phys. stat. sol. (c) 3, No. 11, 3726–3729 (2006) / DOI 10.1002/pssc.200671574

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Pure dephasing of two charge qubits in coupled quantum dots

W. Ben Chouikha1, S. Jaziri*, 2, and R. Bennaceur1

1 Laboratoire de Physique de la Matière Condensée, Faculté des Sciences de Tunis, Tunisie 2 Departement de Physique, Faculté des Sciences de Bizerte, Jarzouna 7021 Bizerte, Tunisie

Received 1 May 2006, accepted 20 June 2006

Published online 24 November 2006

PACS 03.65.Yz, 63.20.Kr, 73.21.La, 73.22.–f

We study the pure dephasing of two electrons confined in vertically coupled quantum dots, in which the

presence of one electron confined in the double dot represents one qubit. We numerically evaluated the de-

phasing rates due to electrons coupling to both acoustic and optical phonons. Our numerically results show

that the pure dephasing rates dependent on the separation between dots and the strength of electrons con-

finement.

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Environment induced decoherence plays a fundamental role in many areas, ranging from quantum meas-

urement [1] to quantum information and quantum computing [2–4]. It also plays an important role in

solid systems, and might suppress quantum tunnelling of defects in crystals [5], spin tunnelling in mag-

netic molecules and nanoparticles [6], or destroy Kondo effect in a dissipationless manner [7]. Environ-

mental decoherence can be defined as the loss of quantum coherence of a quantum system due to its

coupling to an environment [8].

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. The interaction between the environment and a qubit system of interest can lead to deco-

herence. The entanglement, as a nonlocal property of a composed quantum system, should be very frag-

ile under the influence of the environment. This fragility is a main obstacle for the realization of practical

quantum computers.

In contrast to spin qubits, charge qubits in semiconductors have the substantial advantage of being easy

to manipulate and measure since the experimental techniques for measuring single electron charges in

semiconductors are extremely well-developed (the direct controllability via external voltage sources) [9].

Relaxation is not the only way charge qubits can decohered. The decoherence can due to pure dephasing.

In this paper we consider two electrons confined in double dot. The qubit is encoded in the presence of

a single electron confined in vertically coupled quantum dots. We denote localized single particle

states{ }0 , 1 where 0 corresponds to an electron localized on the left dot, while 1 denotes an elec-

tron localized on the right dot . We consider both of acoustic and optical phonons as sources of decoher-

ence. We evaluate the dephasing factor as function of time and of interdot separation.

The Hamiltonian of the two electrons confined in the vertically coupled GaAs quantum dot [10–12]:

2 * 2 2 * 2 2

2 2 2 2 2

0 * 2

1,2 1 2

( ) ( )2 2 8

i z z

i i i

i r

P m m eH x y z d

m d r r

ω α ω

ε

� �

=

Ê ˆ= + + + - +Á ˜Ë ¯ - .

The two dots are separated by a GaAs barrier of variable thickness (2d), α is the anisotropy parameter

determines the strength of the lateral relative to the vertical confinement, *

m = 0.067 is the electron

* Corresponding author: e-mail: sihem.jaziri@fsb.rnu.tn

phys. stat. sol. (c) 3, No. 11 (2006) 3727

www.pss-c.com © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

effective mass in GaAs and the last term in equation (1) represents the Coulomb potential where r

ε =

13.1 is dielectric constant. We consider z

ω� = 20 meV and α = 0.5.

The four lowest eigen states of 0

H , are given as 0 n n n

H Eφ φ= where

1 1 1 2 2 1 2

( ) ( ) ( ) ( )s

r r r rφ γ φ φ γ φ φ�� �� �� ��

+ + - -= - ,

1 2 2 1 2

1( ( ) ( ) ( ) ( ) )

2T

r r r rφ φ φ γ φ φ�� �� �� ��

+ - - += - ,

2 1 2 2 1 2

1( ( ) ( ) ( ) ( ) )

2s

r r r rφ φ φ γ φ φ�� �� �� ��

+ - - += + ,

3 1 1 2 2 1 2

( ) ( ) ( ) ( )s

r r r rφ γ φ φ γ φ φ�� �� �� ��

+ + - -= + ,

where φ±

decribes the eigen states of one electron confined in double dot.

In a polar semiconductor like GaAs, electrons couple to all types of phonons. More specifically, in

GaAs electrons couple to longitudinal acoustic phonons through a deformation potential, to longitudinal

and transverse acoustic phonons through piezoelectric interaction, and to the optical phonons through the

polar interaction [13].

The electron-phonon interaction is given by the following expression: *

int

1,2

( )i iikr ikr

k k k k

i k

H M e a M e a

� �

� �

- ++

=

= +ÂÂ corresponding to emission or absorption of a phonon.

The bulk matrix element for the deformation potentiel coupling is given by

1/2

2k c

s

M DV cσ

Ê ˆ= Á ˜Ë ¯�

where c

D = 8.6 meV is a deformation potentiel, σ = 5.3 g cm–3 is the crystal density of GaAs,

sc =37×10

5 cm/s is the phonon sound velocity.

The bulk matrix element for the piezoelectric potentiel is given by

1/2

14

02

k

r s

eeM

V c kε ε σ

Ê ˆ= Á ˜Ë ¯�

where e14

= 0.16 C/m2 is the piezoelectric coupling constant [14].

