Examples of time-dependent control targets
Objective: Determine a laser pulse which achieves as prescribed
goal that
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
Control the path of the current with laser
left lead
right lead
Control the path of the current with laser
left lead
right lead
Optimal control of time-dependentTHANKS
targets
OUTLINE
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 Castro
Esa Räsänen
Angel Rubio (San Seb)
Kevin Krieger
Jan Werschnik
Ioana Serban
Optimal control of static targets
(standard formulation)
For given target state Φf , maximize the functional:
J1  T   f
2
ˆ T 
 T   f  f T   T  O
Optimal control of static targets
(standard formulation)
For given target state Φf , maximize the functional:
J1  T   f
2
ˆ T 
 T   f  f T   T  O
Ô
Optimal control of static targets
(standard formulation)
For given target state Φf , maximize the functional:
J1  T   f
2
ˆ T 
 T   f  f T   T  O
Ô
with the constraints:
T

2
J 2     dt  t   E 0 
0

E0 = given fluence
Optimal control of static targets
(standard formulation)
For given target state Φf , maximize the functional:
J1  T   f
2
ˆ T 
 T   f  f T   T  O
Ô
with the constraints:
T

2
J 2     dt  t   E 0 
0

T
E0 = given fluence


ˆ  t  t 
J 3 , ,   2 Im  dt t    t  Tˆ  V
0
Optimal control of static targets
(standard formulation)
For given target state Φf , maximize the functional:
J1  T   f
2
ˆ T 
 T   f  f T   T  O
Ô
with the constraints:
T

2
J 2     dt  t   E 0 
0

T
E0 = given fluence


ˆ  t  t 
J 3 , ,   2 Im  dt t    t  Tˆ  V
0
TDSE
Optimal control of static targets
(standard formulation)
For given target state Φf , maximize the functional:
J1  T   f
2
ˆ T 
 T   f  f T   T  O
Ô
with the constraints:
T

2
J 2     dt  t   E 0 
0

T
E0 = given fluence


ˆ  t  t 
J 3 , ,   2 Im  dt t    t  Tˆ  V
0
GOAL: Maximize J = J1 + J2 + J3
TDSE
Set the total variation of J = J1 + J2 + J3 equal to zero:
Control equations
Algorithm
1. Schrödinger equation with initial condition:
 J  0  i  (t )  Hˆ (t ) (t ),  (0)  
Forward propagation

t
2. Schrödinger equation with final condition:
 J  0  it  (t )  Hˆ (t ) (t ),
 (T )  Oˆ  (T )
Backward propagation
3. Field equation:
 J  0 
 (t ) 
1

Im  (t ) ˆ  (t )
New laser field
Algorithm monotonically convergent: W. Zhu, J. Botina, H. Rabitz,
J. Chem. Phys. 108, 1953 (1998))
Control of currents
2
|t| and j (t)
l=1
l = -1
l=0
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:
ˆ  T
J1    T  O
with
ˆ  1ˆ 
O
bound

i
i
i
Standard OCT algorithm (forward-backward propagation) does
not converge:
ˆ before the backward-propagation eliminates the
Acting with O
smooth (numerically friendly) part of the wave function.
Instead of forward-backward propagation, parameterize the laser pulse
to be optimized in the form
  t   f  t  cos  0 t  ,
with ω0 = 0.114 a.u. (λ = 400 nm)


2
2
f  t     fn
c os  n t   gn
sin  n t   ,
T
T
n 1 

N
with ωn = 2πn/T
Choose N such that maximum frequency is 2ω0 or 4ω0 . T is fixed to 5 fs.
Maximize J1 (f1…fN, g1…gN) directly with constraints:
i 
f  0  f  T   0

