Coherent Control of Molecules

Edited by

Benjamin Lasorne

Graham Worth

Coherent Control of Molecules

Edited by

Benjamin LasorneDepartment of ChemistryImperial College LondonSouth KensingtonLondonSW7 2AZUnited Kingdom

and

Graham A. WorthSchool of ChemistryUniversity of BirminghamEdgbastonBirminghamB15 2TTUnited Kingdom

ii

Suggested Dewey Classification: 541.2

ISBN 0-9545289-5-6

Published by

Collaborative Computational Projecton Molecular Quantum Dynamics (CCP6)

Daresbury LaboratoryDaresburyWarringtonWA4 4ADUnited Kingdom

c!CCP6 2006

iii

Preface vi

Ab Initio Design of Laser Pulses for Controlling Molecular MotionGabriel G. Balint-Kurti!, Frederick R. Manby, Qinghua Ren,Shiyang Zou, Maxim Artamonov, Tak-San Ho and Herschel Ra-bitz

1

Ultrafast Control of Coherent Electronic ExcitationMatthias Wollenhaupt, Tim Bayer, Andreas Prakelt, CristianSarpe-Tudoran and Thomas Baumert!

7

Learning from Adaptive Feedback: An Analysis of Two Liquid StateExperimentsPaul Brumer!, Kunihito Hoki and Michael Spanner 12

Laser Control of Nuclear and Electron Dynamics: Bond Selective Pho-todissociation and Electron CirculationIngo Barth, Leticia Gonzalez, C. Lasser, Jorn Manz! and TamasRozgonyi

18

Optimal Control of Dissipative Quantum Phase Space DynamicsKeith H. Hughes! and Meilir Hywel 28

Quantum Direct Dynamics Applied to the Intelligent Control of BenzenePhotochemistryBenjamin Lasorne!, Michael J. Bearpark, Michael A. Robb andGraham A. Worth

34

Optimal Control Theory: Applications to Polyatomic SystemsVolkhard May! 40

Instantaneous Dynamics and Quantum Control: Principles and Appli-cationsPhilipp Marquetand!, Stefanie Grafe, Volker Engel, ChristophMeier

45

Optimal Control of Multidimensional Vibronic Dynamics: AlgorithmicDevelopments and Applications to 4D-PyrazineHans-Dieter Meyer!, Luxia Wang and Volkhard May 50

Optimal Control of Nonadiabatic Photochemical Processes Induced byConical IntersectionsYukiyoshi Ohtsuki!, Mayumi Abe, Yuichi Fujimura, ZhenggangLan and Wolfgang Domcke

56

iv

An Exploration of Optical Control of Quantum Dynamics in the LiquidPhaseStuart A. Rice! 61

Coherent Control and Coherence Spectroscopy of Rotational Wavepack-ets in Dissipative MediaS. Ramakrishna, Adam Pelzer and Tamar Seideman! 69

Tubular Image States and Trapping on the NanoscaleDvira Segal, Petr Kral and Moshe Shapiro! 77

Design of Femtosecond Pulse Sequences to Control Photochemical Re-actionsDavid J. Tannor! 86

Molecular Axis Alignment via Controlled Molecular RotationJonathan G. Underwood! 97

Coherent Control of Decoherence in Diatomic MoleculesMatthijs P. A. Branderhorst, Pablo Londero, Piotr Wasylczyk,Ian A. Walmsley!, Constantin Brif, Herschel Rabitz and RobertL. Kosut

103

Quantum Cartography: Mapping the Control LandscapeNicholas T. Form and Benjamin J. Whitaker! 107

! speakers at the meeting

v

B. Lasorne and G. A. Worth (eds.)Coherent Control of Molecules

c! 2006, CCP6, Daresbury

Preface

This booklet was produced in connection with the CCP6 Workshop on “TheCoherent Control of Molecules” held at the Paragon Hotel, Birmingham, U.K.,2-5 July, 2006. The meeting brought together people from a diverse range ofbackgrounds working in this area, and was sponsored generously by CCP6 andby CoCoChem. CCP6 is the 6th Computational Chemistry Project (“Molec-ular Quantum Dynamics”) of the UK Engineering and Physical Sciences Re-search Council (EPSRC), and has funded a series of such workshops. Detailsof these and of the other activities of CCP6 may be found on the website(www.ccp.ac.uk). CoCoChem is a new EPSRC national network responsiblefor promoting the field of coherent control. There was also a poster sessionsponsored by the Quantum information, Quantum optics & Quantum control(QQQ) group of the IoP.

The use of laser fields to control chemistry is now becoming a reality throughthe use of optimal control and feedback loops. The workshop was a timelymeeting to investigate the present state of the art, and it combined leadingrepresentatives from both theory and experiment. All speakers at the meetingcontributed an article in this booklet and this represents a good overview of thefield.

We should like to warmly thank all these contributors for their e!orts, andmore generally all those who attended the workshop, presented posters, and tookpart in the spirited discussions.

B. Lasorne, LondonG. A. Worth, BirminghamJuly 2006

vi

B. Lasorne and G. A. Worth (eds.)Coherent Control of Molecules

c! 2006, CCP6, Daresbury

Ab Initio Design of Laser Pulses for Controlling MolecularMotion

Gabriel G. Balint-Kurti, Frederick R. Manby, Qinghua Ren, Shiyang ZouSchool of Chemistry, University of Bristol, Bristol BS8 1TS, UK

Maxim Artamonov, Tak-San Ho and Herschel RabitzDepartment of Chemistry, Princeton University, Princeton, New Jersey 08544

I. INTRODUCTION

The concept of using specially designed laser pulses to control chemical pro-cesses began as a theoretical concept [1–3] but has now been realized in anincreasing numbers of successful experiments [4–9] especially those guided byclosed loop techniques [10, 11].

The work described here attempts to apply high level ab initio theory to de-signing laser pulses in a quantitative manner which should be amenable to directexperimental verification. The calculations begin with ab initio electronic struc-ture calculations of the energy of a molecule in a static electric field [12]. Thisis termed as the Electric Nuclear Born-Oppenheimer (ENBO) approximation.In this method the energy of the molecule is computed as a function of: 1) thegeometry of the molecule; 2) the electric field strength and 3) the orientationof the molecule with respect to the electric field direction. The validity of thisapproximation is discussed in detail in Ref. 12 and it is shown to be applicableto infrared excitation of H2. The ENBO approximation takes account of the in-teraction of the molecule with the electric field to all orders of the field strength,including the two lowest order terms of the dipole moment and the polarizability.

The application of the ENBO approximation, in conjunction with optimalcontrol theory and the numerical solution of the time-dependent Schrodingerequation, can then be used to design laser pulses to achieve the desired controlgoals. Model calculations, in which the H2 molecule was aligned with the electricfield direction, were reported in Ref. 12 while more realistic calculations in whichthe a full 3-dimensional treatment is used and where the rotation of the moleculeis fully taken into account have been reported in Ref. 13.

Based on these successful optimal control theory calculations we have beenable to formulate an analytic method for the design of IR laser pulses for thevibrational-rotational excitation of homonuclear diatomic molecules [14]. The

1

method is applicable to systems and situations where the energy level spacingis greater than the Rabi frequency. As the Rabi frequency is proportional tothe square of the field strength, it is in principle always possible to satisfy thiscondition by lowering the field strength su!ciently.

Another recent application of the ENBO approach coupled with optimal con-trol theory has been the demonstration that it is possible to e!ciently deexcitea homonuclear diatomic molecule from its highest vibrational state to its groundvibrational state using a sequence of IR laser pulses [15]. This may permit theuse of such a sequence of optimally designed IR laser pulses in the stabilizationof molecular Bose-Einstein condensates.

II. OPTIMAL CONTROL THEORY

Optimal control theory proceeds by defining an objective functional, J , whichis maximised by varying the control parameters [16]. For our purposes theobjective functional is written in the following form [3]:

J(!) = |!"(T )|""|2 # #0

! T

0[!(t)]4 dt

# 2 Re

"! T

0!$(t)| %

%t+ iH(R, !(t))|"(t)"dt

#, (1)

where " is the wave function of the target state, "(T ) is the wavefunction of thesystem at time T and T is the time at the end of the laser pulse. The first termin Eq. (1) is the main one expressing the fact that we want the wavefunction ofthe system to be equal to that of the target wavefunction at the end of the pulse.The second term tries to minimise the electric field strength and the final termconstrains the wavefunction to obey the time-dependent Schrodinger equation(see Ref. 12).

Functional variation of this objective function provides us with an expressionfor the derivative of J with respect to the electric field, !(t), at time t. Armedwith this expression we can find the time varying electric field which maximisesJ [12].

Fig. 1 shows the results of the optimal control calculations for the designof a laser pulse to achieve the excitation process H2(v = 0, j = 2) $ H2(v! =1, j! = 2). The top panel shows the optimised electric field, the middle panelshows the corresponding frequency spectrum and the bottom panel displays thechange of population in the two vibrational states as a function of time. Wesee that the electric field displayed consists mainly of two frequencies. Thesecorrespond approximately to half the di#erence in energy between the two levelsinvolved and to 1.5 times this frequency. There is also a small contribution from

2

-0.04

-0.02

0

0.02

0.04

0 0.3 0.6 0.9 1.2 1.5

!(t)

/ a

.u.

time / ps

0

20

40

60

80

100

0 0.5 1 1.5 2 2.5 3 3.5

!(")|

/ a

rbitra

ry u

nit

" / 1014

Hz

0

0.2

0.4

0.6

0.8

1

0 0.3 0.6 0.9 1.2 1.5

Popula

tion

time / ps

v=0, j=2

v=1, j=2

FIG. 1: (a.) The optimised electric field as a function of time for the H2(v = 0, j =2) ! H2(v

! = 1, j! = 2) excitation. (b.) Absolute value of the Fourier transform of theoptimised electric field. (c.) The change in populations of the ground state and targetexcited state shown as a function of time.

3

a frequency roughly equal to the transition frequency. The interpretation ofthis spectrum is that the excitation mechanism consists mainly of two processes,a two photon absorption and a Raman processes [12]. We have shown thatdi#erent starting electric fields and di#erent total times can yield many di#erentoptimised fields and di#erent excitation mechanisms.

III. ANALYTIC PULSE DESIGN FOR VIBRATIONAL-ROTATIONALEXCITATION OF HOMONUCLEAR DIATOMICS

Our underlying model for the analytic design of IR laser pulses for vibrational-rotational excitation of homonuclear diatomics is a two state model consistingof a ground state |g" and an excited state |e". The states are coupled by a laserfield,

!(t) = !A0f(t) cos"(t), (2)

where ! is the polarization vector, A0 is the field strength, f(t) is the normalizedpulse envelope, and "(t) is the temporal phase of the field. The Hamiltonian ofthis two level model may be written as

H =$

Eg # #gg!2(t)/2 ##eg!2(t)/2##ge!2(t)/2 Ee # #ee!2(t)/2

%(3)

where Ei is the field-free energy of state |i" and #ij = !i|! · ·!|j" is a matrixelement of the polarizability tensor #. If the pulse envelope f(t) and laserfrequency &(t) % d"/dt are both slowly varied in comparison to the fast opticaloscillations, we can invoke the two-photon rotating wave approximation (RWA)and transform Eq. (3) into a Floquet representation [17]. The correspondingFloquet Hamiltonian may be written as:

H =$

Eg + 2!&(t) #!$ge(t)/2#!$eg(t)/2 Ee

%(4)

where Ei = Ei # #iiA20f

2(t)/4 is the polarization shifted energy of state |i" and$eg(t) = #egA2

0f2(t)/4! represents the e#ective laser-molecular coupling (Rabi

frequency) between the ground and excited states. The diabatic energies, i.e. thediagonal elements of H , cross at the polarization shifted two-photon resonance&2r = (Ee # Eg)/2!, and the adiabatic energies, i.e. the eigenvalues of full H,cannot cross. An avoided crossing arises at & = &2r with energy gap proportionalto $eg(t), which induces a (localized) non-adiabatic transition between the twoadiabatic states.

If the instantaneous frequency of the field, &(t), is modulated so as to continu-ally maintain degeneracy between the two diabatic Floquet levels. This enables

4

0

0.25

0.5

0.75

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Am

plit

ud

e (

A0)

(a)

61.9

62

62.1

62.2

62.3

62.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Fre

qu

en

cy (

TH

z)

Time (ps)

(b)

FIG. 2: (a) pulse envelope; (b) time-modulated frequency of analytic pulse for thecomplete population inversion of H2 molecules between the vibrational levels v = 0, j =0, m = 0 and v = 1, j = 0, m = 0.

us to analytically solve the coupled time-dependent Schrodinger equation for thesystem. We find that the excitation probability Peg is seen to be

Peg = sin2 '

2(5)

and complete excitation is achieved when

' =#egA2

0(s0

4! = ) (6)

where s0 =& !0 f2(t)dt/( depends on the shape of the pulse envelope. Equation

(6) shows that the polarization-shifted resonant two-photon absorption dependson only one parameter, A2

0(s0, which we can loosely refer to as the “pulse area”.Complete excitation can be achieved by adjusting the field strength A0, pulseduration ( and pulse envelope s0.

Figure 2 shows the pulse envelope (upper panel) together with the time-dependent modulated frequency required to achieve complete excitation for the

5

transition v = 0, j = 0, m = 0 $ v = 1, j = 0, m = 0 in H2 [14]. Numerical solu-tion of the coupled time-dependent Schrodinger equation shows that we attainan excitation probability of 98.4% using the pulse predicted analytically in thisway.

Acknowledgments

The Bristol group is grateful to the EPSRC for a research grant. This workwas supported in part by a grant from the United States Department of Energy.QR is grateful to the UK Department for Education and Skills for an ORSASaward. FRM is grateful to the Royal Society for funding under the UniversityResearch Fellowship Scheme.

[1] D. J. Tannor and S. A. Rice, J. Chem. Phys. 83, 5013 (1985).[2] P. Brumer and M. Shapiro, Chem. Phys. Lett. 126, 541 (1986).[3] S. Shi and H. Rabitz, J. Chem. Phys. 92, 364 (1990).[4] A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Seyfried, M. Strehle,

and G. Gerber, Science 282, 919 (1998).[5] R. J. Levis, G. M. Menkir, and H. Rabitz, Science 292, 709 (2001).[6] R. Bartels, S. Backus, E. Zeek, L. Misoguti, G. Vdovin, I. P. Christov, M. M.

Murnane, and H. C. Kapteyn, Nature 406, 164 (2000).[7] S. Vajda, A. Bartelt, E. Kaposta, T. Leisner, C. Lupulescu, S. Minemoto,

P. Rosendo-Francisco, and L. Woste, Chem. Phys. 267, 231 (2001).[8] T. Hornung, R. Meier, and M. Motzkus, Chem. Phys. Lett. 326, 445 (2000).[9] T. C. Weinacht, J. L. White, and P. H. Bucksbaum, J. Phys. Chem. A 103, 10166

(1999).[10] H. Rabitz and S. Shi, in Advances in Molecular Vibrations and Collision Dynamics,

edited by J. Bowman (JAI press, 1991), vol. 1, p. 187.[11] R. S. Judson and H. Rabitz, Phys. Rev. Lett. 68, 1500 (1992).[12] G. G. Balint-Kurti, F. Manby, Q. Ren, M. Artamonov, T. Ho, and H. Rabitz, J.

Chem. Phys. 122, 084110 (2005).[13] Q. Ren, G. G. Balint-Kurti, F. Manby, M. Artamonov, T. Ho, and H. Rabitz, J.

Chem. Phys. 124, 014111 (2006).[14] S. Zou, Q. Ren, G. Balint-Kurti, and F. Manby, Phys. Rev. Lett. 96, 243003

(2006).[15] Q. Ren, G. G. Balint-Kurti, F. Manby, M. Artamonov, T. Ho, and H. Rabitz, (in

press).[16] S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics (Wiley Inter-

science, New York, 2000).[17] C. Zhu, Y. Teranishi, and H. Nakamura, Adv. Chem. Phys. 117, 127 (2001).

6

7

B. Lasorne and G. A. Worth (eds.)

Coherent Control of Molecules

© 2006, CCP6, Daresbury

Ultrafast Control of Coherent Electronic Excitation

Matthias Wollenhaupt, Tim Bayer, Andreas Präkelt, Cristian Sarpe-

Tudoran and Thomas Baumert Universität Kassel, Institut für Physik and CINSaT, Heinrich-Plett-Str. 40,

D-34132 Kassel, Germany

With the advent of ultrashort femtosecond laser pulses the temporal

aspect of the interplay of light and molecular dynamics came to the fore

and became experimentally accessible. The beauty of femtochemistry lies

in our ability to observe [1] and to manipulate [2, 3] ultrafast processes as

they occur. Shaped femtosecond optical laser pulses [4] are the suitable

tools to exert microscopic control on molecular dynamics at the quantum

level. The combination of pulse-shaping techniques with closed loop

adaptive feedback learning algorithms [5–8] allows to optimize virtually

any conceivable observable as reviewed for example in [9, 10]. However, it

is not always possible to deduce the underlying physical mechanism from

the electrical fields obtained by this procedure. Therefore, the need to

bridge the gap between the efficient ‘black box’ closed loop optimal

control methods and detailed understanding of the physical processes

especially in strong laser fields is quite evident. To that end we combine

femtosecond laser techniques with atomic-/ molecular beam techniques

and photoelectron-/ ion detection techniques [11]. So far we have extended

weak field methods to free electrons [12]. New techniques making use of

polarization control in molecular multi-photon excitation [13] and shaped

intense laser pulses for molecular alignment [14] open further dimensions

in this field.

An exciting new strong field control scenario is based on ultrafast

control of electronic coherent excitation. This approach makes explicit use

of the manipulation of the temporal phase of a pulse sequence with

attosecond precision [15]. Experimentally we make use of a 1+2 REMPI

process on potassium atoms. An intense fs-laser couples coherently the 4s

– 4p level and at the same time ionizes the system in a two photon process

8

(see Fig.1). The shape of the photoelectron spectra reflects the temporal

phase of the excited state amplitude c4p(t) [11].

FIG. 1: (a) Schematic of the excitation scheme (potassium-atoms): The bare states are

indicated with thin lines. Thick lines illustrate the dressed state splitting during the

interaction giving rise to a symmetric Autler Townes splitting (left). Selective population of

a dressed state with a tailored pulse train is shown in the right panel, leading to a strongly

asymmetric Autler Townes doublet. (b) Schematic of experimental set-up: tailored pulse

trains are created via applying a phase mask in the Fourier plane of our pulse shaper [16]. In

the case discussed here, the spectrum of our femtosecond laser pulse (785 nm, 30 fs, 0.35–2

µJ) is phase-modulated in frequency domain with a sinusoidal phase function !(") = A

sin[(" – "0) T + #] with A = 0.2, T = 170 fs and "0 = 2.40 fs–1 to produce a sequence of

pulses in time domain separated by T. The pulses are focused on a potassium atomic beam.

The resulting photoelectrons are detected with a magnetic bottle Time of Flight

photoelectron spectrometer.

a

b

9

FIG. 2: The Selective Population of Dressed States (SPODS) is directly mapped into the

measured photoelectron spectra by variation of the phase #. The maximum of the

asymmetric photoelectron distribution alternates between 0.33 eV and 0.52 eV. These results

are obtained at a laser energy of W = 0.5 µJ. A section through the distribution along the

energy axis at # = 0.7 $ – indicated with a trajectory – yields the photoelectron spectrum

(A) where the lower dressed state is selectively populated as depicted in the inset to (A).

Fringes in the spectrum with an energy separation of h/T arise from the interference of the

free electron wave packets [12] launched during the different pulses. Selective population of

the upper dressed state is achieved at # = 1.7 $ as indicated with a trajectory and plotted in

spectrum (B). The signal of the slow photoelectrons at 0.33 eV (S) and the fast

photoelectrons at 0.52 eV (F) – as indicated with the bars – is obtained as a function of the

phase # by taking a section through the distribution along the phase coordinate. The

contrast of F and S, i.e. (F–S)/(F+S) as shown in (C) is a measure of the selectivity of

dressed state population. The phases corresponding to the highest selectivity for population

of the lower dressed state – spectrum (A) – and the upper dressed state – spectrum (B) – are

indicated with arrows.

10

In particular, the photoelectron spectra map the dressed state

population. During the time evolution, the dressed states are characterized

by a time-dependent energy splitting giving rise to the observed Autler-

Townes (AT) splitting [17] in the photoelectron spectra. Employing two-

photon ionization as the non-linear probe step precludes averaging over

the intensity distribution within the laser focus since the ionization

probability is highest in the spatial region of highest laser intensity. This

technique permits us to overcome the common problem of washing out

intensity dependent strong field effects. Making use of adaptive feedback

learning algorithms we are able to control the dressed state population by

more than 90% as seen by the corresponding suppression of one AT

component [18]. With the help of tailored pulse trains we demonstrate

that this Selective Population of Dressed States (SPODS) is highly

selective, tunable (up to 250 meV) and robust [19]. In Figure 2

experimental results – obtained with a pulse train created by applying a

sine mask in the Fourier plane of the pulse shaper (see Fig. 1) – are

displayed.

Since switching between selective population of either dressed states

occurs within a few femtoseconds, this technique is also interesting for

applications in the presence of decoherence processes. SPODS can be

realized with very different pulse shapes making use of diverse physical

mechanisms ranging from Photon Locking [15, 19, 20] to Rapid Adiabatic

Passage [21]. Because SPODS combines high selectivity and tunability

with efficient population transfer, relevant applications to chemistry – so

far investigated theoretically [19, 22] – are within reach.

Acknowledgments

We want to thank the Deutsche Forschungsgemeinschaft – DFG – for financial support.

[1] A. H. Zewail, J. Phys. Chem. 104, 5660 (2000).

[2] M. Shapiro and P. Brumer, in Principles of the Quantum Control of Molecular

Processes, 1 ed. (John Wiley & Sons, Hoboken, New Jersey, 2003).

[3] S. A. Rice and M. Zhao, in Optical control of molecular dynamics. (Wiley, New York,

2000).

[4] A. M. Weiner, Rev. Sci. Instr. 71, 1929 (2000).

[5] R. S. Judson and H. Rabitz, Phys. Rev. Lett. 68, 1500 (1992).

11

[6] T. Baumert, T. Brixner, V. Seyfried, M. Strehle, and G. Gerber, Appl. Phys. B 65, 779

(1997).

[7] D. Meshulach, D. Yelin, and Y. Silberberg, Opt. Comm. 138, 345 (1997).

[8] C. J. Bardeen, V. V. Yakolev, K. R. Wilson, S. D. Carpenter, P. M. Weber, and W. S.

Warren, Chem. Phys. Lett. 280, 151 (1997).

[9] T. Brixner, T. Pfeifer, G. Gerber, M. Wollenhaupt, and T. Baumert, in Femtosecond

Laser Spectroscopy, edited by P. Hannaford (Kluwer Series, 2004), Chap. 9.

[10] H. Rabitz, R. de Vivie-Riedle, M. Motzkus, and K. Kompa, Science 288, 824 (2000).

[11] M. Wollenhaupt, V. Engel, and T. Baumert, Ann. Rev. Phys. Chem. 56, 25 (2005).

[12] M. Wollenhaupt, A. Assion, D. Liese, C. Sarpe-Tudoran, T. Baumert, S. Zamith, M.

A. Bouchene, B. Girard, A. Flettner, U. Weichmann, and G. Gerber, Phys. Rev. Lett. 89,

173001 (2002).

[13] T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt, O. Graefe,

C. Horn, D. Liese, and T. Baumert, Phys. Rev. Lett. 92, 208301 (2004).

[14] C. Horn, M. Wollenhaupt, M. Krug, T. Baumert, R. de Nalda, and L. Banares, Phys.

Rev. A 73, 031401 (2006).

[15] M. Wollenhaupt, A. Assion, O. Bazhan, C. Horn, D. Liese, C. Sarpe-Tudoran, M.

Winter, and T. Baumert, Phys. Rev. A 68, 015401 (2003).

[16] A. Präkelt, M. Wollenhaupt, A. Assion, C. Horn, C. Sarpe-Tudoran, M. Winter, and

T. Baumert, Rev. Sci. Instr. 74, 4950 (2003).

[17] S. H. Autler and C. H. Townes, Phys. Rev. 100, 703 (1955).

[18] M. Wollenhaupt, A. Präkelt, C. Sarpe-Tudoran, D. Liese, and T. Baumert, J. Opt. B 7,

S270 (2005).

[19] M. Wollenhaupt, D. Liese, A. Präkelt, C. Sarpe-Tudoran, and T. Baumert, Chem. Phys.

Lett. 419, 184 (2006).

[20] M. Wollenhaupt, A. Präkelt, C. Sarpe-Tudoran, D. Liese, and T. Baumert, J. Mod. Opt.

52, 2187 (2005).

[21] M. Wollenhaupt, A. Präkelt, C. Sarpe-Tudoran, D. Liese, and T. Baumert, Appl. Phys.

B 82, 183 (2006).

[22] M. Wollenhaupt and T. Baumert, J. Photochem. Photobiol. A 180, 248 (2006).

B. Lasorne and G. A. Worth (eds.)Coherent Control of Molecules

c! 2006, CCP6, Daresbury

Learning from Adaptive Feedback: An Analysis of TwoLiquid State Experiments

Paul Brumer, Kunihito Hoki and Michael SpannerChemical Physics Theory Group, Department of Chemistry,and Center for Quantum Information and Quantum Control,University of Toronto, Toronto, Ontario, Canada, M5S 3H6

I. INTRODUCTION

Current approaches to the quantum control of molecular dynamics can beroughly divided into two distinct philosophies: coherent control and optimalcontrol [1, 2]. Although both rely upon the interaction of lasers and matter tointroduce controllable interference contributions, they do so in philosophicallydi!erent ways. Historically, coherent control has endeavored to construct scenar-ios that explicitly incorporate quantum interference e!ects in well defined ways.The result is a clear relationship between the resultant control and the quantuminterference contributions. By contrast, optimal control, as currently practiced[3], takes advantage of engineering tools designed to optimize the system vari-ables so as to maximize the target cross section within specific constraints. Theresult often calls for complicated laser pulses, and the mechanisms that underliethe resultant control are di"cult to discern. This is particularly true in themodern experimental implementation of adaptive feedback control.

Within the framework of optimal control-adaptive feedback we note an ad-ditional distinction, that between formal theoretical studies and experimentalimplementations. The former assume wide ranging attributes to the apparatusthat constructs the incident laser field, i.e. the field is variable over a large rangeof frequencies and time scales. By contrast, the experimental results are limitedby practicalities, a situation that imposes serious constraints in optimal con-trol. These experimental limitations are far from trivial, and have considerableconsequences, as discussed in specific cases below.

Regardless of the approach, and regardless of the experimental limitations, onewould hope that the currently popular adaptive feedback experimental approachwould expose underlying mechanisms through which control is operative. Evenmore significant would be the discovery of a set of simple control scenarios thatunderlie the e!ect of the complex pulse shapes. Indeed, such a set of simplescenarios, a ”control tool box” if you will, has been developed within the coherent

12

control approach. Scenarios such as [1] bichromatic control, N-photon vs. M-photon control, incoherent interference control, etc., as well as the original [4]Tannor-Rice pump-dump scenario, are sample examples. A thorough analysis ofadaptive feedback experiments would, we hope, expose these elementary controlsteps as part of the overall control scheme.

Of greatest interest to Chemistry is the liquid phase environment. Since theessence of quantum control is to utilize quantum interference e!ects to manip-ulate product cross sections, one may expect that such e!orts are thwarted bycollisional and decoherence e!ects (which destroy quantum features) when ap-plied to liquid phase dynamics. Consequently, e!orts have been ongoing sincethe inception of quantum control to design scenarios that allow for control inliquids [5]. (See also the discussion of liquid phase scenarios in this conference,by Stuart Rice). We have undertaken the analysis of two liquid phase adaptivefeedback experiments, which both had the goal of coherently controlling liquidphase dynamics, in an e!ort to extract their underlying mechanisms. In bothcases, results were surprising. Specifically, neither of the experimental resultsseemed to imply control via coherent quantum interference e!ects, and neitherof the experiments required other than simple qualitative explanations. Thereasons for this can be traced back to the particular experimental apparatus,and the particular optimization target chosen for the feedback algorithm.