For optical phonons we have 2

2

2

2 1 1

k LO

r

eM

k V

Πω

ε ε•

Ê ˆ= -Á ˜Ë ¯� where

k Loω ω= = 36.25 meV and ε

•

=

10.89 is the high frequency dielectric constant. Electron–phonon interaction not only causes relaxation between qubit states, it can also cause dephas-

ing between them if the electron in a superposition state [15]. If the energy difference difference between

the two charge states fluctuates, phase information will get lost and decoherence occurs, such as pure

dephasing( in the sense that no transistion occurs between the two charge states) due to bosonic bath.

In this work, we study the pure dephasing for two charges qubits due to both acoustic and opyical pho-

nons, in order to show his effect on entangled states of two qubits.

The density operator of two electrons in a boson bath can be written in a general expression as

2 2 2

1 1 2 1 3

2 2 2

1 2 3

2 2 2

2 1 2 2 3

2

3 1

( ) ( ) ( )

1 1 1 1 2 1 3

( ) ( ) ( )

1 2 3

( ) ( ) ( )

2 1 2 2 2 2 3

(

3 1

(0) (0) (0) (0)

(0) (0) (0) (0)

(0) (0) (0) (0)

(0)

s T s s s s

Ts Ts Ts

s s s T s s

s s

B t B t B t

s s s T s s s s

B t B t B t

Ts TT Ts Ts

B t B t B t

s s s T s s s s

B

s s

e e e

e e e

e e e

e

ρ ρ ρ ρ

ρ ρ ρ ρ

ρ ρ ρ ρ

ρ

- - -

- - -

- - -

-

2 2

3 3 2) ( ) ( )

3 3 2 3 3(0) (0) (0)s T s st B t B t

s T s s s se eρ ρ ρ- -

Ê ˆÁ ˜Á ˜Á ˜Á ˜Á ˜Á ˜Ë ¯

1 2 3

1 2 3

, , ,

, , ,

n s T s s

n s T s s

E E E E E

φ φ φ φ φ

=ÏÌ

=Ó

3728 W. Ben Chouikha et al.: Pure dephasing of two charge qubits in coupled QDs

© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.pss-c.com

where the exponent function nmj

2

22 3 2

2 3 2( ) ( ) sin ( )coth( )

2 2

k k k

nm

k B

V M tB t d k I k

k T

ω ω

Π ω= Ú

�

�and ( )

nmI k is

defined by 1 2 1 21

( ) ( )2

i k r i k r i k r i k r

nm n n m mI k e e e eφ φ φ φ= + - +

� �� � ��� � �� � ���

.

Figure 1 presents the time dependence of dephasing between the two charges qubits states due to de-

formation potential. We have four states so we obtain six dephasing terms. For symmetry raisons we

obtain 2

2TsB = 0 and

2

3

2

32

2

1

2

21 TsssTsssBBBB === . An interesting feature of curves in Fig. 1 is that

they rapidly increase for the first 5 ps. The very fast time dependence of dephasing is due to the trigono-

metric dependence on phonons frequencies. The initial rise of dephasing is mostly determined by the

higher frequency phonons because their density of state is much higher. After 5 ps we show very fast

time dependence until 30 ps but after this last time we observe oscillation of the dephasing terms.

Fig. 1(left) Dephasing rates due to deformation potentiel as function of time t, d = 14.5 nm.

Fig. 2(right) Dephasing rates due to deformation potentiel and piezoelectric interaction, d = 14.5 nm. In Fig. 2 we explore the dephasing rates due to deformation potential and piezoelectric interaction.

We remark the dephasing due to piezoelectric interaction is more intense.

In Figs. 3a and b we further explore the dephasing rates as a function of the interdot distance. If the two

quantum dots are well separated, the overlap integrals go down quickly, so that dephasing should also be

strongly suppressed. This is exacty the behaviour we observe on both Figs. 3a and b. We remark that the

Fig. 3 Dephasing factor as a function of the half interdot distance d. The time t is chosen to 50 ps. a) Left part: 2

1s TB ,

b) right part: 2

1 3s sB

0 20 40 60 80 100

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

B2

(t)

t(ps)

B2

s1T= B

2

s2s3= B

2

Ts2

B2

s1s3

0 20 40 60 80 10

0.0

4.0x10-5

8.0x10-5

1.2x10-4

B2

s1T(t)

t(ps)

Deformation

Piezoelectric

80 100 120 140 160 180

0,00

0,01

0,02

0,03

0,04

B2

s1T(t)

d(A0

)

Deformation

Piezoelectric

LO

80 100 120 140 160 180

0,00

0,02

0,04

0,06

0,08

0,10

0,12

0,14

0,16

Deformation

Piezoelectric

LO

Deformation

Piezoelectric

LO

B2

s1s3

d(A0

)

phys. stat. sol. (c) 3, No. 11 (2006) 3729

www.pss-c.com © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

contribution of optical phonons is smaller than the acoustic phonon effect. This difference is more im-

portant for 2

1 3s sB . We note that optical phonons do not contribute significantly to electron relaxation, due

to their large energies.

In conclusion, we have investigated the decoherence of two electrons confined in double dot due to

electron phonon coupling. We have evaluated electrons dephasing through interaction with both acoustic

and optical phonons, because of the absence of energy conservation requirement, all phonon modes con-

tribute to dephasing.

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