Bose-Einstein Condensation of two-dimensional magnetoexcitons on the superposition state - art. no. 672632

Abstract

The Bose-Einstein Condensation (BEC) of the two-dimensional magnetoexcitons on the superposition state is studied. The superposition of two excitonic states formed by electron and hole on the lowest Landau levels (0, 0) and on the first excited Landau levels (1, 1) was considered. The generalized Bogoliubov u-v transformation was deduced. The criterion on the exciton concentration was established. The influence of the BEC on the absorption band shape is discussed.

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Bose-Einstein condensation of two-
dimensional magnetoexcitons on the
superposition state
Moskalenko, S., Liberman, M., Podlesny, Ig., Kiselyova, E.
S. A. Moskalenko, M. A. Liberman, Ig. V. Podlesny, E. S. Kiselyova, "Bose-
Einstein condensation of two-dimensional magnetoexcitons on the
superposition state," Proc. SPIE 6726, ICONO 2007: Physics of Intense and
Superintense Laser Fields; Attosecond Pulses; Quantum and Atomic Optics;
and Engineering of Quantum Information, 672632 (19 July 2007); doi:
10.1117/12.751827
Event: The International Conference on Coherent and Nonlinear Optics, 2007,
Minsk, Belarus
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Bose-Einstein Condensation of two-dimensional magnetoexcitons on
the superposition state
S.A. Moskalenkoa*, M.A. Libermanb, Ig.V. Podlesnya and E.S. Kiselyovac
aInstitute of the Applied Physics of the Academy of Sciences of Moldova, 5, Academy str.,
MD-2028, Chisinau, Republic of Moldova.
bDepartment of Physics, Uppsala University, Box 530, SE-751 21, Uppsala, Sweden
cMoldova State University, 60, A. Mateevich str., MD-2012, Chisinau, Republic of Moldova.
ABSTRACT
The Bose-Einstein Condensation (BEC) of the two-dimensional magnetoexcitons on the superposition state is studied.
The superposition of two excitonic states formed by electron and hole on the lowest Landau levels (0, 0) and on the first
excited Landau levels (1, 1) was considered. The generalized Bogoliubov u-v transformation was deduced. The criterion
on the exciton concentration was established. The influence of the BEC on the absorption band shape is discussed.
Keywords: Bose-Einstein Condensation, Magnetoexcitons.
1. INTRODUCTION
In the Refs1, 2 the influence of the Coulomb scattering processes of electrons and holes between different Landau levels
on the energy spectrum and on the collective properties of the two-dimensional magnetoexcitons was investigated. It was
shown that the virtual quantum transitions of the Coulomb interacting particles from the lowest Landau levels to excited
Landau levels with arbitrary numbers n and m and their transitions back to the lowest Landau levels in the second order
of the perturbation theory result in the indirect attraction between the particles supplementary to their Coulomb
interaction. These investigations were fulfilled in pure electron-hole description as well as in the excitonic approach,
which permitted to determine the wave function of the lowest magnetoexciton band. The more exact exciton wave
function as well as the exciton creation operator were represented in the superposition form composed by the wave
function and creation operators of the bare exciton states. The existence of the ground state superposition state will
influence on the character of the megnetoexciton BEC, changing the distribution of the particles no the lowest and
excited Landau levels. The BEC of magnetoexcitons influences on their absorbtion band shape. To this end we will study
below the band shape of the combined optical quantum transitions involving the creation of one magnetoexciton and the
simultaneous excitation of an electron from the lower to upper Landau levels. The experimental results in this direction
were published in the papers[3, 4, 5]. Both aspects of the magnetoexciton physics are strongly correlated with the
knowledge concerning the influence of the excited Landau levels accumulated in the frame of two earlier published
papers[1, 2].
The paper is organized as follows. In the section 2 the BEC of magnetoexcitons in the superposition state is studied. In
section 3 the absorbtion band shape for the combined quantum transition with the creation of an magnetoexciton and the
simultaneous excitation of an electron from the lower to upper Landau levels is deduced. The influence of the BEC of
magnetoexcitons on the band shape is discussed. The conclusions form the section 4.
* Fax: (373 22) 738149. E-mail: exciton@phys.asm.md
ICONO 2007: Physics of Intense and Superintense Laser Fields; Attosecond Pulses;
Quantum and Atomic Optics; and Engineering of Quantum Information, edited by M. V. Fedorov,
et al., Proc. of SPIE Vol. 6726, 672632, (2007) · 0277-786X/07/$18 · doi: 10.1117/12.751827
Proc. of SPIE Vol. 6726 672632-1
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2. BOSE-EINSTEIN CONDENSATION OF MAGNETOEXCITONS ON THE SUPERPOSITION
STATE
For the begining we will define the magnetoexciton creation operators †
,,nmk
X

