esa räsänen - sourceforgeelk.sourceforge.net/cecam/rasanen_optimal_control.pdfoptimal control and...
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Esa Räsänen
Nanoscience Center, Department of Physics, University of Jyväskylä, Finland
Optimal control and quantum dynamics
CECAM, July 23, 2011
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Some motivation
TDDFT I Side remark: “Initialstate dependence”
Optimal control theory (OCT) and applications
Control of excitations
Control of 1electron ionization
Control of 2electron ionization (OCT & TDDFT)
Outline
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Longterm objectives
lasercontrol of moleculeselectromagnetic control of
lowdimensional systems
4th generation solar cells
Laarmann et al. (2007)
Goldman (2007)
Wagner (2009)
Delft Qubit Project
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TDDFT: Initialstate dependence
system classicism all the trouble!
Reminder: we propagate individual particles exposed to
density atprevious times
initialKS wavefunction
initialmanybodywave function
N. T. Maitra and K. Burke, Phys. Rev. A 63, 042501 (2001)
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B
Example: Two 2D Gaussian wave packets in magnetic field
initially at rest interaction effects at t > 0: => repulsion => Lorentz force => “flowerlike” motion
initial wave function
initial density
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Quiz
1. How to construct the initial KohnSham orbitals?
(a) orthonormalize the wavepacket orbitals and use them(b) simply use the wave packets as initial KohnSham orbitals(c) take the square root of the exact density (divided by two)(d) they cannot be properly constructed in this case
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Quiz
1. How to construct the initial KohnSham orbitals?
(a) orthonormalize the wavepacket orbitals and use them(b) simply use the wave packets as initial KohnSham orbitals(c) take the square root of the exact density (divided by two)(d) they cannot be properly constructed in this case
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Quiz
1. How to construct the initial KohnSham orbitals?
(a) orthonormalize the wavepacket orbitals and use them(b) simply use the wave packets as initial KohnSham orbitals(c) take the square root of the exact density (divided by two)(d) they cannot be properly constructed in this case
2. Are there alternative choices for orthonormal orbitals that give the exact initial density?
(a) no(b) yes one other choice(c) yes infinitely many choices
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Quiz
1. How to construct the initial KohnSham orbitals?
(a) orthonormalize the wavepacket orbitals and use them(b) simply use the wave packets as initial KohnSham orbitals(c) take the square root of the exact density (divided by two)(d) they cannot be properly constructed in this case
2. Are there alternative choices for orthonormal orbitals that give the exact initial density?
(a) no(b) yes one other choice(c) yes infinitely many choices => Harriman construction
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Harriman construction
J. E. Harriman, Phys. Rev. A 24, 680 (1980):
“For any nonnegative, normalized density an arbitrary number of orthonormal orbitals can be constructed with squares which sum to the given density.”
Harriman orbitals:
with any set of
and with
It is straightforward to show that
(1)
(2)
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classical
Harriman (ALDA)
exact
ALDA
Timepropagation (animation follows...)
Fig: Initial density for the timepropagation with different methods.
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Gateway Arch in St. Louis, Missouri, USA
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Optimal control: Overview
from H. Rabiz et al., Science 288, 824 (2000)
General goals: (i) control of chemical reactions (e.g. molecular design),(ii) coherent control of spin/charge operations (qubits)
Classical control since 1697
“Traditional” control in chemistry: Learningloop experiments
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Quantum optimal control theory (OCT)
Key question: What is the external timedependent field that drives the system into a predefined goal?
control functions
Usually the control function is an electric field (laser pulse)
Most commonly the objective is the transition probability to a target state
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Formulation of OCT Find the extremal points of the functional
targetfunctional
fulfillment ofthe TDSE
field constraint
here is a projection operator
(with fixed fluence)
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Control equations
Forward propagation for
Backward propagation for
Solution field:
These selfconsistent equations are solved iteratively (various algorithms).
with
For a review, see J. Werschnik and E.K.U. Gross, J. Phys. B: At. Mol. Opt. Phys. 40, R175R211 (2007).
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Application 1: Control of current in a quantum ring
Ihn et al. (2001)Lorke et al. (2000)
30 nm
Experiments:
Model:
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Coherent spinswitch / singlequbit gate
l = 1 l = 1
l = 0
E.R., A. Castro, J. Werschnik, A. Rubio, and E.K.U. Gross, Phys. Rev. Lett. 98, 157404 (2007)
“singlequbit gate”
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Application 2: Enhancing ionization through pulse shaping
Optimize this!
Target operator:
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Pulse constraints
Representation in the basis
Sumrule constraint:
Endpoints:
Cutoff frequency:
where the initial frequency (before optimization) is a typical value for frequencydoubled Ti:S lasers with
Pulse length: eight cycles corresponding to
Pulse strength (fluence) fixed:
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Effect of pulse optimization
(a) parallel polarization (b) perpendicular polarization
A. Castro, E. R., A. Rubio, and E.K.U. Gross, Europhys. Lett. 87, 53001 (2009)
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Application 3: Enhancing ionization in a twoparticle system
1D model with softCoulomb interaction=> exactly solvable (on a 2D grid x and y as electron coordinates)=> in TDDFT with 1DLDA
Main idea:
Use the density (outside the molecule) as the target for ionization within TDDFTOCT. Compare the result with the exact case.
For formal “marriage” of OCT and TDDFT, seeA. Castro, J. Werschnik, and E.K.U. Gross, arXiv:1009.2241
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Optimal control theory (OCT) is a powerful tool to achieve quantum mechanical targets via optimization of the external field.
Ionization of small molecules can be remarkably enhanced by pulse shaping. A density target can be used to describe manyelectron ionization, and optimization within ALDA produces pulses that work well
in a real system (exact solution / experiment!)
Summary
OCTOPUS code (realspace & realtime DFT/TDDFT) freely available at: www.tddft.org A. Castro et al., Phys. Stat. Sol. (b) 243, 2465 (2006)
charge & spinionization
HHG
dissociationcurrent
Thanks to: Ville Kotimäki, Tobias Kramer, Alberto Castro, Jan Werschnik, Maria Hellgren, Angel Rubio, E.K.U. Gross
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Thank you