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Entangled Bell states of two electrons in coupled quantum dots—phonon decoherence
ARTICLE IN PRESS
Physica E 24 (2004) 234–243 www.elsevier.com/locate/physe
Entangled Bell states of two electrons in coupled quantum dots—phonon decoherence A. Hichria, S. Jazirib,*, R. Ferreirac a b
Laboratoire de Physique de la Mati"ere Condens!ee, Facult"e des Sciences de Tunis, Tunisia D!epartement de Physique, Facult!e des Sciences de Bizerte, 7021 Zarzouna, Bizerte, Tunisia c Laboratoire de Physique de la Mati"ere Condens!ee, ENS, F-75005, Paris, France
Received 4 December 2003; received in revised form 25 February 2004; accepted 16 April 2004 Available online 19 June 2004
Abstract We demonstrate coupling and entangling of quantum states in a pair of vertically aligned self assembled quantum dots by studying the dynamics of two interacting electrons driven by external electric field. The present entanglement involves the spatial degree of freedom for the two electrons system. We show that system of two interacting electrons initially delocalized (localized each in one dot) oscillate slowly in response to electric field, since the strong Coulomb repulsion makes them behaving so. We use an explicit formula for the entanglement of formation of two qubit in terms of the concurrence of the density operator. In ideal situations, entangled quantum states would not decohere during processing and transmission of quantum information. However, real quantum systems will inevitably be influenced by surrounding environments. We discuss the degree of entanglement of this system in which we introduce the decoherence effect caused by the acoustic phonon. In this entangled states proposal, the decohering time depends on the external parameters. r 2004 Elsevier B.V. All rights reserved. PACS: 03.67.a; 71.10.Li; 71.35.y Keywords: Quantum dots; Entanglement; Phonon decoherence
1. Introduction Recently, solid state realizations of the entanglement have received increasingly attention due to the fact that semiconductor nanostructures such as quantum dots and double quantum dots with well *Corresponding author. Tel.: +21672591906; fax: +21672590566. E-mail address: [email protected] (S. Jaziri).
defined atom-like and molecule-like properties have been fabricated and studied [1–13]. Entanglement is an essential ingredient in quantum information processing like information cryptography and quantum computation, and therefore it is a problem of great current interest to find or design systems where entanglement can be manipulated [14]. Most of the theoretical and experimental activity until now has been associated with atomic and quantum-optic systems. Two [15,16],
1386-9477/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2004.04.036
ARTICLE IN PRESS A. Hichri et al. / Physica E 24 (2004) 234–243
three [17], and four particle [18] entanglement have been successfully demonstrated experimentally in trapped ions, Rydberg atoms, and cavity. Semiconductor quantum dots have their own advantages as a candidate of the basic building blocks of solid state based quantum logic devices, due to the existence of an industrial base for semiconductor processing and the ease to integration with existing device [19,20]. The primary motivation for creating entangled states was to test Bell’s inequality [21] which was derived by using the hidden variables theory. The original Bell inequalities and reformulated versions are extremely useful tools for exploring the nonlocal and nonclassical [22] character of entangled systems. The state of an entangled system is represented as a nonfactorizable superposition of product states; whereas, that of unentangled systems is represented as a product state. Various schemes based on electron spins and electron–hole pairs have been proposed to implement quantum computer hardware architectures [6–9,20]. Although there have been some numerical studies on interacting electron systems driven by an AC field [11,23], there is still little theoretical understanding of the observed effects beyond the phenomenology level. Imamoglu et al. [20] have considered a quantum computer model based on both electron spins and cavity which is capable of realizing controlled interactions between two distant quantum dots. Zhang and Zhao [10,11], have presented a scheme based on dynamic localization and quantum entanglement of two interacting electron in coupled quantum dots. Quiroga and Johnson [12] have suggested that the resonant transfer interaction between spatially separated excitons in quantum dots can be exploited to produce maximally entangled Bell states. Here, we consider a generic example of quantum control consisting of two interacting electrons in coupled quantum dots driven by a time-dependent electric field: The quantum bits (qubit) are the states of individual carriers which can be either on the lower dot or the upper dot. The different dot positions play the same role as a ‘‘spin’’. Quantum mechanical tunnelling between the dots leads to a superposition of the quantum dot states. The spatial degree of freedom of the two particles form entangled states. The entangle-
235
ment can be controlled by applying an electric field along the heterostructure growth direction. Given the central status of entanglement, the task of quantifying the degree to which a state is entangled is important, and several measures have been proposed to quantify it. It is worth remarking that even for the smallest Hilbert space capable of exhibiting entanglement, i.e., the two qubit systems, there are aspects of entanglement that remain to be explored. However, many measures of entanglement proposed in the past have relied on either the Schmidt decomposition [24] or decomposition in a ‘‘magic basis’’ [25,26]. In this work, we use the concurrence to measure the degree of entanglement for pure bipartite states of two qubits. Coherent control of these quantum systems can be achieved by the application of external electric field [27]. An account of decoherence in the quantum evolution of a small system interacting with an environment (phonons) [12,13], became a problem of crucial importance in quantum computing. Since, decoherence caused by acoustic phonon–electron interaction is also studied in this paper. As a result we can relate the quantum behavior with the electric field and therefore establish how the decoherence rate of the two particles states can be determined. The main purpose of the present paper is to focus on the influence of the phonons environment on the entangled states. The paper is organized as follows: in Section 2, we study the coherent control of the quantum system of two interacting electrons in coupled quantum dots. We describe the general Hamiltonian and solution scheme for the electrons confined in a vertically aligned double GaAs quantum dot. We derive a solution of the master equation for the density operator in the presence of an oscillatory electric field. With a specific choice of the initial state, we introduce the entangled Bell states for the measurements of the degree of entanglement. In Section 3, we consider the rates for carriers scattering by acoustical phonons decoherence. Transition probabilities from initial state to all possible final states are evaluated. We then come to elucidate the effect of decoherence on the master equation for the density operator and the degree of entanglement. Finally, we conclude in Section 4.