We summarize these results below, along with comments on the simplificationsthat emerge from complex environments. Since both studies appear in recentarticles [6–8] we focus solely upon the essential qualitative features learned fromthese investigations.

II. TRANS-CIS ISOMERIZATION IN NK88

Vogt et al. [9] examined control of trans to cis isomerization of ”NK88” (i.e.3,3’-diethyl-2,2’-thiacyanine iodide) in liquid methanol. Using adaptive feed-back, they demonstrated the ability to optimize the yield of the cis productby shaping a 60 nm wide fs laser source. The process takes place by excita-tion from the ground state to the excited state, followed by isomerization. Twocases were studied: (“maximizing”) pulses that increased the extent of tran-cisisomerization and (“minimizing”) pulses that minimized the amount of tran-cis isomerization. In both cases the resultant optimal pulses [9] consisted of anumber of peaks as a function of time, with an overall time width of 4 ps. Fur-ther, the minimizing pulse showed a far more complex structure, suggesting [9]complicated coherent dynamics designed to guide the dynamics away from theproduct cis state.

These seminal results motivated our subsequent study of the dynamics. Wewere able to successfully model these studies using a simple one-dimensionalmodel representing the cis-trans isomerization reaction coordinate, plus a de-

13

coherence term (within a Redfield framework) modeling the remainder of theNK88 degrees of freedom plus the liquid methanol environment. The ability todo so, emphasized later below, can be qualitatively justified from the theory ofdecoherence, as applied to the case of rapid decoherence.

Our studies showed that the experimental results were consistent with controlexercised through alteration of the probability of excitation, rather than con-trol over the dynamics. That is, in the case of maximization the pulse createssignificantly more of the excited state, whereas the minimization pulse aims todecrease the excitated population. Thus, although one would have hoped thatthese results constitute evidence of active control over the coherent motion ofthe cis to trans process, our results show that the experiment can be explainedin a far simpler fashion.

What then of the ”tool box” of control scenarios that we anticipate seeing?For example, control of isomerization by pump-dump [4] or by the CW version[10] applied directly to isomerization, seems ”natural” for isomerization control.Why is it not manifest? A second study that we performed [6] shows thatthe answer relates to the practical restrictions involved in the experiment. Inparticular, the state-of-the-art laser used [9] had a 60 nm wide profile. However,in order to successfully carry out pump-and-dump from the trans to cis wouldhave required a laser with a width of at least 30,000 cm!1. In particular, byrepeating the optimization computationally with an unrestricted laser width wefound that the optimal control scheme was indeed pump-dump, which gave farlarger control than that obtained with the pulse of restricted width. Specifically,the unconstrained pulse showed a final cis population of 0.36, as compared tothe 0.16 result obtained with the 60 nm wide pulse. Of some interest is thefact that the decoherence here is su"ciently rapid so as to result in a modifiedpump-dump scenario wherein the second pulse operates well after the excitedwave packet has decohered.

Decoherence, often viewed (in Chemistry) as arising from a thermal bath, isindeed far more general. Consider, in particular, an N degree-of-freedom (dof)system. If one is interested in a property of only M < N degrees of freedom, i.e.if one ignores the remaining (N !M) dof, then the ignored dof serve to decoherethe observed M dof. In the case of the experiment under consideration, one isactually measuring only one degree-of-freedom, i.e. whether the system is cisor trans. This being the case, the remaining degrees of freedom are correctlytreated, due to the nature of the measurement, as decoherence. Naturally, thefully correct approach would be to compute the full quantum dynamics andthen compute the quantity of interest, which would then entail an averagingover the unobserved degrees of freedom. Here, however, we succeed with amaster equation approach (i.e., Redfield theory), which essentially pre-averagesover the unobserved degrees of freedom. We believe that the results are in goodaccord with experiment due to the rapid decoherence in the system.

14

III. CONTROLLED STOKES EMISSION VS. VIBRATIONALEXCITATION IN METHANOL

A series of experiments [11–15] carried out by the Michigan group consideredcontrol of stimulated Raman scattering (SRS) in liquid methanol. The basicidea was to selectively excite one of two closely-spaced Raman active modes ofmethanol associated with the symmetric and antisymmetric C-H stretch modes.The Raman modes were driven by a strong pump pulse with 150 fs bandwidth.This bandwidth is larger than the spacing of the excited Raman modes butsmaller than the energies of the vibrational states themselves, thus placing theexperiment in the non-impulsive, or transient, regime where the molecules haveenough time to vibrate during the pulse. By using an adaptive feedback algo-rithm to shape a strong pump pulse, significant control over the relative peakheights of the two Raman modes in the Stokes emission was demonstrated.Pulses were found that could selectively enhance the emission from either mode,enhance both modes together, or complete suppress all emission. By assumingthat the intensities of the peaks in the Stokes emission was directly proportionalto the associated vibrational modes, it was concluded that selective vibrationalexcitation had been achieved.

In the absence of a firm theoretical footing, these experiments are being toutedby many [11–20] as an example of quantum or coherent control of mode selectivevibrational excitation in the liquid phase. This view had been temporarily up-held by a proposal for a coherent mechanism of Raman control in a closely relatedexperiment using a double-Gaussian shaped pump pulse [14]. Unfortunately, thisproposed mechanism has been shown to be faulty [21]. We undertook a detailedtheoretical analysis of these experiments, including both molecular excitationand nonlinear optical propagation of the pump and Stokes pulses, in order toextract the underlying control mechanisms.

Our studies [7, 8] revealed many control mechanisms capable of a!ecting therelative intensity of two Stokes peaks in the emitted Raman spectrum. Allthe identified mechanisms are third-order nonlinear optical e!ects, and includetransient stimulated Raman scattering in a collisional environment, saturatedRaman scattering, self- and cross-phase modulation, and focused-beam e!ects.In hindsight this is maybe not surprising since SRS is itself a third-order non-linear optical e!ect, and having a pump pulse strong enough to drive SRS isthen necessarily strong enough to drive competing third-order nonlinear pro-cesses. All mechanisms have the same clear physical interpretation: shaping thepump pulse controls the nonlinear optical response of the medium, which in turncontrols the Stokes emission.

With relation to the vibrational excitation, we were able to draw a number ofconclusions. First, although it was found that the vibrational populations area!ected by the same control mechanisms that a!ect the Stokes spectra, the ratioof the Stokes spectra peak heights does not directly reflect the ratio of the level

15

populations, as was assumed in the experiment. Second, the control was foundto be completely incoherent at the molecular level. The reason for the incoher-ence was that no amplitude at the Stokes wavelengths was present in the initialpump pulse, hence forcing the Stokes mode to build up from noise (spontaneousemission or collisional excitation). These seeding processes are themselves inco-herent, and prevented any chance for coherent dynamics. (Note that coherentdynamics could be induced if both the pump and Stokes frequencies have sig-nificant amplitude in the initial shaped pulse, suggesting again that bandwidthlimitations seriously hampered the range of possible solutions available to theadaptive feedback algorithm). Finally, no coherent quantum interference e!ectsare needed to explain the observed control. Indeed, classical models that treatthe vibrational modes as classical oscillators give the identical equations of mo-tion for the vibrational excitations and the pump and Stokes fields. Interferingpathways of the type considered in coherent control would only contribute tofifth-order nonlinear optical emission, which contributes negligibly to the emis-sion at the Stokes wavelengths.

IV. OUTLOOK FOR COHERENT CONTROL IN LIQUIDS

In light of the conclusion that no coherent mechanisms were present in thetwo liquid phase experiments considered, one is lead to question the viability ofcoherent control in liquids. We anticipate, however, that the lack of coherentcontributions or of traditional control schemes in the exposed mechanisms aredue, in large part, to experimental limitations. That is, the ”optimal” solutionsfound by adaptive feedback loops are not globally optimal, but severely restrictedby the limitations of the state-of-the-art experimental apparatus. Although thisis a seemingly obvious statement, our studies have shown that these restrictionsmay in fact exclude from the very start the possibility of accessing traditionalcoherent control mechanisms, which could be manifest with experimental setupsof wider generality.

[1] M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Pro-cesses (Wiley, N.Y., 2003).

[2] S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics (Wiley, N.Y.,2000).

[3] R. S. Judson and H. Rabitz, Phys. Rev. Lett. 68, 1500 (1992).[4] D. J. Tannor and S. Rice, J. Chem. Phys. 83, 5013 (1985).[5] M. Shapiro and P. Brumer, J. Chem. Phys. 90, 6179 (1989).[6] K. Hoki and P. Brumer, Phys. Rev. Lett. 95, 168305 (2005).[7] M. Spanner and P. Brumer, Phys. Rev. A 73, 023809 (2006).

16

[8] M. Spanner and P. Brumer, Phys. Rev. A 73, 023810 (2006).[9] G. Vogt, G. Krampert, P. Niklaus, P. Nuernberger, and G. Gerber, Phys. Rev.

Lett. 94, 068305 (2005).[10] D. Gruner, M. Shapiro, and P. Brumer, J. Phys. Chem. 96, 281 (1999).[11] T. Weinacht, J. White, and P. Bucksbaum, J. Phys. Chem. A 103, 10166 (1999).[12] B. Pearson, J. White, T. Weinacht, and P. Bucksbaum, Phys. Rev. A 63, 063412

(2001).[13] T. Weinacht and P. Bucksbaum, J. Opt. B: Quantum Semiclass. Opt. 4, R35

(2002).[14] B. Pearson and P. Bucksbaum, Phys. Rev. Lett. 92, 243003 (2004).[15] J. White, B. Pearson, and P. Bucksbaum, J. Phys. B: At. Mol. Opt. Phys. 37,

L399 (2004).[16] H. Rabitz and W. Zhu, Acc. Chem. Res. 33, 572 (2000).[17] J. Geremia and H. Rabitz, Phys. Rev. Lett. 89, 263902 (2002).[18] I. Walmsley and H. Rabitz, Phys. Today 56, 43 (2003).[19] M. Dantus and V. Lozovoy, Chem. Rev. 104, 1813 (2004).[20] T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt,

O. Graefe, C. Horn, D. Liese, and T. Baumert, Phys. Rev. Lett. 92, 208301 (2004).[21] B. Pearson and P. Bucksbaum, Phys. Rev. Lett. 94, 209901(E) (2005).

17

B. Lasorne and G. A. Worth (eds.)Coherent Control of Molecules

© 2006, CCP6, Daresbury

Laser Control of Nuclear and Electron Dynamics: BondSelective Photodissociation and Electron Circulation

Ingo Bartha, Leticia Gonzaleza, C. Lasserb, Jorn Manza and Tamas Rozgonyica Institut fur Chemie und Biochemie, Freie Universitat Berlin,

Takustrasse 3, 14195 Berlin, Germanyb Fachbereich Mathematik, Freie Universitat Berlin,

Arnimallee 6, 14195 Berlin, Germanyc Institute of Structural Chemistry, Chemical Research Center,

Hungarian Academy of Sciences 1025 Budapest, Pusztaszeri ut 59-67, Hungary

I. INTRODUCTION

After the first fundamental approaches to quantum control of chemical reac-tions [1–4], the field has been developed to a powerful tool for driving reactantsto specific products, beyond traditional chemical kinetics or photochemistry, seee.g. the reviews [5, 6] and the monographies [7, 8]. The early methods employedso-called “analytical” laser pulses which can be expressed by means of formulaswith few parameters [1–3, 5–8]. Optimal control [2, 4, 9, 10], and further exten-sions to adaptive design of optimal laser pulses by means of feedback learningalgorithms [11] have introduced much more flexible laser pulses, with myriadsof parameters, stimulating the break-through [12] to impressive experimentalapplications [13–16]. Nevertheless, the original approaches based on analyti-cal laser pulses remain useful, e.g. they may serve as a start, or reference forsubsequent optimizations. Moreover, they allow rather easy interpretations ofthe underlying mechanism of laser control [1, 2, 5, 7, 8], in contrast with mostapplications of optimal laser pulses [12–15]; for an exception, see Ref. [16]. Itremains an important task, therefore, to extend the previous approaches basedon analytical laser pulses to new domains of applications.

The present paper demonstrates two extensions of quantum control based onanalytical laser pulses: First, we shall apply the pump-dump approach of Tannorand Rice [17] to break preferably the stronger bond of a molecule, not the weakerone, for a new scenario where a single laser pulse would break either the weakerbond, or both. For this purpose, we shall design the laser driven path to selectivebond-breaking, which may appear somewhat counter-intuitive, on first glance:The pump laser pulse excites the system to a transient highly excited electronicstate which would induce double bond breaking. This impending disaster will

18

be turned into an advantage, however, i.e. preferential breaking of the strongbond, by means of the dump pulse.

Second, we present new aspects of extensions of quantum control, from nu-clear to electron dynamics. Historically, the first example is the prediction [18]and experimental demonstration [19] of coherent control of uni-directional elec-tron currents in solids by means of interfering one- plus two- or multi-photonexcitations, analogous to coherent control of nuclear dynamics [1, 8]. Here, weinvestigate an alternative extension based on ! pulses (cf. Refs. [3, 20, 21]) or!/2 pulses (cf. Refs. [22, 23]), in order to excite state-selective electronic ringcurrents [24] or electron circulation [25, 26] in ring-shaped molecules, respec-tively. For the previous quantum control of nuclear dynamics, we have designede!cient ! or !/2 pulses with many cycles in the IR spectral domain, and withdurations from few hundred fs to few ps, see also Refs. [5, 6, 23, 27, 28]; forearly applications to non-reactive processes, see also the pioneering paper [29].In contrast, analogous quantum control of electron dynamics calls for ! or !/2laser pulses with few cycles and with re-optimized parameters in the UV do-main, and with durations from several hundred as to few fs [24–26], see alsoRefs. [21, 30]. Moreover, linear polarizations of the laser pulses [30] are extendedto circularly polarized ones [24–26], see also Refs. [23, 31, 32]; for complemen-tary applications of elliptically polarized or even arbitrary polarized pulses, seeRefs. [14, 22, 33, 34]. Below, we shall focus on the condition [35]

! !

"!E(t)dt = 0 (1)

for the extended applications of ! or !/2 laser pulses with di"erent shapes. Thiscondition has been disregarded for traditional, rather long pulses because it isalways fulfilled, at least as an excellent approximation, due to e"ects of can-cellations of positive and negative contributions from electric fields with manycycles. However, these cancellations do not apply automatically for few cy-cle laser pulses, see e.g. Ref. [36]. Here we adapt the recipe of Bandrauk andcoworkers [35] in order to design, exemplarily, few cycle !/2 pulses for electroncirculation, in accord with the condition (1).

The two applications below, for extensions of pump-dump or !/2 laser pulses,assume that the molecules have been pre-oriented, e.g. by the methods ofRef. [37].

II. LASER CONTROL OF PHOTODISSOCIATION DYNAMICS

A popular target for coherent control is to maximize or minimize the branchingbetween breaking a weak and strong bond in a polyatomic molecule. CH2BrClis a typical example where feedback learning algorithms [14] have demonstrated

19

an increase in the breaking of the strong C-Cl bond versus the weak C-Br one.Here, we investigate the e!ciency of corresponding pump-dump laser pulses.

Our reduced model is composed of the three lowest singlet adiabatic potentialenergy surfaces (PES) along the two relevant q1 = d(C-Cl) and q2 = d(C-Br)reaction coordinates, as calculated in Refs. [38, 39]. The corresponding statesa1A#, b1A# and c1A# are labelled 1, 2, 3, respectively, and in the Franck-Condonwindow they correspond to the n(Br)! "$(C-Br) and n(Cl)! "$(C-Cl) tran-sitions. The b1A# PES has two dissociation channels along the C-Br and C-Clbond coordinates, but after vertical excitation the potential gradient favoursC-Br dissociation. The c1A# state allows double bond breaking, CH2+X+Y,supported by transitions to the b1A# state close to an avoided crossing near theline q1 " q2 [39].

The nuclear dynamics of the system is simulated with the time-dependentSchrodinger equation

i!

"

##1

#2

#3

$

% =

"

#H11 H12 H13

H21 H22 H23

H31 H32 H33

$

% ·

"

##1

#2

#3

$

% . (2)

The time-dependent matrix elements of Hij are in semiclassical dipole approxi-mation written as Hij(t) = H0

ij #dij · E(t). The molecular Hamiltonian H0ij con-

tains the kinetic energy T , the adiabatic PES V , and the non-Born-Oppenheimercoupling, which as a first approximation is neglected. The coupling with thelaser field E(t) is described with the transition dipole operator dij , here taken as|d| = 0.5 ea0 for each possible transition. The electric field comprises a pump-dump sequence E(t) = Ep(t) + Ed(t) with time delay tdel = tp # td, and each ofthe linearly polarized pulses (l = p, d) is given by

E l(t) = E0,l cos(#l(t # tl) + $l)sl(t # tl)el (3)

where el, E0,l, #l, $l and tl denote the polarization, the field amplitude, the carrierfrequency, the phase, and the time of the peak maximum, respectively, and cos2shape

sl(t) = cos2(!t/tp,l) for # tp,l/2 $ t $ tp,l/2, (4)

with total pulse duration tp,l. Phases $l are set to zero for simplicity. The electricfields yield intensities Il(t) = %0c|E l(t)|2 (l = 1, 2) with peak values Imax,l =max Il(t) and full width at half maximum (FWHM) &l, Il (±&l/2) = Imax,l/2.Averaging over the rapid (#l) cycles, Il(t) = %0c|E l(t)|2. The correspondingmaximum intensities are then defined as Imax,l = max Il(t) = 1/2%0cE2

0,l.Since the Franck-Condon region lies entirely in the q1 < q2 side, an excitation

resonant to state b1A# results exclusively in C-Br bond breaking, in accordancewith experiments and chemical intuition. In contrast, a pump pulse resonant

20

22.22.42.62.83

1.8 22.2

2.42.6 2.8 3

3

4

5

6

7

8

q (C-Cl) / Å q (C-Br) / Å

E / eV

1 2

FIG. 1: Pump-dump control scenariofor breaking the strong C-Cl bond ofCH2BrCl. The first, pump pulse createsa wave packet in the upper excited elec-tronic state, which evolves in time (blackdotted line). After crossing the q1 ! q2

(solid) line, the second pulse dumps pop-ulation to the lower excited electronicstate, triggering dissociation of the targetbond.

to the c1A# state (!#p=7.17 eV, tp,p=27.5 fs corresponding to 10 fs at FWHM,E0,p=3.89 GV/m, and peak intensity Imax,p=2 TW/cm2) creates a wave packetwhich crosses the avoided crossing line in the C-Cl direction, and then it turnsto double dissociation CH2 + Br + Cl, while broadening very quickly.

In order to maximize the fragmentation of the strong C-Cl bond we proposethe pump-dump control scenario as shown schematically in Fig. 1: The pumppulse excites the molecule to the c1A# state, and because of the shape of thePES (vide supra) the wave packet gets momentum in the C-Cl direction. Aftercrossing the avoided crossing line q1 " q2, the dump or control pulse de-excitesthe system to the b1A# state in the q2 < q1 region, where it dissociates towardsC-Cl. This pump-dump mechanism profits from a ”delicate” choice of the delaybetween both pulses. On one hand, the quick broadening of the wavepacket inthe c1A# state requires an early de-excitation, i.e. shortly after pumping. On theother hand, de-excitation should be late enough so that most of the wavepacketresides on the q2 < q1 side of the crossing line and with a dominant momentumstill in the C-Cl direction and not yet along the q1 " q2 line, i.e. the appropriateposition and momentum is obtained. As a compromise out of this dilemma, weuse a very short dump pulse with optimal timing.

The optimized pump-dump pulse sequence and the time evolution of the pop-ulations in the corresponding states are shown in Fig. 2. The frequency of thedump pulse, !#d=1.0 eV, has been determined after investigating the path ofthe center of mass of the single-pulse-excited wave packet in the state c1A#. Theduration of the dump pulse and the time delay between the pulses have beenoptimized to achieve maximal depumping in the region of the PES which leadsto C-Cl dissociation. More specifically, we have searched for i) maximizing thedepumping from b1A# to c1A#, i.e. the maximum of P2/(P2 + P3), which callsfor early dump, ii) maximizing the amount of dumped population which dis-sociates along the C-Cl coordinate; this is given by P2(q1 > q2)/P2, and callsfor a later dump pulse, and iii) the control e!ciency defined as the productP2/(P2 + P3) · P2(q1 > q2)/P2 = P2(q1 > q2)/(P2 + P3) for goals (i) and (ii).Here, Pi (i = 1, 2, 3) denote the asymptotic values of the time-dependent popu-

21

FIG. 2: The left figure shows the electric field for a pump pulse of !!p=7.17 eV,tp,p=27.5 fs (FWHM = 10 fs), E0,p=3.89 GV/m, Imax,p=2 TW/cm2, and a dumpor control pulse of !!d=1.0 eV, tp,d=60 fs (FWHM = 22 fs), E0,p=6.15 GV/m, andImax,d=5 TW/cm2. The time delay tdel=25 fs. The right figure shows the time-evolution of the populations of di!erent electronic states. P1, P2 and P3 are thepopulations of states a1A!, b1A! and c1A! respectively. P2|q1 > q2 is the amount of P2

which is on the q1 > q2 side of the crossing line, breaking the strong bond.

lations

Pi(t) =! !

|#i(q1, q2, t)|2dq1dq2 (5)

integrated over all the space. Likewise,

P2(q1 > q2) =! !

|#2(q1, q2, t)|2dq1dq2 (6)

integrated for the domain q1 > q2 accounts for the fraction of the population P2,which dissociates preferably to the C-Cl direction. The wave functions #i arenormalized, so that P1(t) + P2(t) + P3(t) = 1 for any time t .

Using these criteria (i)-(iii), we determine the optimal parameters of the dumppulse. For example, increasing the time delay from 20 fs to 30 fs, decreases thee!ciency of depumping P2/(P2 + P3) but increases the ratio P2(q1 > q2)/P2.The optimum delay tdel is chosen at 25 fs. Likewise, depumping P2/(P2 + P3)increases but the ratio P2(q1 > q2)/P2 decreases with the duration of the dumppulse. The resulting laser parameters for the optimal compromise are listed inthe Figure legend 2. With these parameters the overall depumped population,P2/(P2 + P3), measured at t=70 fs is 47.1 %, the portion of P2 in the (q1 > q2)region, P2(q1 > q2)/P2, is 86.2 % and the control e!ciency P2(q1 > q2)/(P2+P3)is 40.6 % (see Fig. 2).

In summary, we have shown that a pump-dump sequence can be tailoredto induce significant fragmentation of a strong bond in a polyatomic molecule.

22

This is achieved by means of the apparent “detour” via the second excited statewhich would lead to double bond breaking, and optimal de-excitation to the firstexcited state which leads to the target product. Refinements of the model canbe accomplished by including coordinate dependent transition dipole momentsand non-adiabatic couplings. Moreover, the use of chirp [40], other polarizations,selective momentum along the C-Cl bond [41], or ultimately feedback controlledoptimization [11] can certainly help improving the control yield.

III. LASER CONTROL OF ELECTRON DYNAMICS

In this section, laser control of nuclear dynamics is extended to electron dy-namics for oriented molecules driven by (sub-)fs circularly polarized ! or !/2pulses. The right circularly polarized laser pulses propagating along the z-direction adapted from Refs. [24–26] may be rewritten as

E1(t) = E0,1s1(t)

"

#cos(#1t + $1)sin(#1t + $1)

0

$

% (7)

with amplitude E0,1, carrier frequency #1, carrier envelope phase $1, cos2 shapefunction denoted s1(t), equivalent to Eqn. (4), and duration tp,1. For circularlypolarized few cycle laser pulses (7), the condition (1) cannot be satisfied. Ac-cording to Ref. [35], we suggest, therefore, a more flexible ansatz based on thevector potential A2(t) such that the condition (1) is satisfied automatically,

A2(t) = #A0,2s2(t)

"

#sin(#2t + $2)

# cos(#2t + $2)0

$

% (8)

with amplitude A0,2 = cE0,2/#2. As shape function, we employ

s2(t) = cos20 (!t/tp,2) for # tp,2/2 $ t $ tp,2/2 (9)

which is very similar to a Gaussian [42], and therefore called “Gaussian” below.From expression (8), we derive the electric field E2(t) = #A2(t)/c

E2(t) = E0,2s2(t)

"

#cos(#2t + $2)sin(#2t + $2)

0

$

% +E0,2

#2s2(t)

"

#sin(#2t + $2)

# cos(#2t + $2)0

$

% (10)

For comparison of the results, we employ two laser pulses E1(t) and E2(t) withthe same durations &E = &E,1 = &E,2 (= &1) of the squares of the envelopes s1(t)and s2(t), s2

k (±&E,k/2) = 1/2.

23

Mg-porphyrin

x

y

z

FIG. 3: Few cycle right circularly polar-ized laser pulse propagating along the z-direction (thick arrow) in order to inducethe unidirectional right electron circula-tion in Mg-porphyrin, pre-oriented in thex/y-plane. The thin arrows indicate thetime evolution of the electric field act-ing on the molecule, which appears clock-wise (“right”) when viewed along the z-direction of propagation, from t = "2 fs(•) to t = 2 fs (#), compare Fig. 4.

Subsequently, we assume that the laser pulses are so short (&k < 1.5 fs) thatthe nuclei are essentially frozen. The laser driven electron dynamics are thendescribed by time-dependent electronic wave functions |#(t)%, solving the time-dependent Schrodinger equation in semiclassical dipole approximation

i!|#(t)% = H(t)|#(t)% = Hel|#(t)% # d · Ek(t)|#(t)% (11)

where Hel denotes the electronic Hamiltonian, and d is the dipole operator.Below, we apply laser pulses with intensities below the ionization thresholds

(Imax.k < 10TW/cm2). The solution of Eqn. (11) may then be expanded interms of electronic eigenstates |#j% with eigenenergies Ej .

|#(t)% =&

j

Cj(t)|#j%e"iEjt/! (12)

Insertion of the ansatz (12) into Eqn. (11) yields the equivalent set of equationsfor the time-dependent coe!cients

i!Cj(t) = #E(t)&

i

Ci(t)&#j |d|#i%e"i(Ei"Ej)t/! (13)

The solutions Cj(t) yield the populations Pj(t) = |Cj(t)|2 of the levels Ej .The subsequent application is for the laser pulse control of selective electron

circulation in the ring-shaped molecule Mg-porphyrin (MgP) by means of acircularly polarized few cycle laser pulse, see Fig. 3.

The corresponding electronic eigenenergies and transition dipole matrix ele-ments are taken from the literature [43, 44], as in Refs. [24–26]. Previously, wehave designed various re-optimized ! pulses of the type E1(t) in order to preparethe system in di"erent excited state with Eu+ (or Eu") symmetry, and rep-resenting states with state selective right (or left) uni-directional ring currents[24]. These ring currents driven by laser pulses turn out to be much stronger,

24

x/GVm

!2

0

2

y/GVm

1.0

4

!4

!2

0

2

4

!4

X 1A1g

2 1Eu+

1.37 fs

0.91 fs

/TWcm

6

2

4

8

0.0

0.2

0.4

0.6

0.8 Po

pu

latio

n

-2 -1 0 1 2

t / fs

-1 0 1 2

t / fs

FIG. 4: Optimized circularly polarized laser pulses for half population transfer from theelectronic ground state X 1A1g to excited target state 2 1Eu+ implying state selectiveelectron circulation in Mg-porphyrin. The figures show the x- and y-components oftwo electric fields E1(t) (Eqn. (7), dashed lines) and E2(t) (Eqn. (10), continuouslines), the corresponding intensities I1(t) and I2(t), and the resulting populations ofthe ground state X 1A1g and excited state 2 1Eu+. The parameters of the laser pulseE1(t) and E2(t) are E0,1 = 5.54 GVm"1, !!1 = 1.94 eV, "1 = 0 [25], and E0,2 = 3.94GVm"1, !!2 = 1.29 eV, "2 = 0, respectively. Note that, the FWHM of the squares ofthe envelopes s1(t)2 and s2(t)2 are the same for both pulses, #E = 0.91 fs. In contrast,the FWHM of the intensities I1(t) and I2(t) are di!erent, #1 = 0.91 fs and #2 = 1.37 fs.

e.g. 84.5 µA than traditional ones, i.e. one would need B " 8000T in order toinduce the same order of ring currents by permanent magnetic fields. Moreover,we have applied di"erent re-optimized !/2 laser pulses E1(t) in order to prepareso-called “hybrid” superposition states

|#i,j(t)% = 1/'

2|#i%e"i(Eit/!+!i) + 1/'

2|#j%e"i(Ejt/!+!j) (14)

These are not eigenstates, i.e. they represent electron circulation, as demon-strated in Refs. [25, 26]. Below we apply the corresponding “Gaussian” laserpulse E2(t) with the same duration &E = &E,2 = &E,1 as for the cos2 shaped pulseE1(t). The optimal laser parameters are listed in the Figure legend 4, and theresults for E2(t) are compared with those of E1(t) (adapted from Ref. [25]) inFig. 4.