N
f
n 1
n
0
 ii  0 dt 2 (t)  E0 .
T
using algorithm NEWUOA (M.J.D. Powell, IMA J. Numer. Analysis 28, 649 (2008))
of initial pulse
of initial pulse
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.
Control of electron localization in double quantum dots:
t = 0 ps
t = 1.16 ps
t = 2.33 ps
t = 3.49 ps
t = 4.66 ps
t = 5.82 ps
E. Räsänen, A. Castro, J. Werschnik, A. Rubio, E.K.U.G., Phys. Rev. B 77, 085324 (2008).
target state:
f = first excited state
(lives in the well on the right-hand side)
Optimization results
Optimized pulse
1  (T )
2
 99.91%
Occupation numbers
Spectrum
3
0
1
2
1
0
2
1
0 3
OCT finds a combination of
several transition processes
E
algorithm
Forward propagation of TDSE  (k)
Backward propagation of TDSE  (k)
new field:
~ k 1 t    1 Im  k  t  ˆ  k  t 

(W. Zhu, J. Botina, H. Rabitz, J. Chem. Phys. 108, 1953 (1998))
algorithm
Forward propagation of TDSE  (k)
Backward propagation of TDSE  (k)
~ k 1 t    1 Im  k  t  ˆ  k  t 

new field:
(W. Zhu, J. Botina, H. Rabitz, J. Chem. Phys. 108, 1953 (1998))
With spectral constraint:
k 1

filter function:
or
t  : F f  F ~ k 1 t 
2
2
f ω  exp  γω  ω0   exp  γω  ω0  
2
2
f ω  1  exp  γω  ω0   exp  γω  ω0  
J. Werschnik, E.K.U.G., J. Opt. B 7, S300 (2005)
E
Frequency constraint:
Only direct transition frequency 0 allowed
Spectrum of optimized pulse
1  (T )
2
 0.9997
occupation numbers
Time-Dependent Density
Frequency constraint:
Selective transfer via intermediate state 2
E
ω0 2
ω2 1
0 
2 
1
Spectrum of optimized pulse
occupation numbers
Time-Dependent Density
Frequency constraint:
Selective transfer via intermediate state 3
ω0 3
ω3 1
0 
3 
1
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
Maximize
J  J1  J 2  J 3
T
1
ˆ t  t 
J1     dt t  O
T0
T

2
J 2     dt  t   E 0 
0

T


ˆ  t  t 
J 3 , ,   2 Im  dt t    t  Tˆ  V
0
Set the total variation of J = J1 + J2 + J3 equal to zero:
Control equations
Algorithm
1. Schrödinger equation with initial condition:
 J  0  i  (t )  Hˆ (t ) (t ),  (0)  
Forward propagation

t
2. Schrödinger equation with final condition:
Inhomogenous TDSE :
i ˆ
 J  0  
ˆ 
i t  H (t )   (t )   T O(t ) (t ),
 (T )  0
Backward propagation
3. Field equation:
 J  0 
 (t ) 
1

Im  (t ) ˆ  (t )
New laser field
Y. Ohtsuki, G. Turinici, H. Rabitz, JCP 120, 5509 (2004)
I. Serban, J. Werschnik, E.K.U.G. Phys. Rev. A 71, 053810 (2005)
Control of path in Hilbert space
ˆ t   t  t 
O
t   0 t e
with
 0 t 
2
o t
0  1 t e
given target occupation, and
1t
 t 
1
2
1
1
Goal: Find laser pulse that reproduces |αo(t)|2
I. Serban, J. Werschnik, E.K.U.G. Phys. Rev. A 71, 053810 (2005)
 t 
0
2
Control path in real space
ˆ t   r  r0 t  
O
1
 r  r  t  
e
2
2
2
0
2 2
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(113) (ISBN: 9780819464040, doi: 10.1117/12.680065); also on
arXiv:0707.1874
Control of time-dependent density of hydrogen atom in laser pulse
Control of charge transfer along selected pathways
Trajectory 1
Trajectory 2
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:
H     dte
it
2
d
dt t


 3

  d r z   r, t 




2
Lmax
Maximize:
J1   L max H  L
L 1
0 
To optimize the 7th harmonic of ω0 , choose coefficients as, e.g.,
α7= 4, α3 = α5 = α9 = α11 = -1
Harmonic generation of helium atom (TDDFT calculation in 3D)
3
5
7
9
11 13 15 17 19 21
Enhancement of 7th harmonic
xc functional used: EXX
Research&Training Network
SFB 450
SFB 658
SPP 1145