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z ¼ a and þa; given by [29]
2. Model and method 2.1. The Hamiltonian Following the work of Burkard et al. [9], we model the two electrons in two coupled quantum dots by the following Hamiltonian [9–11]: X H¼ h07a ð~ ri ; ~ p i Þ þ HCoul þ Hc ðtÞ; ð1aÞ i¼1;2
h07a ð~ r; ~ pÞ ¼
~ p2 þ Vjj ð~ r Þ þ V> ð~ r Þ; 2m
ð1bÞ
the single particle Hamiltonian h07a describes a single electron confined in the upper (lower) dot of the double dot system. For the lateral confinement we choose the parabolic potential, mw2z a2 2 ðx þ y2 Þ; ð2Þ 2 where we have introduced the anisotropy parameter a determines the strength of the vertical relative to the lateral confinement, _wz is the quantization energy, and m is the electron effective mass. In describing the confinement V> along the inter-dot axis, we have used a (locally harmonic) double well potential of the form Vjj ðx; yÞ ¼
mw2z 2 ðz a2 Þ2 ð3Þ 8a2 which, in the limit of large inter-dot distance abRz ; separates (for zE7a) into two harmonic pffiffiffiffiffiffiffiffiffiffiffiffiffiffi wells with characteristic dot radius Rz ¼ _=mwz : Here a is half the distance between the centers of the dots. The Coulomb interaction is included by HCoul ¼ e2 =kj~ r1 ~ r 2 j; with a relative dielectric constant k: Finally, the interaction Hc ðtÞ ¼ P ez F ðtÞ represents the coupling by a timei i¼1;2 dependent electric field F ðtÞ applied along the z direction. In the calculations we use the GaAs parameters m ¼ 0:067m0 with m0 the free electron mass, and k ¼ 13:1: We use the Hund–Mulliken method [28] of molecular orbitals to describe the low lying energy levels of our system. This approach accounts for double occupation and is therefore suitable for investigating the questions at issue here. The oneparticle Hamiltonian has the ground state solution of the two isolated dots, centred, respectively, on V> ðzÞ ¼
j7a ðx; y; zÞ a 1=2 1 1=4 ¼ pR2z pR2z 1 exp 2 faðx2 þ y2 Þ þ ðz8aÞ2 g 2Rz
ð4Þ
and correspond to the ground state energy e7 ¼ _wz ð1 þ 2aÞ=2: The two ground states are not orthogonal and their overlap is Z 2 ð~ S ¼ d3 rjþa r Þja ð~ r Þ ¼ eða=Rz Þ ; ð5Þ a nonvanishing overlap S implies that the electrons can tunnel between the dots. From these nonorthogonal states, we construct the orthonormalized one-particle wave function 1 f7 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjþa 7ja Þ: 2ð17SÞ
ð6Þ
In this work we shall be concerned with nanometric quantum dots, for which the one-electron ground state of each dot taken as isolated is well separated from the excited ones (large lateral and vertical confinement energies). In this case, the two low lying single-electron states of the double dot are roughly the symmetric fþ and antisymmetric f linear combinations of the two isolated dots ground states. Thus, the eigenstates of H can be constructed from the four (two per electron) oneparticle molecular orbital states. We end-up with four two-particle states: three-singlet states jFþ ð~ r 1 ÞFþ ð~ r 2 ÞS; jFp ð~ r 1ffiffiffiÞF ð~ r 2 ÞS;fjFþ ð~ r 1 ÞF ð~ r 2 ÞS þjF ð~ r 1 ÞFþ ð~ r 2 ÞSg= 2 and one p triplet state ffiffiffi fjFþ ð~ r 1 ÞF ð~ r 2 ÞS jF ð~ r 1 ÞFþ ð~ r 2 ÞSg= 2: Let us consider initially the time-independent solutions for two interacting electrons in a tunnelcoupled double dot structure ðF ðtÞ ¼ 0Þ: The matrix elements of the two-electrons Hamiltonian in the orthonormal basis is given by: H 0 B B ¼B @
2eþ þ Jþ
X
0
0
X
2e þ J
0
0
0 0
0 0
eþ þ e þ Vþ 0
0 eþ þ e þ V
1 C C C; A
ð7Þ
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where
50
e7 ¼ /F7 ð~ r 1 ÞF7 ð~ r 2 Þjh0i jF7 ð~ r 1 ÞF7 ð~ r 2 ÞS;
48
237
E E E E
46
J7 ¼ /F7 ð~ r 1 ÞF7 ð~ r 2 ÞjHCoul jF7 ð~ r 1 ÞF7 ð~ r 2 ÞS;
44
E(meV)
V7 ¼ /Fþ ð~ r 1 ÞF ð~ r 2 Þ7F ð~ r 1 ÞFþ ð~ r2Þ jHCoul jFþ ð~ r 1 ÞF ð~ r 2 Þ7F ð~ r 1 ÞFþ ð~ r 2 ÞS;
42 40 38
X ¼ /Fþ ð~ r 1 ÞFþ ð~ r 2 ÞjHCoul jF ð~ r 1 ÞF ð~ r 2 ÞS:
36
In Eq. (7), the J7 and V7 are direct Coulombic couplings whereas X is an exchange contribution involving the singlet states jFþ Fþ S and jF F S: The eigenenergies of H can be easily solved as: Jþ þ J ES7 ¼ ðeþ þ e Þ þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7
ðeþ e þ ðJþ þ J Þ=2Þ2 þ X 2 ;
ET ¼ eþ þ e þ V
34 32 25
50
75
100
125
150
175
200
a(Å)
Fig. 1. Variation of the energies of the interacting two-electrons states as a function of the inter-dot distance. Two identical dots ( with a_oz ¼ 8 meV and Rz ¼ 85 A:
ES0 ¼ eþ þ e þ Vþ
and
with the corresponding eigenstates are given as jcS S ¼ d1 jFþ ð~ r 1 ÞFþ ð~ r 2 ÞS d2 jF ð~ r 1 ÞF ð~ r 2 ÞS;
ð8aÞ
1 jcT S ¼ pffiffiffifjFþ ð~ r 1 ÞF ð~ r 2 ÞS 2 r 1 ÞFþ ð~ r 2 ÞSg; jF ð~
ð8bÞ
1 jcS0 S ¼ pffiffiffifjFþ ð~ r 1 ÞF ð~ r 2 ÞS 2 þ jF ð~ r 1 ÞFþ ð~ r 2 ÞSg;
ð8cÞ
jcSþ S ¼ d2 jFþ ð~ r 1 ÞFþ ð~ r 2 ÞS þ d1 jF ð~ r ÞF ð~ r 2 ÞS; ð8dÞ q1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 and where d1 ¼ X = X 2 þ ðESþ 2eþ Jþ Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 ¼ ESþ 2eþ Jþ = X 2 þ ðESþ 2eþ Jþ Þ2 : Fig. 1 shows the behavior of the electronic energy levels as a function of the inter dot distance, for a system comprising two equal dots with vertical confinement energy _wz ¼ 16 meV and horizontal confinement energy a _wz ¼ 8 meV with Rz ¼ ( It appears that, when the distance 2a 85 A: between the two quantum dots is large ðabRz Þ; the energy spectrum presents two series of two nearly degenerate states. This ‘‘tight-binding’’ limit can be easily recovered by using the approximation jF7 ð~ r i ÞSEp1 ffiffi ðj0S7j1SÞ; where 2
j0S ¼ jja S; j1S ¼ jjþa S and SE0: Note that d1;2 would be 0 or 1 in absence of the exchange term X in Eq. (8), while jd1 jEjd2 jE1=O2 in the tight-binding limit when X a0: We find in this limit: ES EET E2E0 and ES0 EESþ E2E0 þ l0 with l0 ¼ /00jHCoul j11S=2: The eigenstates are correspondingly given by a superposition of Bell states pffiffiffi jcS SEjCþ S ¼ ðj01S þ j10SÞ= 2; ð9aÞ pffiffiffi jcT SEjC S ¼ ðj01S j10SÞ= 2; ð9bÞ pffiffiffi jcS0 SEjw S ¼ ðj00S j11SÞ= 2; ð9cÞ pffiffiffi jcSþ SEjwþ S ¼ ðj00S j11SÞ= 2; ð9dÞ the first (second) label holds for the first (second) electron coordinates. Therefore, we can see from Eq. (9) that due to the strong Coulomb repulsion, the low energy states jcS S and jcT S contain mainly single dot occupancies j01S and j10S; while the excited states jcS0 S and jcSþ S contain mainly double dot occupancies j00S and j11S: The four states in Eq. (9) reflect this single or double occupancy features. Each electron is, of course, equally present in the two identical dots, but the low and high energy levels differ by a single or double occupation of the dots. Note finally that the tight-binding two-electron states can be written
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in terms of the four states in Eq. (9)
80
jcS SE fðd1 d2 Þjwþ S
70
pffiffiffi þ ðd1 þ d2 Þjcþ Sg= 2;
60
ð10aÞ
E E E
jcT S ¼ jc S;
ð10bÞ
jcS0 S ¼ jw S;
ð10cÞ
jcSþ SE fðd1 þ d2 Þjwþ S ðd1 d2 Þjcþ Sg=
E(meV)
50 40 30 20
pffiffiffi 2:
10
ð10dÞ
We shall be concerned with states in Eq. (10) below, when discussing the entanglement characteristics of the two-electron states driven by the electric field.
