ARTICLE IN PRESS
Physica E 40 (2008) 1862–1864 www.elsevier.com/locate/physe
Inﬂuence of valence band spin–orbit coupling on the entanglement of excitons in coupled quantum dots J.I. Climentea,, M. Korkusinskib, G. Goldonia,c, P. Hawrylakb a
CNR-INFM National Center on nanoStructures and bioSystems at Surfaces (S3), Via Campi 213/A, 41100 Modena, Italy b Institute for Microstructural Sciences, National Research Council of Canada, Ottawa, Canada K1A 0R6 c Dipartimento di Fisica, Universita` di Modena e Reggio Emilia, Via Campi 213/A, 41100 Modena, Italy Available online 6 September 2007
Abstract We study the effect of valence band spin–orbit interaction on the exciton entanglement in vertically stacked quantum dots, using a multi-band k p theory for holes. It is shown that the spin–orbit interaction reduces the exciton entanglement, establishing an intrinsic upper limit. For usual InGaAs nanostructures, however, this effect is generally small and does not pose a challenge for current attempts for the development of quantum information. r 2007 Elsevier B.V. All rights reserved. PACS: 73.21.La; 03.67.Mn Keywords: Coupled quantum dots; Entanglement; Spin–orbit; Exciton
Excitons in vertically coupled quantum dots (CQDs) constitute a reliable system for the preparation of one- and two-qubit entangled states [1–3]. The information is carried by the dot occupation index (or isospin) : using appropriate electric ﬁelds, one can keep the particles (electron and hole) localized in individual quantum dots (isospin index 0 or 1) or let them hybridize forming coherent superpositions (j0i j1i) . Tunnel coupling between the dots enables single-qubit rotation, while the exciton Coulomb interaction is responsible for the formation of two-qubit entangled states [1,6]. For the practical development of quantum computation it is critical that the entanglement of the exciton states is as high as possible, exceeding a threshold set by quantum error correction algorithms . A number of theoretical works have investigated the inﬂuence of several factors present in state-of-the-art CQD structures on the exciton entanglement, such as strain ﬁelds and broken symmetry [8,9]. Here we address another factor which is present in these systems and has been largely Corresponding author.
E-mail addresses: [email protected]
(J.I. Climente), [email protected]
(P. Hawrylak). 1386-9477/$ - see front matter r 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2007.08.042
disregarded to date, namely the spin–orbit (SO) coupling present in the valence band of semiconductors. We show that it leads to a reduction of the entanglement as compared to previous estimates neglecting it [6,9]. However, explicit calculations for InGaAs CQDs suggest that this reduction is small for most ranges of interdot distance. The double dot structure we study is composed by two vertically coupled quantum disks. The disks are identical, so that the structure has well-deﬁned reﬂection symmetry in the z direction and rotational symmetry in the plane. The exciton Hamiltonian reads H eh ¼ H e þ H h
e2 , jre rh j
where H e is the electron Hamiltonian, H h the hole Hamiltonian, and the last term is the Coulomb attraction between electron and hole, with e as the electron charge and as the effective dielectric constant. In Eq. (1), H e ¼ p2 =2m þ V e ðr; zÞ, where m stands for the electron effective mass and V e ðr; zÞ is the conﬁning potential. We deﬁne V e ðr; zÞ ¼ V ðrÞ þ V e ðzÞ, where V ðrÞ is a hard-wall conﬁnement in the radial direction (zero inside the disk, inﬁnite outside it) and V e ðzÞ is a square-well potential in the vertical direction (zero inside the disk, V ec outside it, V ec