Accordingly, “Gaussian” laser pulses derived from corresponding ultra-shortvector potential (&E = 910 as) may be applied to induce state selective electroncirculation in oriented ring-shaped molecules. This is advantageous becauseexperimental “Gaussian” type laser pulse can be prepared more easily than cos2shaped ones. Gratifyingly, the present e"ects of “Gaussian” laser pulses, which

25

satisfy condition (1) [35], confirm less rigorous previous results based on laserpulses with cos2 type shapes.

The present examples for exclusive quantum control of nuclear or electrondynamics point to a challenge, i.e. propagatation of the joint wave packet dy-namics of electrons and nuclei driven by laser pulses, see e.g. Refs. [45, 46], withextended applications to quantum control.

Acknowledgments

JM would like to thank Prof. A. D. Bandrauk, Sherbrooke, for stimulatingdiscussions of condition (1). Financial support by Berliner Forderprogramm(LG), Deutsche Forschungsgemeinschaft (project Ma 515/23-1), and Fonds derChemischen Industrie (JM) are also gratefully acknowledged.

[1] P. Brumer and M. Shapiro, Chem. Phys. Lett. 126, 541 (1986).[2] D. J. Tannor, R. Koslo!, and S. A. Rice, J. Chem. Phys. 85, 5805 (1986).[3] T. Joseph and J. Manz, Molec. Phys. 58, 1149 (1986).[4] S. Shi, A. Woody, and H. Rabitz, J. Chem. Phys. 88, 6870 (1988).[5] J. Manz, in Femtochemistry and Femtobiology: Ultrafast Reaction Dynamics at

Atomic Scale Resolution, Nobel Symposium, edited by V. Sundstrom (ImperialCollege Press, London, 1997), vol. 101, pp. 80–318.

[6] M. V. Korolkov, J. Manz, and G. K. Paramonov, Adv. Chem. Phys. 101, 327(1997).

[7] S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics (Wiley, NewYork, 2000).

[8] M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Pro-cesses (Wiley-VCH, Weinheim, 2003).

[9] R. Koslo!, S. A. Rice, P. Gaspard, S. Tersigni, and D. J. Tannor, Chem. Phys.139, 201 (1989).

[10] W. Jakubetz, J. Manz, and H.-J. Schreier, Chem. Phys. Lett. 165, 100 (1990).[11] R. S. Judson and H. Rabitz, Phys. Rev. Lett. 68, 1500 (1992).[12] A. Assion, T. Baumert, M. Bergt, T. Brixner, B.Kiefer, V. Seyfried, M. Strehle,

and G. Gerber, Science 282, 919 (1998).[13] R. J. Levis, G. M. Menkir, and H. Rabitz, Science 292, 709 (2001).[14] N. Damrauer, C. Dietl, G. Kramert, S.-H. Lee, K.-H. Jung, and G. Gerber, Eur.

Phys. J. D. 20, 71 (2002).[15] T. Brixner and G. Gerber, Chem. Phys. Chem. 4, 418 (2003).[16] C. Daniel, J. Full, L. Gonzalez, C. Lupulescu, J. Manz, A. Merli, S. Vajda, and

L. Woste, Science 299, 536 (2003).[17] D. J. Tannor and S. A. Rice, J. Chem. Phys. 83, 5013 (1985).[18] G. Kurizki, M. Shapiro, and P. Brumer, Phys. Rev. B 39, 3435 (1989).

26

[19] E. Dupont, P. B. Corkum, H. C. Liu, M. Buchanan, and Z. R. Wasilewski, Phys.Rev. Lett. 74, 3596 (1995).

[20] M. Sargent III, M. O. Scully, and W. E. Lamb (Jr.), Laser Physics (Addison-Wesley, London, 1974).

[21] J. Cao, C. J. Bardeen, and K. R. Wilson, Phys. Rev. Lett. 80, 1406 (1998).[22] Y. Fujimura, L. Gonzalez, K. Hoki, D. Kroner, J. Manz, and Y. Ohtsuki, Angew.

Chem. Int. Ed. 39, 4586 (2000).[23] K. Hoki, D. Kroner, and J. Manz, Chem. Phys. 267, 59 (2001).[24] I. Barth, J. Manz, Y. Shigeta, and K. Yagi, J. Am. Chem. Soc. 128, 7043 (2006).[25] I. Barth and J. Manz, Angew. Chem. Int. Ed. 45, 2962 (2006).[26] I. Barth and J. Manz, in Femtochemistry VII: A Conference Devoted to Funda-

mental Ultrafast Processes in Chemistry, Physics, and Biology, edited by J. A. W.Castleman and M. L. Kimble (Elsevier, Amsterdam, 2006), (in press).

[27] M. Holthaus and B. Just, Phys. Rev. A 49, 1950 (1994).[28] L. Gonzalez, D. Kroner, and I. R. Sola, J. Chem. Phys. 115, 2519 (2001).[29] G. K. Paramonov and V. A. Savva, Phys. Lett. A 97, 340 (1983).[30] P. Krause, T. Klamroth, and P. Saalfrank, J. Chem. Phys. 123, 074105 (2005).[31] M. Kitzler, K. O’Kee!e, and M. Lezius, J. Mod. Opt. 53, 57 (2006).[32] S. X. Hu and L. A. Collins, Phys. Rev. A 73, 023405 (2006).[33] Y. Fujimura, L. Gonzalez, K. Hoki, J. Manz, and Y. Ohtsuki, Chem. Phys. Lett.

310, 1 (1999), Corrigendium, ibid. 310, 578 (1999).[34] L. Polachek, D.Oron, and Y. Silberberg, Opt. Lett. 31, 631 (2006).[35] S. Chelkowski and A. D. Bandrauk, Phys. Rev. A 71, 053815 (2005).[36] N. Elghobashi, L. Gonzalez, and J. Manz, J. Chem. Phys. 120, 8002 (2004).[37] H. Stapelfeldt and T. Seideman, Rev. Mod. Phys. 75, 543 (2003).[38] L. Gonzalez and T. Rozgonyi, J. Phys. Chem. A 106, 11150 (2002).[39] L. Gonzalez and T. Rozgonyi, J. Phys. Chem. A (2006), (in press).[40] C. J. Bardeen, V. V. Yakovlev, K. R. Wilson, S. Carpenter, P. M. Weber, and

W. Warren, Chem. Phys. Lett. 280, 151 (1997).[41] N. Elghobashi, P. Krause, J. Manz, and M. Oppel, Phys. Chem. Chem. Phys. 5,

4806 (2003).[42] I. Barth and C. Lasser, (in preparation).[43] M. Rubio, B. O. Roos, L. Serrano-Andres, and M. Merchan, J. Chem. Phys. 110,

7202 (1999).[44] D. Sundholm, Chem. Phys. Lett. 317, 392 (2000).[45] A. D. Bandrauk, S. Chelkowski, and H. S. Nguyen, Int. J. Quant. Chem. 100, 834

(2004).[46] G. K. Paramonov, Chem. Phys. Lett. 411, 350 (2005).

27

B. Lasorne and G. A. Worth (eds.)Coherent Control of Molecules

c! 2006, CCP6, Daresbury

Optimal Control of Dissipative Quantum Phase SpaceDynamics

Keith H. Hughes and Meilir HywelSchool of Chemistry, University of Wales Bangor,

Bangor, Gwynedd LL57 2UW, UK

I. INTRODUCTION

In the pioneering days of quantum mechanics, Wigner [1] developed a quan-tum pseudo-distribution function that possesses many of the properties of thecorresponding classical phase space distribution function that is widely used instatistical mechanics. Since then, the Wigner distribution [2] has been used tostudy a wide range of phenomena in science and engineering and the literatureis rich with examples of its applicability. In particular, the Wigner distributionis indispensable in describing the dissipative processes of an open quantum sys-tem. In the following, a phase space approach to the optimal control of quantumdynamics influenced by dissipation is described and some preliminary results arepresented for i) a quantum sub-system coupled directly to the environment, andii) a quantum sub-system coupled to a classical sub-system that undergoes dis-sipation – the quantum sub-system then undergoes delayed dissipation.

II. THEORY

A. Dissipative systems

The dynamics of a system coupled to a bath can be represented in the formof a master equation

!"s

!t= ! i

!Ls"s + D"s (1)

where "s is the reduced density matrix of the system, Ls is the Liouvillian ofthe system and D is a dissipative operator that characterises the influence of thebath. Numerous methods have been proposed for the treatment of open quantumsystems, and extensive reviews of these methods are provided in Refs. [3, 4].The method chosen in this work is based on the Caldeira-Leggett [5] model of a

28

system linearly coupled to a bath of oscillators. In the Caldeira-Leggett modelthe influence of the bath is reduced to a single parameter – the friction term #.

D = ! i#2! [x, [p, "s]+] ! #mkBT

!2[x, [x, "s]] (2)

The Caldeira-Leggett model is valid in the high temperature limit where kBT !!!, ! being the cut-o" frequency for the spectral density. Some assumptionsof the Caldeira-Leggett model are i) factorised initial conditions, ii) Markovianapproximation – this assumes that the system evolves on a much shorter timescale than the bath. However, when considering ultra-fast processes that occuron the sub-picosecond scale the system evolves on similar time-scales to thebath and a Markovian approach is not justified. In such cases a non-Markovianapproach should be used.

In the following models, the dynamics of i) a quantum sub-system under-going Caldeira-Leggett type dissipation as a result of direct coupling to theenvironment, and ii) a quantum sub-system coupled to a classical oscillator thatundergoes Caldeira-Leggett type dissipation [6] are presented. For the mixedquantum-classical model of the reduced system ii) the quantum sub-system thenundergoes delayed (non-Markovian) dissipation due to the coupling to the clas-sical Brownian oscillator.

The models are described in the Wigner phase space representation. An ad-vantage of working in phase space is that it is easier to implement a mixedquantum-classical scheme. The phase space representation of the Caldeira-Leggett equation is given by

!W

!t= ! i

!LW W + DW W (3)

where W = W (q,p, t) is the Wigner phase space distribution function; q =(q, Q) are the position variables and p = (p, P ) the momentum variables. LW isthe phase space Liouvillian of the system where

LW W = HT W ! WT H (4)

with

T = ei!!q,p/2 (5)

and

#q,p ="!#q

!$#p !"!#p

!$#q

The direction of the arrows represent the direction of derivative operation. Forthe case of a quantum sub-system coupled to a classical sub-system the T oper-ator is given by

T = ei!!q,p/2(1 +12

!#Q,P ) (6)

29

The dissipative part of Eq. (3) is given by

DW W = #!pW

!p+ #mkBT

!2W

!p2(7)

for a quantum sub-system coupled directly to the environment, and

DW W = #!PW

!P+ #MkBT

!2W

!P 2(8)

for a quantum sub-system coupled to a classical oscillator undergoing dissipation.

B. Optimal control scheme in phase space

The goal of quantum control theory is to find the optimal laser pulse, or pulses,that takes some initial state of the quantum system to a desired target state. Inoptimal control theory this is done by maximising the target functional,

J = (2$!)!NTr[T"(tf )] =!

WT (q,p)W (q,p, tf )dqdp, (9)

where WT is the target phase space distribution function and W (q,p, tf ) is thephase space distribution function at some final time tf . Also shown in Eq. (9)for comparison is the corresponding functional expressed in terms of the morefamiliar density operators, where T is the target density operator.

The functional J is subject to the constraint that Eq. (3) is solved and that thetotal energy of the laser pulse is limited. Using Lagrange multipliers W!(q,p, t)and % a functional of the form,

J =!

WT (q,p)W (q,p, tf )dqdp ! %

! tf

0s(t)|&|2dt

!! tf

0

!W!

" !

!t+

i!LW !DW

#Wdqdpdt, (10)

allows an unconstrained optimisation of J , where % is a scalar Lagrange multi-plier and W! is a phase space function Lagrange multiplier. The time-dependentterm s(t) = sin2($t/tf ) ensures a smooth switch on/o" to the laser pulse. Vari-ation of J with respect to W, W! , and the real and imaginary parts of &(t) andsetting 'J = 0 leads to the following equations of motion

!W

!t= ! i

!LW W + DW W, W (q,p, t0) = W0 (11)

!W!

!t= ! i

!(L†)W W! + (D†)W W! , W!(q,p, tf ) = WT (q,p) (12)

30

For the control pulse

$& = ! i2%!

!W!MWdqdp

where M = [µ, •]

III. APPLICATION TO A TEST CASE

In this study the control scheme is applied to a 2 electronic state system witha system Hamiltonian given by

H =$

K + Vg !µ&(t)!µ&"(t) K + Ve

%(13)

where K is the kinetic energy term and Vg/e are the potential energy termsfor the ground and excited electronic state respectively. Within the Condonapproximation the dipole moment operator µ = constant is taken as independentof the spatial coordinate.

For a 2-state system with a Hamiltonian given by Eq. (13) we define the follow-ing Wigner functions W11(q,p), W12(q,p), W21(q,p), W22(q,p) as the Wignertransform of the corresponding density matrix terms, "11(x, x#), "21(x, x#) etc.

The ground state potential is taken to be an asymmetric double well potentialof the form Vg = V2q2 + V3q3 + V4q4. As an initial condition the quantumsub-system is assumed to be in the ground v = 0 vibrational eigenstate, %0,of this potential. The goal is to drive the system from the v = 0 state of theleft-hand well to the v = 1 state that is localised in the right-hand well. Dueto the asymmetry of the potential function the probability of tunnelling fromone potential well to another is negligible. The excited electronic potential istaken to be harmonic Ve = V #q2 + ' and is separated from the ground electronicstate (at q = 0) by ' = 16000 cm!1. The optimal control scheme described inSec. II B is used to drive the system from the v = 0 to the v = 1 vibrationalstate of the ground electronic state via the excited electronic state. The initialphase space distribution function is a Wigner transform of the v = 0 eigenstateof the ground electronic state,

W11(q, p, t0) =1

2$!

! $

!$< q +

r

2|%0 >< %0|q !

r

2> e!i rp

! dr, (14)

and the target WT is given by the Wigner transform of the v = 1 eigenstate

WT (q, p) =1

2$!

! $

!$< q +

r

2|%1 >< %1|q !

r

2> e!i rp

! dr. (15)

31

0 5 10 15 20 25 30Iterations

0

0.2

0.4

0.6

0.8

1

Ob

ject

ive

ov

erla

p

Non-Dissipative

Dissipative

FIG. 1: Objective versus the number of iterations.

0 1000 2000 3000 4000 5000

0

Re !(

t)

0 1000 2000 3000 4000 5000t (a.u.)

-0.5

0

0.5

1

popula

tions/

coher

ence

W11

W22

Re W21

a)

b)

FIG. 2: In b) the populations/coherence are correspond toR !"!

R !"! Wij(q, p, t)dqdp.

For the case of the quantum subsystem coupled directly to the environmentat a temperature T = 600 K the convergence of the control scheme is illustratedin Fig. 1 for 30 iterations. Also shown in this figure is the convergence of thecontrol scheme for the corresponding non-dissipative case. Due to the influenceof decoherence and energy dissipation the objective overlap for the dissipativecase (0.8) is less than the overlap for the non-dissipative case (0.95). The optimalcontrol pulse for the dissipative case is depicted in Fig. 2a). The time variationof the populations and coherences due to the control scheme depicted in Fig. 2b)are typical of a pump-dump sequence.

For the delayed dissipative (non-Markovian) case the quantum sub-system isbilinearly coupled to a classical harmonic oscillator that undergoes dissipation.

32

0 1000 2000 3000 4000 5000t (a.u.)

-0.5

0

0.5

1

po

pu

lati

on

s/co

her

ence

W11

W22

Re W21

FIG. 3: The time variation of the populations and coherences.

The time variation of the populations and coherences are depicted in Fig. 3.

IV. SUMMARY

An optimal control scheme was developed in a phase space representationand applied to i) a quantum sub-system undergoing dissipation as a result ofdirect coupling to the environment, and ii) a quantum sub-system undergoingdissipation as a result of coupling to classical Brownian oscillator.

Acknowledgments

The authors thank their collaborators Irene Burghardt and Klaus B. Møllerfor discussions related to the dissipative aspects of this work.

[1] E. Wigner, Phys. Rev. 40, 749 (1932).[2] M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, Phys. Rep. 106, 121

(1984).[3] D. Cohen, C. C. Marston, and D. J. Tannor, J. Chem. Phys. 107, 5236 (1997).[4] U. Weiss, Quantum Dissipative Systems (World Scientific, Singapore, 1993).[5] A. O. Caldeira and A. J. Leggett, Physica A 121, 587 (1983).[6] I. Burghardt, K. B. Møller, and K. H. Hughes, in Workshop on Quantum Dynamics

of Complex Molecular Systems, edited by D. Micha and I. Burghardt (Springer-Verlag, 2006), chapter 17.

33

34

B. Lasorne and G. A. Worth (eds.)

Coherent Control of Molecules

© 2006, CCP6, Daresbury

Quantum Direct Dynamics Applied to the Intelligent Control

of Benzene Photochemistry

Benjamin Lasorne, Michael J. Bearpark, Michael A. Robb Department of Chemistry, Imperial College London, London SW7 2AZ, UK

and Graham A. Worth School of Chemistry, University of Birmingham, Birmingham B15 2TT, UK

I. INTRODUCTION

Optimal control experiments of molecular processes use closed loop

techniques to achieve maximum efficiency when designing the laser pulse

that will drive the system to a targeted outcome [1–3] through a ‘black-

box’ mechanism. Simulations based on optimal control theory [4, 5] can

monitor the wavepacket evolution, but they give very complicate pulse

shapes, the effect of which is not easy to decipher. Few attempts have

been made to connect experiments and simulations in this context [6].

Especially for large molecules the level of quantum chemistry calculations

may not be accurate enough to expect ready-to-use ab initio designed laser

pulses, as can be done for small systems [7]. Intelligent control of

reactivity is thus a challenge that should make possible a rationalisation of

experiment using theory if rather simple laser pulses can be decoded in

terms of potential energy surface (PES) exploration.

We present here an application of the direct dynamics variational

multi-configuration Gaussian wavepacket (DD-vMCG) method to the non-

adiabatic photochemistry of benzene. Based on the multi-configuration

time-dependent Hartree (MCTDH) algorithm [8–11], it is designed to treat

quantum effects in large molecules with on-the-fly calculation of the

potential energy surface performed by an interfaced quantum chemistry

program. Our aim is to identify realistic non-radiative decay pathways

that lead to alternative photochemical reactivity and to find

corresponding targets that can be reached by optimal control experiments.

35

II. THEORY

The vMCG method has already been applied to the non-adiabatic

photochemistry of the butatriene radical cation [12], and the algorithm

has been improved in some ways since then. It derives from a more

sophisticated method [13–15] designed to treat molecular motion within a

hierarchical model. The general working equations of the vMCG method

can be found in references [12, 13, 16] and more details are to be given in

future works. The vMCG ansatz for the wavepacket component on the

electronic state s and describing f nuclear degrees of freedom reads

!s

x1,…,xf,t( ) = A

j(s) t( )gj

(s) x1,…,xf,t( )

j

" . (1)

The evolution of the g-functions comes from the variational optimisation

of a small number of time-dependent parameters. Here, we use frozen and

normalised Gaussian wavepackets (GWPs). Each one-dimensional function

of the product is defined by two real parameters

xj

t( ),pjt( ){ } , the mean

position and momentum of the Gaussian function

g x; xj

t( ),pjt( ){ }!

"#$%&

=1

2!"

e

'x'x

jt( )

2"

!

"

###

$

%

&&&

2

+ip

jt( )x

!. (2)

The real phase is constrained to preserve normalisation, while the

imaginary phase is kept to zero for numerical stability. All additional

phase information is shifted into the A-coefficients.

The equations of motion for the coefficients and the Gaussian

parameters read as a set of coupled dynamical equations [12, 13] that

show that all the basis functions are coupled to each other, both directly

and through the expansion coefficients. Therefore, by contrast with the ab

initio multiple spawning method [17], the GWPs follow non-classical

trajectories, which lead to faster convergence. Solving these equations

requires the evaluation of the Hamiltonian matrix and its derivatives with

respect to Gaussian parameters, for which a local harmonic approximation

(LHA) is used to expand the potential energy function around the centre

of each GWP. This approximate potential energy function is calculated

on-the-fly by an interfaced quantum chemistry program – here, a

36

development version of the GAUSSIAN program [18]. For two coupled

electronic states, the dynamics is carried out in a diabatic representation,

which makes the LHA easier to define. This step is parallelised and a

database is used to store and recycle expensive ab initio calculations. The

algorithm has been implemented in a development version of the

Heidelberg MCTDH package [19].

III. RESULTS

We are interested here in the ‘channel 3’ process of the S1

photochemistry of benzene for which the fluorescence vanishes beyond a

vibrational excess of 3000 cm–1 (see [20] and references therein). In

addition, the quantum yield of the primary photoproduct (benzvalene)

increases with increasing excess energy but stays very low. The

mechanism has been rationalised by theoretical studies of Robb and co-

workers [20]. Starting from the Franck-Condon region, the system must

overcome a transition barrier before reaching the conical intersection

minimum. Internal conversion leads then to a prefulvenoid plateau on S0,

which is connected back to benzene and can also lead to benzvalene. The

same authors showed also that parts of the S1/S0 seam of intersection

correspond to geometries that are precursors of Dewar benzene. Our

purpose here is double. First, we want to know which deformations must

be excited to enhance internal conversion. Then, we want to control the

selectivity of benzvalene vs. Dewar benzene by targeting specific regions of

the seam.

We used normal modes of the D6h benzene minimum on S0 as

coordinates for the dynamics. They are transformed on-the-fly into

Cartesian coordinates in an Eckart frame for the frequency calculations,

which are performed with a complete active space self-consistent field

(CASSCF) of six electrons spread over six ! molecular orbitals at the 6-

31G* level. The dominant modes to activate have been identified by

various approaches, including semi-classical trajectories based on

molecular mechanics valence bond (MMVB) calculations [21]. We used

first a 5-dimensional (5D) model including the breathing mode 1a1g (!1)

(main component of the Franck-Condon gradient), the Kékulé mode 1b2u

(!15) (main component of the derivative coupling) and three out-of-plane

37

modes, 1b2g (!4), 1–2e2u (!16). The latter three lead respectively to chair,

boat and twist geometries that can be combined to keep five C nuclei in a

plane and one out. They are needed to describe prefulvenoid electronic

structures implied in the state crossing (a 3-electron cyclopropenoid ring

coupled to an allylic moiety). We used also a more refined 9-dimensional

(9D) model, including in addition four in-plane modes, namely: 1–2e2g (!6),

the quinoidic rectangular deformation and its antisymmetrical counterpart

– involved in the Herzberg-Teller effect that makes the forbidden

electronic transition from S0 (1A1g) to S1 (

1B2u) vibronically induced [22] –,

and the 1–2e1u (!18) trapezoidal modes which enable the formation of the

prefulvenoid CC bond. N.B.: the !-nomenclature is that of Wilson.

In the 5D simulations, we used Franck-Condon Gaussian wavepackets

placed on S1 and multiplied it by a spatial phase factor exp i k !Q( ) ,

giving thus an additional mean momentum !k . By varying the k vector

components along the Q normal coordinates, we showed that it was

possible to control the enhancement of non-adiabatic transitions.

Increasing the magnitude of k is of course in favour of a more diabatic

behaviour. More interesting is the influence of its direction. For a given

mean energy, we showed that the best efficiency is reached when exciting

together 1a1g, 1b2g, 1e2u and 2e2u in a cooperative way. A case with a ratio

of (–1.5):3.2:1:1 between the corresponding k components and a mean

energy of 11.1 eV above the S0 minimum gives a population transfer of

65% between 15 and 25 fs. The electronic populations stay stationary

afterwards with a 0.45:0.55 splitting. With a (–1):1:1:1 ratio and the same

mean energy, the non-adiabatic transitions happen at the same time but

their instantaneous efficiency is only of 25% and there is no longer

permanent transfer. In all cases, internal conversion occurs in a time range

when deformations reach their maxima almost simultanesously.

The 9D model is more flexible and it enables an approximate

description of the reaction coordinate leading to products. Exploratory

simulations revealed that targeting the intersection seam required passage

through the transition barrier in a first step in order to keep the mean

energy sufficiently low. To target this region, we followed two different

strategies. First, we looked for a single initial kick able to drive the

wavepacket to such geometries directly. We achieved it in 6 fs by giving a

specific momentum corresponding to 15.2 eV above the S0 minimum (see

38

Fig. 1). In this case, efficient non-adiabatic transitions occur from 7 to 9

fs, leading to a 0.6:0.4 population splitting. Beyond the transition barrier,

the breathing (1a1g) and trapezoidal (1e1u) modes lose dramatically their

harmonicities and show a monotonous behaviour, thus following the path

towards the prefulvenoid plateau (mainly ring contraction and CC bond

formation). Such a high excess energy is not accessible experimentally. So,

we changed our strategy and opted for a sequence of delayed kicks, each

of them corresponding to the excitation of a single mode, starting with the

Herzberg-Teller rectangular mode (1e2g). This is a work in progress and we

have succeeded in driving all the normal modes except 1a1g to their values

at the transition barrier after 42 fs. However, the breathing mode is either

too early or late, mostly because excited out-of-plane modes tend to make

it deviate strongly.

FIG. 1: ‘Quantum trajectory’ on the

diabatic state connecting the Franck-

Condon region (S1) to the prefulvenoid

plateau (S0). The five normal coordinates

represented are those initially excited by

the impulsional kick. Frequency-mass-

weighted dimensionless coordinates were

used. The transition barrier is reached at

6 fs. The diabatic population changes

from 1 to 0.4 between 7 and 9 fs.

IV. CONCLUSIONS

In the context of intelligent control, theory must help experiment by

finding an easy and relevant target early on in the mechanism. Only then

can we expect to rationalise optimal control in large systems. Our

preliminary study shows that it is possible to identify which motions have

to be excited in benzene to reach specific geometries that may serve as

targets. We now have a database of structural information made up of

about 9200 records for now. The database enables the direct dynamics

simulations to become faster and contains valuable information on the

explored non-adiabatic energy landscape. We must now find conditions for

the simulations that can be realistically achieved from an experimental

viewpoint.

39

Acknowledgments

Special thanks are due to Dr. I. Burghardt for a fruitful collaboration in theoretical and

algorithmic aspects of this work. The authors are indebted to Dr. M. Boggio-Pasqua and

Prof. M. Desouter-Lecomte for helpful discussions. This work is part of a joint

experiment/theory project. Laser pulse experiments are carried out at University College

London by Dr. R. E. Carley, D. S. N. Parker, A. D. G. Nunn, T. M. Borders and Prof. H. H.

Fielding. Evolution algorithms are developed at the University of Wales by Dr. R. D.

Burbridge and Prof. R. D. King. Financial support by EPSRC is gratefully acknowledged

(Grant No: GR/T20311/01).

[1] S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics. (Wiley, New York,

2000).

[2] M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes.

(Wiley, New York, 2003).

[3] V. May and O. Kühn, Charge and Energy Transfer Dynamics in Molecular Systems.

(Wiley-VCH, Berlin, 2004).

[4] D. J. Tannor and S. A. Rice, J. Chem. Phys. 83, 5013 (1985).

[5] R. S. Judson and H. Rabitz, Phys. Rev. Lett. 68, 1500 (1992).

[6] C. Daniel, J. Full, L. González, C. Lupulescu, J. Manz, A. Merli, S. Vajda, and L.

Wöste, Science 299, 536 (2003).

[7] S. Zou, Q. Ren, G. Balint-Kurti, and F. Manby, Phys. Rev. Lett. 96, 243003 (2006).

[8] H.-D. Meyer, U. Manthe, and L. S. Cederbaum, Chem. Phys. Lett. 165, 73 (1990).