0 0
2
4
6
8
10
12
F(kV/cm)
Fig. 2. Energy spectrum of the two electrons states as a function of the strength of a constant electric filed applied along ( the growth axis, for the double dot in Fig. 1 ða ¼ 170 AÞ:
2.2. Equations of motion We introduce in the following the electric field. Obviously, Hc ðtÞ does not mix singlet and triplet states, and thus jcT S is insensitive to the applied field within our truncated basis. The matrix representation H1 ðtÞ of the two-electrons Hamiltonian in the singlet basis fjcS S; jcS0 S; jcSþ Sg; reads: 0 1 0 0 ES B C ES0 0 A H1 ðtÞ ¼ @ 0 0
0 0
ESþ
0 B þ F ðtÞq*z@ d1 d2 0
d1 d2 0 d1 þ d2
1 0 C d1 þ d2 A ; 0
ð11Þ pffiffiffi where z* ¼ 2=ð1 S 2 Þa: We present in Fig. 2, the energies of the double dot eigenstates in the presence of a constant electric field F ðtÞ ¼ F0 for ( (note that the energy a dot separation 2a ¼ 340 A diagram is symmetrical with respect to the sign of the field since we consider two identical quantum dots). For this dot separation the tight-binding limit roughly pffiffiffi applies (see Fig. 1). In this case, d1 Ed2 E1= 2 and thus, to the lowest order, the field couple only the excited states jcSþ S and jcS0 S; which are nearly degenerated at zero field. This explains the linear splitting of these levels with F0 in Fig. 2. The small energy anti-crossing
with jcS S around FC E1:9 kV=cm results from the small d1 d2 value in Eq. (10). For F bFC ; the ground state is Ep1 ffiffi ðjcSþ S þ jcS0 SÞEj00S 2
and the high energy state is Ep1 ffiffi ðjcSþ S 2
jcS0 SÞEj11S; which both comprise two electrons in the same dot (the decrease in electrostatic energy for the two electrons compensates for their mutual repulsion) and are roughly disentangled states (E product of two one-electron states). In conclusion, it is important to keep in mind the two following features related to the electric field couplings in the tight-binding limit: (i) the field couples only weakly states with different dot occupancies and (ii) it breaks the left-right symmetry and is able to induce a simultaneous localization of both electrons on the same quantum dot and correspondingly to disentangle the two-electrons states. We shall come back to these results further below, in relation with a timedependent field. We assume in the following an oscillatory electric field F ðtÞ ¼ F0 cosðotÞ: The evolution of any initial state jcð0ÞS can be expressed as jCðtÞS ¼ C1 ðtÞjcS S þ C2 ðtÞjcS0 S þC3 ðtÞjcSþ SE C1 ðtÞjcþ S þC2 ðtÞjw S þ C3 ðtÞjwþ S corresponding to the density operator rðtÞ ¼ jcðtÞS/cðtÞj with rð0Þ ¼ jcð0ÞS/cð0Þj: The master equation of the density operator is i_dr=dt ¼ ½H; r ; with H is the
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two-electron Hamiltonian (in the basis of the 3 singlet states, r is a 3 3 matrix and H1 is given in Eq. (11)). The diagonal terms of the density matrix are the probabilities of finding the two particles in the basis states, while its off-diagonal matrix elements (the ‘coherencies’) describe the linear superposition of these states induced by the applied electric field. Let us consider the dynamics of the system starting with the unperturbed state jcð0ÞS ¼ jcS S: A nontrivial time evolution is possible because of the term proportional to d1 d2 ; which couples jcS S to the excited singlet states. We plot in Fig. 3 the time evolution of Bell the Bell probabilities rBell 11 ¼ /cþ jrjcþ S; r22 ¼ Bell /w jrjw S and r33 ¼ /wþ jrjwþ S: The oscillatory field has amplitude F0 ¼ 1:5 kV=cm and frequency o ¼ ðES0 ES Þ=_ðZoE5:44 meV at zero field ( We can see from Fig. 3 that and for a ¼ 170 A). the two electrons oscillate between the ground and the two excited singlet states. The oscillation pattern is complex but shows two main features: (i) a slow periodic exchange of populations between the ground (initial) and the two excited states, and (ii) fast oscillations between the populations of the two excited levels. The first feature indicates a slow field-induced variation of the mean dot occupancy, while the second one indicates an important field-induced coupling of the two excited singlet states, in accordance with the previous results for a static field (see Fig. 2).
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In order to quantify the degree of entanglement, we will adopt the concurrence C defined by [25,26]. The concurrence varies from C ¼ 0 for an unentangled state to C ¼ 1 for a maximally entangled state. The concurrence may be calculated explicitly from the density matrix r : CðrÞ ¼ maxf0; l1 l2 l3 l4 g where the quantities li are the square roots of the eigenvalues B in decreasing order of the matrix r ¼ rðsA y #sy Þ A B r ðsy #sy Þ; where r denotes the complex conjugation of r in the standard basis. The most general pure states in the case of the two-qubit model can be written as jcS a1 j00S þ P¼ 4 2 a2 j01S þ a3 j10S þ a4 j11S; where i¼1 jai j ¼ 1: The concurrence of the pure state is simply given by [25] C ¼ 2ja2 a3 a1 a4 j:
ð12Þ
Thus, the state is entangled if and only if a1 a4 aa2 a3 : Fig. 4 shows that the degree of entanglement varies in time between zero for nonentangled state and 1 for maximally entangled states. In order to understand this result, it is important to recall two points: (i) the zero-field eigenstates are maximally entangled, since it is impossible to decompose them into simple products of one-electron orbital, and (ii) an applied field is able to mix the zero-field entangled states and generate non-entangled solutions, as discussed in connection with the results shown in Fig. 2. Correspondingly, Fig. 4 shows the
1.0 1.0
0.8 0.8
0.6
ρBell
0.6
C
0.4 ρ
0.4
ρ ρ
0.2
0.2
0.0 0
1
2
3
4
5
6
7
8
9
10
11
12
t (ps) Bell Fig. 3. Time evolution of the Bell probabilities rBell 11 ; r22 and rBell 33 under the influence of a sinusoidal electric field for the ( double dot in Fig. 1 ða ¼ 170 AÞ:
0.0
0
1
2
3
4
5
6
7
8
9
10
11
12
t (ps)
Fig. 4. Degree of entanglement corresponding to the time evolution in Fig. 3.
