n @n because of the fact that the thickness of the loop is much smaller than rm @h n can be replaced by R @hn its radius. Noting the aforementioned facts and employing the bosonic creation (annihilation) operators, a (a), the Hamiltonian of Eq. (6), in the strong magnetic field limit, may be cast into the form,

  2 2 1 1 h k 1 h  xc m þ H¼ h  xc ay a þ þ hkxc þ g rz 2 2 2 2 4 m0 2m R h  xc ‘B ih y ak þ pffiffiffi ½ðe a þ eih aÞrx  iðeih ay  eih aÞry  þ ðrx cos h þ ry sin hÞ; R 4 2 ‘so where ‘B ¼

qffiffiffiffiffiffiffiffiffi h  m xc

ð7Þ

2

is the magnetic length with xc ¼ meB c (the cyclotron frequency) and ‘so ¼ 2mh  a is the

length scale associated with the Rashba SO interaction. It is observed that the second and third terms

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on the right-hand-side of Eq. (7) contribute a constant phase to the time evolution operator and thus they are legitimately omitted in what follows. The last term in the Hamiltonian of Eq. (7), stemming ! from Rashba SO interaction, indicates that it is a build-in Zeeman type interaction, 12 g lB B f  ~ r, due to the radial fictitious magnetic field,

! 2ak ^r Bf ¼ g lB R

ð8Þ

In the presence of a uniform external magnetic field normal to the loop, the effective magnetic field,

! ! B eff ¼ B^z þ B f

ð9Þ

! interacts with the magnetic moment of the electron’s spin. Since the direction of B eff depends on electronic location in the nanoloop, the corresponding torque changes direction and therefore leads to different spin states along the loop. It is also evident that the last term of Eq. (7) plays as an entangler because the spin precession about the effective magnetic field produces more spin states. The fifth term of Eq. (7) does the same through an interaction between spin and subbands. These two effects indeed lead to spin-subband entanglement. 3. Evolution of the spin-subband entanglement In this section the manner of computing the time evolution of spin-subband entanglement, for the present system, is outlined. In general, for any bipartite system, consisting of systems A and B, described by a state jWiAB, the entanglement may be quantified by the von Neumann entropy. Defining the reduced density operators as, qAðBÞ ¼ Tr BðAÞ ðjWiAB AB hWjÞ, the degree of entanglement is given by

X   q  q  k EðqÞ ¼ Tr qA log2 A ¼ Tr qB log2 B ¼  ki log 2i

ð10Þ

i

where ki denotes the eigenvalues of qA(B). In view of the physical interpretation of the entropy, the significance of von Neumann entropy, which follows, is apparent. When E(q) vanishes full information about the subsystems is available (local information), while for the maximal value of E(q)(=1) information is evenly distributed amongst the subsystems (nonlocal information). Evidently, if initially the density operator is formed by an eigenstate of the total Hamiltonian then E(q) of Eq. (10) remains the same at all times. However, the system is usually prepared in a basis (or superpositions) of the total Hilbert space, H ¼ HA  HB . In that case the density operator evolves with time,

b b y ðtÞ; U qðtÞ ¼ UðtÞj0ih0j

ð11Þ

b giving rise to time variation of E(q) and thus the degree of entanglement. In Eq. (11) UðtÞ , ¼ exp iHt h  with H being the total Hamiltonian, given in Eq. (7), represents the time evolution operator. The Hilbert space of the system aforementioned in the previous section, for a given subband index, n, is spanned by j1i jn + 1, +i, j2i jn, +i, j3i jn, i, and j1i jn  1, +i. Since these bases correspond to the eigenstates of the unperturbed Hamiltonian (first and forth terms) in Eq. (7), the matrix representation of H with respect to these bases is not diagonal. However, it is convenient to use a representation in which H is diagonal in order to evaluate the time evolution operator. It is clear that in diagonalizing H one encounters a forth-degree algebraic equation, with no straightforward analytical solutions, for the eigenvalues. Therefore in evaluating the eigenvalues, the eigenvectors, the transformation matrix, etc., one has to resort to numerical methods. When such numerical methods are performed, the time evolution matrix in the aforementioned bases,

b UðtÞ ¼ ½M expði½Kt=hÞ½M1 ;