[1, 2] characterized by the number n of the
electron Landau level (LL) and by the number m of the hole LL, as well as by the two-dimensional (2D) wave vector
(
)
,
xy
kk k

()
†2††
,, ,,2
2
1exp x
xk
k
y
nmk nt
mt
t
Xiktlab
N+−
=−
∑
 (1)
Here †
,,np np
aa and †
,,mq mq
bb are the creation and annihilation operators of electrons and holes correspondingly. In the
Landau gauge the quantum numbers p and q are unidimensional wave vectors. The wave function of the bare
magnetoexciton state can be obtained acting with the operator (1) on the vacuum state 0 determined as
,,
000
np mq
ab==
.
†
,, ,, 0
nmk nmk
XΨ=

(2)
Following the papers [1, 2] the main contribution in the ground state superposition state is due to two states. One of them
is (0, 0) state with n=m=0 and another one is (1, 1) state, which corresponds to the values n=m=1. There are also another
two involved states (1, 0) and (0, 1), but their contributions are considerable smaller and here they were neglected.
In such a way in the frame of this two-level model the magnetoexciton creation operator has the form [1, 2, 6]
†††
12
, , 0,0, 1,1,nmk k k
XaXaX=+

, 22
12
1aa+= (3)
The unitary transformation
()
ˆ
ex
DN
(
)
(
)
†
ˆˆˆ
exp
ex ex kk
DN N X X
⎡
⎤
=−
⎣
⎦

