phys. stat. sol. (c) 1, No. 3, 598–602 (2004) / DOI 10.1002/pssc.200304048

© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Decoherence and quantum entangled exciton states in semiconductor quantum dot molecule

A. Hichri1, S. Jaziri*, 2, and R. Ferreira3

1 Laboratoire de Physique de la Matière Condensée, Faculté des Sciences de Tunis, Tunisie 2 Département de Physique, Faculté des Sciences de Bizerte, 7021 Zarzouna, Bizerte, Tunisie 3 Laboratoire de Physique de la Matière condensée, ENS, 75005, Paris, France

Received 15 September 2003, accepted 18 September 2003 Published online 2 February 2004

PACS 03.67.–q, 63.22.+m, 73.21.La

Recently a pair of vertically aligned quantum dots has been suggested as optically driven quantum gate. The later is built when two different particles, an electron and a hole, are created optically. The two parti-cles form entangled states. We study theoretically the dynamics of a single heavy hole exciton confined in a small vertically coupled quantum dot. We use harmonic potentials to model the confining of electrons and holes and calculate the exciton’s energy spectrum using the Hund-Mulliken approach, and including the Coulomb interaction. The entanglement may be controlled by the application of external electric field. At certain times of dynamical evolution one can generate a maximally entangled coherent exciton state by initially preparing the quantum dots in a superposition of coherent state. In ideal situations, entangled quantum states would not decohere during processing and transmission of quantum information. How-ever, a real quantum system is inevitably influenced by its environment. Since, we study the decoherence effect caused by the interaction of the confined carriers with the acoustic phonon effects.

© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction One of the key distinguishing features of quantum mechanics not found in classical physics is the possibility of entanglement between subsystems. The significance of this phenomenon is now unquestioned, as it lies at the core of several of the most important achievements of quantum infor-mation science [1], such as quantum communication and quantum computation [2]. The experimental realization of quantum entangled state may be traced back to the mid 1960’s, when entangled photon pairs created from cascade emission [3]. The primary motivation for creating entangled states was to test Bell’s inequality [4], which was derived by using the hidden variables theory. Creating entangled states is the first step toward studying any effects related to entanglement. Semi- conductor quantum dots (QDs) 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 case to integration with existing device [5]. Experimental realization of optically induced entanglement of excitons in a single QD [6] and theoretical study on coupled QDs [7] was re-ported most recently. It has suggested that a pair of vertically aligned QDs could be used as the optically driven quantum gate: The quantum bits (qubits) 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 QD states. The quantum gate is constructed when an electron and a hole are injected into the coupled QDs, which can be considered to some extent as artificial molecules. The two particles form entangled states. The entanglement can be controlled by applying an electric field along the heterostructure growth direction. In this contribution

* Corresponding author: e-mail: Sihem.Jaziri@fsb.rnu.tn, Phone : +216 72 591 906, Fax: +216 72 590 566

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results will be presented which support the feasibility of this proposal: We have studied electron-hole pair confined in a QD molecule. Quantum coherence needs to be maintained for a quantum computer to maximize its capability; it has to satisfy stringent requirements [8]. One requirement is that the qubit should have long coherence time (i.e. slow decoherence). Indeed, decoherence can be loosely defined as irreversibly losing information from the operational Hilbert space into the environment, which is the rest of the universe from the per-spective of a quantum computer system. Furthermore, the requirement of slow decoherence needs to be satisfied not only when the qubits are standing alone, but also when the qubits are interacting with their environment includes the underlying crystal lattice in terms of phonons. However, the decoherence caused by acoustic phonon-exciton interaction is the main purpose here. 2 Model To begin, we consider the Hamiltonian (1) describing electron-hole pair confined in vertically coupled QDs.

0

,

( )coul ce h

H h H H tη

η=

= + +∑ (1)

2 2, ,0

, ,, ,

( ) ( )2 2

zz

z

p ph V V z

m mρ η η

η ρ η η η η

ρ η η

ρ= + + + is the single particle Hamiltonian for the electron ( eη = ) or

hole ( hη = ) in three dimensions using ( , , zρ θ ) cylindrical coordinates. , 00.067em mρ

= and

, 00.03hm mρ

= are the electron and hole masses for the in-plane motions ( 0m is the free electron mass),

, 00.067z em m= and , 00.7z hm m= are the electron and hole masses for the z motions. The potential 2 2

, , ,( ) / 2zV m wρ η η ρ η η

ρ αρ= describes the lateral confinement, whereas 2 2 2 2 2, , , ( ) / 8z z zV m w z a aη η η= −

models the vertical double well structure, which in the limit of large inter dot distance ,za Rη