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dynamical aspects of the role played by the fieldinduced couplings on the nature of the twoelectron eigenstates. Note that the entanglement approaches its maximal value anytime the population of one of the zero-field singlet states approaches unity (compare with Fig. 3). When the system is principally in the excited states, the entanglement oscillates quickly, in agreement with the fast oscillations in Fig. 3 between the populations of the two excited singlet states. Fig. 4 shows that the degree of entanglement of the system can be dynamically monitored by the application of an oscillating electric field. In addition, the control presents a 100% contrast, since completely entangled and completely disentangled configurations can be dynamically generated (at different times, of course) in this way.
3. Phonon decoherence 3.1. Acoustic phonon scattering In the last section, we have discussed how the application of an oscillatory electric field allows controlling the degree of entanglement of the twoelectron states. In order to study the influence of environment on the dynamical evolution of the degree of entanglement, we consider in the following the coupling of the electrons with acoustic phonons [30]. Let us focus initially on the zero-field case and use the electron–phonon interaction described by the deformation potential mechanism. Within this model, the coupling of two-electrons states precede via the single-electrons interactions and transitions between singlet and triplet states are forbidden. The electron– phonon is given by [31] He2ph ¼ PHamiltonian P i~ q~ ri þ fað~ q Þe b þ ccg; where bþ q q is the usual i¼1;2 q creation operator for an acoustical phonon in ~ þ~ ~ the in-plane component mode ~ q¼Q q z with Q of ~ q and jað~ q Þj2 ¼ D2c _q=2rcs V : We have used in the calculations an isotropic linear phonon dispersion oq ¼ cs q and the GaAs material parameters: conduction band deformation potential Dc ¼ 8:6 eV; density r ¼ 5300 kg=m3 and longitudinal velocity of sound cs ¼ 3700 m=s: Transition probabilities from an initial state jci S with energy Ei to
all possible final states jcf S with energy Ef are evaluated within the Born approximation X _ ¼ G¼ Nq7 ðEq Þj/cf jHe2ph jci Sj2 2pt q dðEf Ei 7Eq Þ;
ð13Þ
where Eq ¼ _oq is the phonon energy. The sum over all the q vectors can be expressed as an integral over qx ; qy ; and qz : The scattering rate involving two-electron levels reads Gi ¼
1 D2c 7 2 2 N ðEQm ÞjQ2m jeR Qm =2a 2p_ 2rc2s Qm Z Qm 2 2 eR qz =2a jgif ðqz Þj2 dqz ;
ð14Þ
Qm
where Qm ¼ jEf Ei j=_cs is the wave vector of the exchanged phonon; Nq7 ðEq Þ ¼ fexpðEq =kB TÞ 1g1 þ 1=271=2 is the occupation factor, T is the lattice temperature and the upper (lower) sign corresponds to emission (absorption) processes. Finally, the form-factors gif ðqz Þ involve integrals of the kind Is;s0 ðqz Þ ¼ /Fs ðz1 ; z2 Þjexpðiqz z1 Þ þ expðiqz z2 ÞjFs0 ðz1 ; z2 ÞS;
ð15Þ
where Fs ðz1 ; z2 Þ is the z-dependent part of the twoparticle total wavefunction. In the tight-binding limit, as discussed above, these later are linear combinations involving either single dot occupancies j01S and j10S or double dot occupancies j00S and j11S (see e.g. Eq. (9)). We can thus show that only transitions involving the excited states jcSþ S and jcS0 S will not be negligible in this limit. Fig. 5 shows the calculated rates for phonon assisted absorption ðG23 Þ and emission ðG32 Þ processes between the two excited states jcS0 S and jcSþ S as a function of both the inter-dot distance (Fig. 5a) and the vertical confinement (Fig. 5b) at T ¼ 77 K: The scattering rates involving the ground singlet state are found to be orders of magnitude smaller than those of Fig. 5. The resonant-like profiles in Fig. 5 follow from the particular dependence of the deformation potential coupling upon the exchanged phonon energy (the energy difference between the initial and final states): it is very weak for both very small and very large Eq values. Thus, the scattering rate displays a
ARTICLE IN PRESS A. Hichri et al. / Physica E 24 (2004) 234–243 0.5
0.4
1/τ 1/τ -1
1/τi j (ps )
0.3
0.2
0.1
0.0 140
150
160
(a)
170
180
190
200
a(Å)
0.5
-1
1/τi j (ps )
0.4
1/τ 1/τ
0.3
0.2
0.1
0.0 10
(b)
12
14
16
18
20
22
h wz (meV)
Fig. 5. Scattering rate due to acoustical phonon for a GaAs quantum dots as a function of the inter dot distance for _wz ¼ ( (b). 16 meV (a), and of the vertical confinement for a ¼ 170 A ( in both cases. Rz ¼ 85 A
Let us neglect initially the applied electric field ðF0 ¼ 0Þ: When the initial state is the ground singlet one, it follows that r11 E1 at all times, since acoustical phonons are unable to couple this state to the excited ones. Fig. 6 shows the populations evolutions in the case where the initial state is jcS0 S: We see that an effective population exchange is possible between the two excited singlet, since G23 and G32 are not vanishing. At long times, the populations reach a stationary solution, obtained by imposing qri;j =qt ¼ 0 in the presence of the damping terms. Let us consider now to which extent the phonon-induced decoherence effects are detrimental to the dynamical control (i.e., by an external field F ðtÞ ¼ F0 cosðotÞÞ of the populations and of the degree of entanglement. Fig. 7 shows the time evolution of the Bell probabilities for the same system, initial conditions and field parameters as in Fig. 3. The initial evolution is a transient regime in a time interval given basically by the characteristic phonon scattering rates. At long times, the probabilities show a driven behaviour, as demonstrated by the fact that they present oscillations with half (because the energies in Fig. 2 do not depend on the sign of the perturbing field) the period T ¼ 2p=o related to the driven field. In the same way, the coherencies ri;jai ðtÞ display an initial transient behavior followed by a driven long-time solution The long-time solution of the
maximum when varying Eq ¼ ESþ ES0 by either changing the inter-dot distance or the intra-dot vertical frequency.
1.0
0.8
3.2. Equation of motion of the reduced matrix
ð17Þ
ρ Bell
0.6
In order to study the influence of the phonon assisted interactions on the time evolution of the two-electron system, we add in the master equation for qr=qt the damping terms: X qrii ¼ ðGj-i rjj Gi-j rii Þ; ð16Þ qt dec jai qrii 1X ¼ ðGi-k þ Gj-k Þrij : qt dec 2 kai
241
ρ ρ ρ
0.4
0.2
0.0 0
2
4
6
8
10
t (ps) Bell Fig. 6. Time evolution of the Bell probabilities rBell 11 ; r22 and rBell under the influence of the phonon decoherence in absence 33 ( of applied field. Identical dots with _wz ¼ 16 meV; Rz ¼ 85 A; ( and T ¼ 77 K: a ¼ 170 A
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double dot parameters, the phonon-induced decoherence rates can be made very small, allowing a roughly coherent time-evolution of the driven twoelectron system.
1.0 ρ ρ
0.8
ρ
ρ Bell
0.6
References
0.4
0.2
0.0 0
1
2
3
4
5
6
7
8
9
10
11
12
t (ps) Bell Fig. 7. Time evolution of the Bell probabilities rBell 11 ; r22 and Bell r33 under the influence of the phonon decoherence and of a sinusoidal field. Same parameters as in Fig. 6.
density matrix evolution is, of course, independent upon the initial conditions. In particular, the amplitudes of the driven solutions depend only on the double dot and field parameters and on the scattering rates. However, according to Fig. 5, the inter-level scattering rates can be made very small. Thus, by properly choosing the parameters of the double dot system, the transient regime transforms into a quasi-stationary one. This characterizes a rather robust (with respect to the environment influences) system, allowing to explore the quantum nature of its time evolution and, more importantly, to monitor this later evolution by means of an external tool, like the application of an oscillatory electric field.