ð12Þ

where [M] represents the transformation matrix and [K] is a diagonal matrix composed of the eigenvalues of H, is determined. The transformation matrix [M], is, of course, formed by elements of the b obtained in this manner, the density operator at a time t can be eigenvectors of H. Using UðtÞ computed. In order to find the reduced density matrix, partial tracing of q(t) over the two spin states

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is more convenient. We can obtain the degree of entanglement between the spin states and subband excitations, as a function of time, by Eq. (10). In the next section we use the procedure outlined in this section to investigate the temporal behavior of spin-subbands entanglement of an electron located in different points of the nanoloop when it is prepared in different initial states. 4. Numerical results and discussions In this section, we present space dependency of dynamical behavior of spin-subbands entanglement for different electronic initial conditions. We use the procedure outlined in the previous section to compute the von Neumann entropy (entanglement). We take the initial state as a direct product of the state of subband excitations and that of the electronic spin. Therefore subband excitations and the electronic spin states are initially uncorrelated. In our previous work [28], we have investigated the entanglement for different spin initial conditions in just one point of the loop; however, in the present work, it has been studied for all electronic positions in the loop. For the electronic spin itself we consider two types of initial conditions which give rise to significantly different time evolutions of the entanglement, depending on electronic locations. In one the spin points upwards along the z-direction and in the other it is along the positive x-axis (a maximally coherent superposition of Sz states). From the discussion given at the end of Section 2 and the radial fictitious magnetic field, it is clear that for j0i1 = jni  j + i, the spin precession occurs about ^r , independent of electronic position. On the other hand, for j0i2 = jni  jSx, +i the axis of spin precession varies at different locations along the loop. It is then concluded that the generation of spin states, with direct consequences in the time-evolution, is independent of electronic location for the first of initial conditions, in contrast to the second one. To support these conclusions, variations of the degree of entanglement against the angle h, for the two initial states, is illustrated in Fig. 1. This Figure and the following ones are drawn using the parameters given in references [32,27] for InAs. Moreover, From the structure of time evolution operator, it is evident that the degree of spinsubband entanglement involves a superposition of periodic functions of time. Since the eigenvalues of H is independent of electronic location, the frequencies of these periodic functions is also independent of it. However, their amplitudes (through the elements of the transformation matrix) depend upon the electronic location. It is then clear that the number of periodic functions in this superposition (or, equivalently the number of frequencies) depends on the electronic location. Furthermore, for b jwðtÞi1 ¼ UðtÞj0i 1 , the amplitudes are such that they cancel out in the density operator, giving a position-independent entanglement. This points are also verified from the inset in Figs. 1 and 2. We

0.030 0.025

E

0.020 0.015 0.010 0.005 0.000

0

1

2

3

4

5

6

rad

Fig. 1. Degree of entanglement versus the angle h at a fixed time t = 1.5  1015 s. The dashed and solid curves correspond to j0i1 and j0i2, respectively.

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a

b 0.025

0.020

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0

1. 10

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t s

14

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14

t s

Fig. 2. Degree of entanglement versus time for j0i1 = j500i  j + i, at h = 0. (a) indicates the presence of one dip per cycle, while (b) shows the interference pattern.

b

a 0.025

0.020

0.020

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0.015 E

E

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d

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0.000 0

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1. 10 t s

14

1.5 10

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0

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15

1. 10 t s

Fig. 3. The degree of entanglement versus time for j0i2 = j500i  jSx, +i, at different locations. Parts (a–d) correspond to h ¼ 0; p4 ; p2 and 34p, respectively. The inset in part (c) indicates the occurrence of two dips.