(4)
depends on the full number of magnetoexcitons ex
N, and permits to transform the starting Hamiltonian
ˆ
H
to the form
†
ˆ
DHD . This transformation leads to breaking of the gauge symmetry and generalizes the Bogoliubov u-v
transformation as follows
() ()
†2†
0, 1 0, 1 0,
cos sin exp ;
2x
x
pp p y kp
k
Da D ga a ga ik p l bα
−
⎡⎤
⎛⎞
== − −−
⎜⎟
⎢⎥
⎝⎠
⎣⎦
() ()
†2†
0, 1 0, 1 0,
cos sin exp ;
2x
x
pp p y kp
k
Db D ga b ga ik p l aβ−
⎡⎤
⎛⎞
== + −−
⎜⎟
⎢⎥
⎝⎠
⎣⎦
() ()
†2†
1, 2 1, 2 1,
cos sin exp ;
2x
x
pp p y kp
k
Da D ga a ga ik p l bγ−
⎡⎤
⎛⎞
== − −−
⎜⎟
⎢⎥
⎝⎠
⎣⎦
(5)
() ()
†2†
1, 2 1, 2 1,
cos sin exp .
2x
x
pp p y kp
k
Db D ga b ga ik p l aδ−
⎡⎤
⎛⎞
== + −−
⎜⎟
⎢⎥
⎝⎠
⎣⎦
Here the denotations were introduced
2
2
ex
glnπ= ; ex
ex
N
n
S
=, (6)
where S is the layer surface area and l is the magnetic length.
The new BCS-type ground state wave function has the expression
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()
()
() ()
2†† ††
12
0, 0, 1, 1,
22 22
ˆ0 cos sin exp
xx xx
gex ykkkk
tt tt
t
kDN g g iktlaa b aab
+− +−
⎛⎞
⎡
⎤
⎡⎤
Ψ= = + − +
⎜⎟
⎢
⎥
⎣⎦
⎜⎟
⎣
⎦
⎝⎠
∏ (7)
It plays the role of the vacuum state for the operators p
α, p
β, p
γ, p
δ.
The average number of electrons or holes equals to the average number of excitons ex
N
(
)
(
)
(
)
(
)
(
)
†† 2 2
00 11 1 2
sin sin
gttttg
t
kaaaa kN
g
a
g
a
⎡
⎤
Ψ+Ψ=+
⎣
⎦
∑ (8)
Owing to the definition of
g
(6) one arrives to the concentration relation
(
)
(
)
22 2
12
sin singgaga=+ (9)
In the case 11a= and 20a= we obtain the previous expression Eq. (36) of Ref. [6], namely 22
singg=, what can
be satisfied only in the case 1g<. In another special case 12
1
2
aa== the condition (9) obtains the form
2
2
sin
22
gg
⎛⎞
=⎜⎟
⎝⎠
(10)
what implies 2g<. This relation generalizes the description of the BEC of the magnetoexciton gas on its ground
superposition state with the electrons and holes residing in the both LLLs and first excited Landau levels [FELLs].
The BEC of magnetoexcitons will influence on their optical absorption band shape and could be evidenced as its
supplementary structure. This influence will be discussed below. As an example, the combined optical quantum
transition with the creation of one magnetoexciton accompanied by the excitation of one electron from the lower to upper
Landau level will be considered.
3. COMBINED MAGNETOEXCITON – ELECTRON QUANTUM TRANSITIONS.
The combined magnetoexciton – electron quantum transitions will be calculated in the second order of the perturbation
theory. The Hamiltonians of the electron-radiation and electron-electron (e-e) Coulomb interactions are listed below.
()
()
{
()
()
()
}
,,
,,
0
†† ††
33
,, ,,
', , ', ,
22
†
†
33
,, ,,
', , ', ,
22
2,;', ;
,;', ;
xyz
xx
xx
erad x y
pll
cv lp lp
lk p lk p
xyvc lp lp
lk p lk p
e
Hlplp
mV
PCa b C a b
lpl p P C b a C b a
κκ κ κ κ
κκ
κκ
π
κκ
ω
κκ
−
⊕−
↑↓
−− −
⊕−
↑↓
−− −
⎛⎞
=− Φ − ×
⎜⎟
⎝⎠
⎡⎤
×++
⎢⎥
⎣⎦
⎡⎤
+Φ − − +
⎢⎥
⎣⎦
∑∑






(11)
where only the resonant terms are included and only the heavy holes are involved. The following notations were
introduced
(
)
(
)
(
)
(
)
*2 2
'
,;', ; yy
iR
xy l y l y x y
lpl p R pl R p l e dR
κ
κκ ϕ ϕ κΦ−=− −−
∫ (12)
,,xy
CCiC
κκ
κ±=±
○ ;
()
()
()
0
*
,0 ,0
0
1
vc v c
v
PdU iU
vρ
ρ
ρρ=−∇
∫




They are expressed through the wave functions of electrons in the strong perpendicular magnetic field
()
()
2
,
x
ipR
np n y
e
RRpl
LϕΨ= −

In the Landau gauge they are characterized by the quantum number n of Landau quantization in one in-plane direction
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and by the unidimensional wave vector p in the perpendicular in-plane direction.
The Coulomb e-e interaction describing the quantum transitions of two electrons from the lowest Landau levels (LLLs)
to the excited LLs with the numbers n` and m` has the form
()
††
0, ', 0, ', ', ' ', ', ' ',
'''' '
10, ';0, '; ', ' '; ', ' ' . .
2
Coul e e pqmqsnps
pqs nm
H
Fpqnpsmqsaaa a hc
−↓↓+↓−↓
=−+ +
∑∑ (13)
where
()
(
)
(
)
(
)
(
)
***
0, ' 1 ', ' ' 1 12 0,' 2 ',' ' 2 1 2
0, '; 0, '; ', ' '; ', ' '
ee p np s q mq s
F
pqnpsmqs R RV R RdRdR
−−+
−+=ΨΨ ΨΨ
∫∫

