objective: determine a laser pulse which achieves as prescribed goal that
DESCRIPTION
Examples of time-dependent control targets. Objective: Determine a laser pulse which achieves as prescribed goal that. the wave function follows a given path in Hilbert space (i.e. a given TD wave function) the density should follow a given classical trajectory r(t) - PowerPoint PPT PresentationTRANSCRIPT
Objective: Determine a laser pulse which achieves as prescribed goal that
Examples of time-dependent control targets
a) the wave function follows a given path in Hilbert space (i.e. a given TD wave function)
b) the density should follow a given classical trajectory r(t)
c) a given peak in the HHG spectrum is enhanced
left lead right lead
Control the path of the current with laser
left lead right lead
Control the path of the current with laser
OUTLINE
• Optimal Control Theory (OCT) of static targets -- OCT of current in quantum rings -- OCT of ionization -- OCT of particle location in double well with frequency constraints
• Optimal Control of time-dependent targets -- OCT of path in Hilbert space -- OCT of path in real space -- OCT of harmonic generation
THANKS
Alberto CastroEsa RäsänenAngel Rubio (San Seb)Kevin KriegerJan WerschnikIoana Serban
Optimal control of time-dependent targets
OUTLINE THANKS
TOTTTTJ f f 2
f 1
Optimal control of static targets(standard formulation)
For given target state Φf , maximize the functional:
Optimal control of static targets(standard formulation)
TOTTTTJ f f 2
f 1
Ô
For given target state Φf , maximize the functional:
Optimal control of static targets(standard formulation)
TOTTTTJ f f 2
f 1
Ô
0
T
0
22 EtdtJ E0 = given fluence
with the constraints:
For given target state Φf , maximize the functional:
T
0
t3 t tVT tdtIm2,,J
Optimal control of static targets(standard formulation)
TOTTTTJ f f 2
f 1
Ô
0
T
0
22 EtdtJ E0 = given fluence
with the constraints:
For given target state Φf , maximize the functional:
Optimal control of static targets(standard formulation)
TOTTTTJ f f 2
f 1
Ô
0
T
0
22 EtdtJ E0 = given fluence
with the constraints:
TDSE
T
0
t3 t tVT tdtIm2,,J
For given target state Φf , maximize the functional:
Optimal control of static targets(standard formulation)
TOTTTTJ f f 2
f 1
Ô
0
T
0
22 EtdtJ E0 = given fluence
with the constraints:
TDSE
T
0
t3 t tVT tdtIm2,,J
For given target state Φf , maximize the functional:
GOAL: Maximize J = J1 + J2 + J3
Control equations
1. Schrödinger equation with initial condition:
2. Schrödinger equation with final condition:
3. Field equation:
ˆ( ) ( ) ( ), (0)ti t H t t
ˆˆ( ) ( ) ( ), ( ) ( )ti t H t t T O T
1ˆ( ) Im ( ) ( )t t t
0J
0J
0J
Set the total variation of J = J1 + J2 + J3 equal to zero:
Algorithm
Forward propagation
Backward propagation
New laser field
Algorithm monotonically convergent: W. Zhu, J. Botina, H. Rabitz, J. Chem. Phys. 108, 1953 (1998))
Control of currents
l = -1 l = 1
l = 0
|t||t|2j (t)j and
I ~ A
E. Räsänen, A. Castro, J. Werschnik, A. Rubio, E.K.U.G., PRL 98, 157404 (2007)
OCT of ionization
• Calculations for 1-electron system H2+ in 3D
• Restriction to ultrashort pulses (T<5fs)
nuclear motion can be neglected
• Only linear polarization of laser (parallel or
perpendicular to molecular axis)
• Look for enhancement of ionization by pulse-shaping
only, keeping the time-integrated intensity (fluence)
fixed
Control target to be maximized:
1ˆJ T O T
with bound
iii1O
Standard OCT algorithm (forward-backward propagation) does not converge:
Acting with before the backward-propagation eliminates the smooth (numerically friendly) part of the wave function.
O
Instead of forward-backward propagation, parameterize the laser pulse to be optimized in the form
0t cot s t , f
N
n nn
n1
n
2 2cos t sin t ,
T Tt
ff g
Maximize J1 (f1…fN, g1…gN) directly with constraints:
N
nn 1
T 200
i f 0 f T 0 f 0
ii dt (t) E .
using algorithm NEWUOA (M.J.D. Powell, IMA J. Numer. Analysis 28, 649 (2008))
with ωn = 2πn/T
with ω0 = 0.114 a.u. (λ = 400 nm)
Choose N such that maximum frequency is 2ω0 or 4ω0 . T is fixed to 5 fs.