ARTICLE IN PRESS J.I. Climente et al. / Physica E 40 (2008) 1862–1864
being the conduction band-offset between the dot and barrier materials). Owing to the rotational symmetry, the electron states have a well-deﬁned envelope azimuthal quantum number mz . Likewise, the inversion symmetry in the vertical direction allows to classify the states into symmetric and antisymmetric. For our purposes, we focus on the lowest energy symmetric and antisymmetric states, j þ ie ¼ jmz ¼ 0ie ðj0ie þ j1ie Þ and j ie ¼ jmz ¼ 0ie ðj0ie j1ie Þ, respectively. Here jmz i represents the in-plane part of the wavefunction, and the isospin notation is used to describe the vertical part, i.e. j0i (j1i) indicates upper (lower) dot occupation. The hole Hamiltonian is H h ¼ H LK þ V h ðr; zÞ, where the conﬁnement potential is the same as for electrons, but using the valence band-offset V hc . H LK is the (Luttinger– Kohn) four-band k p Hamiltonian , which we solve following Rego et al. . This is the simplest Hamiltonian including the valence-band SO coupling between heavy hole (HH) and light hole (LH) subbands. SO interaction mixes the valence band spin quantum number (s ¼ 12) with the microscopic angular momentum (l ¼ 1), so that hole states have a total Bloch angular momentum J ¼ l þ s, where ðJ; J z Þ ¼ ð32 ; 32Þ for HHs and ð32 ; 12Þ for LHs . The eigenfunctions of H LK are the so-called Luttinger spinors, four-component objects with Bloch functions jJ ¼ 32 ; J z i. Owing to the rotational symmetry of the CQD structure, the individual components of the Luttinger spinor have a well-deﬁned envelope azimuthal quantum number mz . However, different components of the Luttinger spinor have different mz , and only the total angular momentum F z ¼ mz þ J z is conserved. Similarly, owing to the inversion symmetry in the vertical direction, the individual components have a given parity (symmetric or antisymmetric) but, due to SO coupling, the Luttinger spinor includes two symmetric and two antisymmetric components . Still, since H LK commutes with the three-dimensional inversion operator, one may classify Luttinger spinors by their chirality . Thus, the two lowest-lying hole states of the CQD are the chirality up and down Luttinger spinors with F z ¼ 32: 0
c"ð#Þ þ3=2 jmz ¼ 0ih ðj0ih j1ih ÞjJ z ¼ þ3=2i
C B "ð#Þ C Bc B 1=2 jmz ¼ 2ih ðj0ih j1ih ÞjJ z ¼ 1=2i C C, B j " ð#Þi ¼ B "ð#Þ C B cþ1=2 jmz ¼ 1ih ðj0ih j1ih ÞjJ z ¼ þ1=2i C A @ c"ð#Þ 3=2 jmz ¼ 3ih ðj0ih j1ih ÞjJ z ¼ 3=2i (2) where the upper and lower sign of the isospin conﬁguration applies to the chirality " and # state. The simplest basis set for the exciton problem is composed by the four electron-hole Hartree products jþij "i , jij #i, jþij #i, jij "i. It can be shown that the two former states are optically active and the two latter optically inactive. Coulomb scattering does not mix these two subspaces. For our analysis, we shall focus on the optically active states (equivalent ﬁndings hold for the
inactive ones): Ceh 1 ¼ ajþij "i þ bjij #i;
Ceh 2 ¼ bjþij "i ajij #i. (3)
We may rewrite the exciton states factorizing the terms with identical isospin conﬁguration: a b Ceh j ¼ k j ðj0; 0i þ j1; 1iÞ þ kj ðj0; 1i þ j1; 0iÞ
þ kcj ðj0; 0i j1; 1iÞ þ kdj ðj1; 0i j0; 1iÞ.
Here jie ; ih i ¼ jie ijih i gives the isospin indexes of electron (ie ) and hole (ih ) components. The kij factors are ka1 ¼ aX " þ bX # ;
ka2 ¼ bX " aX # ,
kb1 ¼ aX " bX # ;
kb2 ¼ bX " þ aX # ,
kc1 ¼ aY " þ bY # ;
kc2 ¼ bY " aY # ,
kd1 ¼ aY " bY # ;
kd2 ¼ bY " þ aY # ,
where X "ð#Þ ¼ jmz ¼ 0ie ðc"ð#Þ þ3=2 jmz ¼ 0ih jJ z ¼ þ3=2i þ c"ð#Þ 1=2 jmz ¼ 2ih jJ z ¼ 1=2iÞ, Y "ð#Þ ¼ jmz ¼ 0ie ðc"ð#Þ þ1=2 jmz ¼ 1ih jJ z ¼ þ1=2i þ c"ð#Þ 3=2 jmz ¼ 3ih jJ z ¼ 3=2iÞ.