 
(14)
and
()
12
12
(,)
01 2
1
xy
iRR
VeV
RR
κ
κ
κκ κ
ε
−
==
−∑





 (15)
We will discuss the case of quantum transitions when the initial i, intermediary 1
u and final
F
states of the
perturbation theory are
††
0
00
QT
iC a
−↓
=
○
†††
13
00
2
0
fh
g
uaba
↓↓
=
(
)
††
,,
ˆ,0
ex
nR
Fa k
↓−
=Ψ

 (16)
()
2
†
,0 † †
3
,, ,
2
22
1
ˆ,y
xx
ik tl
m
ex k k
mt t
t
keab
N+↓ −
Ψ−=
∑

○
Here the spin oriented electrons and heavy holes with the projection 3
2
z
j= are considered. They interact with
circularly polarized light in one definite direction.
In the initial state one electron is on the LLL n=0 with wave vector T, whereas in the final state it has a quantum number
n and wave number R.
The exciton creation operator
(
)
†
,0
ˆ,
m
ex kΨ−

○ is characterized by the quantum numbers (m, 0) for electron-hole pair and
by circular polarization in a definite direction.
The energies of the mentioned states are equal to
1
2
iQg ce
EEωω=++
; ci
i
eH
mc
ω=
1
11
2
22
ug c ce
EE µ
ωω=+ +
; ccechµ
ωωω=+
()
,0
11
2
22
m
Fg ce cex
EEmn Ik
µ
ωω
⎛⎞
=+++ + −
⎜⎟
⎝⎠
 (17)
() ()
,0 ,0
1
2
mm
ex g ce c ex
EkEm Ik
µ
ωω=+ + −
The first order matrix elements 1er
iH u and 1Coul
uH F
are
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()
()
()
()
()( )
1
0
20, ; 0, ; ,
,0,;0,;,,
er vc x y kr
Q
kr x x y kr kr x
e
iH u P
ff
QQ Th
mV
g
Q
f
hhQ Q
f
T
g
Qh
πδ
ω
δδδ
⎛⎞ ⎡
=− Φ − − ×
⎜⎟ ⎣
⎝⎠
⎤
×−−Φ−− −
⎦

()
()( )()
2
2
1
10, ; 0, ; , ; ,
2
,0,;0,;,;,,
x
y
k
ik g l
Coul e e x
kr x e e x kr x
uH F e F f hnRmk g
N
fRk gh F h fnRmk g hRk gfδδ
⎛⎞
−
⎜⎟
⎝⎠
−
−
=−×
⎡
⎣
×−−−− − −−−
⎤
⎦
(18)
whereas the second order matrix element equals to
()
()
()()
2
11
11 2
2, 0,;0,;
0, ; 0, ; , ; , 0, ; 0, ; , ; , ;
x
yx
k
ik Q f l
er Coul
kr x x x y
u f
iu
ee x x ee x x
iH u u H F ATQkRe f fQQ
EE
FfTnRmkQfFTfnRmkQf
δ
⎛⎞
−+
⎜⎟
⎝⎠
−−
=++ Φ −−×
−
×−+−−+
⎡⎤
⎣⎦
∑∑
(19)
where
0
21
1
2
2
vc
Q
Qg c
P
e
A
mV NEµ
π
ω
ωω
⎛⎞
=−
⎜⎟ ⎛⎞
⎝⎠ −−
⎜⎟
⎝⎠


