Creating Molecular Entanglement in Functionalized Semiconductor Nanostructures
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Mr. Sabas Abuabara
Sabas Abuabara, Luis G.C. Rego and Victor S. BatistaDepartment of Chemistry, Yale University, New Haven, CT
06520-8107
Quantum Information and Quantum Control ConferenceJuly 19-23, 2004 - University of Toronto, The Fields Institute, Toronto (BA1170)
Dr. Luis G.C. Rego*
*Current Address:
Physics Department, Universidade Federal do Parana, CP 19044, Curitiba, PR, Brazil, 81531-990
Creating Molecular Entanglement in Functionalized Semiconductor Nanostructures
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Photo-Excitation and Relaxation Processes
Add energy diagram here
Interfacial electron transfer
Hole relaxation dynamics
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Aspects of Study
Interfacial Electron Transfer Dynamics Relevant timescales and mechanisms Total photo-induced current Dependence of electronic dynamics on the
crystal symmetry and dynamicsHole Relaxation Dynamics Decoherence timescale. Effect of nuclear dynamics on quantum
coherences, coherent-control and entanglement.
Luis G.C. Rego and Victor S. Batista, J. Am. Chem. Soc. 125, 7989 (2003);
Victor S. Batista and Paul Brumer, Phys. Rev. Lett. 89, 5889 (2003), ibid. 89, 28089 (2003);
Samuel Flores and Victor S. Batista, J. Phys. Chem. B, 108, 6745 (2003).
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Model System – Unit Cell
TiO2-anatase nanostructure functionalized by an adsorbed catechol molecule
124 atoms:
32 [TiO2] units = 96catechol [C6H6-202] unit = 1216 capping H atoms = 16
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Mixed Quantum-Classical Dynamics Propagation Scheme
)0()(ˆ)( tUt
and withq
qq ttBt )()()(
tHi
etUˆ
)(ˆ
, where
p
pqqp
i
pq tttEtE
tBtB e )()()]()([
)()( 2
Short-Time Propagation Scheme
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i
iqiq tKtCt )()()( ,
)()()()()( tEtCtStCtH
which are obtained by solving the Extended-Huckel generalized eigenvalue equation :
..
where H is the Extended Huckel Hamiltonian in the basis of Slater type atomic orbitals (AO’s), including 4s, 3p and 3d AO’s of Ti 4+ ions, 2s and 2p AO’s of O 2- ions, 2s and 2p AO’s of C atoms and 1s AO’s of H atoms (i.e., 596 basis functions per unit cell). S is the overlap matrix in the AO’s basis set.
..
Time-Dependent Molecular Orbitals
)(tKi
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Time-Dependent Propagation Scheme cont’d
For a sufficiently small integration time step ,
)()()()( 2)(
2 tqetBttUq
tEi
q
q
)()(
)(),(
2)(
2
tqetB
tttU
tEi
q
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Time-Dependent Propagation Scheme cont’d
Setting equal to and multiplying by a MO at the iterated time gives the expression
)()()()( 2))()((
tptqetBtBp
tEtEi
pq
qp
when 0,
2))()((
)()(
tEtE
i
qp
etBtB
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Ab Initio DFT-Molecular Dynamics Simulations VASP/VAMP simulation package
Hartree and Exchange Correlation Interactions: Perdew-Wang functional
Ion-Ion interactions: ultrasoft Vanderbilt pseudopotentials
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Model System – Mixed Quantum-Classical Simulations
Three unit cells along one planar directions with periodic boundary conditions in the other.
