Download Entangled Bell states of two electrons in coupled quantum dots

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Physica E 24 (2004) 234–243
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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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236
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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238
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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A. Hichri et al. / Physica E 24 (2004) 234–243
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
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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A. Hichri et al. / Physica E 24 (2004) 234–243
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
ARTICLE IN PRESS
A. Hichri et al. / Physica E 24 (2004) 234–243
242
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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