Each isospin conﬁguration appearing in Eq. (4) corresponds to a maximally entangled (Bell) state. However, the admixture between them introduces a departure from pure Bell states, thus reducing the entanglement. When SO coupling is neglected, the hole ground state is just the HH component of the Luttinger spinor with J z ¼ þ 32. In this limit, only the terms involving jJ z ¼ þ 32i survive in Eqs. (5). Thus, the kcj and kdj terms of Eq. (4) disappear and the optically active exciton state is a linear combination of ðj0; 0i þ j1; 1iÞ and ðj0; 1i þ j1; 0iÞ isospin conﬁgurations, as discussed in Ref. . By contrast, when the ﬁnite SO coupling is taken into account, admixture with J z a þ 32 components is present, kc and kd terms appear in Eq. (4) and the entanglement decreases. To quantify the deviation from the Bell states, we calculate numerically the density probability of the isospin eh conﬁgurations which constitute Ceh 1 and C2 for a typical InGaAs/GaAs CQD. The disks have 10 nm radius and 2 nm thickness, and the interdot distance is changed over a wide range of values. For the electron, we take m ¼ 0:05 m0 and V ec ¼ 500 meV, while for the hole we take the Luttinger parameters g1 ¼ 11:01, g2 ¼ 4:18, g3 ¼ 4:84 and V hc ¼ 380 meV . ¼ 12:9 is assumed. The results are plotted in Fig. 1. As can be seen, for large interdot distance the exciton state is almost a pure Bell state, either ðj0; 0i þ j1; 1iÞ or ðj0; 1i þ j1; 0iÞ. However, as the distance decreases, these two isospin conﬁgurations become comparable and their admixture reduces the entanglement. This result is essentially the same as predicted in the SO free limit: nearly pure Bell states are achieved at long interdot distances, when Coulomb interaction dominates over tunneling
ARTICLE IN PRESS J.I. Climente et al. / Physica E 40 (2008) 1862–1864
0.8 |0.0> + |1.1>
|0.1> + |1.0>
0.6 0.4 |0.1> + |1.0>
|0.0> + |1.1>
0.2 0 2
4 6 8 Interdot distance (nm)
8 4 6 Interdot distance (nm)
Fig. 1. Probability density of the isospin conﬁgurations composing the optically active exciton states of a CQD, as a function of the interdot distance. Solid lines correspond to ðkaj Þ2 , dashed lines to ðkbj Þ2 , dashed-dotted line to ðkcj Þ2 and dotted lines to ðkdj Þ2 .
energy, but the entanglement decreases for small distances, as tunneling starts prevailing . Most importantly, Fig. 1 reveals that, except for very small interdot distances, the SO-induced isospin conﬁgurations, ðj0; 0i j1; 1iÞ and ðj1; 0i j0; 1iÞ are much smaller than the dominant ones. Therefore, they do not reduce the entanglement signiﬁcantly. This is in spite of the fact that SO coupling is overestimated in our simulations, as we neglect the strain ﬁelds present in self-assembled CQDs, which tend to reduce the mixing between HH and LH subbands . The insets in Fig. 1 zoom in on the probability density of the minor isospin conﬁgurations induced by SO coupling. It can be seen that their behavior is qualitatively similar to that of the dominant conﬁgurations: at small interdot distance ðj0; 0i j1; 1iÞ and ðj1; 0i j0; 1iÞ have comparable weight, and at large distance one of the two prevails. In summary, we have shown that valence band SO interaction introduces a departure from the maximally entangled exciton states in CQDs. This effect is general for all CQD structures, regardless of their geometry, as the strong SO coupling is an inherent property of the semiconductor valence band. However, for usual CQDs, the entanglement reduction is only moderate and it should not pose a major obstacle for the realization of qubits.
We acknowledge support from FIRB-MIUR Italy-Canada RBIN06JB4C, PRIN 2006022932, Canadian Institute for Advanced Research and Quantum Works, MEIF-CT-2006023797 (J.I.C.), and Cineca Calcolo Parallelo 2007.
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