[9] U. Manthe, H.-D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 97, 3199 (1992).

[10] M. H. Beck, A. Jäckle, G. A. Worth, and H.-D. Meyer, Phys. Rep. 324, 1 (2000).

[11] H.-D. Meyer and G. A. Worth, Theor. Chem. Acc. 109, 251 (2003).

[12] G. A. Worth, M. A. Robb, and I. Burghardt, Farad. Discuss. 127, 307 (2004).

[13] I. Burghardt, H.-D. Meyer, and L. S. Cederbaum, J. Chem, Phys. 111, 2927 (1999).

[14] G. A. Worth and I. Burghardt, Chem. Phys. Lett. 368, 502 (2003).

[15] I. Burghardt, M. Nest, and G. A. Worth, J. Chem. Phys. 119, 5364 (2003).

[16] B. Lasorne, M. J. Bearpark, M. A. Robb, and G. A. Worth, Chem. Phys. Lett. 432,

604 (2006).

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Pittsburgh PA, 2003).

[19] G. A. Worth, M. H. Beck, A. Jäckle, and H.-D. Meyer, The MCTDH Package,

Development Version 9.0 (University of Heidelberg, 2006).

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116, 670 (2006).

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B. Lasorne and G. A. Worth (eds.)Coherent Control of Molecules

c! 2006, CCP6, Daresbury

Optimal Control Theory:Applications to Polyatomic Systems

Volkhard MayInstitut fur Physik, Humboldt-Universitat zu Berlin,Newtonstraße 15, D-12489 Berlin, F. R. Germany

I. INTRODUCTION

Laser pulse guided molecular dynamics within closed loop control experimentsrepresents one frontier in ultrafast optical spectroscopy (for a recent overview see[1, 2]). The whole research field is based on the vision to tailor femtosecond laserpulses in the optical and infrared region in order to drive the molecular wavefunction in a particular way. One tries to remove certain parts of the molecule(laser pulse driven dissociation), one switches the molecule between di!erentconformations, or one guides electron as well as excitation energy transfer.

Theoretical simulations of such experiments are mainly carried out in theframework of the Optimal Control Theory (OCT) (see, for example, [3]). Butwhile the studies in closed loop control experiments concern multi–dimensionalwave packet dynamics in polyatomic and condensed phase systems, OCT basedsimulations are restricted to models with at most two nuclear coordinates (forexample, [4, 5], but note the single exception [6]). However, to understand themolecular dynamics which underly the particular control mechanism realized inthe experiment, more sophisticated models are often necessary. One way outwould be the application of dissipative quantum dynamics methods.

II. OPTIMAL CONTROL THEORY

OCT gives the formal framework to compute the laser pulse field–strength(the control field) E which optimizes the observable O measured in the particularcontrol experiment. To obtain E one searches for an extremum of O under theconstraint of a finite control field intensity I0. Accordingly, the following controlfunctional is introduced

J [E] = O[E] ! !!12

tf"

t0

dt E2(t) ! I0

#, (1)

40

with the second term representing the constraint (! is as Lagrange multiplier).The field pulse E resulting in an extremum of O[E] is called the optimal field,and the value of O[E] obtained after optimization is named control yield.

In its most basic form O is defined via a particular state "tar of the ac-tive system, the so–called target state, which should be realized in the ex-ternal field driven molecular system at a particular time tf . This results inO[E] = |""tar|"(tf ;E)#|2 (here, "(tf ;E)# denotes the laser pulse driven molecu-lar wave function). The open system dynamics generalization of O[E] is givenby trS{O#(tf )} where the operator O characterizes the target to be reached and#(tf ) is the reduced density operator (RDO) of the open molecular system S(trS{...} denotes the trace take in the state space of S). A more general expres-sion of O which will be of interest in the following is given by

O[E] =!"

t0

dtf

"dp trS{O(tf ; p)#(tf ; p;E)} , (2)

where p is a certain parameter or set of parameters, which should refer to aparticular property changing among the individual molecules. Therefore, O[E]accounts for a distribution of the operator O (target state) in the space of theparameters p, and, additionally, in time. Expressions (2) is ready to describepump pulse optimization in a pump probe experiment [7] or inhomogeneousbroadening present in the considered molecular system [8].

The optimal field can be calculated from $J/$E = 0. It yields a functionalequation which solution fixes the temporal behavior of the optimal field

E(t) =i

!!

"dp

!"

t

dtf trS{O(tf ; p)U(tf , t; p;E)[µ, #(t; p;E)]} . (3)

The expression contains the commutator of the molecular dipole operator µ withthe RDO at time t. It is further propagated up tf symbolized by the action ofthe time–propagation superoperator U(tf , t;E) (in dependence on the parameterset p and at the presence of the field E). Notice also the integration with respectto tf which is originated by the time distribution of O. To solve Eq. (3) onerearranges the trace expression in order to replace

$ !t dtf O(t; p)U(tf , t; p;E)

by the auxiliary density operators %(t; p;E) propagated backwards in time [9,10]. The control task is solved by a combined iterative forward and backwardpropagation [11].

III. CONTROLLING EXCITONIC WAVE PACKET MOTION

One major application of femtosecond spectroscopy represents the investiga-tion of electronic excitations in chromophore complexes known as Frenkel ex-

41

citons (for a recent introduction in this field see [3]). Of particular interesthave been studies of light harvesting antennae belonging to the photosyntheticapparatus of bacteria or higher plants (see the overview in [12, 13]). Althoughfemtosecond laser pulse control techniques are widely used meanwhile (see [1, 2]),there only exist a single example where these techniques have been applied tochromophore complexes. Ref. [14] describes such an application to discriminatebetween internal conversion and excitation energy transfer taking place amonga carotenoid and bacteriochlorophyll (BChl) molecule in the light harvestingantenna LH2 of purple bacteria.

This concept of guiding excitation energy into one of two particular transferchannels has been put into a more general frame in Refs. [8, 15]. There, a the-oretical analysis has been presented of laser pulse controlled excitation energymotion and localization in systems of strongly coupled chromophores like theFMO–complex or the PS1 (for both antenna systems see Ref. [12, 13]). Refs.[16, 17] put emphasis on polarization shaping what is of particular interest sincerecent pulse shaping technology allows for a simultaneous and independent ma-nipulation of the two di!erent polarization directions of the laser beam [18].

When studying Frenkel excitons one is faced with spatially delocalized excitedstates with the basic electronic excitations, however, completely localized at theindividual chromophores of the complex. The respective state vector for such alocalized excitation will be denoted as |"m# with the index m indicating whichchromophore is in the excited state. The strong coupling among di!erent chro-mophores results in the formation of delocalized exciton states |&# =

%m C!(m)

|"m# with energy !"!. This energy is much larger than the room temperaturethermal energy, and if the coupling to vibrational coordinates remains weak, asit is often the case, excitation energy transfer may proceed coherently up to some100 fs.

The particular control task which has been discussed in [8, 15–17] aims atan excitation energy localization at a single chromophore at a definite time.The localization has to be achieved against the tendency to form delocalizedstates. It requires the photoinduced formation of an excitonic wave packet,i.e. the time–dependent superposition

%! A!(t)|&# of the various exciton states

in such a way that at the final time tf of the control task the superpositioncorresponds to excitation energy localization at a particular chromophore m,i.e.

%! A!(t = tf )|&# = |"m#. To form the superposition state

%! A!(t)|&#

all exciton states in a control task have to be addressed, therefore the oscillatorstrength should be distributed over all exciton states |&# (what excludes theuse of highly symmetric complexes for such studies). For an appropriate non–regular structure, however, di!erent transition dipole moments d! may alsoposses di!erent spatial orientations. This would favor the use of polarizationshaped control fields. The spatial orientation of E(t) (perpendicular to thepropagation direction) would increases the flexibility for putting the variouscoupling expressions d!E(t) in the right order of magnitude at the right time

42

0.5

0.4

0.3

0.2

0.1

0

2-2

0 100 200 300 400 500

Po

pu

latio

n

E in

MV

/cm

t in fs

target site

other sites

x-polarized

0.5

0.4

0.3

0.2

0.1

2-2

2-2

0 100 200 300 400 500

Po

pu

latio

n

E in

MV

/cm

t in fs

target site

other sites

y-polarized

x-polarized

FIG. 1: Disorder averaged chromophore populations in the FMO complex at the targetsite (mtar = 7) and at all other sites (for more details see [8]). Left panel: use of alinear polarized control pulse, right panel: the optimization covered the independenttwo polarization directions of the field. An ensemble of 10 complexes with randomlychosen spatial orientation and fluctuating excitation energies (! = 100 cm!1) has beenconsidered. The temporal evolution of the field components is shown in the upper partof both panels.

interval to achieve the proper wave packet formation.The description of excitation energy localization discussed so far has to be

generalized to the inclusion of exciton relaxation and dephasing originated bythe presence of exciton–vibrational coupling. This requires an approach basedon open system dynamics techniques, i.e. the (reduced) exciton density matrix#!"(t) has to be calculated. Moreover, one has to account for structural andenergetic disorder. Such a combination of density matrix propagation, polar-ization shaping, and disorder to solve the control task using the OCT has beenpresented for the first time in Refs. [16, 17]. It uses Eq. (3) to calculate theoptimal field.

Focusing on energy localization at a particular time tf , the operator O intro-duced in Eq. (2) and characterizing the target of the control task is given by theprojector |"mtar#""mtar | times $(tf !'f ). Then, O[E] of Eq. (2) can be identifiedwith the population of the target state Ptar at t = 'f and can be calculated bythe exciton density matrix according to

Ptar('f ) =1

Nmol

&

p

&

!,"

Cp(&; mtar)C"p ((; mtar)#p(&, (; 'f ) . (4)

Fig. 1 displays respective results for excitation energy localization at chro-mophore 7 of the monomeric FMO complex. The presence of disorder results ina rather low control yield (left panel Fig. 1). Fortunately, this ine#ciency canbe compensated when changing to laser pulse control with polarized light. Thepossibility of an independent control of both polarizations of the field–strengthincreases the control yield considerably (see the right panel of Fig. 1). As

43

demonstrated also in Fig. 1 the time interval where the two components (x–and y–component) of the field act becomes longer compared to the control witha linearly polarized field (the di!erent sub pulses of the two components act outof phase indicating strong elliptic polarization of the control field).

Acknowledgments

Financial support by the Deutschen Forschungsgemeinschaft through Sfb 450is gratefully acknowledged.

[1] J. L. Herek, ed., Coherent Control of Photochemical and Photobiological Processes,vol. 180 of J. Photochem. Photobiol. (2006), special issue.

[2] T. Halfmann, ed., Quantum Control of Light and Matter, vol. 264 of Opt. Comm.(2006), special issue.

[3] V. May and O. Kuhn, Charge and Energy Transfer Dynamics in Molecular Systems(Wiley-VCH, Berlin, 2000), second edition 2004.

[4] K. Sundermann and R. de Vivie-Riedle, J. Chem. Phys. 110, 1896 (1999).[5] M. Abe, Y. Ohtsuki, Y. Fujimura, and W. Domcke, J. Chem. Phys. 123, 144508

(2005).[6] L. Wang, H.-D. Meyer, and V. May, J. Chem. Phys. 125, 014102 (2006).[7] A. Kaiser and V. May, J. Chem. Phys. 121, 2528 (2004), Chem. Phys. Lett. 405,

339 (2005), Chem. Phys. 320, 95 (2006).[8] B. Bruggemann and V. May, J. Phys. Chem. B 108, 10529 (2004), Gerald F.

Small Festschrift.[9] T. Mancal and V. May, Euro. Phys. J. D 14, 173 (2001).

[10] T. Mancal, U. Kleinekathofer, and V. May, J. Chem. Phys. 117, 636 (2002).[11] Y. Ohtsuki, W. Zhu, and H. Rabitz, J. Chem. Phys. 110, 9825 (1999).[12] H. van Amerongen, L. Valkunas, and R. van Grondelle, Photosynthetic Excitons

(World Scientific, Singapore, 2000).[13] T. Renger, V. May, and O. Kuhn, Phys. Rep. 343, 137 (2001).[14] J. L. Herek, W. Wohlleben, R. J. Cogdell, D. Zeidler, and M. Motzkus, Nature

417, 533 (2002).[15] B. Bruggemann and V. May, Chem. Phys. Lett. 400, 573 (2004).[16] B. Bruggemann, T. Pullerits, and V. May, in [1], p. 322.[17] B. Bruggemann and V. May, in Analysis and Control of Ultrafast Photoinduced

Reactions, edited by O. Kuhn and L. Woste (Springer, 2006).[18] T. Brixner, G. Krampert, T. Pfeifer, R. Selle, G. Gerber, M. Wollenhaupt,

O. Graefe, C. Horn, D. Liese, and T. Baumert, Phys. Rev. Lett. 92, 208301 (2004).

44

B. Lasorne and G. A. Worth (eds.)Coherent Control of Molecules

© 2006, CCP6, Daresbury

Instantaneous Dynamics and Quantum Control:Principles and Applications

Philipp Marquetanda, Stefanie Grafeb, Volker Engela and Christoph Meierc

a Institut fur Physikalische Chemie,Universitat Wurzburg, Am Hubland, Wurzburg, Germany

b Stacie Institute for Molecular Sciences, NRC, Ottawa, Canada andc Laboratoire Collisions, Agregats, Reactivite,Universite Paul Sabatier, Toulouse, France

I. INTRODUCTION

Most of the theoretical schemes to control molecular processes by externalfields, such as the Brumer-Shapiro scheme [1–3], the Tannor-Koslo!-Rice scheme[4, 5], optimal control theory [6, 7] or feedback control [8] are based on the opti-mization of a predefined objective which is given at some final time. In contrast,it is also possible to construct external control fields based on the instantaneousmolecular dynamics, in designing the fields at any time such, that the system isdriven in a desired way. This idea was initially proposed by Koslo! et al. [9]and extensively developped by Tannor and coworkers [10–14]. Extensions of thisapproach can be found in Ref. [15–18]. One of the interesting applications ofthe general theory is the laser cooling of molecular vibrational motion on theelectronic ground state [10, 11]. Furthermore, this approach has been used for ageneralization of stimulated Raman adiabatic passsage (STIRAP) to a N -levelquantum system [13].

Here, we show that this algorithm yields electric fields which allow for a clearphysical interpretation for a number of complex control tasks, like the selec-tive mode excitation, excitation and stabilisation of isomerisation processes ormultiphoton electronic and vibrational excitation [19, 20].

II. THEORY

The general idea is to devise a value of the external field at any instant intime and based on the instantaneous system dynamics itself, in such a waythat the expectation value of a specific operator A increases or decreases intime. Additionally, the objective can be more complex, i.e. one can change thestrategy at di!erent times or as a function of other expectation values. Supposea system Hamiltonian H0 which interacts with external fields within the dipole

45

!"#$%#&'()$%$*+,

distance

A

X

!- . -

#&/$0!12

.

.34

.35

.36

.37

-

!- . -

#&/$)0!12

!!"#

!!"$

!!"%

$"&&'()!%&

!"%

!"$

!"#

$($8#*&8)9&$(:

!"!;('#&"%)<=>?

PX PA

FIG. 1: Excitation of Na2 using a laserpulse to achieve an monotonous increase of theA-state population [19].

approximation, yielding a total Hamiltonian of the form

H(t) = H0 + W (t) = H0 ! µE(t) (1)

For a time-independent operator A, the rate of change of "A# is proportional tothe expectation value of the commutator [H(t), A]. If the observable of interestA commutes with the unperturbed Hamiltonian but does not commute with theinteraction term W (t), one has:

d"A#dt

= ! i

!E(t)"!| [µ, A] !# (2)

which clearly shows that E(t) can be adjusted to control d!A"dt , as long as

"!| [µ, A] !# is nonzero. In the case that A commutes with H0, an additionalterm appears in Eq. (2) and the design of the control field has to be adjustedaccordingly. In what follows, we use the relationship (2) not to impose a specificvalue of d!A"

dt , but only to ensure its sign. Hence we can simply use:

E(t) = !i"(t)"!| [µ, A] !# (3)

leaving enough liberty to impose additional constraints, like overall pulse lengthof total absorbed energy, controlled by a specific choice of "(t).

III. APPLICATIONS

A. Electronic population transfer

The formalism given above is, in a straightforward manner, applicable to thecontrol of population transfer between electronic states in atoms and molecules.

46

To this end, we consider the total molecular wavefunction to be expanded intoa set of electronic states |k# according to |!(t)# =

!k |#k(t)#|k# The total pop-

ulation in the electronic state |n# can be expressed as the expectation value ofthe projector onto this state, hence A = |n#"n|. Using Eq. (3), for the electricfield

E(t) = !"(t)"

k

Im"#n|µnk|#k# , (4)

yields a population transfer in state |n# at all times. As an example, in Fig. 1,we show the controlled excitation of the electronic A-state of Na2. Hence, theoperator A is chosen to be the projector onto the electronic A-state. The modelcontains a total of eleven states to have a description as realistic as possible. Ifthere is initially no population in the A-state, Eq. (4) yields E(t) = 0, hencethe process needs to be started by a weak, short resonant pulse starting theA $ X transition. Figure 1 shows the populations in the electronic states andthe control field as a function of time. The A-state population increases insteps which is accompanied by a decrease of the population in the electronicground state. The dynamically constructed field consists of a sequence of pulseswhich are separated by the vibrational period in the A state and is modulatedwith faster frequencies approximately proportional to the electronic transitionenergies. If the wave packet !A, prepared in the A-state moves out of the Franck-Condon region for the A $ X transition, the field becomes zero which is readilyseen by inspection of Eq. (4) containing the overlap integral "#k|µkn|#n(t)#.The field intensity increases again if, after one round-trip, the wave packet re-enters the Franck-Condon window. Then, obviously the already existing wavepacket interferes constructively with those parts which are newly transfered tothe excited electronic state. This is very much the scenario where populationsin excited states are influenced via sequences of phase-locked pulses [21, 22].Further exemples can be found in Ref. [19].

B. Selective mode excitation

Here, we regard a model of the active site of carboxy-hemoglobine. Thissystem has been studied extensively for many years, both theoretically and ex-perimentally using a wide range of theoretical methods or experimental tech-niques. For a recent review of the theoretical approaches, see Ref. [23] and refer-ences therein. Here, we choose the six-coordinated iron-porphyrin-imidazole-CO(FeP(Im)-CO) as active site model, which recently has also been used for thestudy of the excited states [24].

Inspired by recent experimental results on multi-photon IR excitation of the C-O stretch [25], we are specifically interested in the anharmonicity of this mode aswell as its coupling to the other modes. The vibrational dynamics is investigatedwithin a 2D model, comprising the CO vibration (q0) with harmonic energy $0

and the most strongly coupled mode, the Fe-CO stretch vibration (q1, energy $1).

47

-0.5 0.0 0.5time [ps]

-1.0 0.0 1.0time [ps]

2100

2000

1900

$%$*+,)08/))2

@-

FIG. 2: Model of the active site of carboxy hemoglobine (left) and Wigner plots ofoptimized laser pulses with 1 ps (middle) and 1.5 ps overall pulse duration [26].

For a detailed description of the Hamiltonian, we refer to Ref. [26]. Consistently,the dipole moment µ(q0, q1) is also expressed as a function of these two modedisplacements. To selectively excite the CO stretch, the operator entering intothe local control scheme is chosen as A = !!0

2"2

"q20

+ V (q0). According to Eq.(3), a condition for the electric field at any instant is obtained. In this case, "(t)is chosen to ensure an overall pulse duration of 1 ps and 1.5 ps respectively, witha total integrated intensity of 1.33 µJ.

The pulses obtained lead to a monotonic increase of the vibrational energy inthe CO stretch, up to a total energy of % 12000 cm#1 (15000 cm#1) for the 1ps (1.5 ps) pulse (vibrational levels of up to % 13 (% 15))[26]. The electric fieldsconstructed in this way are further analysed and shown as Wigner distribution[27] in Fig. 2. In the case of a 1 ps pulse (middle panel), one clearly sees howthe optimized pulse form requires a change in the excitation frequency fromabout 2080 cm#1 at early times to 1900 cm#1 at later times. This behaviour iscommonly known as ”chirp”, induced by a quadratic phase of the complex fieldsaround the central frequency.

This result shows that, for su"ciently short pulses, the local control algorithmautomatically finds the well-known chirped pulse strategy of multi-photon vibra-tional excitation. This is indeed the strategy which has been used experimentallyfor a wide range of molecular systems (see, e.g. [28–33], including the HbCO,considered in this work [25]. For longer pulses, (keeping the same overall laserintensity), we find a completely di!erent behaviour: the beginning of the pulseis very similar to the result obtained when imposing the short pulse consideredbefore. However, in contrast to the previous situation, for this longer pulse, thealgorithm does not continue this initial chirp to lower energies, but creates newpulses at energies of % 2080 cm#1 , forming a chain parallel to the initial chirp.This structure can be viewed as a second chirped pulse, more structured thanthe initial one, but with the same chirp parameter, determined by the slope of

48

the structures in the time-frequency plane. This suggests, that in this case, it ismore e"cient to excite the ’leftovers’ in fairly low-lying states in a second sweepthan to continue exciting to even higher levels.

With the latest pulse shaping techniques, the experimental verification of thismultiple sweep strategy for e"cient energy pumping in specified modes shouldbe possible.

[1] P. Brumer and M. Shapiro, Chem. Phys. Lett. 126, 541 (1986).[2] P. Brumer and M. Shapiro, Ann. Rev. Phys. Chem. 43, 257 (1992).[3] M. Shapiro and P. Brumer, Rep. Prog. Phys. 66, 859 (2003).[4] D. Tannor and S. A. Rice, J. Chem. Phys. 83, 5013 (1985).[5] D. Tannor, R. Koslo!, and S. A. Rice, J. Chem. Phys. 85, 5805 (1986).[6] A. P. Peirce, M. A. Dahleh, and H. Rabitz, Phys. Rev. A 37, 4950 (1988).[7] R. Koslo! et al., Chem. Phys. 139, 201 (1989).[8] R. S. Judson and H. Rabitz, Phys. Rev. Lett. 68, 1500 (1992).[9] R. Koslo!, A. D. Hammerich, and D. J. Tannor, Phys. Rev. Lett. 69, 2172 (1992).

[10] A. Bartana, R. Koslo!, and D. J. Tannor, J. Chem. Phys. 99, 196 (1993).[11] A. Bartana, R. Koslo!, and D. J. Tannor, Chem. Phys. 267, 195 (2001).[12] D. Tannor, in Molecules in Laser Fields, edited by A. D. Bandrauk (Marcel Dekker,

New York, 1994).[13] V. Malinovsky, C. Meier, and D. J. Tannor, Chem. Phys. 221, 67 (1997).[14] V. Malinovsky and D. J. Tannor, Phys. Rev. A 56, 4929 (1997).[15] P. Gross et al., Phys. Rev. A 47, 4593 (1993).[16] M. Sugawara, J. Chem. Phys. 118, 6784 (2003).[17] M. Sugawara, S. Yoshizawa, and S. Yabushita, Chem. Phys. Lett. 350, 253 (2001).[18] Y. Ohtsuki, H. Kono, and Y. Fujimura, J. Chem. Phys. 109, 9318 (1998).[19] S. Grafe et al., Chem. Phys. Lett. 398, 180 (2004).[20] S. Grafe, C. Meier, and V. Engel, J. Chem. Phys. 122, 184103 (2005).[21] H. Metiu and V. Engel, J. Opt. Soc. Am. 7, 1709 (1009).[22] N. F. Scherer, J. Chem. Phys. 95, 1487 (1991).[23] C. Rovira, J. Phys.: Condens. Matter 15, S1809 (2003).[24] A. Dreuw, B. D. Duniez, and M. Head-Gordon, J. Am. Chem. Soc. 124, 12070

(2002).[25] C. Ventalon et al., Proc. Natl. Acad. Sci. USA 101, 13261 (2004).[26] C. Meier and M.-C. Heitz, J. Chem. Phys. 123, 044504 (2005).[27] C. Bardeen, J. Cau, F. L. H. Brown, and K. R. Wilson, Chem. Phys. Lett. 302,

405 (1999).[28] D. J. Maas et al., Chem. Phys. Lett. 290, 75 (1998).[29] V. D. Kleiman et al., Chem. Phys. 233, 207 (1998).[30] L. Windhorn et al., Chem. Phys. Lett. 357, 85 (2002).[31] M. Bonn, C. Hess, and M. Wolf, Phys. Rev. Lett. 85, 4341 (2000).[32] T. Witte et al., J. Chem. Phys. 118, 2021 (2003).[33] L. Windhorn et al., J. Chem. Phys. 119, 641 (2003).

49

B. Lasorne and G. A. Worth (eds.)Coherent Control of Molecules

c! 2006, CCP6, Daresbury

Optimal Control of Multidimensional Vibronic Dynamics:Algorithmic Developments and Applications to

4D-Pyrazine

Hans-Dieter MeyerTheoretische Chemie, Universitat Heidelberg,

Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany

Luxia Wang and Volkhard MayInstitut fur Physik, Humboldt-Universitat Berlin,

Newtonstrasse 15, D-12489 Berlin, Germany

I. INTRODUCTION

It is of ongoing interest to steer molecular dynamics by specially tailored fem-tosecond laser pulses. Simulations are usually based on optimal control theory(OCT) [1, 2]. These simulations, however, become very expensive when poly-atomic molecules with several active modes are to be studied. OCT requires tosolve a non-linear time-dependent Schrodinger equation several times, becausethe algorithm is iterative. It is hence important that the numerical algorithmsused are e!cient. The multiconfiguration time-dependent Hartree (MCTDH)method is an e!cient algorithm for solving the time-dependent Schrodingerequation of multi-dimensional systems. In the present paper OCT is combinedwith MCTDH. This approach is then applied to a four-dimensional model ofpyrazine.

II. MULTICONFIGURATION TIME-DEPENDENT HARTREE(MCTDH)

The wavepacket propagation calculations reported in this article are performedwith the Heidelberg MCTDH package [3]. MCTDH [4–7] is a general algorithmto solve the time-dependent Schrodinger equation for distinguishable particles.The MCTDH method uses a complete active space (CAS) of so–called single–particle functions (SPFs). As the SPFs are time–dependent, they follow thewavepacket and often a rather small number of SPFs su!ces for convergence.

50

The ansatz for the MCTDH wavefunction reads

"(q1, · · · , qf , t) =n1!

j1

· · ·nf!

jf

Aj1,··· ,jf (t)f"

!=1

!(!)j!

(q!, t) (1)

=!

J

AJ #J ,

where f denotes the number of degrees of freedom. There are n! SPFs for the"’s particle. The AJ ! Aj1...jf denote the MCTDH expansion coe!cients andthe configurations, or Hartree-products, #J are products of SPFs, implicitlydefined by Eq. (1). The SPFs are finally represented by linear combinations oftime-independent primitive basis functions.

The MCTDH equations of motion are derived by applying the Dirac-Frenkelvariational principle [8, 9] to the ansatz Eq. (1). After some algebra, one obtains

iAJ =!

L

"#J |H |#L#AL , (2)

i!(!) =#1 $ P (!)

$ #"(!)

$!1"H#(!)!(!) , (3)

where a vector notation, !(!) = (!(!)1 , · · · , !(!)

n! )T , is used. Here "H#(!) denotesa matrix of mean-fields (for the "-th degree of freedom), "(!) is a one-particledensity matrix and P (!) denotes a projector onto the set of SPFs. Details onthe derivation, as well as more general results, can be found in Refs. [5–7].

The MCTDH equations conserve the norm and, for time-independent Hamil-tonians, the total energy. This follows directly [6] from the variational prin-ciple. MCTDH contains Time-Dependent Hartree (TDH) and the standardmethod (i.e. propagating the wavepacket on the primitive basis) as limitingcases. MCTDH simplifies to TDH when setting all n! = 1. Increasing the n!

recovers more and more correlation, until finally, when n! equals the numberof primitive functions, the standard method is used. It is important to note,that MCTDH uses variationally optimal SPFs, because this ensures fast conver-gence. The MCTDH equations of motion are non-linear and complicated, butbecause there are comparatively few equations to be solved, MCTDH can bevery e!cient.

The application discssed below deals with a non-adiabatic system where thewave packet evolves on two coupled diabatic electronic states. The formalismmust hence be extended to include electronic states [10, 11]. The wavefunction" and the Hamiltonian H are expanded in the set {|##} of electronic states:

|"# ="!

