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

SFB 450SFB 658SPP 1145

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