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 Castro Esa Räsänen Angel Rubio (San Seb) Kevin Krieger Jan Werschnik Ioana Serban Optimal control of time-dependent targets OUTLINE THANKS

Optimal control of static targets (standard formulation) For given target state Φ f, maximize the functional:

Optimal control of static targets (standard formulation) Ô For given target state Φ f, maximize the functional:

Optimal control of static targets (standard formulation) Ô E 0 = given fluence with the constraints: For given target state Φ f, maximize the functional:

Optimal control of static targets (standard formulation) Ô E 0 = given fluence with the constraints: For given target state Φ f, maximize the functional:

Optimal control of static targets (standard formulation) Ô E 0 = given fluence with the constraints: TDSE For given target state Φ f, maximize the functional:

Optimal control of static targets (standard formulation) Ô E 0 = given fluence with the constraints: TDSE For given target state Φ f, maximize the functional: GOAL: Maximize J = J 1 + J 2 + J 3

Control equations 1. Schrödinger equation with initial condition: 2. Schrödinger equation with final condition: 3. Field equation: Set the total variation of J = J 1 + J 2 + J 3 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  | 2 j (t)j and I ~  A E. Räsänen, A. Castro, J. Werschnik, A. Rubio, E.K.U.G., PRL 98, (2007)

OCT of ionization Calculations for 1-electron system H 2 + 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: with 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.

Instead of forward-backward propagation, parameterize the laser pulse to be optimized in the form Maximize J 1 (f 1 …f N, g 1 …g N ) directly with constraints: using algorithm NEWUOA (M.J.D. Powell, IMA J. Numer. Analysis 28, 649 (2008)) with ω n = 2πn/T with ω 0 = a.u. (λ = 400 nm) Choose N such that maximum frequency is 2ω 0 or 4ω 0. 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

E. Räsänen, A. Castro, J. Werschnik, A. Rubio, E.K.U.G., Phys. Rev. B 77, (2008). t = 0 pst = 1.16 pst = 2.33 ps t = 3.49 pst = 4.66 pst = 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 pulseOccupation numbers

Spectrum OCT finds a combination of several transition processes E

algorithm Forward propagation of TDSE   (k) Backward propagation of TDSE   (k) new field: (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: (W. Zhu, J. Botina, H. Rabitz, J. Chem. Phys. 108, 1953 (1998)) With spectral constraint: filter function: 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

Time-Dependent Density

Frequency constraint: Selective transfer via intermediate state E Spectrum of optimized pulse occupation numbers

Time-Dependent Density

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 Maximize

Control equations 1. Schrödinger equation with initial condition: 2. Schrödinger equation with final condition: 3. Field equation: Set the total variation of J = J 1 + J 2 + J 3 equal to zero: Algorithm Forward propagation Backward propagation New laser field Inhomogenous TDSE : I. Serban, J. Werschnik, E.K.U.G. Phys. Rev. A 71, (2005) Y. Ohtsuki, G. Turinici, H. Rabitz, JCP 120, 5509 (2004)

Control of path in Hilbert space with given target occupation, and I. Serban, J. Werschnik, E.K.U.G. Phys. Rev. A 71, (2005) Goal: Find laser pulse that reproduces |α o (t)| 2

Control path in real space with given trajectory r 0 (t). Algorithm maximizes the density along the path r 0 (t): I. Serban, J. Werschnik, E.K.U.G. Phys. Rev. A 71, (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: , doi: / ); also on arXiv:

Control of time-dependent density of hydrogen atom in laser pulse

Trajectory 2 Trajectory 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: Maximize: To optimize the 7 th harmonic of ω 0, choose coefficients as, e.g., α 7 = 4, α 3 = α 5 = α 9 = α 11 = -1

Enhancement of 7 th harmonic Harmonic generation of helium atom (TDDFT calculation in 3D) xc functional used: EXX

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