#=1

"(#) |## (4)

51

and

H ="!

#,$=1

|##H(#$)"$ | , (5)

where each state function "(#) is expanded in MCTDH form Eq. (1). Theequations of motion read

iA(#)J =

"!

$=1

!

L

"#(#)J |H(#$) |#($)

L #A($)L , (6)

i!(#,!) =#1 $ P (#,!)

$#%(#,!)

$!1 "!

$=1

"H#(#$,!)!($,!) , (7)

with mean-fields

"H#(#$,!)jl = ""(#,!)

j |H(#$) |"($,!)l # . (8)

The superscripts # and $ denote to which electronic state the functions andoperators belong. A fuller derivation of these equations is given in Ref. [11].

III. OPTIMAL CONTROL

In its standard version laser pulse control of molecular dynamics is formulatedas the task to maximize an observable O at a particular final time tf by radi-ation field excitation [12–14]. In the present case of pure–state dynamics theobservable is obtained as

O(tf ;E) =< "(tf ;E)|O|"(tf ;E) > , (9)

where O is the operator corresponding to the observable and "(tf ;E) denotesthe laser–pulse driven system wave function at final time. Assuming a singlestate (target state) "tar which should be reached at tf , O would become theprojector |"tar >< "tar|.

The maximization of O(tf ) may be achieved by applying the OCT based onthe functional

J(tf ;E) = O(tf ;E) $ 12

tf%

t0

dt &(t)E2(t) . (10)

This so–called control functional contains a constraint to guarantee finite fieldintensity, where the respective Lagrange parameter &(t) includes a time depen-dence to switch the laser field on and o$ smoothly, &(t) = &0/ sin2('t/tf ).

52

Once the extremum of J(tf ;E) has been determined the laser pulse can becalculated which solves the control task, usually named optimal pulse. It has tobe deduced from the following functional equation

E(t) = $ 2!&(t)

Im < %(t;E)|µ|"(t;E) > , (11)

where the wave function "(t;E) obeys the standard time–dependent Schrodingerequation (including the laser pulse and with the initial value |"0 >)

i! (

(t|"(t) >= (Hmol $ E(t)µ)|"(t) > . (12)

The function %(t) results from a similar equation

i! (

(t|%(t) >= (Hmol $ E(t)µ)|%(t) > , (13)

but propagated backwards in time starting at tf and ending at t0. The respective”initial” value reads

|%(tf ) >= O|"(tf) > . (14)

In Refs. [1, 2] an iteration scheme to determine the optimal pulse has beensuggested which combines this forward and backward propagation iteratively.The key point in this connection is the replacement of the field–strength inEqs. (12, 13) by the right–hand side of Eq. (11) resulting in coupled nonlinearSchrodinger equations for forward and backward propagation. When iteratingthese equations one guesses an initial pulse E(0)(t) to compute the zeroth–orderwave function "(0)(t) using directly Eq. (12). For the subsequent backwardpropagation (calculation of %(0)) one replaces E(t) in Eq. (13) by Eq. (11) eval-uated with %(0) and "(0). The nonlinear Schrodinger equation for %(0) can besolved and the whole procedure is repeated until convergence is obtained. Then’th forward iteration uses the result of the (n $ 1)’th backward iteration andthe n’th backward iteration uses the result of the n’th forward iteration. Withinevery iteration step (finalized after the computation of "(n)) an approximateoptimal pulse E(n)(t) can be calculated from the right–hand side of Eq. (11).

At present, however, the existing MCTDH–code [3] does not allow for thesolution of nonlinear equations. Therefore, the iteration scheme discussed sofar to solve Eq. (11) for the optimal pulse cannot be directly transferred tothe MCTDH method. In order to circumvent this obstacle we proceed in thefollowing way. Instead of introducing the field E(t) (Eq. (11)) into the equationsfor "(t) and %(t), we use previous values of the field–strength and extrapolatethose to the future. This predicted field is used for the propagation step andis afterwards updated. To get the n’th iteration of the field at time t + &t weevaluate the field at the current time t according to Eq. (11) using %(n!1)(t)

53

0 20 40 60 80 100

Time (fs)

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Lev

el P

op

ula

tio

n

0 10 20 30

Time (fs)

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

Lev

el P

op

ula

tio

n

FIG. 1: Laser pulse controlled maximization of the overall S1–state population (upperpanel, tf = 100 fs) and of the S2–state population (lower panel, tf = 30fs). Solid line:S0–state population, dashed line: S2–state population, dashed-dotted line: S1–statepopulation. The thin solid line displays the temporal behavior of the respective optimalpulse (the unit 0.1 at the axis corresponds to an electric field–strength of about 2 · 107

V/cm).

and "(n)(t). (Note that %(n!1)(t) is known for all 0 % t % tf ). Using E(t)and the previous values E(t$&t), E(t$ 2&t), the E-field is then quadratically

54

extrapolated up to t + &t and the wave packet is propagated from t to t + &t.The process is iterated till the final time t = tf is reached. A similar algorithmapplies for the backward propagation. The error one introduces by this form oflinearisation is the extrapolation error. This error vanishes as &t & 0.

IV. APPLICATION: 4D-PYRAZINE

The pyrazine model is characterized by three electronic states, the electronicground state S0 and two electronically excited states S1 and S2. The latter twoare strongly coupled vibronically via a conical intersection. Only the S2 state isbright, i.e. the field couples S0 and S2 exclusively. As an example, we maximisethe population of the state S1 or S2, respectively. Fig. 1 displays the results ofour calculations (note the di$erent time scales). Because S1 is populated via theconical intersection after S2 is populated by the light pulse, a rather large tf ishelpful for obtaining a high population (see Fig. 1, upper part). On the otherhand, when S2 is to be populated, a rather short tf is of advantage to avoid thatthe state is depopulated by the vibronic coupling (see Fig. 1, lower part). Thefigures are taken from Ref. [15].

[1] W. Zhu, J. Botina, and H. Rabitz, J. Chem. Phys. 108, 1953 (1998).[2] W. Zhu and H. Rabitz, J. Chem. Phys. 109, 385 (1998).[3] G. A. Worth, M. H. Beck, A. Jackle, and H.-D. Meyer, The MCTDH Package,

Version 8.2, (2000). H.-D. Meyer, Version 8.3 (2002). See http://www.pci.uni-heidelberg.de/tc/usr/mctdh/.

[4] H.-D. Meyer, U. Manthe, and L. S. Cederbaum, Chem. Phys. Lett. 165, 73 (1990).[5] U. Manthe, H.-D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 97, 3199 (1992).[6] M. H. Beck, A. Jackle, G. A. Worth, and H.-D. Meyer, Phys. Rep. 324, 1 (2000).[7] H.-D. Meyer and G. A. Worth, Theor. Chem. Acc. 109, 251 (2003).[8] P. A. M. Dirac, Proc. Cambridge Philos. Soc. 26, 376 (1930).[9] J. Frenkel, Wave Mechanics (Clarendon Press, Oxford, 1934).

[10] J.-Y. Fang and H. Guo, J. Chem. Phys. 101, 5831 (1994).[11] G. A. Worth, H.-D. Meyer, and L. S. Cederbaum, J. Chem. Phys. 105, 4412

(1996).[12] S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics (Wiley, New

York, 2000).[13] M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Pro-

cesses (Wiley, New York, 2003).[14] V. May and O. Kuhn, Charge and Energy Transfer Dynamics in Molecular Systems

(Wiley-VCH, Weinheim, 2004).[15] L. Wang, H.-D. Meyer, and V.May, J. Chem. Phys. 125, 014102 (2006).

55

B. Lasorne and G. A. Worth (eds.)Coherent Control of Molecules

c! 2006, CCP6, Daresbury

Optimal Control of Nonadiabatic Photochemical ProcessesInduced by Conical Intersections

Yukiyoshi Ohtsukia,b), Mayumi Abea), Yuichi Fujimuraa)

Zhenggang Lanc) and Wolfgang Domckec)

a) Department of Chemistry, Graduate School of Science,Tohoku University, Sendai 980-8578, Japan

b) CREST, Japan Science and Technology Agency (JST),4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japanc) Department of Chemistry, Technical University

of Munich, D-85747 Garching, Germany

I. INTRODUCTION

Nonadiabatic transitions are ubiquitous in photochemical processes [1]. Inorder to successfully achieve coherent control of photochemical processes, it istherefore essential to design laser pulses that steer a wave packet from an initialstate to an objective state through nonadiabatic transitions [2–6]. From an ex-perimental viewpoint, closed-loop experiments are known to be e!cient for thispurpose because the feedback loop can design control pulses with minimal knowl-edge of molecular Hamiltonians [7]. As the closed-loop experiments are optimalcontrol experiments, the theoretical counterpart of the closed-loop experimentsis optimal control theory. Both optimal control experiments and theory can worktogether to attain the overall best molecular control achievements. For example,optimal control simulations with appropriate model potential-energy surfaces(PESs) provide information about control mechanisms that can be useful forunderstanding molecular dynamics underlying experimental outputs.

Among many nonadiabatic transitions, here we focus on those induced byconical intersections (CIs), where two Born-Oppenheimer PESs intersects witheach other at a point or seam. A CI is characterized by the divergence of thederivative coupling between two Born-Oppenheimer PESs. One of the conse-quence associated with a CI is that it provides an e"ective region of very strongnonadiabatic coupling, which is often attributed to ultrafast nonadiabatic pho-tochemical processes. Another consequence is termed the geometric (Berry)phase.

In the present talk, we discuss coherent control mechanisms of nonadiabaticphotochemical processes induced by CIs by means of optimal control simulation.

56

For this purpose, we conduct two case studies, in which either one of the above-mentioned characteristics of CIs plays an important role.

In the first case study [5], we apply optimal control simulation to the cis-transisomerization of retinal in rhodopsin within a two-dimensional, two-electronic-state model with a CI, whose potential parameters are proposed by Hahn andstock. This model consists of a reaction coordinate that connects a trans isomerwith a cis isomer via a CI, and a coupling mode that corresponds largely tothe stretching coordinate of the polymer chain. We will show that optimallydesigned pulses work cooperatively with a CI that acts as a wave-packet cannon.The optimal pulses largely consist of shaping subpulses that prepare the wavepacket, which is localized along a reaction coordinate and has little energy in thecoupling mode, through multiple electronic transitions. This shaping process isessential for achieving a high target yield although the detailed structures of thepulse envelopes depend on the local topography of the PESs around the CI.

In the second case study [6], we consider the geometric phase e"ects in thecoherent control of the branching ratio of photodissociation products of phenolwithin a two-dimensional, three-electronic-state model with two CIs.

II. THEORY

We consider a molecule interacting with a linearly polarized laser pulse, E(t)within the electric dipole approximation. The Hamiltonian of the system Ht

is given by Ht = H0 ! µE(t), where H0 is a field-free Hamiltonian and µ isthe electric dipole moment operator. We consider that a physical objective isspecified by a target operator W that is chosen so that it gives the maximumexpectation value when a molecular system reaches an objective state. Theoptimal pulse is the pulse that maximizes the target expectation value at aspecified final time, < W (tf ) > subject to minimal pulse fluence. According toa standard optimal control procedure, the optimal pulse is expressed as

E(t) = !AIm < !(tf ) | WU(tf , t)µ | !(t) > , (1)

where A is a positive constant (or a time-dependent positive function) thatweighs the physical significance of the penalty due to the pulse fluence. Thewave function, | !(t) >, obeys the Schrodinger equation,

i! "

"t| !(t) >= [H0 ! µE(t)] | !(t) > , (2)

with the initial condition, | !(t) >=| !0 >. In Eq. (1), U(tf , t) is the timeevolution operator associated with Eq. (2).

57

0.5 1.0 1.5 2.0 2.5 3.0

0

20000

40000

60000

80000

potential en

erg

y (cm

-1

)

O-H bond length (Å)

0( )S !!

1( *)S !!

2( *)S !"

2( *)S !!

1( *)S !"

1( )S !!

0( *)S !"

2

H(1 )s! !

2

H(1 )s" !

FIG. 1: Adiabatic potential profiles along the OH stretching coordinate r for plannarphenol (! = 0). The adiabatic electronic states are referred to as the S0 (solid line),S1 (dotted line) and S2 (dashed line) states in order of increasing energy. [Taken fromRef. 6]

If we introduce an auxiliary function (the Lagrange multiplier) defined by| #(t) >= U(t, tf)W | !(tf ) >, we can numerically solve these coupled pulsedesign equations by the iteration procedure. When we deal with the control ofunbounded molecular dynamics, we modify the solution algorithm, according toour previous paper, in order to treat the wave packet components that spreadsbeyond the grid region [8].

III. RESULTS

We summarize the results concerning the second case study [6]. The pho-todissociation of phenol is described by a two-dimensional, three-electronic-statemodel [9]. The PESs are constructed from accurate ab initio calculations. Theadiabatic electronic states are referred to as the S0, S1 and S2 states in orderof increasing energy. This reduced-dimensionality model that consists of theOH-stretching coordinate r and the CCOH dihedral angle $. The former is thereaction coordinate and the latter is the coupling mode of the S2!S1 and S1!S0

CIs. Profiles of the adiabatic PESs of phenol as a function of r at $ = 0 areshown in Fig. 1. The S0 state is the ground electronic state, with %% characterfor r " 2.0 A and %&# character at a longer distance (r > 2.0 A). We use thenotations S0(%%) and S0(%&#) , etc., when referring to the electronic characterof a given state. The control target concerned here is the increase in one of thedissociation products. We solve the pulse design equations using a monotonicallyconvergent iteration algorithm [8, 10].

Figure 2 (left) shows the results when the maximization of the 2& + H(1s) ischosen as a target, while Fig. 2 (right) shows the results when the maximization

58

0 50 100 150 200

0.0

0.1

0.2

0.3

0.4

0.5

-6

-4

-2

0

2

4

6

population

time (fs)

(b)

(a)

"3

electric field (109

Vm

-1

)

2

H(1 )s" !

2

H(1 )s! !

*!!

0 50 100 150 200

0.0

0.1

0.2

0.3

0.4

0.5

-6

-4

-2

0

2

4

6

"3

(b)

population

time (fs)

electric field (109

Vm

-1

)

(a)

2

H(1 )s" !

2

H(1 )s! !

*!!

FIG. 2: (Left figure) The enhancement of the S1 [2" + H(1s)] dissociation is chosen asa target. (Right figure) The enhancement of the S0 [2# +H(1s)] dissociation is chosenas a target. (a) Optimal control pulses. (b) Dissociation probabilities associated withthe 2" + H(1s) product (dotted line) and the 2# + H(1s) product (solid line). Thedashed line shows the time-dependent population of the diabatic ##! state. [Takenfrom Ref. 6]

of the 2% + H(1s) is chosen as a target. The former (latter) control pulse isreferred to as the S1 (S0) dissociation pulse. In the left figure, the S1 dissociationpulse causes 47.1% (34.9%) dissociation in the 2&+H(1s) (2%+H(1s)) channel.The optimal pulse has a simple structure with a central frequency of 44380 cm!1,which is estimated from a power spectrum. This is higher in energy than the CI(41930 cm!1).

In the right figure, on the other hand, 35.3% (43.8%) of the population dis-sociate through the 2& + H(1s) (2% + H(1s)) channel. The total dissociationprobability is similar to that caused by the S1 dissociation pulse. The optimalpulse changes the branching ratio [2% + H(1s)]/[2& + H(1s)]) from 0.81 (leftfigure) to 1.35 (right figure). Except for slight modulations, the S0 dissociationpulse again has a simple structure with a central frequency of 43980 cm!1. Theseresults strongly suggest that the control achievements are largely attributed toinherent properties of phenol, i.e., the PES structure.

From the time-dependent population of the diabatic %%# state, we see that theS1 dissociation pulse transfers more population to the S1(%%#) state than theS0 dissociation pulse. That is, the optimal pulse with a higher (lower) centralfrequency transfers more population to the S1 (S2) state. This may be coun-

59

terintuitive because the S1 potential is lower in energy than the S2 potentialaround the equilibrium configuration, which can be explained by the vibroniccoupling mechanism. If we simply add one quantum of the OH stretching vibra-tion of the S0 potential, then the optical excitation energy is estimated to be43300 cm!1, which is similar to the S2(%&#) $ S0(%%) vertical excitation en-ergy (43148 cm!1). This strongly suggests that there exists a vibronic couplingmanifold between the S1 and S2 states in this energy range, which can explainthe reduction in the %%# population when excited by the S0 dissociation pulse.On the other hand, as the S1 dissociation pulse has a higher central frequencythan this manifold, it can avoid the S2 ! S1 coupling and selectively create awave packet in the S1(%%#) state.

The S1 dissociation pulse creates a wave packet on the S1 PES. This wavepacket moves adiabatically on the S1 PES and it is bifurcated at the first CI intotwo components with opposite phases because of the geometric phase e"ects (seeFig. 1). The destructive interfaces as well as the shape of the S1 PES reduce theprobability density around $ = 0, which suppress nonadiabatic transitions at thesecond CI, resulting in an enhancement of the 2&+H(1s) dissociation. Thus theS1 dissociation pulse employs the adiabatic control path. On the other hand,the S0 dissociation pulse utilizes the diabatic control path. That is, it creates awave packet on the S2 PES via intensity borrowing e"ects. This wave packet istransferred to the S1 state by nonadiabatic transitions at the first CI, while beingsqueezed by the funnel structure of the CI. Because of this squeezing e"ects, thewave packet has a finite probability density near $ = 0, which enhances thenonadiabatic transition at the second CI and thus 2% + H(1s) dissociation.

[1] W. Domcke, D. R. Yarkony, and H. Koppel, eds., Conical Intersections (WorldScientific, Singapore, 2004).

[2] D. Geppert, A. Hofmann, and R. de Vivie-Riedle, J. Chem. Phys. 119, 5901(2003).

[3] M. Sukharev and T. Seideman, Phys. Rev. Lett. 93, 093004 (2004), Phys. Rev. A71, 012509 (2005).

[4] P. S. Christopher, M. Shapiro, and P. Brumer, J. Chem. Phys. 123, 064313 (2005).[5] M. Abe, Y. Ohtsuki, Y. Fujimura, and W. Domcke, J. Chem. Phys. 123, 144508

(2005).[6] M. Abe, Y. Ohtsuki, Y. Fujimura, Z. Lan, and W. Domcke, J. Chem. Phys. 124,

224316 (2006).[7] R. J. Levis, G. M. Menkir, and H. Rabitz, Science 292, 709 (2001).[8] K. Nakagami, Y. Ohtsuki, and Y. Fujimura, J. Chem. Phys. 117, 6429 (2002).[9] Z. Lan, W. Domcke, V. Vallet, A. L. Sobolewski, and S. Mahapatra, J. Chem.

Phys. 122, 224315 (2005).[10] Y. Ohtsuki, G. Turinici, and H. Rabitz, J. Chem. Phys. 120, 5517 (2004), and

references therein.

60

61

B. Lasorne and G. A. Worth (eds.) Coherent Control of Molecules

© 2006, CCP6, Daresbury

An Exploration of Optical Control of

Quantum Dynamics in the Liquid Phase

Stuart A. Rice Department of Chemistry and The James Franck Institute, The University of Chicago,

Chicago, IL 60637, USA

I. INTRODUCTION

First principles approaches to achieving active control of the quantum

dynamics of an isolated molecule using shaped light fields were developed about twenty years ago and in the past ten years several different control methodologies have been experimentally validated [1, 2]. The theme that is common to these methodologies is quantum interference, specifically, field-generated manipulation of the phase of the molecular excitation in a fashion that permits control of transfer of population from a specified initial state to a specified final state. Because of the central role of interference effects in all of the control methodologies, it at first sight appears to be futile to seek to control the quantum dynamics of a molecule in the liquid phase wherein very frequent interactions are known to generate rapid dephasing. There are, however, two classes of observations that mitigate this pessimism. First, studies of vibrational relaxation in the liquid state show that, in the absence of strong specific interactions such as hydrogen bonding, population relaxation occurs on the time scale of tens of picoseconds whilst phase relaxation occurs on the time scale of picoseconds [3], implying the existence of a time window for the application of short shaped pulses that can control population transfer between selected states of a solute molecule. Second, there are several experimental demonstrations [4–7] of active control of molecular dynamics in solution, arguably the most amazing of which is the work of Gerber et al. [7] on control of the cis-trans isomerization of a large diazo dye molecule.

62

The initial, and still the most common, approach to theoretical analysis of control of the quantum dynamics of a molecule in the liquid phase starts with the full system Hamiltonian, so that in principle the interactions between all the molecules and between the molecules and the applied field are accounted for [8, 9]. If the analysis could be carried out this approach would yield the most complete and most accurate information concerning different control methodologies. However, this N-body problem cannot be addressed without drastic simplifications, usually introduced via modeling the solute-solvent interaction and reduction of the degrees of freedom included in the treatment of the internal solute molecule dynamics. Consequently, although examples of these approaches lead to very interesting predictions vis a vis the possibilities for achievement of active control of the solute molecule dynamics, the validity of the predictions is subject to uncertainty. A different approach to developing control methodologies valid in the liquid phase can be based on exploitation of adiabatic population transfer between states of the solute. Because electronic dephasing is very rapid, on the time scale of femtoseconds, we seek control of molecular dynamics on the ground electronic state, within the manifold of vibrational states. The key problems that must be resolved are:

(1) What is an appropriate representation of the states of the solute molecule?

(2) What is an appropriate representation of the influence of the solvent on the states of the solute molecule?

(3) Given answers to (1) and (2), how robust is controlled population transfer in solution to perturbations of various sorts?

As to (1), we adopt the states defined by the transitions in the measured spectrum of the molecule. In the isolated molecule these are, except for radiation, the rovibrational eigenstates of the molecular Hamiltonian, and they generate a diagonal representation free of mechanical couplings. That these eigenstates can have a complicated representation in some chosen set of orthonormal basis states, say, harmonic oscillator-rigid rotator states, is not relevant. In the liquid phase the corresponding states are vibrational pseudo-eigenstates which have lifetimes sufficiently long that they generate a useful short time diagonal representation. The price that is paid for this simplification includes limitation of the approach to cases in

63

which the vibrational states of the solute are well resolved in the liquid solution, and the requirement that the character of each state, e.g. to what it evolves, must be determined from a different calculation or experiment. As to (2), we assume that the effects of the liquid can be represented in the time interval of interest for control of the molecular dynamics as fluctuations of the vibrational levels of the solute molecule, as in the Kubo theory of line shape [10]. The distribution and amplitudes of these fluctuations must be determined using input information from other calculations or experiments.

We illustrate the usefulness of the approach outlined by considering the achievability of STIRAP [11] (stimulated Raman adiabatic passage) generated transfer of population between levels of a solute molecule in a liquid [12]. Although the theoretical calculations have been extended to several more complicated situations [13–15], including cases with more levels and with level degeneracy, for simplicity we focus attention on transfer of population between levels in a three-level system embedded in a liquid.

II. STIRAP IN A LIQUID SOLUTION

We consider first STIRAP transfer of population between three levels

of an isolated molecule; these three levels define the molecular subsystem of interest. Levels 1 and 2, and 2 and 3, are connected by nonzero transition dipole moments while direct transitions between levels 1 and 3 are forbidden. Our goal is to transfer 100% of the population from level 1 to level 3. This is achieved by first applying a strong field pulse that couples levels 2 and 3 (Stokes field), and subsequently a second strong field pulse that couples levels 1 and 2 (pump field), with temporal overlap of the applied fields. One of the eigenstates of the coupled field-molecule system, when represented in terms of the states of the isolated molecule, has no component of molecular state 2. When the Stokes and pump field pulses are turned on and off adiabatically, all of the population of molecular state 1 is transferred to molecular state 3.

Consider now three vibrational levels of a molecule in solution. Can STIRAP generated transfer of population be achieved? We assume that the rate of depopulation by vibrational energy transfer is much smaller

64

than the rate of dephasing and that the influence of the surrounding solvent on the states of the solute can be represented by Gaussian random fluctuations of its energy levels. The population dynamics is then described by a set of six equations for the real and imaginary parts of the time dependent amplitudes of the molecular states; these are stochastic differential equations because of the fluctuations. Numerical solution of the equations of population dynamics yields the results shown below [12].

Here the yield refers to the population of level 3, the delay time refers to separation of the peaks of the Stokes and pump pulses, the time unit (TU) is the pulse width, the distribution of level fluctuation amplitude had width of 150 cm–1 and (a), (c) and (f) correspond to level fluctuation frequencies of 1000, 10 and 0.01 (TU)–1, respectively. Efficient population transfer can be achieved when the frequency of fluctuations is small and large relative to the Rabi frequencies of the Stokes and pump pulses; it degrades when these frequencies are equal. The efficiency of population transfer when the fluctuation frequency is small is to be expected for a nearly static environment, and when it is large is a consequence of only requiring two–photon resonance that can be satisfied with many pairs of detuning amplitudes.

The preceding analysis can be extended to the case that the target state has a finite lifetime, e.g. a predissociating state [13]. The intuitive expectation is that control of population transfer is very difficult in this case. Analysis shows that the field molecule eigenstates are complex and each has a contribution from the decaying state. The remarkable finding is that if the decay rate constant is large compared with the bandwidth of the Stokes and pump pulses but small compared with the peak Rabi frequencies the dephasing induced non-adiabaticity associated with the population transfer can be suppressed with the result that, irrespective of the correlation time of the stochastic energy fluctuations, complete population transfer to the decaying target state is possible. An example is shown below [13].

65

The left panel displays the STIRAP generated population transfer when state 3 has zero decay rate, the right panel when it decays with a rate constant of 15 ps–1. In this example the Stokes and pump pulse widths are T = 2 ps, the delay time 1 ps, the peak Rabi frequencies are 60 ps–1, the variance of the energy fluctations 5 ps–1 and the correlation times of the fluctuations (a) 8 fs, (b) 40 fs and (c) 200 fs.

A version of STIRAP that generates control of branching to two

different reaction products, represented by two states in a five state set, was developed by Kobrak and Rice [16]; it is the strong field limit of the pulsed incoherent interference scheme of Chen, Shapiro and Brumer [17]. State 1 is coupled to an intermediate state, 2, which is coupled to degenerate product states, 3 and 4, that are not coupled to state 1. Control of the ratio of population transfers from 1 to 3 and 4 is achieved by coupling 3 and 4 to “branch” state, 5, which is not coupled to states 1 or 2. The ratio of populations transferred to states 3 and 4 is found to be inversely proportional to the ratio of transition moments connecting 3 and 5 and 4 and 5. Using the same model for the influence of solvent on the molecular energy levels, control of the product branching ratio remains achievable in the limits that the fluctuation frequency is small and large relative to the Rabi frequencies, just as in three-level STIRAP. Even when strong dephasing limits control of that branching ratio, selectivity can be re-established by continuous measurement of the branch state, although

66

some of the population of the branch state is absorbed away thereby limiting the absolute amount of product formed [13]. Using the slowly varying continuum approximation, continuous measurement of the

population of state 5 can be described with a rate constant ! and tuned via the system-probe interaction. An example is shown below [13].

Panel (a) shows the populations of states 3 and 4 when the population of 5 is not measured, while panels (b), (c) and (d) show those populations when the rate of measurement of state 5 corresponds to !T = 3, 6 and 9, respectively. The peak Rabi frequencies are !12T = 20, !23T = 50, !24T = 40, !53T = 15, !54T = 75.

All of the preceding, though promising, depends on the accuracy of the

representation of the effect of the liquid on the energy levels of the solute molecule. How good is that representation? Demirplak and Rice [18] have carried out a simulation study of HCl in dense fluid Ar, using the very accurate PICKABACK propagator [19] to describe the HCl vibration; the HCl rotation and translation and the Ar translations were described with classical mechanics. Some of the results obtained are shown below.

67

The top-left pair of panels shows, for HCl in Ar with a reduced density of 0.85 at reduced temperature 1.0, that the distributions of modulations of the energies of the ground and first vibrational states are different in magnitude and non-Gaussian. The top-right panels show that STIRAP generated population transfer from v = 0 to v = 2, with two different driving fields (0.05 and 0.10 V/Å), is severely degraded from 100%. This degradation is primarily a consequence of transition frequency overlap arising from power broadening of the slightly separated v = 0 " v = 1 and v = 1 " v = 2 transitions. If the same non-Gaussian modulation statistics are used with the original three-level model of Demirplak and Rice [12], which has well separated levels, the STIRAP generated population transfer is of order 80%, as shown in the bottom-left panel.