4. Conclusion We have shown how the entanglement of two interacting electrons in a double quantum dot system can be dynamically manipulated by an external electric field. Decoherence processes represent the most problematic issue pertaining to most quantum computing processing. In the present work we have considered the role of acoustical phonons on the time evolution of the density matrix (the populations and the coherences) describing two electrons confined in a double dot structure. By properly choosing the
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Physica E 24 (2004) 234–243 www.elsevier.com/locate/physe
Entangled Bell states of two electrons in coupled quantum dots—phonon decoherence A. Hichria, S. Jazirib,*, R. Ferreirac a b
Laboratoire de Physique de la Mati"ere Condens!ee, Facult"e des Sciences de Tunis, Tunisia D!epartement de Physique, Facult!e des Sciences de Bizerte, 7021 Zarzouna, Bizerte, Tunisia c Laboratoire de Physique de la Mati"ere Condens!ee, ENS, F-75005, Paris, France
Received 4 December 2003; received in revised form 25 February 2004; accepted 16 April 2004 Available online 19 June 2004
Abstract We demonstrate coupling and entangling of quantum states in a pair of vertically aligned self assembled quantum dots by studying the dynamics of two interacting electrons driven by external electric field. The present entanglement involves the spatial degree of freedom for the two electrons system. We show that system of two interacting electrons initially delocalized (localized each in one dot) oscillate slowly in response to electric field, since the strong Coulomb repulsion makes them behaving so. We use an explicit formula for the entanglement of formation of two qubit in terms of the concurrence of the density operator. In ideal situations, entangled quantum states would not decohere during processing and transmission of quantum information. However, real quantum systems will inevitably be influenced by surrounding environments. We discuss the degree of entanglement of this system in which we introduce the decoherence effect caused by the acoustic phonon. In this entangled states proposal, the decohering time depends on the external parameters. r 2004 Elsevier B.V. All rights reserved. PACS: 03.67.a; 71.10.Li; 71.35.y Keywords: Quantum dots; Entanglement; Phonon decoherence
1. Introduction Recently, solid state realizations of the entanglement have received increasingly attention due to the fact that semiconductor nanostructures such as quantum dots and double quantum dots with well *Corresponding author. Tel.: +21672591906; fax: +21672590566. E-mail address: [email protected] (S. Jaziri).
defined atom-like and molecule-like properties have been fabricated and studied [1–13]. Entanglement is an essential ingredient in quantum information processing like information cryptography and quantum computation, and therefore it is a problem of great current interest to find or design systems where entanglement can be manipulated [14]. Most of the theoretical and experimental activity until now has been associated with atomic and quantum-optic systems. Two [15,16],
1386-9477/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2004.04.036
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three [17], and four particle [18] entanglement have been successfully demonstrated experimentally in trapped ions, Rydberg atoms, and cavity. Semiconductor quantum dots have their own advantages as a candidate of the basic building blocks of solid state based quantum logic devices, due to the existence of an industrial base for semiconductor processing and the ease to integration with existing device [19,20]. The primary motivation for creating entangled states was to test Bell’s inequality [21] which was derived by using the hidden variables theory. The original Bell inequalities and reformulated versions are extremely useful tools for exploring the nonlocal and nonclassical [22] character of entangled systems. The state of an entangled system is represented as a nonfactorizable superposition of product states; whereas, that of unentangled systems is represented as a product state. Various schemes based on electron spins and electron–hole pairs have been proposed to implement quantum computer hardware architectures [6–9,20]. Although there have been some numerical studies on interacting electron systems driven by an AC field [11,23], there is still little theoretical understanding of the observed effects beyond the phenomenology level. Imamoglu et al. [20] have considered a quantum computer model based on both electron spins and cavity which is capable of realizing controlled interactions between two distant quantum dots. Zhang and Zhao [10,11], have presented a scheme based on dynamic localization and quantum entanglement of two interacting electron in coupled quantum dots. Quiroga and Johnson [12] have suggested that the resonant transfer interaction between spatially separated excitons in quantum dots can be exploited to produce maximally entangled Bell states. Here, we consider a generic example of quantum control consisting of two interacting electrons in coupled quantum dots driven by a time-dependent electric field: The quantum bits (qubit) are the states of individual carriers which can be either on the lower dot or the upper dot. The different dot positions play the same role as a ‘‘spin’’. Quantum mechanical tunnelling between the dots leads to a superposition of the quantum dot states. The spatial degree of freedom of the two particles form entangled states. The entangle-
235
ment can be controlled by applying an electric field along the heterostructure growth direction. Given the central status of entanglement, the task of quantifying the degree to which a state is entangled is important, and several measures have been proposed to quantify it. It is worth remarking that even for the smallest Hilbert space capable of exhibiting entanglement, i.e., the two qubit systems, there are aspects of entanglement that remain to be explored. However, many measures of entanglement proposed in the past have relied on either the Schmidt decomposition [24] or decomposition in a ‘‘magic basis’’ [25,26]. In this work, we use the concurrence to measure the degree of entanglement for pure bipartite states of two qubits. Coherent control of these quantum systems can be achieved by the application of external electric field [27]. An account of decoherence in the quantum evolution of a small system interacting with an environment (phonons) [12,13], became a problem of crucial importance in quantum computing. Since, decoherence caused by acoustic phonon–electron interaction is also studied in this paper. As a result we can relate the quantum behavior with the electric field and therefore establish how the decoherence rate of the two particles states can be determined. The main purpose of the present paper is to focus on the influence of the phonons environment on the entangled states. The paper is organized as follows: in Section 2, we study the coherent control of the quantum system of two interacting electrons in coupled quantum dots. We describe the general Hamiltonian and solution scheme for the electrons confined in a vertically aligned double GaAs quantum dot. We derive a solution of the master equation for the density operator in the presence of an oscillatory electric field. With a specific choice of the initial state, we introduce the entangled Bell states for the measurements of the degree of entanglement. In Section 3, we consider the rates for carriers scattering by acoustical phonons decoherence. Transition probabilities from initial state to all possible final states are evaluated. We then come to elucidate the effect of decoherence on the master equation for the density operator and the degree of entanglement. Finally, we conclude in Section 4.