Three unit cells extending the system in [-101] direction
System extened in the [010] direction
[010]
[-101]
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Phonon Spectral Density
O-H stretch, 3700 cm-1
(H capping atoms)
C-H stretch
3100 cm-1
TiO2 normal modes
262-876 cm-1
C-C,C=C stretch
1000 cm-1,1200 cm-1
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Electronic Density of States (1.2 nm particles)
HOMO
LUMO,LUMO+1
Valence Band Conduction BandBand gap =3.7 eV
ZINDO1 Band gap =3.7 eVExp. (2.4 nm) = 3.4 eVExp. (Bulk-anatase) = 3.2 eV
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Therefore, we can calculate the wavefunction and electronic density for all t>0 and we can also define the survival probability for the electron to be found on the initially populated adsorbate molecule
SYS
j
jij
MOL
iiMOL StCtCtP
,
,,,
,
*, )()()(
Propagation Scheme cont’d
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Injection from LUMO
TiO2 system extended in [-101] direction with PBC in [010] direction
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Injection from LUMO (frozen lattice, 0 K)
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LUMO Injection cont’d
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LUMO Injection (frozen lattice) cont’d
PM
OL(
t)
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Injection from LUMO+1
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Injection from LUMO+1 (frozen lattice, 0 K)
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LUMO+1 Injection (frozen lattice) cont’d
PM
OL(
t)
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LUMO Injection at Finite Temperature (100 K)
0 K
100 K
0 K
100 K
PM
OL(
t)
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t=0 ps
t=15 ps
Hole-Relaxation Dynamics in the Semiconductor Band Gap
Super-exchange hole transfer
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0 20 40 60 80 1000.0
0.2
0.4
0.6
0.8
1.0
SU
RV
IVA
L P
RO
BA
BIL
ITY
TIME (PS)
Coherent Hole-Tunneling Dynamics
);(sin22
2
22/
)( tjk
jk
jkj tP
22 )2/()/( jkjkjk
RCLC
*5.4 *2.1LCRC
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)0()0()0(
)()0(2)()(
t
tt
CB
VB
t= k*
= 200 fs,
A1A2
superexchange
TiO2 semiconductor
Adsorbate molecules (A1, A2,…)
Train of 2- pulses : Agarwal et. al. Phys. Rev. Lett. 86, 4271 (2001)
Investigation of Coherent-Control
…
2- pulses
HOMO
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0 20 40 60 80 1000.0
0.2
0.4
0.6
0.8
1.0
S
UR
VIV
AL
PR
OB
AB
ILIT
Y
TIME (PS)
Investigation of Coherent-Control cont’d
2- pulses (200 fs spacing)
14 ps 60 ps
Time, ps
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0 20 40 60 80 1000.0
0.2
0.4
0.6
0.8
1.0
S
UR
VIV
AL
PR
OB
AB
ILIT
Y
TIME (PS)
Investigation of Coherent-Control
2- pulses (200 fs spacing) 2 ps 42 ps
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torSemiconducCoupling
CouplingAdsorbates
Reduced Density Matrix: Hole-States
ttt p
Index indicates a particular initial nuclear configuration
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Occupancy Notation
001
010
100
R
C
L
HOMO
HOMO
HOMO
Thus these kets describe the state of all three adsorbates -- at once, i.e., the state of the hole as
distributed among all three adsorbates.
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Example: In the occupancy representation,
kC
CCC
TiOk
t
100
2
001010100001010100
0
2
10001
0010 CCphysical state of
maximal entanglement , e.g.,
between the center and right adsorbates.
Investigation of Entanglement cont’d
a state defined by only two nonzero expansion coefficients represents a
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Investigation of Coherences cont’d
Compute the subspace density matrix explicitly
2
001*010001
*100001
*001010
2
010*100010
*001100
*010100
2
100
CCCCC
CCCCC
CCCCC
Ads
t
Ads
tAdst p
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Investigation of Coherences cont’d
Off-diagonal elements are indicative of decoherence
iAds eRRt)(,
if nuclear motion randomizes the phases, i.e, becomes a random quantity and the average
The system will no longer be in a coherent superposition of adsorbate states.
0
ieRR
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Investigation of Coherences cont’d
Diagonal Elements of the Reduced Density MatrixT=100 K
Off-Diagonal Elements of the Reduced Density MatrixT=100 K
Time, ps Time, ps
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Investigation of Coherences cont’d
Measure of decoherence:
2
tTr
2 +
+ +
+
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Decoherence Dynamics cont’d
100 fs
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•We have investigated interfacial electron transfer and hole tunneling relaxation dynamics according to a mixed quantum-classical approach that combines ab-initio DFT molecular dynamics simulations of nuclear motion with coherent quantum dynamics simulations of electronic relaxation.
•We have investigated the feasibility of creating entangled hole-states localized deep in the semiconductor band gap. These states are generated by electron-hole pair separation after photo-excitation of molecular surface complexes under cryogenic and vacuum conditions.
•We have shown that it should be possible to coherently control superexchange hole-tunneling dynamics under cryogenic and vacuum conditions by simply applying a sequence of ultrashort 2-pulses with a frequency that is resonant to an auxiliary transition in the initially populated adsorbate molecule.
•We conclude that large scale simulations of quantum dynamics in complex molecular systems can provide valuable insight (into the behavior of the quantum coherences in exisiting materials), which might be essential to bridge the gap between the quantum information and quantum control communities.
Conclusions
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•NSF Nanoscale Exploratory Research (NER) Award ECS#0404191•NSF Career Award CHE#0345984•ACS PRF#37789-G6•Research Corporation, Innovation Award•Hellman Family Fellowship•Anderson Fellowship•Yale University, Start-Up Package•NERSC Allocation of Supercomputer Time • Organizing Committee at The Fields Institute, University of Toronto: Paul Brumer, Daniel Lidar, Hoi-Kwong Lo and Aephraim Steinberg.
Thank you !
Acknowledgments