1. Quantum Information and Quantum Control Conference
July 19-23, University of Toronto, The Fields Institute, Toronto (BA1170)
Creating Molecular Entanglement in Functionalized Semiconductor Nanostructures
Sabas Abuabara, Luis G.C. Rego and Victor S. Batista
Department of Chemistry, Yale University, New Haven, CT
Mr. Sabas Abuabara
Dr. Luis G.C. Rego*
*Current Address:
Physics Department, Universidade Federal do Parana, CP 19044, Curitiba, PR, Brazil,

2. Photo-Excitation and Relaxation Processes
Add energy diagram here
Interfacial electron transfer
Hole relaxation dynamics

3. 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.
There is some controversy over the appropriate timescales associated with the electron injection mechanism. There are experimental results of short (~100 L.G.C. Rego and V.S. Batista, JACS 125, 7989 (2003)
) and ultrashort(~10 fs) timescales associated with the electron transfer investigated here. Furthermore, there are reports of anisotropy in the transport properties of electrons in the TiO2 bulk and the investigation of whether this effect is manifested in surface reactions is important. How long the initial electron wavepacket remains coherent is also a basic question with ramifications for the robustness of the photocurrent in photovoltaic devices with respect to the temperature and presence of impurities in the crystal structure.
In addition to electron transfer being relatively poorly understood, these studies have important ramifications for the manufacture of photovoltaic cells.
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, (2003);
Samuel Flores and Victor S. Batista, J. Phys. Chem. B, 108, 6745 (2003).

4. 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
Catechol molecule adsorbed onto (010) surface of anatase crystal.

5. Mixed Quantum-Classical Dynamics Propagation Scheme
, where
and with
Short-Time Propagation Scheme

6. 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 (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.

7. Time-Dependent Propagation Scheme cont’d
For a sufficiently small integration time step t,

8. Time-Dependent Propagation Scheme cont’d
Setting equal to and multiplying by a MO at the iterated time gives the expression
when t 0,
The approximation here amounts to an infinitessimal time increment where the MOs are essentially orthogonal (since they computed by diagonalizing the Hamiltonian at a particular time in the basis of AOs, the eigenvectors at that time are orthogonal).

9. Ab Initio DFT-Molecular Dynamics Simulations VASP/VAMP simulation package Hartree and Exchange Correlation Interactions: Perdew-Wang functional Ion-Ion interactions: ultrasoft Vanderbilt pseudopotentials

10. Model System – Mixed Quantum-Classical Simulations
Three unit cells along one planar directions with periodic boundary conditions in the other.
[-101]
Three unit cells extending the system in
[-101] direction
Due to finite size effects in results from unit cell dynamics calculations, for the purposes of calculations, the actual system consisted of typically 3 unit cells arranged as shown with periodic boundary conditions in the non-extended planar direction.
The PBCs are only used when constructing the Hamiltonian and Overlap Matrices in the extended Huckel formalism.
[010]
System extened in the [010] direction

11. Phonon Spectral Density
O-H stretch, 3700 cm-1
(H capping atoms)
C-C,C=C stretch
1000 cm-1,1200 cm-1
C-H stretch
3100 cm-1
TiO2 normal modes
cm-1

12. Electronic Density of States (1.2 nm particles)
LUMO,LUMO+1
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

13. Propagation Scheme cont’d
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

14. Injection from LUMO
TiO2 system extended in [-101] direction with PBC in [010] direction

15. Injection from LUMO (frozen lattice, 0 K)

16. LUMO Injection cont’d

17. LUMO Injection (frozen lattice) cont’d
PMOL(t)
This plot suggests the presence of two distinct timescales for the process of interfacial electron transfer from the catechol LUMO: initial electron injection, followed by anisotropic delocalization within the crystal.

18. Injection from LUMO+1

19. Injection from LUMO+1 (frozen lattice, 0 K)

20. LUMO+1 Injection (frozen lattice) cont’d
PMOL(t)

21. LUMO Injection at Finite Temperature (100 K)
PMOL(t)
0 K
100 K

22. Hole-Relaxation Dynamics in the Semiconductor Band Gap
t=0 ps
t=15 ps
Super-exchange hole transfer

23. Coherent Hole-Tunneling Dynamics

24. Investigation of Coherent-Control
t = 200 fs, w12
t= k*t
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)

25. Investigation of Coherent-Control cont’d
2-p pulses (200 fs spacing)
14 ps
60 ps
Time, ps

26. Investigation of Coherent-Control
2-p pulses (200 fs spacing)
2 ps
42 ps

27. Reduced Density Matrix: Hole-States
Index x indicates a particular initial nuclear configuration

28. 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.

29. Investigation of Entanglement cont’d
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.

30. Investigation of Coherences cont’d
Compute the subspace density matrix explicitly

31. Investigation of Coherences cont’d
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.

32. Investigation of Coherences cont’d
Diagonal Elements of the Reduced Density Matrix
T=100 K
Off-Diagonal Elements of the Reduced Density Matrix
T=100 K
Time, ps
Time, ps

33. Investigation of Coherences cont’d
Measure of decoherence: +

34. Decoherence Dynamics cont’d
100 fs

35. 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.

36. Acknowledgments Thank you !
NSF Nanoscale Exploratory Research (NER) Award ECS#
NSF Career Award CHE#
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 !