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z ¼ a and þa; given by [29]
2. Model and method 2.1. The Hamiltonian Following the work of Burkard et al. [9], we model the two electrons in two coupled quantum dots by the following Hamiltonian [9–11]: X H¼ h07a ð~ ri ; ~ p i Þ þ HCoul þ Hc ðtÞ; ð1aÞ i¼1;2
h07a ð~ r; ~ pÞ ¼
~ p2 þ Vjj ð~ r Þ þ V> ð~ r Þ; 2m
ð1bÞ
the single particle Hamiltonian h07a describes a single electron confined in the upper (lower) dot of the double dot system. For the lateral confinement we choose the parabolic potential, mw2z a2 2 ðx þ y2 Þ; ð2Þ 2 where we have introduced the anisotropy parameter a determines the strength of the vertical relative to the lateral confinement, _wz is the quantization energy, and m is the electron effective mass. In describing the confinement V> along the inter-dot axis, we have used a (locally harmonic) double well potential of the form Vjj ðx; yÞ ¼
mw2z 2 ðz a2 Þ2 ð3Þ 8a2 which, in the limit of large inter-dot distance abRz ; separates (for zE7a) into two harmonic pffiffiffiffiffiffiffiffiffiffiffiffiffiffi wells with characteristic dot radius Rz ¼ _=mwz : Here a is half the distance between the centers of the dots. The Coulomb interaction is included by HCoul ¼ e2 =kj~ r1 ~ r 2 j; with a relative dielectric constant k: Finally, the interaction Hc ðtÞ ¼ P ez F ðtÞ represents the coupling by a timei i¼1;2 dependent electric field F ðtÞ applied along the z direction. In the calculations we use the GaAs parameters m ¼ 0:067m0 with m0 the free electron mass, and k ¼ 13:1: We use the Hund–Mulliken method [28] of molecular orbitals to describe the low lying energy levels of our system. This approach accounts for double occupation and is therefore suitable for investigating the questions at issue here. The oneparticle Hamiltonian has the ground state solution of the two isolated dots, centred, respectively, on V> ðzÞ ¼
j7a ðx; y; zÞ a 1=2 1 1=4 ¼ pR2z pR2z 1 exp 2 faðx2 þ y2 Þ þ ðz8aÞ2 g 2Rz
ð4Þ
and correspond to the ground state energy e7 ¼ _wz ð1 þ 2aÞ=2: The two ground states are not orthogonal and their overlap is Z 2 ð~ S ¼ d3 rjþa r Þja ð~ r Þ ¼ eða=Rz Þ ; ð5Þ a nonvanishing overlap S implies that the electrons can tunnel between the dots. From these nonorthogonal states, we construct the orthonormalized one-particle wave function 1 f7 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjþa 7ja Þ: 2ð17SÞ
ð6Þ
In this work we shall be concerned with nanometric quantum dots, for which the one-electron ground state of each dot taken as isolated is well separated from the excited ones (large lateral and vertical confinement energies). In this case, the two low lying single-electron states of the double dot are roughly the symmetric fþ and antisymmetric f linear combinations of the two isolated dots ground states. Thus, the eigenstates of H can be constructed from the four (two per electron) oneparticle molecular orbital states. We end-up with four two-particle states: three-singlet states jFþ ð~ r 1 ÞFþ ð~ r 2 ÞS; jFp ð~ r 1ffiffiffiÞF ð~ r 2 ÞS;fjFþ ð~ r 1 ÞF ð~ r 2 ÞS þjF ð~ r 1 ÞFþ ð~ r 2 ÞSg= 2 and one p triplet state ffiffiffi fjFþ ð~ r 1 ÞF ð~ r 2 ÞS jF ð~ r 1 ÞFþ ð~ r 2 ÞSg= 2: Let us consider initially the time-independent solutions for two interacting electrons in a tunnelcoupled double dot structure ðF ðtÞ ¼ 0Þ: The matrix elements of the two-electrons Hamiltonian in the orthonormal basis is given by: H 0 B B ¼B @
2eþ þ Jþ
X
0
0
X
2e þ J
0
0
0 0
0 0
eþ þ e þ Vþ 0
0 eþ þ e þ V
1 C C C; A
ð7Þ
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where
50
e7 ¼ /F7 ð~ r 1 ÞF7 ð~ r 2 Þjh0i jF7 ð~ r 1 ÞF7 ð~ r 2 ÞS;
48
237
E E E E
46
J7 ¼ /F7 ð~ r 1 ÞF7 ð~ r 2 ÞjHCoul jF7 ð~ r 1 ÞF7 ð~ r 2 ÞS;
44
E(meV)
V7 ¼ /Fþ ð~ r 1 ÞF ð~ r 2 Þ7F ð~ r 1 ÞFþ ð~ r2Þ jHCoul jFþ ð~ r 1 ÞF ð~ r 2 Þ7F ð~ r 1 ÞFþ ð~ r 2 ÞS;
42 40 38
X ¼ /Fþ ð~ r 1 ÞFþ ð~ r 2 ÞjHCoul jF ð~ r 1 ÞF ð~ r 2 ÞS:
36
In Eq. (7), the J7 and V7 are direct Coulombic couplings whereas X is an exchange contribution involving the singlet states jFþ Fþ S and jF F S: The eigenenergies of H can be easily solved as: Jþ þ J ES7 ¼ ðeþ þ e Þ þ 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7
ðeþ e þ ðJþ þ J Þ=2Þ2 þ X 2 ;
ET ¼ eþ þ e þ V
34 32 25
50
75
100
125
150
175
200
a(Å)
Fig. 1. Variation of the energies of the interacting two-electrons states as a function of the inter-dot distance. Two identical dots ( with a_oz ¼ 8 meV and Rz ¼ 85 A:
ES0 ¼ eþ þ e þ Vþ
and
with the corresponding eigenstates are given as jcS S ¼ d1 jFþ ð~ r 1 ÞFþ ð~ r 2 ÞS d2 jF ð~ r 1 ÞF ð~ r 2 ÞS;
ð8aÞ
1 jcT S ¼ pffiffiffifjFþ ð~ r 1 ÞF ð~ r 2 ÞS 2 r 1 ÞFþ ð~ r 2 ÞSg; jF ð~
ð8bÞ
1 jcS0 S ¼ pffiffiffifjFþ ð~ r 1 ÞF ð~ r 2 ÞS 2 þ jF ð~ r 1 ÞFþ ð~ r 2 ÞSg;
ð8cÞ
jcSþ S ¼ d2 jFþ ð~ r 1 ÞFþ ð~ r 2 ÞS þ d1 jF ð~ r ÞF ð~ r 2 ÞS; ð8dÞ q1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 and where d1 ¼ X = X 2 þ ðESþ 2eþ Jþ Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d2 ¼ ESþ 2eþ Jþ = X 2 þ ðESþ 2eþ Jþ Þ2 : Fig. 1 shows the behavior of the electronic energy levels as a function of the inter dot distance, for a system comprising two equal dots with vertical confinement energy _wz ¼ 16 meV and horizontal confinement energy a _wz ¼ 8 meV with Rz ¼ ( It appears that, when the distance 2a 85 A: between the two quantum dots is large ðabRz Þ; the energy spectrum presents two series of two nearly degenerate states. This ‘‘tight-binding’’ limit can be easily recovered by using the approximation jF7 ð~ r i ÞSEp1 ffiffi ðj0S7j1SÞ; where 2
j0S ¼ jja S; j1S ¼ jjþa S and SE0: Note that d1;2 would be 0 or 1 in absence of the exchange term X in Eq. (8), while jd1 jEjd2 jE1=O2 in the tight-binding limit when X a0: We find in this limit: ES EET E2E0 and ES0 EESþ E2E0 þ l0 with l0 ¼ /00jHCoul j11S=2: The eigenstates are correspondingly given by a superposition of Bell states pffiffiffi jcS SEjCþ S ¼ ðj01S þ j10SÞ= 2; ð9aÞ pffiffiffi jcT SEjC S ¼ ðj01S j10SÞ= 2; ð9bÞ pffiffiffi jcS0 SEjw S ¼ ðj00S j11SÞ= 2; ð9cÞ pffiffiffi jcSþ SEjwþ S ¼ ðj00S j11SÞ= 2; ð9dÞ the first (second) label holds for the first (second) electron coordinates. Therefore, we can see from Eq. (9) that due to the strong Coulomb repulsion, the low energy states jcS S and jcT S contain mainly single dot occupancies j01S and j10S; while the excited states jcS0 S and jcSþ S contain mainly double dot occupancies j00S and j11S: The four states in Eq. (9) reflect this single or double occupancy features. Each electron is, of course, equally present in the two identical dots, but the low and high energy levels differ by a single or double occupation of the dots. Note finally that the tight-binding two-electron states can be written
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in terms of the four states in Eq. (9)
80
jcS SE fðd1 d2 Þjwþ S
70
pffiffiffi þ ðd1 þ d2 Þjcþ Sg= 2;
60
ð10aÞ
E E E
jcT S ¼ jc S;
ð10bÞ
jcS0 S ¼ jw S;
ð10cÞ
jcSþ SE fðd1 þ d2 Þjwþ S ðd1 d2 Þjcþ Sg=
E(meV)
50 40 30 20
pffiffiffi 2:
10
ð10dÞ
We shall be concerned with states in Eq. (10) below, when discussing the entanglement characteristics of the two-electron states driven by the electric field.