68

III. CONCLUSIONS

These model calculations suggest that control of population transfer in

a liquid can be achieved in some circumstances using variants of adiabatic population transfer. Experimental tests of the proposed control schemes should be carried out.

[1] S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics. (Wiley, New York,

2000). [2] M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Processes.

(Wiley, New York, 2003). [3] L. K. Iwaki and D. D. Dlott, J. Phys. Chem. A 104, 9101 (2000). [4] C. J. Bardeen, V. V. Yakovlev, K. R. Wilson, S. D. Carpenter, P. M. Weber, and W.

S. Warren, Chem. Phys. Lett. 280, 151 (1997). [5] T. Brixner, N. H. Damrauer, P. Niklaus, and G. Gerber, Nature (London) 414, 57

(2001). [6] J. Herek, W. Wohlleben, R. J. Codgell, D. Zeidler, and M. Motzkus, Nature 417, 533

(2002). [7] C. Vogt, G. Krampert, P. Nicklaus, P. Neuernberger, and G. Gerber, Phys. Rev. Lett.

94, 068305 (2005). [8] See, for example, Y. Yan and R. Xu, Ann. Rev. Phys. Chem. 56, 187 (2005). [9] See, for example, J. Cao, M. Messina, and K. R. Wilson, J. Chem. Phys. 106, 5239

(1997); K. Hoki and P. Brumer, Phys. Rev. Lett. 95, 168305 (2005). [10] R. Kubo, J. Math. Phys. 4, 174 (1963). [11] K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys. 70, 1003 (1998). [12] M. Demirplak and S. A. Rice, J. Chem. Phys. 116, 8028 (2002). [13] J. Gong and S. A. Rice, J. Chem. Phys. 120, 3777 (2004). [14] J. Gong and S. A. Rice, J. Chem. Phys. 120, 5117 (2004). [15] J. Gong and S. A. Rice, J. Chem. Phys. 120, 9984 (2004). [16] M. N. Kobrak and S. A. Rice, Phys. Rev. A 57, 2885 (1998); J. Chem. Phys. 99, 1 (1998). [17] Z. Chen, M. Shapiro, and P. Brumer, Chem. Phys. Lett. 288, 289 (1994); J. Chem. Phys.

102, 5683 (1995); Phys. Rev. A 52, 2225 (1995). [18] M. Demirplak and S. A. Rice, to be submitted to J. Chem. Phys. [19] P. Nettlesheim, F. A. Bornemann, B. Schmidt, and C. Schutte, Chem. Phys. Lett. 256,

581 (1996).

B. Lasorne and G. A. Worth (eds.)Coherent Control of Molecules

c! 2006, CCP6, Daresbury

Coherent Control and Coherence Spectroscopy ofRotational Wavepackets in Dissipative Media

S. Ramakrishna, Adam Pelzer and Tamar SeidemanDepartment of Chemistry, Northwestern University,2145 Sheridan Road, Evanston, IL. 60208-3113 USA

I. INTRODUCTION

Nonadiabatic molecular alignment by short intense pulses has been the topic ofrapidly growing activity during the past few years [1]. This activity owes bothto the fascinating fundamental physics associated with rotational wavepacketdynamics and to a variety of already demonstrated and projected applicationsin fields ranging from molecular spectroscopy and laser optics [2] through re-action dynamics and stereochemistry [3], to quantum storage and informationprocessing [4]. In this approach, a moderately intense laser pulse of durationshort with respect to the rotational periods aligns the most polarizable axis ofa polyatomic molecule to the field polarization vector. Nonadiabaticity (rapidturn-o! as compared to the system time scales) guarantees that the alignmentwill survive subsequent to the pulse turn-o!, under field free conditions. Whereasa linearly polarized pulse induces 1D order, leaving the rotation about the fieldand molecular axes free, an elliptically polarized pulse aligns the three axes ofthe molecule to three axes defined in space [5]. Although most of the researchin this area has focused on isolated molecules in the rotationally-cooled molec-ular beam environment [1], recent research has illustrated the applicability ofshort-pulse induced alignment to dissipative media, including dense gas cells [6],solutions [7], and matrices [8].

Here we consider the possibility of using the unique coherence properties ofrotationally-broad wavepackets as a coherence spectroscopy, to probe the dissi-pative properties of interesting media. We regard this work as part of a moregeneral e!ort to apply di!erent coherent control strategies as a means of explor-ing material properties. In the next section we briefly summarize our approachto describing alignment in dissipative media within a quantum mechanical den-sity matrix formalism. Section III first (III A) illustrates that the experimentalobservables of alignment disentangle decoherence [9] from population relaxatione!ects, providing independent measures of the relaxation and the decoherencedynamics that go beyond rate measurements. To make connection between the

69

present and other contributions to this booklet, we illustrate, in Sec. III B, thatfurther insight into the system-bath interaction can be derived from applicationof feedback control strategies to the alignment problem. The final section brieflyconcludes.

II. THEORY

We consider a linear molecule interacting with a dissipative medium and sub-ject to a linearly polarized, moderately-intense laser pulse. The system Hamilto-nian (Hs) consists of the molecular part (Hmol) and the field matter interaction(Hint(t)), and we assume a Markovian quantum master equation for the reduceddensity operator, !(t). Thus,

"

"t! = ! i

!Ls!(t) !D!(t), (1)

where L and D represent the system Liouville and the dissipative superoperators[10, 11], respectively:

Ls!(t) " [Hmol + Hint(t), !(t)]!, (2)

D!(t) = !!

J,M,J!,M !

"12[KJMJ!M ! |JM#$JM |, !(t)]+ ! KJMJ!M ! |J "M "#$J "M "|

%$JM |!(t)|JM# + #(pd)JMJ!M ! |JM#$JM |!(t)|J "M "#$J "M "|

#. (3)

In Eq. (3), $$, %|JM# = YJM ($, %) are spherical harmonics, ($, %) are the polarand azimuthal Euler angles, respectively, KJMJ!M ! is the rate of populationtransfer from state |JM# to state |J "M "#, #(pd)

JMJ!M ! is the pure decoherence[9] rate between the states |JM# and |J "M "#, and the dissipation dynamics isapproximated within the multilevel Bloch model.

Optimal control theory of molecular dynamics determines the spectral compo-sition of the laser pulse that will realize a desired goal (e.g., optimize a specificobservable) at the end of the laser pulse, t = tf . In the context of molecularalignment, the observable most often used to quantify the degree of alignmentand study its time evolution is the expectation value of cos2 $. In order tomaximize $cos2 $#(t = tf ) we seek the extremum of the functional A,

A = Tr{cos2 $!(tf )}! 12

tf$

t0

dt&(t)E2(t), (4)

70

where the second term on the right-hand side upper bounds the field intensity,and the penalty function, &, is taken to be time dependent, to avoid suddenswitch on and switch o! of the control field [12]. With alterations in the func-tional (4), other objectives of interest in the alignment context can be realized(and are discussed elsewhere) but for the purpose of the present study the sim-plest case scenario su"ces.

III. RESULTS AND DISCUSSION

In the first part of this section we point to the information content of the ob-servables of alignment experiments regarding the system-bath interactions. Forconcreteness we consider the case of nonresonance interaction and a standardGaussian form of the excitation pulse. Our conclusions, however, are indepen-dent of the observable, the mode of excitation of the wavepacket and the pulseshape. In Sec. III B we explore the further insights that can be gained by sub-jecting the system to a shaped pulse that is iteratively structured to maximizethe alignment at a specific instance of time. To complement the discussion ofSec. III A, we focus here on the case of near-resonance excitation of the rota-tional wavepacket, but other modes of rotational excitation would furnish similarinformation.

A. Intense laser alignment as a route to solvent dynamics

We begin by partitioning the alignment measure as

$cos2 $# " $cos2 $#tot = $cos2 $#p + $cos2 $#c, (5)

where $cos2 $#p "%

J,M !JMJM (t)VJMJM and $cos2 $#c "%J #=J!,M !JMJ!M (t)VJMJ!M =

%JJ!=J±2,M !JMJ!M (t)VJMJ!M , with

VJMJ!M ! = $JM | cos2 $|J "M "#. While formally the partitioning of a rep-resentation of ! into diagonal and o!-diagonal terms is always possible, inthe context of alignment Eq. (5) carries interesting physical significance andimportant practical consequences. Its power owes to the fact that $cos2 $#pprovides a direct experimental measure of the population elements, whereas$cos2 $#c measures directly the time evolution of the coherence elements of thedensity matrix. This property can be illustrated by expressing the VJMJ!M interms of analytical functions, to provide explicit forms for the time-evolutionof $cos2 $#p and $cos2 $#c. During the pulse each of the thermally populatedlevels {Ji, Mi} is excited into a broad superposition of levels {J, M = Mi},$cos2 $#p undergoes a sharp rise from 1/3 to its maximum value (& 1/2 inthe strong field limit), and $cos2 $#c raises from zero to a maximum valuethat approaches 1

4

%JM (!JMJ+2M + !JMJ!2M ) with increasing intensity.

71

0 2 4 6

t/!rot

-0.2

0

0.2

0.4

0.6

0.8

<co

s2"

>x

FIG. 1: The averaged alignment !cos2 !" (solid curve) and its components !cos2 !"p(dashed curve and circles) and !cos2 !"c (dotted curve) versus time, measured in unitsof the rotational period "!/Be. The interaction strength is ! = !(t0)/Be = 291 andthe pulse duration is 0.011. The dot-dashed curve traces the envelop of !cos2 !"c.

In the isolated molecule case, $cos2 $#p is constant after the pulse turn-o!whereas $cos2 $#c oscillates with a constant amplitude at the rotational period.In a dissipative environment, $cos2 $#p = 1

3

%JM

&1 + 2 J(J+1)!3M2

(2J+3)(2J!1)

'!JMJM

describes a smooth baseline that decays from max{$cos2 $#p(t)} & 12 to its initial

(thermal) value of 13

%J

%JM=!J

&1 + 2 J(J+1)!3M2

(2J+3)(2J!1)

'!JMJM = 1

3 , tracingaccurately the decay of population due to inelastic (J-, or J- and M -changing)collisions. It follows that $cos2 $#p presents an experimental measure of thedynamics of population relaxation and is independent of decoherence, whereas$cos2 $#c responds solely to decohering processes.

It is pertinent at this juncture to point out that the description of dissipationbriefly summarized in Sec. II, and the decomposition of the alignment observable$cos2 $# in terms of $cos2 $#p and $cos2 $#c, are general. In particular, they areequally valid for gaseous media and for solutions. System details are incorporatedvia the transition rates, KJMJ!M ! , whose form and magnitude depend on theinteraction of the molecular system with the bath.

Figure 1 provides a pictorial illustration of the discussion surrounding Eq.(5), showing $cos2 $# (solid curve) along with its two components. The timeevolution of $cos2 $#p (dashed curve and circles), which describes the population,is clearly reproduced by the (observable) baseline of $cos2 $#. The time-evolutionof $cos2 $#c (dotted curve), which provides the coherences, is reproduced by the

72

(observable) di!erence curve, $cos2 $# ! $cos2 $#p. We find that the partitioningof the signal in Eq. (5) translates the experimental observable into a direct probeof the timescales involved in the evolution of the system density matrix.

We proceed by defining independent measures of inelastic and elastic scat-tering events as the baseline of $cos2 $# and the ratio of the oscillation envelop(dot-dashed curve in Fig. 1) and the baseline, respectively. The former measure,denoted $cos2 $#inel, provides the evolution of $cos2 $#p and is sensitive only to in-elastic collisions. The latter measure, denoted $cos2 $#elas, traces the ratio of theamplitude of $cos2 $#c and $cos2 $#p and reflects solely the dynamics of elastic col-lisions. Thus, strong-laser-induced alignment signals enable one to decouple andprobe within a single measurement the dynamics of population relaxation and de-coherence. In the specific case where both the population relaxation and the de-coherence can be approximated by a single exponential decay, we obtain the timeconstants of population relaxation (T1), decay of coherence (T2), and pure de-coherence (T $

2 ) from the observable $cos2 $# as, $cos2 $#inel = $cos2 $#p ' e!t/T1 ,$cos2 $#c ' e!t/T2 " e!t/(T1+T"

2 ) and $cos2 $#elas = $cos2 $#c/$cos2 $#p ' e!t/T"2 .

The general case is discussed in [6, 11].

B. An optimal control approach to coherence spectroscopy

Focussing next on the information content of iteratively generated controlfields, we first remark that in the isolated molecule limit, optimal controltheory is capable of precisely timing the occurrence of field-free alignmentin molecules of any symmetry provided that the field matter interaction isallowed su"cient time as compared to the natural system time scales [13].(In order to manipulate the wave-packet to attain a specific phase relationat a given time, the individual constituents of the wave-packet must haveenough time for their phases to di!er significantly from unity. Because thespectrum of Hmol involves level spacings of order of the rotational constant Be,pulses short with respect to 'rot = (/Be are unable to control these interactions.)

For tf values not remote from 2'rot, we observed, in isolated molecules, a clearsignature of a “double-kick” mechanism, where the pulse cleanly separates intotwo sub-pulses, the second timed to the revival of the wavepacket created by thefirst. This result generalizes to the case of multiple kicks, given su"ciently longinteraction times tf , but is restricted to systems that exhibit classically regularrotational dynamics and hence a periodic revival structure (linear or symmetrictops). In the presence of a dissipative bath this result holds as long as theobservation times tf is small with respect to the dissipation time-scale.

The last statement is quantified in Fig. 2, where we show the short-timeFourier transforms of fields that were optimized to time the alignment of COmolecules at di!erent pressures of a gaseous Ar bath. In the dissipation-free

73

(zero pressure) limit, Fig. 2a, the optimal pulse consists of a five subpulsesequence, in which the subpulses gradually shift to higher energy pulse centersas the rotational excitation progresses and the rotational level spacing increases.At finite pressures, the multiple kick scheme with increasing frequency becomesine"cient, as the phase relation established by the early sub-pulses is quicklydisrupted by inelastic collisions with the bath atoms (elastic collisions, and hencepure decoherence, play at most a minor role in this system [14]). As the pressure,and hence the relaxation rate, increase, the subpulses migrate to later times,thereby skipping the low energy early “pre-pulse” (Fig. 2b). In the limit offast relaxation, the rotational excitation mechanism switches to a single, lateintensive pulse that is su"ciently broad in energy to span the range of rotationallevel spacing that our penalty function allows (Fig 2c).

Figure 2d shows the evolving alignment induced by an optimized field andcontrasts the population relaxing bath (solid curve) with a purely relaxing bath(dashed curve). The dissipation rate is 40 ps in both media. Our goal of max-imizing the alignment at tf = 40 ps is clearly attained in both the populationrelaxing and the purely decohering bath. We note, however, the distinct di!er-ence in the mechanism by which peak alignment is attained. In the case of apopulation relaxing bath (solid curve), the rotational excitation is rapidly builtjust before the final time, while the phase relation among the rotational com-ponents is optimized to establish the peak at the target time tf = 40 ps. Thebaseline of $cos2 $# therefore remains close to the isotropic value of 1/3 untiljust before tf . In the case of the purely decohering bath (dashed curve), thealgorithm points to a more e"cient mechanism, where rotational population isbuilt early in the pulse and the phase relation among the rotational componentsis established subsequently. As a consequence, the baseline of $cos2 $# reachesits final value of ca. 0.45 already at the midpoint of the pulse. We note alsothat subsequent to t = tf the baseline of $cos2 $# gradually decays on a 40 pstimescale in the relaxing bath, whereas in the purely decohering bath it remainsconstant.

The alignment strategy adapted to best handle pure decoherence is furtherclarified by examining the short time Fourier Transform of the pulse optimizedto establish alignment at tf = 40 ps for di!erent pure decoherence rates. Oneclear distinction from the analogues analysis of the relaxing bath is that themultiple kick scheme favored in the isolated molecule limit is not replaced by asingle intense pulse just before tf , although its structure mirrors the changingproperties of the bath as the decoherence rate grows. In particular, as all J-levels decohere at the same rate, the isolated molecule strategy of increasingthe frequency with each pulse remains e"cient in the purely decohering bath.Finally, in contrast to the relaxing bath case, the main peak of the optimal pulseis timed to roughly the center of the pulse in the decohering counterpart, wheremuch of the population excitation is established. Here, however, rephasing ofthe rotational components occurs rather late in the pulse as compared to the

74

0 20 40 60Time (ps)

0.3

0.4

0.5

0.6

<co

s2"!

d

FIG. 2: (a–c) Short-time Fourier transforms of optimal tf = 40 ps pulses in the absenceof pure decoherence at (a) 0 Torr, (b) 100 Torr, and (c) 200 Torr. (d)The averagedalignment induced by an optimal field in the presence of population relaxation (solidcurve) and pure decoherence (dashed curve). The dissipation timescale is 40 ps in bothmedia.

isolated molecule limit, so as to best handle the phase disrupting e!ects of elasticcollisions with the bath constituents. Other phenomena, which space limitationprecludes from including here, are discussed in [13]

IV. CONCLUSIONS

Rotational wavepackets provide a fascinating probe of the coherence and dissi-pative properties of dense environments, which expresses itself in the observablesof alignment experiments. Here we noted a partitioning of $cos2 $# in (unopti-mized) alignment experiments that disentangles decoherence from populationrelaxation timescales and provides independent measures of both processes. Wefound also that the time and energy characteristics of the laser fields that maxi-mize the alignment at a predetermined instance provide useful insights into theunderlying bath-system interaction.

75

Acknowledgments

We thank the US Department of Energy (Grant No. DE-FG02-04ER15612)for support.

[1] For reviews see T. Seideman and E. Hamilton, Adv. At. Mol. Phys. 52, 289 (2005);H. Stapelfeldt and T. Seideman, Rev. Mod. Phys. 75, 543 (2003).

[2] R. A. Bartels et al., Phys. Rev. Lett. 88, 013903 (2002).[3] T. Suzuki et al., Phys. Rev. Lett. 92, 133005 (2004).[4] K. F. Lee et al., Phys. Rev. Lett. 93, 233601 (2004).[5] J. J. Larsen et al., Phys. Rev. Lett. 85, 2470 (2000).[6] S. Ramakrishna and T. Seideman, Phys. Rev. Lett. 95, 113001 (2005).[7] J. Ohkubo et al., J. Chem. Phys. 120, 9123 (2004).[8] T. Kiljunen, B. Schnidt, and N. Schwentner, Phys. Rev. Lett. 94, 123003 (2005).[9] We follow the conventions of the gas phase wavepacket dynamics literature, where

“dephasing” is used for change of the relative phases of the wavepacket componentsdue to anharmonicity of the molecular spectrum, whereas “decoherence” is usedfor loss of phase. The former takes place in the isolated molecule limit whereasthe latter requires collisions or photon emmission. It is important to note that thisusage di"ers from that standard in the condensed phase literature.

[10] V. May and O. Kuhn, Charge and Energy Transfer Dynamics in Molecular Systems(Wiley-VCH, Weinheim, 2004).

[11] S. Ramakrishna and T. Seideman, J. Chem. Phys. 124, 34101 (2006).[12] K. Sundermann and R. de Vivie-Riedle, J. Chem. Phys. 110, 1896 (1999).[13] A. Pelzer, S. Ramakrishna, and T. Seideman, to be published.[14] A. Belikov and M. Smith, J. Chem. Phys. 110, 8513 (1999).

76

B. Lasorne and G. A. Worth (eds.)Coherent Control of Molecules

c! 2006, CCP6, Daresbury

Tubular Image States and Trapping on the Nanoscale

Dvira Segal1, Petr Kral2 and Moshe Shapiro3,4

1Department of Chemistry, Columbia University, 3000 Broadway, New York, NY2Department of Chemistry, University of Illinois at Chicago Chicago, IL 60607

3Department of Chemical Physics,Weizmann Institute of Science, 76100 Rehovot, Israel

4Department of Chemistry, University ofBritish Columbia, Vancouver V6T1Z1, Canada

I. INTRODUCTION

Image potential states (IPS) are excited states of electrons that exist in thevicinity of surfaces. They resemble many of the properties of atomic and molec-ular Rydberg states [1]. IPS result from the attractive potential between anelectron outside the surface of a conducting solid and its electrostatic image.When the penetration of the electronic states inside the bulk is prohibited, theimage states form a quantized energy set converging towards the vacuum level.

When an electron exists at a distance z above a flat conductor the attractiveimage potential given as, V (z) = !e2/4z, is identical to the potential producedby a positive (mirror) charge at a distance z inside the material. Because of the1/z attraction, the image potential can support a series of Rydberg like stateswhose energies, given as, En = !0.85 eV/(n + a)2, where n = 1, 2, 3, is theprincipal quantum number, converge as n " # to the “vacuum energy”.

Hofer et al. [2] have used two photon photoemission to populate and probecoherent wave packets of image states above a Cu(100) surface. In this experi-ment one photon excites an electron out of an occupied state below the Fermienergy into an image potential state. A second photon excites the electron toan energy above the vacuum level. By varying the photon energy and detectingthe electronic energy one can obtain “images” of these states.

In flat surfaced metals, the lifetimes of image states is limited to a few picosec-onds due to the annihilation of the electrons on the surface. These lifetimes scalewith n as !n = n3, due to the reduction in overlap of the high n electrons with thesurface. Similar experiments performed above molecular wires laid on surfaces[3], nanoparticles [4], and one and two dimensional liquid Helium [5] demon-strated that extended image states are common in nanoscale systems. However,practical applications of image states above flat metallic surfaces are limited bytheir short lifetimes [6].

77

Recently, a new class of electronic image states existing in the vicinity ofcylindrically shaped objects, such as metallic nanotubes, called “Tubular ImageStates” (TIS), were predicted to exist [7]. These states ought to be much morestable than image states above flat surfaces due to the interplay between theattractive image potential and the repulsive centrifugal potential associated withtheir non zero angular momentum l. The “centrifugal barrier” thus formedprohibits the electron from reaching the surface and be annihilated there, whilethe attractive image potential prevents the electron from escaping the nanotuberegion.

TIS in finite sized systems can also be longitudinally localized along thenanowire main axis by designed inhomogeneities [8], and can be tuned by ex-ternal electric and magnetic fields [9]. In addition, bands of TIS can be formedabove 1D and 2D periodic arrays of nanotubes [10]. The collapse of these stateson the material surface is predicted to be very slow and to be mainly due to theexcitation of circularly polarized phonons [11]. TIS are also an interesting andrich platform for investigating quantum chaos in the nanoscale [12].

The existence of IPS with prolonged lifetime was recently confirmed in multi-walled nanotube. Similarly to the flat metal experiment [2], in this case also theimage states were populated by a two photons photoemission, where one photonphotoexcites an electron out of an occupied states below the Fermi level into animage potential state, and a second photon promotes this image electron to anenergy above the vacuum level. The resulting kinetic energy provides informa-tion on the image states binding energies and lifetimes [13], which nicely agreewith the theoretical predictions [14]. It is expected that similar states could beobserved around single wall nanotubes and nanotubes bundles.

II. THEORY

Following Ref. [7] we consider a charge q0 placed at a distance " relativeto the center of a perfectly conducting, infinitely long, tube of radius a. Thepotential of interaction between the charge and the charges it induces on themetallic surface can be shown to be asymptotically equal to

V (") $ !q20

a

1" ln("/a)

. (1)

The e!ective interaction potential Veff ("), shown in Fig. 1, is formed by addingto the attractive induced potential the repulsive centrifugal potential

Veff (") = V (") +!2(l2 ! 1/4)

2me"2, (2)

where me is the electron mass.

78

For moderate angular momenta (l % 6), the e!ective potential possesses ex-tremely long range wells that can support bound TIS. The high centrifugalbarrier prevents the electron from collapsing onto the surface. The wave func-tions, associated with the TIS-electron motion in this potential are separable inthe cylindrical coordinate. Assuming that the potential along the longitudinalz direction is homogeneous we have that,

"n,l,k(", #, z) = $n,l(") exp(il#)#k(z)/!

2%". (3)

The variation of these wave function with n is depicted in Fig. 2. The corre-sponding eigenenergies are En,l,k = En,l + Ek, where En,l is related with theradial electronic motion, and Ek is the energy for the longitudinal motion alongthe wire. The radial wave function $n,l(") satisfies the Schrodinger equation

"! !2

2me

d2

d"2+ Veff (") ! En,l

#$n,l(") = 0. (4)

The eigenstates corresponding to a few meV of binding energies are highly ex-tended from the surface.

! "! #!!$!

!%!

!

%!

!&'()*

+,--.!/&&'),+* l0%!

l01

% $ 2

!$!!!

!

$!!!

!&'()*

+,--.!/&&'),+*

l0%!

l02

FIG. 1: The e!ective potential between an electron and a conducting nanotube shownfor a number of angular momenta l, and a = 0.68 nm. Long range minima are seenfor l > 6. The inset shows a blowup of the potential near the surface of the nanotubewhere a large potential barrier exists.

79

! 2! 1!!

!3!%

!3!$

!3!2

!&'()*

4"(0%56.!/4$

l01

l07

l0# l08

l=10

l=11

FIG. 2: The n = 1 wave functions (squared) for the set of potentials shown in Fig. 1

.

III. APPLICATION: NANOSCALE PAUL TRAPPING OF A SINGLEELECTRON

We now apply the above formulation to the construction of a “nano Paultrap”[15]. The nano Paul trap is made up of four parallel conducting nanorods,e.g., metallic carbon nanotubes (CNT), spaced symmetrically about a centralaxis. Application of a properly tuned ac field to the nanorods can lead to twodimensional (“radial”) confinement of a single electron at the trap center. Con-finement along the third (“axial”) direction can be attained by manipulating thestructure of the tubes and/or the shapes of the end-caps. In addition to the size,the major di!erences between such a nano-trap and its macroscopic analog arethat the electron must be treated as a quantum object and that its e!ect (“backreaction”) on the trapping device cannot be ignored.

As shown in Fig. 3(a), the typical dimensions of the proposed device are inthe nanometric range: The distance between the tubes is d $ 100 nm, eachtube’s radius is a $ 0.7 nm, and the tubes length is L $ 200 nm. Theserequirements are within present day technology [16]. The proposed device hasto operate at very high frequencies, 100 GHz – 1 THz, which recent studies ofthe ac performance of CNT suggest are realizable [17].

We have calculated numerically the image potential interaction energyV0(x, y, z), including the electrostatic interaction between the rods, using themethod of Ref. [8]. A contour plot of this potential using the typical dimensions

80

!9! !9!

!9!!

9!

!

%!!

$!!

:

;

<&'()*

!:

!:

;

:

=&'()*

.>/

?&'()*

<&'()*

=&'()*

.@/

!%!! ! %!!

!%!!

!

%!!

!%!! ! %!!!%!!

!

%!!

!"!

!

"!

<&'()*

.A/

=&'()*

+:&'),+*

!%!! ! %!!!%!!

!

%!!

!9!

!

9!

<&'()*

.;/

=&'()*+B&'),+*

FIG. 3: (a) A schematic illustration of a nano-trap composed of four parallel tubesaligned in the z direction. The large circle represents the size of the incoming electronicwave packet and the small circle - the size of the trapped wave packet. (b) A contourplot of V0(x, y). (c) A Surface plot of VQ(x, y), for Q/L ! ! = 0.005 e/nm. (d) Thetotal potential energy function for ! = 0.02 e/nm. z = 100 nm in panels (b)-(d).

given above is displayed in Fig. 3(b). The depth of the potential at the centerpoint is $ !0.5 meV, while near the surface of each tube it goes down to !250meV. It can be shown that far from the tubes ends, z $ 100 nm, the axialvariation of the potential is weak in comparison to the radial gradient [8]. Forsu#ciently long tubes it is therefore permissible to neglect the axial dependenceof the potential.

As shown in Fig. 3(a), by connecting the four tubes to an external ac source,oscillating at frequency $, where tube no. 1 and its diagonal counterpart, tubeno. 4, are charged by Q and tube no. 2 and diagonal counterpart, tube no.3, by !Q, we form an oscillating quadrupole. We can calculate VT (x, y, z), thetotal interaction energy of a charged particle with the nano-quadrupole in thepresence of image charges by using a simple extension of our numerical procedure[8]. The total potential, VT , is a sum of two terms,

VT (x, y, t) = V0(x, y) + VQ(x, y) cos($t), (5)

where V0 is the image potential and VQ, the “charging” potential, is the addi-tional potential resulting from the added ac quadrupolar charges. The charging

81

potential has an approximate quadrupolar spatial shape, as shown in Fig. 3(c).Panel (d) presents the combination VT that to a good approximation sustainsthe quadrupolar form. As in conventional linear Paul traps [18] with ac charging,the saddle point at the center provides a point of stability where the electroncan be confined.