b note from Fig. 2 the occurrence of only one dip per cycle. On the other hand, for jwðtÞi2 ¼ UðtÞj0i 2, because it is initially in an Sx state, the cancelations do not occur, giving rise to position-dependent entanglement. This is also seen from Fig. 3, where the degree of entanglement versus time, is drawn for different h’s. Due to the strong symmetry of the loop the same behavior occurs at  h ¼ h þ p and thus we have omitted the corresponding graphs in Fig. 3. From this figure it is observed that the occurrence of dips (no or two) also depends on electronic location in the loop. The number of frequencies involved in the time-evolution of j0i1 (independent of the location in the loop) and j0i2 , obtained from

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Amplitude

0.010

0.008

0.006

0.004

0.002

0.000 0.0

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1.6x10

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Frequency (Hz) Fig. 4. FFT of the temporal behavior of entanglement for j0i1 = j500i  j + i.

(b)

(a) 0.015

0.010

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Amplitude

Amplitude

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(c) 0.012 0.010

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Frequency (Hz)

Fig. 5. FFT of the temporal behavior of entanglement for j0i2 = j500i  jSx, +i. Parts (a–c) correspond to h ¼ 0; p4 and p2 , respectively. The FFT corresponds to h ¼ 34p is similar to that for h ¼ p4 .

an FFT, is illustrated in Figs. 4 and 5, respectively. These frequencies justifies the presence of envelops and inner oscillations. 5. Conclusions In this paper we have presented, in detail, the space-dependency of entanglement dynamics between spin and structural subbands, for an electron in a quasi-1D loop, under the influence of Rashba spin–orbit coupling and a strong magnetic field. The entanglement between the electronic spin states