0 0
2
4
6
8
10
12
F(kV/cm)
Fig. 2. Energy spectrum of the two electrons states as a function of the strength of a constant electric filed applied along ( the growth axis, for the double dot in Fig. 1 ða ¼ 170 AÞ:
2.2. Equations of motion We introduce in the following the electric field. Obviously, Hc ðtÞ does not mix singlet and triplet states, and thus jcT S is insensitive to the applied field within our truncated basis. The matrix representation H1 ðtÞ of the two-electrons Hamiltonian in the singlet basis fjcS S; jcS0 S; jcSþ Sg; reads: 0 1 0 0 ES B C ES0 0 A H1 ðtÞ ¼ @ 0 0
0 0
ESþ
0 B þ F ðtÞq*z@ d1 d2 0
d1 d2 0 d1 þ d2
1 0 C d1 þ d2 A ; 0
ð11Þ pffiffiffi where z* ¼ 2=ð1 S 2 Þa: We present in Fig. 2, the energies of the double dot eigenstates in the presence of a constant electric field F ðtÞ ¼ F0 for ( (note that the energy a dot separation 2a ¼ 340 A diagram is symmetrical with respect to the sign of the field since we consider two identical quantum dots). For this dot separation the tight-binding limit roughly pffiffiffi applies (see Fig. 1). In this case, d1 Ed2 E1= 2 and thus, to the lowest order, the field couple only the excited states jcSþ S and jcS0 S; which are nearly degenerated at zero field. This explains the linear splitting of these levels with F0 in Fig. 2. The small energy anti-crossing
with jcS S around FC E1:9 kV=cm results from the small d1 d2 value in Eq. (10). For F bFC ; the ground state is Ep1 ffiffi ðjcSþ S þ jcS0 SÞEj00S 2
and the high energy state is Ep1 ffiffi ðjcSþ S 2
jcS0 SÞEj11S; which both comprise two electrons in the same dot (the decrease in electrostatic energy for the two electrons compensates for their mutual repulsion) and are roughly disentangled states (E product of two one-electron states). In conclusion, it is important to keep in mind the two following features related to the electric field couplings in the tight-binding limit: (i) the field couples only weakly states with different dot occupancies and (ii) it breaks the left-right symmetry and is able to induce a simultaneous localization of both electrons on the same quantum dot and correspondingly to disentangle the two-electrons states. We shall come back to these results further below, in relation with a timedependent field. We assume in the following an oscillatory electric field F ðtÞ ¼ F0 cosðotÞ: The evolution of any initial state jcð0ÞS can be expressed as jCðtÞS ¼ C1 ðtÞjcS S þ C2 ðtÞjcS0 S þC3 ðtÞjcSþ SE C1 ðtÞjcþ S þC2 ðtÞjw S þ C3 ðtÞjwþ S corresponding to the density operator rðtÞ ¼ jcðtÞS/cðtÞj with rð0Þ ¼ jcð0ÞS/cð0Þj: The master equation of the density operator is i_dr=dt ¼ ½H; r ; with H is the
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two-electron Hamiltonian (in the basis of the 3 singlet states, r is a 3 3 matrix and H1 is given in Eq. (11)). The diagonal terms of the density matrix are the probabilities of finding the two particles in the basis states, while its off-diagonal matrix elements (the ‘coherencies’) describe the linear superposition of these states induced by the applied electric field. Let us consider the dynamics of the system starting with the unperturbed state jcð0ÞS ¼ jcS S: A nontrivial time evolution is possible because of the term proportional to d1 d2 ; which couples jcS S to the excited singlet states. We plot in Fig. 3 the time evolution of Bell the Bell probabilities rBell 11 ¼ /cþ jrjcþ S; r22 ¼ Bell /w jrjw S and r33 ¼ /wþ jrjwþ S: The oscillatory field has amplitude F0 ¼ 1:5 kV=cm and frequency o ¼ ðES0 ES Þ=_ðZoE5:44 meV at zero field ( We can see from Fig. 3 that and for a ¼ 170 A). the two electrons oscillate between the ground and the two excited singlet states. The oscillation pattern is complex but shows two main features: (i) a slow periodic exchange of populations between the ground (initial) and the two excited states, and (ii) fast oscillations between the populations of the two excited levels. The first feature indicates a slow field-induced variation of the mean dot occupancy, while the second one indicates an important field-induced coupling of the two excited singlet states, in accordance with the previous results for a static field (see Fig. 2).
239
In order to quantify the degree of entanglement, we will adopt the concurrence C defined by [25,26]. The concurrence varies from C ¼ 0 for an unentangled state to C ¼ 1 for a maximally entangled state. The concurrence may be calculated explicitly from the density matrix r : CðrÞ ¼ maxf0; l1 l2 l3 l4 g where the quantities li are the square roots of the eigenvalues B in decreasing order of the matrix r ¼ rðsA y #sy Þ A B r ðsy #sy Þ; where r denotes the complex conjugation of r in the standard basis. The most general pure states in the case of the two-qubit model can be written as jcS a1 j00S þ P¼ 4 2 a2 j01S þ a3 j10S þ a4 j11S; where i¼1 jai j ¼ 1: The concurrence of the pure state is simply given by [25] C ¼ 2ja2 a3 a1 a4 j:
ð12Þ
Thus, the state is entangled if and only if a1 a4 aa2 a3 : Fig. 4 shows that the degree of entanglement varies in time between zero for nonentangled state and 1 for maximally entangled states. In order to understand this result, it is important to recall two points: (i) the zero-field eigenstates are maximally entangled, since it is impossible to decompose them into simple products of one-electron orbital, and (ii) an applied field is able to mix the zero-field entangled states and generate non-entangled solutions, as discussed in connection with the results shown in Fig. 2. Correspondingly, Fig. 4 shows the
1.0 1.0
0.8 0.8
0.6
ρBell
0.6
C
0.4 ρ
0.4
ρ ρ
0.2
0.2
0.0 0
1
2
3
4
5
6
7
8
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t (ps) Bell Fig. 3. Time evolution of the Bell probabilities rBell 11 ; r22 and rBell 33 under the influence of a sinusoidal electric field for the ( double dot in Fig. 1 ða ¼ 170 AÞ:
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dynamical aspects of the role played by the fieldinduced couplings on the nature of the twoelectron eigenstates. Note that the entanglement approaches its maximal value anytime the population of one of the zero-field singlet states approaches unity (compare with Fig. 3). When the system is principally in the excited states, the entanglement oscillates quickly, in agreement with the fast oscillations in Fig. 3 between the populations of the two excited singlet states. Fig. 4 shows that the degree of entanglement of the system can be dynamically monitored by the application of an oscillating electric field. In addition, the control presents a 100% contrast, since completely entangled and completely disentangled configurations can be dynamically generated (at different times, of course) in this way.