The dynamics of a single electron subject to this potential is obtained bysolving the time-dependent Schrodinger equation

i!&t"(x, y, t) =$!!2

2me

%&2

x + &2y

&+ VT (x, y, t)

'"(x, y, t), (6)

where me is the electron mass. Unlike the case of a pure quadrupole [18], and dueto the presence of the image charge potential (V0), this equation is non-separablein the x, y coordinates and the analytical results of Ref. [19] cannot be applied.Given Eq. (6), the propagation of an initial wave packet can be computed usinge.g., the unitary split evolution operator propagation method[20].

!#!

!

#!

=&'()*

C0!

!

.>/

#&',D()* C0$39B

.@/

C09B

.A/

C0739B

.;/

!#!

!

#!

=&'()*

!3!!79

.,/ .-/ .E/ .F/

!#!

!

#!

=&'()*

!3!$

.G/ .H/ .I/ .6/

!#! ! #!

!#!

!

#!

=&'()*

<&'()*

!3!9

.)/

!#! ! #!<&'()*

.(/

!#! ! #!<&'()*

.J/

!#! ! #!<&'()*

.K/

FIG. 4: Electron’s probability-density for di!erent charging potentials. (a)-(d) ! = 0;(e)-(h) ! = 0.0075 e/nm; (i)-(l) ! = 0.02 e/nm; (m)-(p) ! = 0.05 e/nm. The calculationis performed in a x-y 250"250 nm2 lattice and for " = 8.3 1011 Hz.

Due to the scaling properties of the Schrodinger equation (6), it is possible

82

to design traps of various spatial dimensions operating at di!erent control con-ditions. Hence, the dynamics is invariant to the scaling up of the spatial andtemporal coordinates ' "

&(', % " (%, provided the frequency, charge

density, and image potential are scaled down as

$ " $/(, ) " )/(, V0 " V0/(. (7)

The scaling of the image potential, which roughly behaves as 1/[" ln("/a)], canbe accomplished approximately by adjusting the tube’s radii.

We are now in a position to demonstrate trapping and focusing of a single elec-tron in the nano-trap. Figure 4 displays the time evolution of an initial (highlylocalized) Gaussian wave packet "(x, y, t = 0) = exp[!(x2 + y2)/2*2]/

&%*2, of

width * = 10.5 nm, subject to di!erent charging potentials and ac modulationfrequency of + ' 1/T ' $/2% = 8.3 1011 Hz. The spatial parameters are as inFig. 3. We see that the wave packet evolves into distinct shapes for di!erentvalues of the applied potential: In the absence of the ac modulation the electronaccumulates in the vicinity of the tubes, as shown in Fig. 4(a-d). For the case ofweak charging potentials, shown in Fig. 4(e-h), the electron is neither trappednor is it strongly attracted to the tubes. Rather, the wave packet slowly spreadswhile filling the entire inter-tube space. Tight focusing occurs when the chargingis high enough in Fig. 4(i-l) with the wave packet oscillating in the middle ofthe trap between its original shape and an elliptical, yet highly focused, shape.When the charging becomes too high, confinement is lost, as shown in Fig. 4(m-p). The electron, though avoiding the tubes, manages to leak out by followingthe x = 0 and y = 0 lines. The wave packed evolution manifest the interplaybetween the image potential and the charging potential. In (a)-(d) the imagepotential dominates, leading to the collapse of the electron onto the nanotubes.In (e)-(h) the two potentials are of the same order, while in (i)-(p) the e!ect ofthe image potential is negligible.

A full characterization of the trap is presented in Fig. 5 where the stabilitydiagram for di!erent frequencies and charging potentials is provided. The dia-gram was generated by propagating for long times ($ 10 T ) initially localizedor unlocalized wave packets and recording whether at the end of the integrationtime the wave function remains confined at the trap center. We ascertained thatthe results are independent of the wave packet’s initial shape. Note that due tothe image potential, the range of stability is decreased. This e!ect ultimatelylimits the operation of very small traps to d > 10 nm. Using Eq. (7) we caneasily generate such diagrams for di!erent spatial dimensions. The trap stabilitycan be also studied as a function of an additional dc voltage, as is usually donein conventional Paul traps [18].

Finally we note that an axial localization of the electron in the trap resultsnaturally from the potential barriers imposed by the tubes end-caps [8]. Thiscauses the electron to oscillate in both radial and axial directions at drasticallydi!erent frequencies. Whereas the radial vibrational frequency is $ 500 GHz,

83

!31 !3# % %3$ %3" %31 %3#

!3!%

!3!2

!3!9

!3!7

$&&%!%$&'L?*

#&',D()*

FIG. 5: A stability map of the nano-trap. Trapping exists in the shaded area. Thedotted line marks the lower border of the stability region when the contribution of theimage potential is removed from the Hamiltonian. This removal has no e!ect on theupper border line.

+z $ 10 GHz for the axial motion [8]. Strong axial confinement can be alsoachieved by applying a dc repulsive field on additional ring shaped electrodes atthe tubes ends [18].

A single trapped electron is a promising candidate for a variety of quantum-information applications utilizing its spin states, the anharmonic quasienergyspectrum [21], and its coupling to the nanowires mechanical modes [22]. Itsdynamics can be manipulated either by magnetic fields or through coherent laserexcitations. Of even greater interest would be arrays of electrons [23] trapped bya 2D lattice [10] of quartets of nano-traps of the type described here. Such arrayscan be used as a set of entangled q-bits, where the degree of entanglement wouldbe controlled by varying the ac charging, thereby allowing a greater or lesserportion of the electronic population to leak into neighboring cells and interactwith other electrons confined there.

Summary

Rydberg-like image states above nanostructures represent a new topic of in-terest with unique potential applications in optics, electronics and surface sci-

84

ence. Their manipulation by light fields can be utilized for quantum informationstorage and retrieval. Study of their localization properties and dynamics canprovide important information about processes occurring on the nanotube sur-face. We have discussed the theoretical reasons for the TIS stability and theiruse in the creation of a nano Paul Trap. Our investigations can be easily gener-alized to include other nanorods structures that can now be are fabricated, andto describe confinement of other charged particles besides electrons.

[1] T. Gallagher, Rydberg Atoms (Cambridge University Press, New York, 1994).[2] U. Hofer, I. L. Shumay, C. Reuß, U. Thomann, W. Wallauer, and T. Fauster,

Science 277, 1480 (1997).[3] J. E. Ortega et al., Phys. Rev. B 49, 13859 (1994).[4] V. Kasperovich et al., Phys. Rev. Lett. 85, 2729 (2000), M. Boyle et al., Phys.

Rev. Lett. 87, 273401, (2001).[5] P. M. Plazman and M. I. Dykman, Science 284, 1967 (1999), P. Glasson et al.,

Phys. Rev. Lett. 87, 176802 (2001).[6] P. M. Echenique, F. Flores, and F. Sols, Phys. Rev. Lett. 55, 2348 (1985), E. V.

Chulkov et al., ibid. 80, 4947 (1998).[7] B. E. Granger, P. Kral, H. R. Sadeghpour, and M. Shapiro, Phys. Rev. Lett. 89,

135506 (2002).[8] D. Segal, P. Kral, and M. Shapiro, Phys. Rev. B 69, 153405 (2004).[9] D. Segal, P. Kral, and M. Shapiro, Chem. Phys. Lett. 392, 314 (2004).

[10] D. Segal, B. E. Granger, H. Sadeghpour, P. Kral, and M. Shapiro, Phys. Rev. Lett.94, 016402 (2005).

[11] D. Segal, P. Kral, and M. Shapiro, Surf. Sci. 577, 86 (2005).[12] D. Segal, P. Kral, and M. Shapiro, J. Chem. Phys. 122, 134705 (2005).[13] M. Zamkov et al., Phys. Rev. Lett. 93, 156803 (2004).[14] M. Zamkov et al., Phys. Rev. B 70, 115419 (2004).[15] D. Segal and M. Shapiro, Nano Lett. 6, 1622 (2006).[16] Y. Tu, Y. Lin, and Z. F. Ren, Nano Lett. 3, 107 (2003), Z. Yu, S. Li, P. J. Burke,

Chem. Mater. 16, 3414 (2004).[17] P. J. Burke, Solid State Electron. 48, 1981 (2004), Z. Yu, P. J. Burke, Nano Lett.

5, 1403 (2005).[18] W. Paul, Rev. Mod. Phys. 62, 531 (1990).[19] L. S. Brown, Phys. Rev. Lett. 66, 527 (1991).[20] N. Balakrishnan, C. Kalyanaraman, and N. Sathyamurthy, Physics Reports 280,

79 (1997).[21] S. Mancini, A. M. Martins, and P. Tombesi, Phys. Rev. A 61, 012303 (2000).[22] L. Tian and P. Zoller, Phys. Rev. Lett. 93, 266403 (2004), W. K. Hensinger et al.,

Phys. Rev. A 72, 041405(R), (2005).[23] G. Ciaramicoli, I. Marzoli, and P. Tombesi, Phys. Rev. Lett. 91, 017901 (2003).

85

B. Lasorne and G. A. Worth (eds.)Coherent Control of Molecules

c! 2006, CCP6, Daresbury

Design of Femtosecond Pulse Sequences to ControlPhotochemical Reactions

David J. TannorDepartment of Chemical PhysicsWeizmann Institute of Science

Rehovot, 76100, Israel

I. INTRODUCTION

In recent years, there has not only been great progress in making femtosecondpulses, but also in being able to shape these pulses, i.e. to give each compo-nent frequency any desired amplitude and phase. Given the great experimentalprogress in shaping and sequencing femtosecond pulses, the inexorable questionis: How is it possible to take advantage of this wide possible range of coherentexcitations to bring about selective and energetically e!cient photochemical re-actions? Many intuitive approaches to laser selective chemistry have been triedin the past twenty years. Most of these approaches have focussed on depositingenergy in a sustained manner, using monochromatic radiation, into a particularstate or mode of the molecule. Virtually all such schemes have failed, due torapid intramolecular energy redistribution.

In 1985, Tannor and Rice [1] (see also [2]) formulated the problem of the searchfor optimal pulses, subject to suitable constraints, as a problem in the calculusof variations. Their formulation was based on a perturbation theory expressionfor the time-dependent wavepacket amplitude and led to a nonlinear integralequation for the optimal pulse(s). A formal and computational breakthroughwas achieved in 1988 by Rabitz et al. [3, 4] who recognized the utility of optimalcontrol techniques, used widely in engineering, for the calculation of optimalpulses. Optimal control theory (OCT) is nothing more than the extension ofthe calculus of variations to problems with di"erential equation constraints. Inthe context of molecular control, this di"erential equation is the time-dependentSchrodinger equation (TDSE), which is introduced via a Lagrange multiplierinto the variational equations. The OCT methodology, in addition to providinga simplified algorithm for obtaining optimal fields numerically, has no di!culty inhandling fields of arbitrary strength. This results in a significant generalizationof the perturbative formalism, not only because greater chemical yields can

86

be obtained with strong fields, but also because many new mechanisms areavailable to strong fields that are not accessible in the weak-field regime. Manyof the components of the TDSE/OCT formalism developed by Rabitz et al. werediscovered independently by Koslo" et al. [5] in the context of control using twoelectronic states. The approach of Koslo" et al. was developed intuitively and assuch is extremely valuable pedagogically as it provides a heuristic explanation forthe optimal control methods, including a physical interpretation for the Lagrangemultiplier wavefunction. Modifications of this approach, with applications toa wide variety of systems, have been developed by other groups. The OCTapproach is reviewed in several places [6, 7].

A second approach, that has grown in popularity recently, is known as ’localoptimization’ [8, 9]. The control method known as ”tracking” is closely related[10]. In these methods, at every instant in time the control field is chosen toachieve monotonic increase in the desired objective (see Fig. 3(b)). Typicallyin these methods, two conditions are used at each time step, one to determinethe phase of the field and one to determine the amplitude. In contrast withOCT, which incorporates information on later time dynamics through forward-backward interation, these methods use only information on the current state ofthe system.

A third approach, that has become very popular in recent years, involvesletting the laser learn to design its own optimal pulse shape in the laboratory[11, 12]. This is achieved by having a feedback loop, such that the increase ordecrease in yield from making a change in the pulse is fed back to the pulseshaper, guiding the design of the next trial pulse. A particular implementationof this approach is the ”genetic algorithm”, in which large set of initial pulses aregenerated; those giving the highest yield are used as ”parents” to produce a new”generation” of pulses, by allowing segments of the parent pulses to combinein random new combinations. Since the implementation of this approach isprimarily driven by experimental considerations, we will not comment on it anyfurther here.

II. INTUITIVE CONTROL CONCEPTS

Consider the ground electronic state potential energy surface in Fig. 1. Thispotential energy surface, corresponding to collinear ABC, has a region of stableABC and two exit channels, one corresponding to to A+BC and one to AB+C.This system is the simplest paradigm for control of chemical product formation:a two degree of freedom system is the minimum that can display two distinctchemical products. The objective is, starting out in a well defined initial state(v = 0 of the ABC molecule) to design an electric field as a function of timewhich will steer the wavepacket out of channel 1, with no amplitude going outof channel 2, and vice versa [1, 13].

87

We introduce a single excited electronic state surface at this point. The moti-vation is severalfold: 1) transition dipole moments are generally much strongerthan permanent dipole moments. 2) the di"erence in functional form of the ex-cited and ground potential energy surface will be our dynamical kernal; with asingle surface one must make use of the (generally weak) coordinate dependenceof the dipole. Moreover, the use of excited electronic states facilitates largechanges in force on the molecule, e"ectively instantaneously, without necessarilyusing strong fields. 3) the technology for amplitude and phase control of opticalpulses is significantly ahead of the corresponding technology in the infrared.

The object now will be to steer the wavefunction out of a specific exit channelon the ground electronic state, using the excited electronic state as an interme-diate. Insofar as the control is achieved by transferring amplitude between twoelectronic states, all the concepts regarding the central quantity µeg introducedabove will now come into play.

A. Pump-Dump Scheme

Consider the following intuitive scheme, in which the timing between a pairof pulses is used to control the identity of products [1]. The scheme is basedon the close correspondance between the center of a wavepacket in time andthat of a classical trajectory (Ehrenfest’s theorem). The first pulse produces anexcited electronic state wavepacket; The time delay between the pulses controlsthe time that the wavepacket evolves on the excited electronic state. the secondpulse stimulates emission. By the Franck-Condon principle, the second stepprepares a wavepacket on the ground electronic state with the same positionand momentum, instantaneously, as the excited state wavepacket. By controllingthe position and momentum of the wavepacket produced on the ground statethrough the second step, on can gain some measure of control over productformation on the ground state. The trajectory originates at the ground statesurface minimum (the equilibrium geometry). At t = 0 it is promoted to the onthe excited state potential surface (a two dimensional harmonic oscillator in thismodel) where it originates at the Condon point, i.e. vertically above the groundstate minimum. Since this position is displaced from equilibrium on the excitedstate, the trajectory begins to evolve, executing a two-dimensional Lissajousmotion. After some time delay, the trajectory is brought down vertically to theground state (keeping both the instantaneous position and momentum it had onthe excited state) and allowed to continue to evolve on the ground state. For onechoice of time delay it will exit into channel 1, for a second choice of time delay itwill exit into channel 2. Note that the position and momentum of the trajectoryon the ground state, immediately after it comes down from the excited state,must be consistent with the values they had before leaving the excited state,and at the same time are ideally suited for exiting out their respective channels.

88

FIG. 1: Stereoscopic view of the ground and excited state potential energy surfaces fora model collinear ABC system with the masses of HHD. The ground state surface hasa minimum, corresponding to to the stable ABC molecule. This minimum is separatedby saddle points from two distinct exit channels, one leading to AB+C the other toA+BC. The object is to use optical excitation and stimulated emission between thetwo surfaces to ’steer’ the wavepacket selectively out one of the exit channels.

A full quantum mechanical calculation based on these classical ideas was per-formed. [13]. The dynamics of the two-electronic state model, was solved, start-ing in the lowest vibrational eigenstate of the ground electronic state, in thepresence of a pair of femtosecond pulses that couple the states. Because thepulses were taken to be much shorter than a vibrational period, the e"ect of thepulses is prepare a wavepacket on the excited/ground state which is almost anexact replica of the instantaneous wavefunction on the other surface. Thus, thefirst pulse prepares an initial wavepacket which is almost a perfect Gaussian, andwhich begins to evolve on the excited state surface. The second pulse transfersthe instantaneous wavepacket at the arrival time of the pulse back to the groundstate, where it continues to evolve on the ground state surface, given its positionand momentum at the time of arrival from the excited state. For one choiceof time delay the exit out of channel 1 is almost completely selective, while fora second choice of time delay the exit out of channel 2 is almost completelyselective. A close correspondence with the classical model is observed.

Before turning to more systematic procedures for designing shaped pulses,

89

we point out an interesting alternative perspective on pump-dump control. Acentral tenet of Feynman’s approach to quantum mechanics was to think ofquantum interference as arising from multiple dynamical paths that lead to thesame final state. The simple example of this interference involves an initial state,two intermediate states, and a single final state, although if the objective is tocontrol some branching ratio at the final energy then at least two final statesare necessary. By controlling the phase with which each of the two intermediatestates contributes to the final state, one may control constructive vs. destructiveinterference in the final states. This is the basis of the Brumer-Shapiro approachto coherent control [14]. It is interesting to note that pump-dump control can beviewed entirely from this perspective. Now, however, instead of two intermediatestates there are many, corresponding to the vibrational levels of the excitedelectronic state. The control of the phase which determines how each of theseintermediate levels contributes to the final state, is achieved via the time delaybetween the excitation and the stimulated emission pulse. This ”interferingpathways” interpretation of pump-dump control is shown in Fig. 2.

III. VARIATIONAL FORMULATION OF CONTROL OF PRODUCTFORMATION

The next step is therefore to address the question: how is it possible to takeadvantage of the many additional available parameters: pulse shaping, multiplepulse sequences, etc — in general an !(t) with arbitrary complexity — to maxi-mize and perhaps obtain perfect selectivity? Posing the problem mathematically,one seeks to maximize

J ! limT!"

< "(T )|P!|"(T ) > (1)

where P! is a projection operator for chemical channel # (here, # takes ontwo values, referring to arrangement channels A+BC and AB+C; in general,in a triatomic molecule ABC, # takes on three values, 1, 2, 3, referring toarrangement channels A+BC, AB+C and AC+B). The time T is understood tobe longer than the duration of the pulse sequence, !(t); the yield, J , is definedas T " #, i.e. after the wavepacket amplitude has time to reach its asymptoticarrangement. The key observation is that the quantity J is a functional of !(t),i.e. J is a function of a function, because "(T ) depends on the whole history of!(t). To make this dependence on !(T ) explicit we may write:

J [!(t)] ! limT!"

< "[!(t)](T )|P!|"[!(t)](T ) >, (2)

where square brackets are used to indicate functional dependence. The problemof maximizing a function of a function has a rich history in mathematical physics,and falls into the class of problems belonging to the calculus of variations.

90

FIG. 2: Multiple pathway interference interpretation of pump-dump control. Sinceeach of the pair of pulses contains many frequency components, there are an infinitenumber of combination frequencies which lead to the same final energy state, whichgenerally interfere. The time delay between the pump and dump pulses controls therelative phase among these pathways, and hence determines whether the interferenceis constructive or destructive. The frequency domain interpretation highlights twoimportant features of coherent control: First, if final products are to be controlled theremust be degeneracy in the dissociative continuum. Second, that a single interactionwith the light, no matter how it is shaped, cannot produce control of final products:at least two interactions with the field are needed to obtain interfering pathways.

In the OCT formulation, the time dependent Schrodinger equation written asa 2 $ 2 matrix in a Born-Oppenheimer basis set, is introduced into the objec-tive functional with a Lagrange multiplier, $(x, t) [5]. The modified objectivefunctional may now be written as:

J ! limT!"

< "(T )|P!|"(T ) >

+ 2Re

! T

0dt < $(t)| %

%t% H

i! |"(t) > % &

! T

0dt|!(t)|2, (3)

91

where a constraint (or penalty) on the time integral of the energy in the electricfield has also been added. It is clear that as long as " satisfies the time dependentSchrodinger equation the new term in J will vanish for any $(x, t). The functionof the new term is to make the variations of J with respect to ! and with respectto " are independent, to first order in '!, i.e. to ’deconstrain’ " and !.

The requirement that "J"# = 0 leads to the equations:

i!%$

%t= H$ (4)

$(x, T ) = P!"(x, T ) (5)

i.e., the Lagrange multiplier must obey the time dependent Schrodinger equa-tion, subject to the boundary condition at the final time T that $ be equalto the projection operator operating on the Schrodinger wavefunction. Theseconditions ’conspire’, so that a change in !, which would ordinarily change Jthrough the dependence of "(T ) on !, does not do so to first order in the field.For a physically meaningful solution it is required that

i!%"

%t= H" (6)

"(x, 0) = "0(x) (7)

Finally, the optimal !(t) is given by the condition that "J"$ = 0 which leads to

the equation:

!(t) =%i

!&[< $a|µ|"b > % < "a|µ|$b >]. (8)

The interested reader is referred to Ref. [5] for the details of the derivation.Equations 4, 5, 6, 7 and 8 form the basis for a double ended boundary value

problem. " is known at t = 0, while $ is known at t = T . Taking a guess for !(t)one can propagate " forward in time to obtain "(t); at time T the projectionoperator P! may be applied to obtain $(T ), which may be propagated backwardsin time to obtain $(t). Note, however, that the above description is not self-consistent: the guess of !(t) used to propagate "(t) forward in time and topropagate $(t) backwards in time is not, in general, equal to the value of !(t)given by Eq. 8. Thus, in general one has to solve these equations iteratively untilself-consistency is achieved. An e!cient way to solve these equations is known asthe Krotov method. The Krotov method guarantees monotonic convergence ofthe objective without requiring a line search as in usual gradient type methods[15, 16]. Optimal control theory has become a widely used tool for designinglaser pulses with specific objectives. The interested reader can consult the reviewin Ref. [6] for further examples.

92

IV. LOCAL VS. GLOBAL IN TIME OPTIMIZATION

As described above, the optimal control equations typically have the structureof five coupled di"erential equations: one for the wavefunction, one for the dualwavefunction, an initial condition on the wavefunction, a final condition on thedual, and finally, an equation for the optimal field, which in turn is expressed interms of the wavefunction and its dual.

Typically, the OCT equations have to be solved via an iterative procedure, in-volving forwards in time propagation of the wavefunction, followed by backwardsin time propagation of the dual, until self-consistency is achieved with respectto the equations for the wavefunction, the dual and the control field. Because ofthe structure of this forwards-backwards propagation, the optimal field ’knows’about the future, i.e. the form of the optimal field at time t takes into accountthe dynamics at time t# > t. Thus, the optimal field may be willing to toleratea non-monotonic increase in the objective during the action of the pulse, since,given knowledge of the future, that may be the best way to attain the highestobjective at the final time (see Fig. 3(a)).

0 1 2 3 4 5 6 7 8 9 10

Time

<|0><0|>

0.0

0.2

0.4

0.6

0.8

1.0

FIG. 3: Representative plot showing the di!erence between global-in-time and local-in-time optimization. (a) In global-in-time optimization, the objective may decrease atintermediate times, but is guaranteed to take on its maximum value at the final time.(b) In local-in-time optimization, the objective increases monotonically in time.

A di"erent class of techniques that have been developed for control of atomicand molecular dynamics is called ’local optimization’ [8, 9]. In these methods, atevery instant in time the control field is chosen to achieve monotonic increase inthe desired objective (see Fig. 3(b)). Typically in these methods, two conditionsare used at each time step, one to determine the phase of the field and one todetermine the amplitude. In contrast with OCT, which incorporates information

93

on later time dynamics through forward-backward interation, these methods useonly information on the current state of the system.

At first glance, one would expect that the solution(s) that come out of anoptimal control calculation would give a higher value of the objective than thatfrom the local optimization: since the approach to the objective in OCT atintermediate times is unconstrained, while in local optimization it is constrainedto increase monotonically in time, one expects a higher yield from OCT sincethe space of allowed solutions includes those from local optimization as a subset(see Fig. 4(a)). However, there is one fallacy with the above argument. Theoptimal control equations generally have multiple solutions; these solutions arein general local, not global maxima in the function space (see Fig. 4(b). Thus,the value of the objective attained with the local-in-time optimization algorithmcan be larger than that obtained with the optimal control algorithm, since thelatter may be stuck in the region of a poor quality local maximum!

Global

Local

a

c b

FIG. 4: Left: Venn diagram of the set of control fields. Fields that yield monotonicallyincreasing solutions are a subset of the set of fields that lead to either an increase or adecrease at intermediate times. Right: The checkerboard is a schematic depiction ofthe space of all allowed control fields. The dots indicate the positions of the optimalfields, i.e. fields that lead to a local maximum of the objective in the function space.The squares show ”basins of attraction”, i.e. neighborhoods of control fields that con-verge to the same optimal field. The one-dimensional function above the checkerboardsymbolizes the value of the objective, and it, (as well as the concentric circles in thefigure on the left) are intended to convey the idea of multiple local maxima (see text).

There are several other attractive features to the local methods. 1) Sincethe increase in yield is continual, these fields are often amenable to immediateinterpretation. 2) Since these methods use only information on the current stateof the system, they could in principle be adapted for laboratory implementation.

94

3) These methods, because they di"er so radically in algorithm from OCT, arecapable of identifying entirely di"erent classes of mechanisms which may beappealing because of other properties, e.g. robustness.

In recent years, there have been a growing number of applications of localoptimization in the literature. For example, in one study the goal was to achieve’Optical Paralysis’ — the locking of excited electronic state population at afixed value, while generating large amplitude vibrational motion in the groundstate. The local optimization procedure discovered a stimulated Raman scat-tering strategy which exploits electronical resonance, but avoids vibrational res-onance [8]. A generalization of this approach led to a strategy for turning o"multiphoton ionization in the presence of intense fields [17]. In a second study,the goal was to discover strategies for complete and robust population transferin N -level systems with sequential coupling. When N = 3, the counterintu-itive STIRAP (Stimulated Raman Adiabatic Passage) pulse sequence, in whichthe dump precedes the pump [18, 19], emerged automatically from the localoptimization procedure. Moreover, the design procedure led to a robust gen-eralization of STIRAP to N -level systems, which we call Straddling-STIRAP[20]. For a review, see [21]. Recent applications include laser control of vibra-tional excitation of the carbonyl group in carboxyhemoglobin [22] and quantumcomputation via local control [23]. Work is in progress to implement local op-timization directly in the laboratory [9] with the hope that this will be moreconvenient for elucidating mechanisms than will the current pulses discoveredby genetic algorithms (see also [24]).

[1] D. J. Tannor and S. A. Rice, J. Chem. Phys. 83, 5013 (1985).[2] D. J. Tannor and S. A. Rice, Adv. Chem. Phys. 70, 441 (1988).[3] A. Peirce, M. A. Dahleh, and H. Rabitz, Phys. Rev. A 37, 4950 (1988).[4] S. Shi, A. Woody, and H. Rabitz, J. Chem. Phys. 88, 6870 (1988).[5] R. Koslo!, S. A. Rice, P. Gaspard, S. Tersigni, and D. J. Tannor, Chem. Phys.

139, 201 (1989).[6] S. A. Rice and M. Zhao, Optical Control of Molecular Dynamics (Wiley, New

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(University Science Books, Sausalito, 2006).[8] R.Koslo!, A. D.Hammerich, and D.Tannor, Phys. Rev. Lett. 69, 2172 (1992).[9] K. Moore, F. Langhojer, T. Brixner, G. Gerber, and D. J. Tannor, to be published.

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[11] R. S. Judson and H. Rabitz, Phys. Rev. Lett. 68, 1500 (1992).[12] J. L. Herek, W. Wohlleben, R. J. Cogdell, D. Zeidler, and M. Motzkus, Nature

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95

[14] M. Shapiro and P. Brumer, Principles of the Quantum Control of Molecular Pro-cesses (Wiley-Interscience, New York, 2003).