>> sepa-

rates into two harmonic wells of frequency ,zwη

. Here a is half the distance between the centers of the

dots, the parameter α determines the strength of the vertical relative to the lateral confinement and

, , ,/z z zR m wη η η= � is the characteristic vertical dot radius. The Coulomb interaction is included by

2 /Coul e hH e r rκ= − −

� �

, with a relative dielectric constant κ (for bulk GaAs, 13.18κ = ). The interac-

tion,

( ) ( )c e hH t q z F t

η ηη==∑ represents the coupling by a time-dependent electric field F(t) applied along

the z direction. In the calculations we use the GaAs QD with , , 35z e z hR R= = Å. We put F=0 here and

discuss the case 0F ≠ below. In this work we shall be concerned with small QDs, for which the one carrier ground state of each dot taken as isolated is well separated from the excited ones (strong lateral and vertical confinement ener-gies). In this case, the two low lying single-carrier states of the double dot are roughly the symmetric

and antisymmetric linear combinations of the two isolated dots ground states ( )0 1 / 2(1 )Sφ±= ± ± ,

where 0 and 1 are the ground solutions of the one-particle (electron or hole) in two isolated dots

centered respectively on z = -a and +a. A non-vanishing overlap 0 1s = implies that the carrier can

tunnel between the dots. If a is large enough, we may expect the coupling between the dots due to the tunnel effect. The eigenstates of H can be constructed from the four (two per carrier) one-particle mo-lecular orbital states. The two first state are dominated with double occupation ( 0.96 0.28χ ψ

+ ++ and

χ−

) and the two excited state are dominated with a single occupation ( 0.28 0.96χ ψ+ +

− + and

ψ−

). We can see that these states are a superposition of the Bell’s states ( )00 11 / 2χ±

= ± and

( )01 10 / 2ψ±

= ± imply that the spatial wave functions of the exciton have been entangled, in the

600 A. Hichri et al.: Decoherence and quantum entangled exciton states in semiconductor quantum dot molecule

© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

usual sense that they are not factorized into single particle states. Therefore, we can deduce that due to the strong Coulomb interaction, the low energy states contain mainly double dot occupancies 00

and 11 , while the excited states contain mainly single dot occupancies 01 and 10 . Each carrier 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. 3 Equation of motion We introduce in the following the electric field. Hc(t) does not mix the first excited state with the other states, and thus χ

−

is insensitive to the applied field within our truncated

basis. In the following we will denote the ground state 1 0.96 0.28ψ χ ψ+ +

= + and the two exited

states as 2ψ ψ−

= and 3 0.28 0.96ψ χ ψ+ +

= − + .

Under the adiabatic approximation [10], the evolution of any initial state (0)ψ can be expressed as

1 1 2 2 3 3( ) ( ) ( ) ( )t C t C t C tΨ ψ ψ ψ= + + ( 1 2 3( ) ( ) ( )C t C t C tχ ψ ψ+ − +

≈ + + ) corresponding to

the density operator ( ) ( ) ( )t t tρ ψ ψ= with (0) (0) (0)ρ ψ ψ= . The master equation of the density

operator is [ ]1,d

i Hdt

ρρ=� , with H1 is the exciton Hamiltonian (in the basis of the 3 exciton states, ρ is

a 3x3 matrix). The diagonal terms of the density matrix are the probabilities of finding the electron-hole pair 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 (0)ψ = 1ψ . We plot in Fig. 1 the time evolution of the Bell

probabilities 11Bell

ρ χ ρ χ+ +

= , 22Bell

ρ ψ ρ ψ− −

= and 33Bell

ρ ψ ρ ψ+ +

= for a system comprising two

equal dots separated by 2a=145Å with vertical confinement energy , 16 meVz hw =� , and 0.5α = . The

oscillatory field has amplitude F0 = 1.5 kV/cm and frequency 2 1( ) /w E E= − � ( w =� 14.5 meV at zero

field). We can see from Fig.1 that the electron-hole oscillates between the ground and the first excited states. The second excited state is inhibited to be occupied due to the strong Coulomb interaction. Note that the entanglement approaches its maximal value anytime the population of one of the states ap-proaches unity.