(
)
2
0, ; 0, ; y
iQ fl
xy
ffQQ e
−
Φ−−= (20)
The Fermi golden rule gives the probability of the quantum transitions
(
)
,,
Q
P
iFω
()
()
11
2
11
2
,, er Coul
QkriF
uiu
iH u u H F
P
iF E E
EE
π
ωδ=−
−
∑
 (21)
Its sum on the final states can be expressed through the response function
()
,
Q
Siω
() ()
2
,, ,
QQ
F
P
iF mS iωω=− ℑ
∑
()
000
111
ˆˆˆˆ
,
Q er Coul Coul er
iii
SiiHHHHi
EH i EH i EH i
ω
δδδ
=
−+ −+ −+
(22)
The averaging on the initial state is necessary in our conditions because the initial electron is situated on the arbitrary
state 0, ,T↓ of the lowest Landau level with the filling factor 21v<. The averaging will expressed by the
procedure
2
T
v
N∑.
The probability transition
(
)
,
P
iF summarized on the final state
F
and averaged on the initial state i at the given
values of n and m has the form
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()
()
()
()
()
()
{
()
()()
2
2
02
,
,
*
*
,,,,
1
0, ; 0, ; , ; , 0, ; 0, ; , ; ,
0, ; 0, ; , ; , 0, ; 0, ; , ; ,
0, ;
yy
xy
QQ
TR
k
ik Q f gl
Rfg
kk k
ee xx xx ee xx xx
ee xx xx ee xx xx
ee
v
Wn PTkR
N
Be
NV
F
f k Q RnRmk Q f F g k Q RnRmk Q g
FkQRfnRmkQfF kQRgnRmkQg
Ff
ωω
−−
−−
−−
−
==
=×
×−+−+ −+−++
+−+ −+ −+ −+−
−
∑∑∑
∑∑∑



()()
()()
}
()
()
*
*
,0
0, ; , ; , 0, ; 0, ; , ; ,
0, ; 0, ; , ; , 0, ; 0, ; , ; ,
1
2
xx xx ee xx xx
ee xx xx ee xx xx
m
Q gap ex ce c
k Q RnRmk Q f F k Q R gnRmk Q g
F kQR fnRmkQ fF gkQRnRmkQg
EI knm µ
δω ω ω
−
−−
−+ −+ −+ −+−
−−+ −+ −+ −+×
⎛⎞
×−+ −+ −
⎜⎟
⎝⎠


(23)
where
()
2
2
22
0
02
2
1
2
vc
Qg c Q
ePv
m
B
Eµ
π
ωωω
⎛⎞
⎜⎟
⎝⎠
=⎛⎞
−−
⎜⎟
⎝⎠

(24)
In more general case we have deal with the initial state
1
††
00
QnT
iC a
−↓
=
○, where the electron is situated on the
Landau level 1
n with quantum number T, with the intermediary state
1
†† †
13
,2
0
lf n h
lg
uab a
↓↓
= and with the final
state
(
)
2
††
ˆ,0
ml
ex n R
Fka=Ψ −

○ with corresponding energies
11
11
222
ug c ce
EEl n
µ
ωω
⎛⎞ ⎛ ⎞
=++ ++
⎜⎟ ⎜ ⎟
⎝⎠ ⎝ ⎠

()
,
2
11
2,
22
ml
F g ce ch c ce ex
EEm l n Ik
µ
ωω ω ω
⎛⎞
=+ + + ++ − −
⎜⎟
⎝⎠

   ○
1
1
2
ig ce Q
EE n ωω
⎛⎞
=++ +
⎜⎟
⎝⎠

(25)
1
1
2
iu Qg c
EE E l µ
ωω
⎛⎞
−= −−+
⎜⎟
⎝⎠

()
()
,
21
1,
2
ml
i f Q g ce ce ch c ex
EE E n n m l I k
µ
ωωωωω−= −− − − − − + −

○
We will have a more general expression
()
()
()
()
2
12 2
,
12
1
,;,;
,;, ; ,;,
xy
yy
l
R
kk k
ik Q fl
ee xx xx
f
WQnmln B
NV
eFnflkQRnRmkQf
−
−
=×
×−+−+−
⎡
⎣
∑∑
∑


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()
()
()
2
12
,
21
,;,;,;,
1
,2
ee xx xx
ml
Q g ex ce ce ch c
F lk Q Rn f n Rmk Q f
EI k nn m l µ
δω ω ω ω ω
−
−−+ −+×
⎤
⎦
⎛⎞
×−+ −−− −−−
⎜⎟
⎝⎠

○
(26)
where l
B equals
()
2
2
22
0
2
2
1
2
vc
l
Qg c Q
ePv
m
B
El µ
π
ωωω
⎛⎞
⎜⎟
⎝⎠
=⎛⎞
⎛⎞
−−+
⎜⎟
⎜⎟
⎝⎠
⎝⎠

