Creating Molecular Entanglement in Functionalized Semiconductor Nanostructures

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Creating Molecular Entanglement in Functionalized
Semiconductor Nanostructures
Quantum Information and Quantum Control
Conference July 19-23, 2004 - University of
Toronto, The Fields Institute, Toronto (BA1170)
Sabas Abuabara, Luis G.C. Rego and Victor S.
Batista
Department of Chemistry, Yale University, New
Haven, CT 06520-8107
Mr. Sabas Abuabara
Dr. Luis G.C. Rego
Current Address Physics Department,
Universidade Federal do Parana, CP 19044,
Curitiba, PR, Brazil, 81531-990
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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 dynamics
• Hole 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 96 catechol
C6H6-202 unit 12 16 capping H atoms 16
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Mixed Quantum-Classical Dynamics Propagation
Scheme
, where
and
with
Short-Time Propagation Scheme
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Time-Dependent Molecular Orbitals
..
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 (AOs),
including 4s, 3p and 3d AOs of Ti 4 ions, 2s
and 2p AOs of O 2- ions, 2s and 2p AOs of C
atoms and 1s AOs of H atoms (i.e., 596 basis
functions per unit cell). S is the overlap matrix
in the AOs basis set.
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Time-Dependent Propagation Scheme contd
For a sufficiently small integration time step t,
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Time-Dependent Propagation Scheme contd
when t 0,
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Ab Initio DFT-Molecular Dynamics Simulations
VASP/VAMP simulation packageHartree 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
-101
010
System extened in the 010 direction
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Phonon Spectral Density
O-H stretch, 3700 cm-1 (H capping atoms)
C-H stretch 3100 cm-1
C-C,CC stretch 1000 cm-1,1200 cm-1
TiO2 normal modes 262-876 cm-1
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Electronic Density of States (1.2 nm particles)
LUMO,LUMO1
HOMO
Band gap 3.7 eV
Conduction Band
Valence Band
ZINDO1 Band gap 3.7 eV
Exp. (2.4 nm) 3.4 eV
Exp. (Bulk-anatase) 3.2 eV
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Propagation Scheme contd
Therefore, we can calculate the wavefunction and
electronic density for all tgt0 and we can also
define the survival probability for the electron
to be found on the initially populated adsorbate
molecule
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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 contd
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LUMO Injection (frozen lattice) contd
PMOL(t)
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Injection from LUMO1
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Injection from LUMO1 (frozen lattice, 0 K)
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LUMO1 Injection (frozen lattice) contd
PMOL(t)
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LUMO Injection at Finite Temperature (100 K)
0 K
100 K
PMOL(t)
0 K
100 K
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Hole-Relaxation Dynamics in the Semiconductor
Band Gap
t0 ps
t15 ps
Super-exchange hole transfer
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Coherent Hole-Tunneling Dynamics
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Investigation of Coherent-Control
t 200 fs, w12
t kt
2-p pulses

A2
A1
CB
HOMO
w12
superexchange
VB
TiO2 semiconductor
Adsorbate molecules (A1, A2,)
Train of 2-p pulses Agarwal et. al. Phys. Rev.
Lett. 86, 4271 (2001)
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Investigation of Coherent-Control contd
2-p pulses (200 fs spacing)
14 ps
60 ps
Time, ps
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Investigation of Coherent-Control
2-p pulses (200 fs spacing)
2 ps
42 ps
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Reduced Density Matrix Hole-States
Index x indicates a particular initial nuclear
configuration
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Occupancy Notation
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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Investigation of Entanglement contd
Example
In the occupancy representation,
a state defined by only two nonzero
expansion coefficients represents a
physical state of maximal entanglement ,
e.g., between the center and right adsorbates.
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Investigation of Coherences contd
Compute the subspace density matrix explicitly
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Investigation of Coherences contd
Off-diagonal elements are indicative of
decoherence
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.
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Investigation of Coherences contd
Diagonal Elements of the Reduced Density
Matrix T100 K
Off-Diagonal Elements of the Reduced Density
Matrix T100 K
Time, ps
Time, ps
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Investigation of Coherences contd
Measure of decoherence
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Decoherence Dynamics contd
100 fs
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Conclusions

• 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
  2p-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.

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Acknowledgments

• NSF Nanoscale Exploratory Research (NER) Award
  ECS0404191
• NSF Career Award CHE0345984
• ACS PRF37789-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 !