0 2 4 6 8 10

0,0

0,2

0,4

0,6

0,8

1,0

ρ11

ρ22

ρ33

ρ

t (ps)

Fig. 1 Time evolution of the Bell probabilities under the influence of a sinusoidal electric field.

phys. stat. sol. (c) 1, No. 3 (2004) / www.pss-c.com 601

© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

60 65 70 75 80 85

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

0,18

0,20

1/τ23

1/τ32

1/τ

i j (

ps

-1)

a (Å)

0 2 4 6 8 10 12 14 16 18

0,0

0,2

0,4

0,6

0,8

1,0

ρ11

ρ22

ρ33

ρ

t (ps)

4 Phonon decoherence 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 exciton with acoustic pho-nons. Let us focus initially on the zero-field case and use the exciton-phonon interaction described by the deformation potential mechanism. The exciton-phonon Hamiltonian is given by [11]

{ },

( ) iiq rex ph q

e h q

H q e b ccη

η

η

β +

−

=

= +∑ ∑ , where qb+ is the usual creation operator for an acoustical phonon in

mode zq Q q= +

�� �

with Q�

the in-plane component of q�

and 2 2

( ) ( )( ) / 2e h c v sq D q c vβ ρ= � . cD and vD are

the deformation potential for conduction and valence bands respectively We have used in the calcula-tions an isotropic linear phonon disper-sion q sw c q= and the GaAs material parameters: conduction

(valence) band deformation potential Dc(v) = 7.2 eV (3.6 eV), density ρ = 5300 kg/m3 and longitudinal velocity of sound cs = 3700 m/s. Transition probabilities from an initial state iψ with energy Ei to all

possible final states fψ with energy Ef are evaluated within the Born approximation

2

( ) ( )2

i f q q f ex ph i f i qqi f

N E H E E EΓ ψ ψ δπ τ

±

− −

−

= = − ±∑�

(2)

where Eq = �ωq is the phonon energy. The sum over all the q vectors can be expressed as an integral over

qρ

, and qz. { }1

( ) exp ( / ) 1 1/ 2 1/ 2q q q BN E E k T−

±

= − + ± is the occupation factor, T is the lattice tempera-

ture and the upper (lower) sign corresponds to emission (absorption) processes. Fig. 2 shows the calculated rates for phonon assisted emission (Γ3-2) and absorption (Γ2-3) processes between the two excited states 2ψ and 3ψ as a function of the inter-dot distance at T = 77 K. The

scattering rates involving the ground exciton state are found to be orders of magnitude smaller than those of Fig. 2. The resonant-like profiles in Fig. 2 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 maximum when varying Eq by changing the inter-dot distance or the intra-dot vertical frequency. In order to study the influence of the phonon assisted interactions on the time evolution of the exciton system, we add in the master equation for ∂ρ /∂t the damping terms:

Fig. 3 Time evolution of the Bell probabilities under the influence of the phonon decoherence and of a sinu-soidal field.

Fig. 2 Scattering rate due to acoustical phonon for a GaAs QDs as a function of the inter dot.

602 A. Hichri et al.: Decoherence and quantum entangled exciton states in semiconductor quantum dot molecule

© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

( )

( )1

2

iij i jj i j ii

j idec

iji k j k ij

k idec

t

t

ρΓ ρ Γ ρ

ρΓ Γ ρ

→ →

≠

→ →

≠

∂= −

∂

∂= + ∂

∑

∑

(3)

Let us consider to which extent the phonon-induced decoherence effects are detrimental to the dynamical control (i.e., by an external field F(t) = F0 cos(ω t)) of the populations and of the coherences. Fig. 3 shows the time evolution of the Bell probabilities for the same system, initial conditions and field pa-rameters as in Fig. 1. The initial evolution is a transient regime in a time interval given basically by the characteristic phonon scattering rates. The phonon energy absorbed by the system allows the population of the last exited state and efficient to destroy the Coulomb interaction. In the same way, the coherencies ρi,j≠i(t) display an initial transient behaviour followed by a driven long-time solution of the density matrix evolution, which is 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, ac-cording to Fig. 2, 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. The entanglement decoherence rate depend on the choice of a parameter. We also emphasize that entangled states will depend on the con-crete form of the interaction between qubits and the environment. We do believe, however, that the fast decay rate of quantum entanglement is a generic feature in a variety of physical processes when decoher-ence is important.

5 Conclusion

We have shown how the entanglement of the exciton in a double QD system can be dynamically ma-nipulated by an external electric field. Decoherence processes represent the most problematic issue per-taining 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 the exciton confined in a double dot structure. By properly choosing the double dot parame-ters, the phonon-induced decoherence rates can be made very small, allowing a roughly coherent time-evolution of the driven exciton system.

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