(27)
4. THE COMBINED ABSORPTION BAND SHAPE
The properties of the combined exciton absorption band can be deduced looking at the expression standing under the
sign of the δ-function. We will introduce the renormalized band gap
g
E

as
21
11
22
g g ce ch
EE nnm lωω
⎛⎞⎛⎞
=+ −++ ++
⎜⎟⎜⎟
⎝⎠⎝⎠

and will transcribe the ionization potential
(
)
,p
ml
ex
Ik

in the form
(
)
(
)
(
)
,,,
0
ml ml ml
ex ex
Ik I Ek=−;
() (
)
()
,
,
,
01
0
ml
ml
ml
ex
Ek
Ek
I
≤= ≤

Introducing the dimensionless frequency detuning
()
,0
Qg
Qml
ex
E
I
ω−
=


 we will find the requirement resulting from the
sign of the δ-function, namely
(
)
,
10
ml
QEk+− =


 with arbitrary k

.
It means that the detuning Q
 can be changed in the interval 10
Q
−< < and the absorption band is concentrated in
the spectral region
(
)
,0
ml
g
ex Q g
EI Eω−≤≤

.
The full exciton energy band
(
)
,ml
Ek participates in the construction of the absorption band shape and is reflected in
its structure due to the participation of an free electron in the initial and final stages of the combined quantum transition.
It leads to the conservation law for the x components of the wave vectors
x
Q,T,
x
k and R as follows
xx
QTk R+= +.
It means that at a given component
x
Q of the photon wave vector Q

, the exciton wave vector component
x
k can be
arbitrary being compensated and supplemented by the corresponding values of the components T and R. The
component y
k is not linked with the component y
Q by the conservation law, but it also influences on the values of the
exciton dispersion law.
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The magnetoexciton with arbitrary values of its energy spectrum
() () ()
,,,
11 0
22
ml ml ml
ex g ce ch ex
EkE m l I Ekωω
⎛⎞ ⎛⎞
=++ ++ − +
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
 can take part in the combined quantum
transition and the absorption band shape has a maximal width
(
)
,0
ml
ex
I. It is true without the participation of another
partners in the quantum transition. The intensity of the combined absorption band in the case when 0l=, 10n= and
2
nn= will be determined by the square power of the electron – electron Coulomb matrix element
(
)
0, ; 0, ; , ; ,
ee
F
pqnpsmqs
−
−+, which has the property
()
2111
0, ; 0, ; , ; , 2!!
ee nm
Fpqnpsmqs nm
−+
−+≈ .
It means that the intensity of the absorption band shape and the probabilities
(
)
,,
Q
Wnmω of the quantum transitions
undergo to the rule
()
111
,, 2!!
Qnm
Ww nm nm
+
≈ (28)
In the presence of the BEC of magnetoexcitons the combined absorption band shape will undergo an essential
transformation due to the presence of a macroscopical large number
0
k
N

of the quasiparticles in the single-particle state
with the wave vector 0
k

. The influence of the BEC of excitons on the forms of their absorption and luminescence bands
was discussed in [7].
We will introduce phenomenologically a supplementary factor
(
)
(
)
00
1,
k
Nkkδ+


in the expression (21), what will
give rise to the final expression
()
(
)
(
)
()()
()
()
0
00
2
2
,0
1
,, 1 ,
0, ; 0, ; , ; , 0, ; 0, ; , ; ,
1
2
Qk
k
ee xx xx ee xx xx
m
Q g ex ce c
WnmB Nkk
NV
F f k Q RnRmk Q f F k Q R f nRmk Q f
EI k nm µ
ωδ
δω ω ω
−−
=+×
× −+ −+− −+ −+ ×
⎛⎞
×−+ −+ −
⎜⎟
⎝⎠
∑




(29)
The supplementary sharp peak of he δ- function type will appear on the background of the band shape described above
in the absence of the magnetoexciton BEC.
The position of the sharp peak corresponds to the frequency
()
()
,0
0
1
2
m
Qg ce cex
Enm Ik
µ
ωωω=++ + −

 (30)
ACKNOWLEDGEMENTS
This work was supported by the common grant 06.05 CRF of the Russian Foundation for Fundamental Research and of
the Academy of Sciences of Moldova.
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