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and subband excitations, as a function of time, is calculated and plotted for specific initial conditions, one formed by a pure Sz state and the other by a maximally coherent superposition of Sz states, at different electronic locations in the loop. Although the results of this article are thoroughly discussed in Section 4, in what follows we summarize the more important aspects of this report. 1. The electronic spin-subband entanglement undergoes periodic oscillations with just one deep local minimum per cycle, independent of electronic location, for the pure Sz initial spin state. 2. The periodic oscillations of the entanglement, for the coherent superposition case, occurs with either two local minima or without local minima depending on the electronic location in the loop. 3. The oscillations of spin-subband entanglement, resulting from superpositions of periodic waves, involves five major frequencies for the initial Sz spin states. The number of involved frequencies differs at different locations for the initial coherent superposition of Sz states. The material presented in this article thus supplements Refs. [27,28] and provides new means for generation and control of entanglement. Acknowledgement This work has been in part supported by a Grant from the Research Council of Shiraz University, under the contract 92-GR-SC-82. References [1] R. Hanson, L.H. van Beveren Willems, I.T. Vink, J.M. Elzerman, M.W.J. Naber, F.H.L. Koppens, L.P. Kouwenhoven, L.M.K. Vandersypen, Phys. Rev. Lett. 94 (2005) 196802. [2] J.D. Sau, R.M. Lutchyn, S. Tewari, S. Das Sarma, Phys. Rev. Lett. 104 (2010) 040502. [3] J.D. Sau, S. Tewari, S. Das Sarma, Phys. Rev. A 82 (2010) 052322. [4] J. Cai, A. Miyake, W. Dür, H.J. Briegel, Phys. Rev. A 82 (2010) 052309. [5] Z. Zhi-Cheng, Chin. Phys. Lett. 28 (2011) 040301. [6] D.L. Feder, Phys. Rev. A 85 (2012) 012312. [7] C.H. Bennett, S.J. Wiesner, Phys. Rev. Lett. 69 (1992) 2881–2884. [8] Z. Shadman, H. Kampermann, C. Macchiavello, D. Bruß, New J. Phys. 12 (2010) 073042. [9] C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895–1899. [10] M. Murao, D. Jonathan, M.B. Plenio, V. Vedral, Phys. Rev. A 59 (1999) 156–161. [11] M. Murao, V. Vedral, Phys. Rev. Lett. 86 (2001) 352–355. [12] A.S.F. Obada, A.A. Eied, G.M. Abd Al-Kader, Int. J. Theor. Phys. 48 (2009) 380–391. [13] J.S. Xu, C.F. Li, M. Gong, X.B. Zou, C.H. Shi, G. Chen, G.C. Guo, Phys. Rev. Lett. 104 (2010) 100502. [14] Q.L. He, Y.Q. Zhang, J.B. Xu, Can. J. Phys. 89 (2011) 753–759. [15] N. Zhao, L. Zhong, J.L. Zhu, C.P. Sun, Phys. Rev. B 74 (2006) 075307. [16] S. Simmons, R.M. Brown, H. Riemann, N.V. Abrosimov, P. Becker, H.J. Pohl, M.L.W. Thewalt, K.M. Itoh, J.J.L. Morton, Nature 470 (2011) 69–72. [17] Y. Li, S.C. Benjamin, New J. Phys. 14 (2012) 093008. [18] D. Loss, D.P. DiVincenzo, Phys. Rev. A 57 (1998) 120–126. [19] G. Burkard, D. Loss, D.P. DiVincenzo, Phys. Rev. B 59 (1999) 2070–2078. [20] G. Burkard, H.-A. Engel, D. Loss, Fortschr. Phys. 48 (2000) 965. [21] J. Schliemann, D. Loss, A.H. MacDonald, Phys. Rev. B 63 (2001) 085311. [22] A. Tan, Y. Wang, X. Jin, X. Su, X. Jia, J. Zhang, C. Xie, K. Peng, Phys. Rev. A 78 (2008) 013828. [23] T.B. Pittman, J.D. Franson, Phys. Rev. A 66 (2002) 062302. [24] D. Bercioux, M. Governale, V. Cataudella, V.M. Ramaglia, Phys. Rev. B 72 (2005) 075305. [25] V.M. Ramaglia, V. Cataudella, G. De Filippis, C.A. Perroni, Phys. Rev. B 73 (2006) 155328. [26] T. Koga, J. Nitta, M. van Veenhuizen, Phys. Rev. B 70 (2004) R161302. [27] R. Safaiee, M.M. Golshan, N. Foroozani, J. Stat. Mech. (2009) P11014. [28] R. Safaiee, M.M. Golshan, Eur. Phys. J. B 83 (2011) 457–463. [29] Y.A. Bychkov, E.I. Rashba, J. Phys. C 17 (1984) 6039–6045. [30] J. Nitta, T. Akazaki, H. Takayanagi, T. Enoki, Phy. Rev. Lett. 78 (1997) 1335–1338. [31] T. Koga, J. Nitta, T. Akazaki, H. Takaynagi, Phys. Rev. Lett. 89 (2002) 046801. [32] M. Trushin, Transport in low dimensional systems with spinorbit coupling, Ph. D. Thesis, University of Hamburg, 2005. [33] P. Stano, Controlling electron quantum dot qubits by spinorbit interactions, Ph. D. Thesis, University of Regensburg, 2007. [34] S. Debald, C. Emary, Phys. Rev. Lett. 94 (2005) 226803. [35] V. Vedral, M.B. Plenio, M.A. Rippin, P.L. Knight, Phys. Rev. Lett. 78 (1997) 2275–2279. [36] T.C. Wei, K. Nemoto, P.M. Goldbart, P.G. Kwiat, W.J. Munro, F. Verstraete, Phys. Rev. A 67 (2003) 022110. [37] S. Bandyopadhyay, M. Cahay, Introduction to Spintronics, CRC Press, Boca Raton, FL, 2008 (and references therein). [38] D. Frustaglia, K. Richter, Phys. Rev. B 69 (2004) 235310.