3. Phonon decoherence 3.1. Acoustic phonon scattering In the last section, we have discussed how the application of an oscillatory electric field allows controlling the degree of entanglement of the twoelectron states. In order to study the influence of environment on the dynamical evolution of the degree of entanglement, we consider in the following the coupling of the electrons with acoustic phonons [30]. Let us focus initially on the zero-field case and use the electron–phonon interaction described by the deformation potential mechanism. Within this model, the coupling of two-electrons states precede via the single-electrons interactions and transitions between singlet and triplet states are forbidden. The electron– phonon is given by [31] He2ph ¼ PHamiltonian P i~ q~ ri þ fað~ q Þe b þ ccg; where bþ q q is the usual i¼1;2 q creation operator for an acoustical phonon in ~ þ~ ~ the in-plane component mode ~ q¼Q q z with Q of ~ q and jað~ q Þj2 ¼ D2c _q=2rcs V : We have used in the calculations an isotropic linear phonon dispersion oq ¼ cs q and the GaAs material parameters: conduction band deformation potential Dc ¼ 8:6 eV; density r ¼ 5300 kg=m3 and longitudinal velocity of sound cs ¼ 3700 m=s: Transition probabilities from an initial state jci S with energy Ei to
all possible final states jcf S with energy Ef are evaluated within the Born approximation X _ ¼ G¼ Nq7 ðEq Þj/cf jHe2ph jci Sj2 2pt q dðEf Ei 7Eq Þ;
ð13Þ
where Eq ¼ _oq is the phonon energy. The sum over all the q vectors can be expressed as an integral over qx ; qy ; and qz : The scattering rate involving two-electron levels reads Gi ¼
1 D2c 7 2 2 N ðEQm ÞjQ2m jeR Qm =2a 2p_ 2rc2s Qm Z Qm 2 2 eR qz =2a jgif ðqz Þj2 dqz ;
ð14Þ
Qm
where Qm ¼ jEf Ei j=_cs is the wave vector of the exchanged phonon; Nq7 ðEq Þ ¼ fexpðEq =kB TÞ 1g1 þ 1=271=2 is the occupation factor, T is the lattice temperature and the upper (lower) sign corresponds to emission (absorption) processes. Finally, the form-factors gif ðqz Þ involve integrals of the kind Is;s0 ðqz Þ ¼ /Fs ðz1 ; z2 Þjexpðiqz z1 Þ þ expðiqz z2 ÞjFs0 ðz1 ; z2 ÞS;
ð15Þ
where Fs ðz1 ; z2 Þ is the z-dependent part of the twoparticle total wavefunction. In the tight-binding limit, as discussed above, these later are linear combinations involving either single dot occupancies j01S and j10S or double dot occupancies j00S and j11S (see e.g. Eq. (9)). We can thus show that only transitions involving the excited states jcSþ S and jcS0 S will not be negligible in this limit. Fig. 5 shows the calculated rates for phonon assisted absorption ðG23 Þ and emission ðG32 Þ processes between the two excited states jcS0 S and jcSþ S as a function of both the inter-dot distance (Fig. 5a) and the vertical confinement (Fig. 5b) at T ¼ 77 K: The scattering rates involving the ground singlet state are found to be orders of magnitude smaller than those of Fig. 5. The resonant-like profiles in Fig. 5 follow from the particular dependence of the deformation potential coupling upon the exchanged phonon energy (the energy difference between the initial and final states): it is very weak for both very small and very large Eq values. Thus, the scattering rate displays a
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Fig. 5. Scattering rate due to acoustical phonon for a GaAs quantum dots as a function of the inter dot distance for _wz ¼ ( (b). 16 meV (a), and of the vertical confinement for a ¼ 170 A ( in both cases. Rz ¼ 85 A
Let us neglect initially the applied electric field ðF0 ¼ 0Þ: When the initial state is the ground singlet one, it follows that r11 E1 at all times, since acoustical phonons are unable to couple this state to the excited ones. Fig. 6 shows the populations evolutions in the case where the initial state is jcS0 S: We see that an effective population exchange is possible between the two excited singlet, since G23 and G32 are not vanishing. At long times, the populations reach a stationary solution, obtained by imposing qri;j =qt ¼ 0 in the presence of the damping terms. Let us consider now to which extent the phonon-induced decoherence effects are detrimental to the dynamical control (i.e., by an external field F ðtÞ ¼ F0 cosðotÞÞ of the populations and of the degree of entanglement. Fig. 7 shows the time evolution of the Bell probabilities for the same system, initial conditions and field parameters as in Fig. 3. The initial evolution is a transient regime in a time interval given basically by the characteristic phonon scattering rates. At long times, the probabilities show a driven behaviour, as demonstrated by the fact that they present oscillations with half (because the energies in Fig. 2 do not depend on the sign of the perturbing field) the period T ¼ 2p=o related to the driven field. In the same way, the coherencies ri;jai ðtÞ display an initial transient behavior followed by a driven long-time solution The long-time solution of the
maximum when varying Eq ¼ ESþ ES0 by either changing the inter-dot distance or the intra-dot vertical frequency.
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3.2. Equation of motion of the reduced matrix
ð17Þ
ρ Bell
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In order to study the influence of the phonon assisted interactions on the time evolution of the two-electron system, we add in the master equation for qr=qt the damping terms: X qrii ¼ ðGj-i rjj Gi-j rii Þ; ð16Þ qt dec jai qrii 1X ¼ ðGi-k þ Gj-k Þrij : qt dec 2 kai
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double dot parameters, the phonon-induced decoherence rates can be made very small, allowing a roughly coherent time-evolution of the driven twoelectron system.
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References
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t (ps) Bell Fig. 7. Time evolution of the Bell probabilities rBell 11 ; r22 and Bell r33 under the influence of the phonon decoherence and of a sinusoidal field. Same parameters as in Fig. 6.
density matrix evolution is, of course, independent upon the initial conditions. In particular, the amplitudes of the driven solutions depend only on the double dot and field parameters and on the scattering rates. However, according to Fig. 5, the inter-level scattering rates can be made very small. Thus, by properly choosing the parameters of the double dot system, the transient regime transforms into a quasi-stationary one. This characterizes a rather robust (with respect to the environment influences) system, allowing to explore the quantum nature of its time evolution and, more importantly, to monitor this later evolution by means of an external tool, like the application of an oscillatory electric field.
4. Conclusion We have shown how the entanglement of two interacting electrons in a double quantum dot system can be dynamically manipulated by an external electric field. Decoherence processes represent the most problematic issue pertaining to most quantum computing processing. In the present work we have considered the role of acoustical phonons on the time evolution of the density matrix (the populations and the coherences) describing two electrons confined in a double dot structure. By properly choosing the
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