[15] D. Tannor, V. Kazakov, and V. Orlov, in Time Dependent Quantum MolecularDynamics, edited by J. Broeckhove and L. Lathouwers (Plenum Press, New York,1992), vol. NATO ASI Ser. B 299.

[16] S. Sklarz and D. J. Tannor, Phys. Rev. A 66, 053619 (2002).[17] V. Malinovsky, C. Meier, and D. J. Tannor, Chem. Phys. 221, 67 (1997).[18] J. Oreg, G. Hazak, and J. H. Eberly, Phys. Rev. A 32, 2776 (1985).[19] K. Bergmann, H. Theuer, and B. W. Shore, Rev. Mod. Phys. 70, 1003 (1998).[20] V. Malinovsky and D. J. Tannor, Phys. Rev. A 56, 4929 (1997).[21] D. Tannor, R. Koslo!, and A. Bartana, Faraday Discuss. 113, 365 (1999).[22] C. Meier and M. C. Heitz, J. Chem. Phys. 123, 044504 (2005).[23] S. Sklarz and D. J. Tannor, Chem. Phys. 322, 87 (2006).[24] M. Wollenhaupt, V. Engel, and T. Baumert, Annual Rev. Phys. Chem. 56, 25

(2005).

96

B. Lasorne and G. A. Worth (eds.)Coherent Control of Molecules

© 2006, CCP6, Daresbury

Molecular Axis Alignment via Controlled MolecularRotation

Jonathan G. UnderwoodDepartment of Physics and Astronomy, The Open University,

Walton Hall, Milton Keynes, MK6 3AQ, UK

I. INTRODUCTION

Experiments on isolated gas phase molecules are routinely carried out on ran-domly oriented ensembles of molecules, frequently in a supersonic molecularbeam expansion. Such measurements often su!er from a loss of informationdue to the orientational averaging that occurs. To ameliorate this situation, itis necessary to define the direction of the molecules in the lab frame prior tomaking measurements. For example, using static electric fields, or through ab-sorption onto a metal surface, it is possible to define the orientation of molecules.However, in these cases, the electric field giving rise to the orientation stronglyperturbs the molecular system, making measurement of the nascent propertiesof the molecule impossible. In this paper we discuss progress towards the use ofstrong non-resonant laser fields to align the axes of molecules in the lab frame,and we show that this may be achieved in a field free manner, allowing mea-surements on an ensemble of unperturbed molecules whose direction is definedin the lab frame.

II. SWITCHED WAVE PACKETS AND FIELD FREE ALIGNMENT

The interaction of the molecular polarisability with a strong, non-resonantlaser field can produce high degrees of molecular axis alignment [1–3]. An adia-batically applied laser field, i.e. a laser pulse whose envelope evolves much slowerthan the period of molecular rotation, will produce strong alignment at the peakof the laser field, and no residual alignment following the pulse. A measurementon the aligned ensemble would then need to take place at the peak of the laserfield. However, this field strongly perturbs the system, distorting the electronicand vibrational structure of the molecule [4] preventing the measurement of in-nate molecular properties. A non-adiabatic change in the interaction with thelaser field is required to produce post-pulse alignment. This may be achieved

97

through the adiabatic application of a laser field followed by rapid truncationthe laser field at its peak [3, 5]. Generating a laser field shaped such that itturns on slowly and turns o! rapidly can be accomplished by using a plasmaswitch to truncate a long laser pulse with a femtosecond laser pulse [3, 5]. Whena molecular gas interacts with a laser field shaped in this manner, at the mo-ment of field truncation the field prepared aligned state (a so called “pendular”state [1]) is projected back onto the field free eigen basis of rotational states.The rotational wave packet created in this way subsequently evolves under thefield free Hamiltonian, exhibiting field-free molecular axis alignment at revivalsof the wave packet. An example is shown in Figure 1, which shows the alignmentof the molecular axes of CO2 molecules to the polarization direction of a linearlypolarized switched laser field. The time evolution of the alignment was probedusing the optical Kerr e!ect (OKE) which is proportional to the expectationvalue !cos2 !", where ! is the angle between the molecular axis and the laserfield polarization direction.

FIG. 1: Experimental demonstration of switched wavepackets. (a) Cross-correlation(C.C.) of the switched 1.064 µm laser pulse with an 80 fs 800 nm pulse. This pulsehas a rise time of !on = 125 ps and a fall time of !sw = 110 fs (b) Optical Kerr E!ectsignal generated by the switched laser pulse from (a), focused (f/30) into 300 torr, 300K CO2 gas. (c) Expanded region from (b) showing rotational wavepacket dynamicsand field-free alignment in CO2.

98

III. FIELD FREE THREE DIMENSIONAL ALIGNMENT

Field-free molecular axis alignment may also be achieved employing a non-resonant laser pulse with a duration which is short compared to the timescale formolecular rotation [2, 6]. Here, the laser pulse causes a torque on the moleculestowards alignment, and alignment is obtained immediately after the laser pulse,and also at subsequent revivals of the rotational wave packet. This is a well estab-lished method for the creation of one dimensional alignment i.e. alignment of onemolecular axis. However, polyatomic molecules are generally asymmetric rotors,with three distinct moments of inertia, leading to complex rotational dynam-ics. While linear and symmetric top rotors exhibit regular classical dynamics,asymmetric top rotors display irregular dynamics (e.g. rotation about the axis ofintermediate inertia is unstable). Quantum mechanically, the rotational energylevel spacings for asymmetric molecules are much less regular than for linearand symmetric top molecules. This complicates the rotational wave-packet evo-lution for asymmetric rotors, reducing alignment at revivals. The degradationof alignment worsens at elevated rotational temperatures due to the incoherentcontributions of more thermally populated rotational states. This problem canlead to complete obfuscation of the rotational revival structure. This is a chal-lenge for experimentalists since rotational cooling is often compromised in orderto produce su"ciently dense molecular beams. A technique which is robust withrespect to temperature is therefore highly desirable.

Figure 2 shows the experimental realization of a scheme for impulsive field-free three dimensional alignment (FF3DA), as measured by coincidence Coulombexplosion imaging [7]. This method is based upon the use of two time separated,perpendicularly polarised, non-resonant, femtosecond laser pulses, with FF3DAproduced after the second laser pulse. The first laser pulse produces post-pulse1D alignment of the most polarizable axis of the molecule, at which point thesecond pulse produces a torque on the second most polarizable axis producingFF3DA after the second pulse. As such, this technique does not rely uponalignment obtained at a revival. This technique is broadly applicable, and it isdemonstrated here for the asymmetric top molecule SO2 [7].

Application of the first linearly polarised non-resonant laser pulse exerts atorque on each molecule. Although the magnitude and direction of this torquedepends on the molecular orientation, it tends to “kick” the most polarisablemolecular axis towards alignment with the laser polarization. For SO2, the mostpolarisable axis is along the O-O direction (called hereafter the O-axis). Wechoose this first pulse to be shorter than the time required for the molecule torotate about the axes perpendicular to the O-axis: maximal alignment of theO-axis thus occurs after the laser pulse. Although the O-axis is confined tothe laser polarization, there is free rotation of the molecules about the O-axis.At this point, the alignment is 1D due to the cylindrically symmetric nature ofthe interaction: only the O-axis direction is constrained. A second laser pulse,

99

FIG. 2: Measured time dependence of SO2 molecular axis alignment. The threecolumns show data for pulse separations of 200 fs (left), 400 fs (middle) and 600 fs(right). The zero of time is defined by the peak of the second laser pulse. The firstand second laser pulses had durations of 50 fs and 180 fs respectively, and an inten-sity of 2 ! 1013 W/cm2. Experimental data points are shown in black with error barscorresponding to the standard error of the mean. The solid lines are the results froma calculation. The dashed line shows the maximum FF3DA achieved.

perpendicularly polarised and applied following the first, breaks the lab framecylindrical symmetry. For molecules whose O-axis is aligned due to the firstpulse, this second laser field additionally exerts a torque about the O-axis which“kicks” the second most polarisable axis towards alignment with the secondlaser polarization. For SO2, this second most polarisable axis (called hereafterthe S-axis) lies in the molecular plane along the C2v axis (bisecting the O-S-Obond angle). The molecules now additionally proceed towards alignment aroundthis new direction and FF3DA is produced after the second laser pulse. Somemolecular O-axes may not be well aligned at the time of application of the secondlaser pulse. These would su!er a torque which misaligns their O-axis. However,the flexibility of this method allows us to choose the duration, intensity andtime separation of the two laser fields so that we can optimize FF3DA of theensemble. In this figure "O is the angle between the first laser pulse polarizationdirection (which defines the lab x-axis) and the projection of the O-axis ontothe lab xy-plane; "S is the angle between the second laser polarization (whichdefines the lab y-axis) and the projection of the S-axis on the yz-plane. Threedimensional alignment is characterized by !cos2 "S" > 0.5 and !cos2 "O" > 0.5occurring simultaneously. Examination of Figure 2 shows optimal FF3DA 500 fsafter the second laser pulse when the pulses are separated by 400 fs.

100

IV. CONCLUSIONS

We have briefly reviewed progress towards achieving field-free alignment ofcomplex molecules through the control of molecular rotation using non-resonant,non-perturbative laser fields. The methods described here suggest the impor-tance of examining the use of more complicated laser pulse shapes and sequencesfor enhancing the degree of alignment obtained. With the now routine laboratoryuse of laser pulse shaping technology, these techniques are readily exploitable.

The methodology reviewed in this paper will benefit the major ongoing ef-fort to develop approaches for probing molecular structures and dynamics usingscattering or di!raction techniques, as well as more traditional spectroscopicmethods. The development of “fourth generation” synchrotron light sourcesbeckons the measurement of molecular structures via the di!raction of short(<100 fs) X-ray light pulses [8, 9]. Complementary techniques include electrondi!raction [10–16], laser-induced electron di!raction [17, 18], high harmonic gen-eration [19] and tomographic imaging [20], and multi-dimensional femtosecondcoincidence spectroscopy [21]. These techniques, when combined with field-free3D molecular axis alignment, will yield much more detailed information aboutmolecular structure and dynamics.

Acknowledgments

I would like to acknowledge the invaluable contributions to the work reviewedhere made by Albert Stolow, Benjamin Sussman, Kevin F. Lee, Paul B. Corkum,David M. Villeneuve, Misha Yu. Ivanov, Michael Spanner and Je! Mottershead.I would also like to acknowledge the award of a British Council-NRC ResearcherExchange Award and support under EPSRC award EP/C530756/1.

[1] B. Friedrich and D. Herschbach, Phys. Rev. Lett. 74, 4623 (1995).[2] H. Stapelfeldt and T. Seideman, Rev. Mod. Phys. 75, 543 (2003).[3] B. J. Sussman, J. G. Underwood, R. Lausten, M. Y. Ivanov, and A. Stolow, Phys.

Rev. A. 73, 053403 (2006).[4] K. Yamanouchi, Science 295, 1659 (2002).[5] J. G. Underwood, M. Spanner, M. Y. Ivanov, J. Mottershead, B. J. Sussman, and

A. Stolow, Phys. Rev. Lett. 90, 223001 (2003).[6] F. Rosca-Pruna and M. J. J. Vrakking, Phys. Rev. Lett. 87, 153902 (2001).[7] K. F. Lee, D. M. Villeneuve, P. B. Corkum, A. Stolow, and J. G. Underwood,

Phys. Rev. Lett. (2006), submitted for publication.[8] J. Miao, K. O. Hodgson, and D. Sayre, Proc. Natl. Acad. Sci. USA 98, 6641

(2001).

101

[9] R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, and J. Hajdu, Nature 406,752 (2000).

[10] J. C. H. Spence and R. B. Doak, Phys. Rev. Lett. 92, 198102 (2004).[11] J. C. Williamson, J. Cao, H. Ihee, H. Frey, and A. H. Zewail, Nature 386, 159

(1997).[12] J. S. Baskin and A. H. Zewail, Chem. Phys. Chem. 6, 2261 (2005).[13] R. Srinivasan, J. S. Feenstra, S. T. Park, S. Xu, and A. H. Zewail, Science 307,

558 (2005).[14] D. W. H. Rankin and H. E. Robertson, Spectroscopic Properties of Inorganic and

Organometallic Compounds 37, 173 (2005).[15] B. J. Siwick, J. R. Dwyer, R. E. Jordan, and R. J. D. Miller, Science 302, 1382

(2003).[16] K. Hoshina, K. Yamanouchi, T. Ohshima, Y. Ose, and H. Todokoro, J. Chem.

Phys. 118, 6211 (2003).[17] H. Niikura, F. Legare, R. Hasbani, A. D. Bandrauk, M. Y. Ivanov, D. M. Vil-

leneuve, and P. B. Corkum, Nature 417, 917 (2002).[18] S. N. Yurchenko, S. Patchkovskii, I. V. Litvinyuk, P. B. Corkum, and G. L. Yudin,

Phys. Rev. Lett. 93, 223003 (2004).[19] S. Baker, J. S. Robinson, C. A. Haworth, H. Teng, R. A. Smith, C. C. Chirila,

M. Lein, J. W. G. Tisch, and J. P. Marangos, Science 312, 424 (2006).[20] J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. Pepin, J. C. Kie!er, P. B.

Corkum, and D. M. Villeneuve, Nature 432, 867 (2004).[21] O. Gessner, A. M. D. Lee, J. P. Sha!er, H. Reisler, S. V. Levchenko, A. I. Krylov,

J. G. Underwood, H. Shi, A. L. L. East, E. t. H. Chrysostom, et al., Science 311,219 (2006).

102

B. Lasorne and G. A. Worth (eds.)Coherent Control of Molecules

© 2006, CCP6, Daresbury

Coherent Control of Decoherence in Diatomic Molecules

Matthijs P. A. Branderhorst, Pablo Londero, Piotr Wasylczyk, Ian A. WalmsleyClarendon Laboratory, University of Oxford, Oxford, OX1 3PU, UK

Constantin Brif, Herschel RabitzDepartment of Chemistry, Princeton University, Princeton, New Jersey 08544

and Robert L. KosutSC Solutions, Sunnyvale, CA, USA

The inevitable interaction of any real system with its environment makes theevolution of the system nonunitary and destroys the coherence of quantum su-perpositions. This process, known as decoherence, is widely regarded as themost important and fundamental obstacle to the practical realization of quan-tum information processing. The measure of coherence of the system is purity.However, the purity of the system is di!cult to measure directly because thatis essentially a two-system measurement and the optimization of purity itself isnot a convex problem.

We have used the concepts of closed loop quantum control to explore the pos-sibility of mitigating decoherence given these di!culties, in particular to excitecoherent molecular states with shaped ultrashort laser pulses manipulating theamplitude and phase profile of the state. The standard model of decoherence isformulated in terms of a system, defined by the observables that can be mea-sured, and a reservoir, defined by those degrees of freedom that are not accessedby observation. The vibrational and rotational degrees of freedom of diatomicmolecules form such a system [1]. The experimental setup for this closed loopcontrol scheme, based on the setup for Molecular Emission Tomography [2, 3],is shown in figure (1). An all–sapphire gas cell containing potassium vapor iskept at 400 !C providing the dimer density of around 1%. A train of 90 fs pulsescentered at 840 nm from a CPA system with a repetition rate of 2 kHz is splitinto two arms, forming the pump and gate pulses that can be delayed with re-spect to each other. The vibrational wavepacket is excited in K2 by transferingpopulation from the X1"+

g ground state to the A1"+u state with a shaped reso-

nant laser pulse of around 1 µJ energy. The emitted fluorescence is collected bya pair of parabolic mirrors and mixed with a gate pulse on a 0.5 mm nonlinearcrystal (BBO) acting as a time filter, the upconverted light is spectrally filteredby a monochromator. The upconverted filtered signal is detected by a photo-

103

PMT

CPA

840nm,90 fs, 2kHz

FX A

P1

P2

C

DL

BSS

GA

FIG. 1: The experimental setup for closed loop control.

multiplier working in photon-counting mode. After measuring the fluorescencethe search algorithm selects a new generation of pulse shapes depending on thefitness value of the former one, this feedback loop makes the setup a closed loopcontrol scheme.

The Hamiltonian of this experimental model does not allow creation ofdecoherence-free subspaces, in which the vibrational system is completely de-coupled from the rotational bath, however the concept of a decoherence-freesubspace can be generalized to a decoherence resistant subspace. In the the deco-herence resistant subspace the evolution of the wavepacket is more robust againstthe rotational dephasing. The optimal control field creating a decoherence re-sistant subspace is found with a genetic search algorithm. The fitness-value ofthe genetic search algorithm is the localization of the wavepacket, evident inthe modulation depth of the fluorescence quantum beats. Localization of thewavepacket is a su!cient, but not necessary condition for coherence, but this isa more feasible experimental measure than purity. We try to optimize the visi-bility in a region where the quantum beats are completely washed out as a resultof the decoherence process. The outcome of the experiment is an improvementfrom no visibility at all to a visibility of around 6%. So the first conclusion isthat a decoherence resistant subspace exists in our experimental model.

The optimal control field found in our experiment is a complicated light pulsewith several identifiable components, the study of which reveals a posteriorithe degrees of freedom in the e#ective control field. The two components ofthe field that are most prominent are negative chirp and a sequence of pulses

104

a) b)

FIG. 2: The lifetime of the quantum beat signal at the outer turning point for severalshaped excitation pulses (a), the trends are supported by simulations (b).

split by the wavepacket vibrational period. The sequence of pulses focussesthe wavepacket at the target time after excitation at which the localization isoptimized. Chirp is the component that makes the wavepacket more robustagainst the ro-vibrational coupling. For this bosonic system, a useful picture ofdecoherence is the phase space dynamics of the wavepacket in which the action ofthe bath is to spread the excitation around the classical phase space trajectory.Therefore the algorithm seeks wavepacket shapes that have small extend in theradial phase space coordinate. The negative chirp of the optimal field e#ectivelysqueezes the Wigner function in radial direction. The optimized field gives animprovement of the coherence lifetime of about 60%.

The experimental outcomes are confirmed by numerical calculations, as shownin Fig (2). We numerically solved the linear Schrodinger equation for the fullro-vibrational density matrix. We calculated the energy eigenvalues using thespectroscopic ABO potentials and the Dunham coe!cients [4]. From this numer-ical model we could calculate the purity, the optimal excitation field does indeedimprove the purity lifetime. In the calculations we were also able to identifythe important internal parameters of the molecule, the only two relevant termsto explain our experimental results are the harmonic vibrational energy levelsand the ro-vibrational coupling term. So the anharmonicity of the electronicpotential is not important, the nonlinearity of the coupling term can be inter-preted as an e#ective anharmonicity and the negative chirp pre-compensates forthe dephasing caused by this term. In addition to this full model we calcu-lated the evolution of the Wigner distribution according to reference [1]. Thismodel provides insight into the phase space dynamics, it shows how a radiallysqueezed Wigner function is more robust against dephasing and makes the finaltotal uncertainty smaller.

105

The conclusion of the study is that by appropriate choice of initial coherencethe system remains in a more pure state for a longer time and that closed loopcontrol is an e#ective way of locating the optimal control field.

[1] C. Brif, H. Rabitz, S. Wallentowitz, and I. A. Walmsley, Phys. Rev. A 63, 063404(2001).

[2] T. Dunn, J. Sweetser, I. A. Walmsley, and C. Radzewicz, Phys. Rev. Lett. 70, 3388(1993).

[3] T. Dunn, I. A. Walmsley, and S. Mukamel, Phys. Rev. Lett 74, 884 (1995).[4] M. R. Manaa et al., J. Chem. Phys. 117, 11208 (2002).

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107

B. Lasorne and G. A. Worth (eds.)

Coherent Control of Molecules

© 2006, CCP6, Daresbury

Quantum Cartography: Mapping the Control Landscape

Nicholas T. Form and Benjamin J. Whitaker School of Chemistry, University of Leeds, Leeds, LS2 9JT, UK

I. INTRODUCTION

The coherent control of physical and chemical processes is a subject of

burgeoning experimental and theoretical interest; particularly as it

promises to control the yield of a chemical reaction by the use of a

tailored optical pulse [1–5]. Several schemes have been proposed to

implement molecular coherent control, although the essential feature of all

such methodologies is the same – quantum interference [6]. The Tannor

and Rice scheme [7] is arguably the conceptually easiest to describe. Here

control is achieved by coherent excitation of a set of vibrational

eigenfunctions to create a vibrational wavepacket, generally on an

electronically excited state potential energy surface (PES), which evolves

in time. A subsequent control pulse is then used to pump (or dump) the

wavepacket to another electronic state, e.g. the ground state, at some

later time. By controlling the spectral amplitude and phase of the

excitation and control pulses the molecule (vibrational wavepacket) can be

driven to particular regions of the PESs and hence to particular products

in, for example, a photochemical dissociation or isomerization reaction.

The general difficulty with such control schemes, however, is that it is

usually impossible to know a priori the temporal form of the electric field

necessary to achieve the desired result because, except in a few cases, e.g.

a diatomic molecule in the gas phase, the PESs are not known with

sufficient accuracy. This difficulty was imaginatively circumvented by

Judson and Rabitz [8] who proposed that optimal control theory could be

combined with evolutionary computational methods in such a way that

the control pulse could be empirically “taught” to the experimentalist by

the sample. The first experimental realisation of such feedback in

molecular quantum control was made by Bardeen et al. [9]. Subsequently,

108

adaptive pulse shaping combined with evolutionary algorithms has been

successfully implemented to control a range of gas phase reactions (see

[10] for a recent review).

Although evolutionary algorithms have proved to be extremely

effective ways of searching the vast parameter space that current pulse

shapers based on spatial light modulators or acousto-optic programmable

dispersive filters (AOPDFs) can operate over the very success of these

methods beg a number of questions. Crucially: What is the nature of the

search space? Is there one clear best solution or many approximately

equally good solutions? Is the topology of the search space convex or

corrugated? How effectively can an evolutionary algorithm converge upon

a solution having typically only sampled a minute fraction of the overall

parameter space? Which variant of evolutionary algorithm is most

effective for quantum control problems? Can other, or additional, learning

strategies improve the speed and effectiveness of the search?

II. CHIRP CONTROL OF 3 PHOTON ABSORPTION IN I2

In order to explore these questions we have recently re-examined 3

photon intrapulse control in I2, in

which linear chirp control of the

D!A band fluorescence yield

following excitation on the

X!B!C!D electronic bands had

been demonstrated in the seminal

paper of Yakovlev et al. [11] as early

as 1998. We use an AOPDF [12] to

systematically vary the group delay

dispersion (GDD) and third order

dispersion (TOD) of the output of a

non-collinear optical parametric

amplifier whose central wavelength

is tuneable between 450–700 nm.

Transform limited (TL) pulses about 30 fs in duration (30 nm bandwidth)

are systematically shaped by the AOPDF and loosely focussed with a

parabolic mirror into a cell containing about 1 Torr of I2 vapour. The

resulting UV fluorescence is detected through a discriminating spectral

FIG. 1: I2 D!A fluorescence quantum

efficiency map for excitation centred at

570 nm.

109

filter at right angles to the laser propagation direction by a PMT.

Typically we restrict the pulse energy to the range 150 to 500 nJ in order

to ensure that we are operating in the perturbative regime. The D!A

fluorescence quantum yield for pulses centred at 570 nm as a function of

GDD and TOD is shown in Fig. 1. We have obtained similar maps over a

range of wavelengths, !0, between 550 and 600 nm. At no value of !0 is

the TL pulse optimal for maximising the fluorescence quantum yield. This

immediately proves that the effects we observe are not due to the laser

intensity effects but rather are due to wavepacket following and intrapulse

interference. A cut taken through

Fig. 1 along the line of zero TOD

agrees very well with the linear chirp

dependence of the D!A fluorescence

efficiency originally reported by

Yakovlev et al. However, they

reported that a linearly up-chirped

pulse at +400 fs2 produced about a

three fold enhancement in

fluorescence quantum yield of the

TL pulse. We find that with the

addition of a small negative TOD of

about –5000 fs3 the enhancement is

increased to over 4.4 times that of

the TL pulse.

The electronic potential energy

surfaces of I2 have been well

characterized by frequency domain

spectroscopy, and are sketched in Fig. 2. As illustrated in the figure, on

the basis of this knowledge one would naively expect a down-chirped pulse

to be the most efficient way of promoting wavepacket motion initially

created on the C electronic surface by two-photon resonantly enhanced

excitation via the B state to the D surface. However, it is clear from Fig. 1

that both up and down chirped pulses appear to be roughly equally

efficient (in agreement with the earlier results of Yakovlev et al.). It is also

apparent in our results that the TOD plays a significant role. TOD is

implicated if two photon resonant excitation pathways are active.

FIG. 2: Sketch of the X, B, C and D

electronic potential energy curves of I2. A

3 photon excitation scheme is illustrated

for light centred at 570 nm.

110

Wigner-Ville and Husimi distributions are convenient ways of

describing the time-frequency correlations in a shaped optical pulse [13].

The Wigner distribution, defined by

S(!,t) = d !!" E(! + !! )E *(!# !! )exp(2i !! t) , (1)

can be generated using the values of the spectral phase, expressed as a

Taylor expansion about the central frequency, programmed into the three

maxima of Fig. 1. The most effective pulse, b, is an up-chirped pulse

stretched from the TL pulse (~30 fs duration) to ~96 fs. This is consistent

with a wavepacket following mechanism in which the low frequency

components of the pulse first excite I2 molecules on the X!B band to

create a wavepacket which moves to longer internuclear distances where

upon the blue shifted components of the pulse are in one photon resonance

with the C state (or perhaps two photon resonance with the D state).

However, pulse c is almost as effective as pulse b. Pulse c is a down-

chirped pulse in which the longer wavelength light arrives after the shorter

wavelengths. Because the X!B Franck-Condon factors favour short

wavelength excitation the unfavourable consequences of shifting the pulse

energy to the red in the latter part of the pulse might be compensated by

the stronger absorption cross-section early on. The time-frequency

distribution exhibited by pulse d suggests a third mechanism by which the

FIG. 3: Wigner-Ville distributions for the TL pulse (a) and the three optimal pulses of Fig.

1. The Taylor coefficients of the optimal pulses are (b) GDD = +600 fs2; TOD = –5000 fs3

(c) GDD = –600 fs2; TOD = –10000 fs3 (d) GDD = –200 fs2; TOD = 30000 fs3.

111

D state may be populated. Strong TOD is indicative of a two photon

resonance via an intermediate state, such that the sum of the two photon

energies !!

1+ !!

2 is constant. We are currently performing wavepacket

simulations to understand these effects better.

III. EVOLUTION STRATEGY

We have also explored the same system using evolutionary algorithms.

Of the myriad variants of evolutionary algorithm (EA) we have selected

for detailed investigation the “evolutionary strategy” (ES) [14], as used

successfully in a similar application by Ziedler et al. [15] and which is

more amenable to analysis than many other EAs. Currently we are

performing a systematic investigation of point based versus population

based ES with a view to maximising the learning speed. In the course of

this we are able to visualise the search landscape by using inter alia the

dimensionality reduction provided by Principal Component Analysis

(PCA). Performing PCA on the set of points in the AOPDF control space

explored by the ES allows us to produce 3-D plots of fitness versus two

orthogonal components capturing maximum variance. Combining such

plots will allow us to build an increasingly detailed map of the landscape;

revealing answers to questions such as those posed in Section 1 above.

IV. CONCLUSIONS

We have re-examined a classic experiment by Yakovlev et al.[ 11] using

an AOPDF pulse shaper to systematically tune the GDD and TOD of

ultrafast optical pulses. In this way we are able to map the 3 photon

excitation efficiency in molecular iodine as a function of the spectral phase

to third order and have been able to demonstrate that TOD plays an

important and hitherto overlooked role in promoting ground state I2

molecules to the fluorescent state. This work forms the basis of an ongoing

study into fitness landscapes in quantum control problems and emerging

quantum technologies.

112

Acknowledgments

This work is supported by EPSRC grant reference S47649. The evolutionary algorithm

employed in this work was designed and written by Robert Burbidge, Jem Rowland and

Ross King of the Department of Computer Science, University of Wales, Aberystwyth, our

collaborators in relation to the machine learning aspects of the project. We are also grateful

to Dr Anjan Barman for his assistance in the early stages of this work.

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