Ionization probability for the initial (circles) and the optimized (squares) pulse as function of the peak intensity of the initial pulse. Pulse length and fluence is kept fixed during the optimization.
of initial pulse of initial pulse
E. Räsänen, A. Castro, J. Werschnik, A. Rubio, E.K.U.G., Phys. Rev. B 77, 085324 (2008).
t = 0 ps t = 1.16 ps t = 2.33 ps
t = 3.49 ps t = 4.66 ps t = 5.82 ps
Control of electron localization in double quantum dots:
target state: f = first excited state(lives in the well on the right-hand side)
Optimization results
Optimized pulse Occupation numbers
21 ( ) 99.91%T
Spectrum
OCT finds a combination of several transition processes
0 12 1
0 2
0 33 1
E
algorithm
Forward propagation of TDSE (k)
Backward propagation of TDSE (k)
new field: tˆtIm
1t~ kk1k
(W. Zhu, J. Botina, H. Rabitz, J. Chem. Phys. 108, 1953 (1998))
algorithm
Forward propagation of TDSE (k)
Backward propagation of TDSE (k)
new field: tˆtIm
1t~ kk1k
(W. Zhu, J. Botina, H. Rabitz, J. Chem. Phys. 108, 1953 (1998))
With spectral constraint:
filter function:
t~f:t 1k1k FF
20
20 ωωγexpωωγexpωf
20
20 ωωγexpωωγexp1ωf or
J. Werschnik, E.K.U.G., J. Opt. B 7, S300 (2005)
Frequency constraint: Only direct transition frequency 0 allowed
E
Spectrum of optimized pulse occupation numbers
21 ( ) 0.9997T
Time-Dependent Density
Frequency constraint: Selective transfer via intermediate state 2
120 2102 ωω
E
Spectrum of optimized pulse occupation numbers
Time-Dependent Density
3
130 3103 ωω
Frequency constraint: Selective transfer via intermediate state
E
Frequency constraint: All resonances excluded
Spectrum of optimized pulse occupation numbers
All pulses shown give close
to 100% occupation at the
end of the pulse
OPTIMAL CONTROL OF TIME-DEPENDENT TARGETS
Maximize321 JJJJ
T
0
1 ttOtdtT
1J
T
0
t3 ttVTtdtIm2,,J
0
T
0
22 EtdtJ
Control equations
1. Schrödinger equation with initial condition:
2. Schrödinger equation with final condition:
3. Field equation:
ˆ( ) ( ) ( ), (0)ti t H t t
1ˆ( ) Im ( ) ( )t t t
0J
0J
0J
Set the total variation of J = J1 + J2 + J3 equal to zero:
Algorithm
Forward propagation
Backward propagation
New laser field
Inhomogenous TDSE :ˆˆ ( ) ( ) ( ) ( ), ( ) 0t
ii H t t O t t T
T
I. Serban, J. Werschnik, E.K.U.G. Phys. Rev. A 71, 053810 (2005)Y. Ohtsuki, G. Turinici, H. Rabitz, JCP 120, 5509 (2004)
Control of path in Hilbert space
tttO
1et0ett t1
t0
1o with
2
0 t given target occupation, and 2 0
2 1 t1t
I. Serban, J. Werschnik, E.K.U.G. Phys. Rev. A 71, 053810 (2005)
Goal: Find laser pulse that reproduces |αo(t)|2
Control path in real space
220 2trr
20 e2
1trrtO
with given trajectory r0(t) .
Algorithm maximizes the density along the path r0(t):
I. Serban, J. Werschnik, E.K.U.G. Phys. Rev. A 71, 053810 (2005)
J. Werschnik and E.K.U.G., in: Physical Chemistry of Interfaces and Nanomaterials V, M. Spitler and F. Willig, eds, Proc. SPIE 6325,
63250Q(1-13) (ISBN: 9780819464040, doi: 10.1117/12.680065); also on arXiv:0707.1874
Control of time-dependent density of hydrogen atom in laser pulse
Trajectory 2Trajectory 1
Control of charge transfer along selected pathways
Time-evolution of wavepacket with the optimal laser pulse for trajectory 1
Trajectory 1: Results
Start
Lowest six eigenstates
Populations of eigenstates
ground state
first excited state
second excited state
fifth excited state
Trajectory 2
Optimization of Harmonic Generation
Harmonic Spectrum:
2
2i t 3
t
dH dte d r r, t
dt
z
Maximize:
To optimize the 7th harmonic of ω0 , choose coefficients as, e.g., α7= 4, α3 = α5 = α9 = α11 = -1
max
0
L
1 LL 1
J max H L
Enhancement of 7th harmonic
3 5 7 9 11 13 15 17 19 21
Harmonic generation of helium atom (TDDFT calculation in 3D)
xc functional used: EXX
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