Optimal pump-dump control: phase-locked versus phase-unlocked schemes

Ž .Chemical Physics 233 1998 191–205

Optimal pump-dump control: phase-locked versus phase-unlockedschemes

YiJing Yan ), Zhen Wen Shen, Yi ZhaoDepartment of Chemistry, Hong Kong UniÕersity of Science and Technology, Kowloon, Hong Kong, China

Received 6 November 1997

Abstract

We report a unified theoretical framework for the study of the pump-dump control with either a single coherent field or apair of phase-unlocked coherent fields in both the strong and weak response regimes, and in terms of both theLiouville-space density matrix dynamics and the Hilbert-space wave function evolution. Shown are also the close relations

Ž .between the pump-dump control kernels in the phase-locked i.e. the single coherent field and the phase-unlocked controlschemes in the strong response regime. These strong field control kernels can further be linearized in the case of pure statecontrol in the weak pump-dump response regime. In this case, the optimal control theory reduces to the eigen problem of acertain specially constructed Hermitian matrix, from which even the globally optimal pump-dump control fields in either thephase-locked or the phase-unlocked control scheme can be identified. The common key quantity in both of the control

Ž X.schemes is a Hilbert-space pump-dump control response function, B t ,t , which shares a great amount of informationmutually with the optical resonant Raman spectroscopies. Numerical examples of pump-dump controlling I vibration onto2

an eigenstate and onto a minimum uncertainty wave packet in the ground electronic X state are presented to further elucidatethe control mechanisms in the phase-locked and phase-unlocked schemes in the weak response regime. q 1998 ElsevierScience B.V. All rights reserved.

1. Introduction

Using light to actively control the outcomes ofmolecular events has been one of the central themesin laser chemistry for decades. The common featurein a variety of quantum active control methods is toexploit the temporal-spectral coherent nature of lightfields to interfere either constructively or destruc-tively with the matter waves. From a theoreticalpoint of view, a quantum dynamic observable associ-ates with an operator, and its experimental outcomerelates to the operator’s expectation value. A control

) Corresponding author. E-mail: [email protected].

theory can therefore generally start by defining anâppropriate target operator A and consider the de-

ˆ² Ž . < < Ž .:sired outcome as c t A c t , the expectationf f

value at the target time t in the presence of controlfŽ .field E t . However,various theoretical control

methods are all constructed under different con-Ž .straints to find the field E t that optimizes the

Ž .outcome of A t .f

Quantum control methods can be largely classi-fied into two categories. One is called the coherentcontrol based on the direct interference among twoor more independent photo-excitation paths. Brumer

w xand Shapiro 1–3 have shown that, in a molecularsystem having a degenerate photo-excitation door-

0301-0104r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0301-0104 97 00362-5

( )Y. Yan et al.rChemical Physics 233 1998 191–205192

way to different products, the selectivity can becontrolled by varying the relative phases and ampli-tudes of several independent but degenerated excita-tion paths. One of their representing control scenariois the interference between one- and three-photonpaths, both of which connect a same bound initialstate of reactant to a same specified product doorwaystate in continuum. The control field in this case isassumed to consist of a pair of phase-locked cw

Ž . < < yi v 1 ty iu 1 < < yi v 3 ty iu 3components, E t s E e q E e ,1 3

where v s3v that assures the same final state be3 1

accessed via both the one v -photon excitation and3w xthe three v -photon excitation processes 1–3 . As1

the result of quantum interference, the final product< 3 <yield will vary with the intensity ratio E rE and1 3

the relative phase 3u yu between the two field1 3

components. The experimental verification ofBrumer-Shapiro’s coherent scheme has been carried

w xout in some atomic and molecular systems 4–6 .An alternative approach to quantum control is by

the use of optimal control theory. Tannor and Ricew x7,8 have demonstrated, in the weak response regime,how to calculate the shape of an optimal dump pulsethat is phase-unlocked to a given pump field. Thefull version of optimal quantum control via a single

Ž .coherent field E t has been formulated by Rabitzw xand coworkers 9,10 in terms of the Hilbert-space

w xwave function dynamics. Kosloff et al. 11 havefurther applied the optimal control theory to molecu-lar systems with multiple electronic surfaces. Later,

w xYan et al. 12 formulated the optimal control via asingle coherent field in terms of the Liouville-spacedensity matrix dynamics. Recently, Yan and cowork-

w xers 13–15 developed general formulation of opti-mal control via a pair of phase-unlocked fields, thusgeneralized the original Tannor-Rice pump-dump

w xcontrol scheme 7,8 to a simultaneous optimizationof the pair of phase-unlocked pump-dump fields inboth the strong and the weak response regimes.

Mathematically, finding optimal fields can be for-mulated as a problem of functional maximization in

Ž .calculus. In general an optimal field E t may con-sist of a single pulse, or a sequence of phase-lockedsubpulses. In most pump-dump experiments, how-ever, the pump field and the dump field carry consid-erably different center frequencies, which makes itnearly impossible for the two fields to be phase-locked experimentally. In this case, the optimal sin-

gle coherent pump-dump field may be complicatedcomparing to the pair of phase-unlocked optimalfields.

As a mathematical tool in searching for the func-tional maximum, the optimal control theory leads ingeneral to highly nonlinear equations which require anumerically expensive approach to solve. The result-ing optimal fields are however not only locally opti-mal but also too complicated to be experimentallyrealizable. To facilitate this problem, there has beena considerable effort on the linearization of control

w xequations 12–20 . The simplest and also generallylinearizable system is the control of one-photon

w xachievable target 12,17–22 . In this case, the opti-mal control in the weak response regime can bereduced to an eigen-problem of a second order Liou-ville-space control response function, whose eigen-function and eigenvalue are the optimal field and

w xyield, respectively 12,20,21 . The globally optimalfield in the one-photon control scheme can thus beunambiguously identified. Moreover, we have found,at least for the systems of study, that the globallyoptimal field is reasonably simple and robust, andretains as a good control field as its intensity scales

w xup to moderately strong region 23 . An examplew xwhich successfully correlates theory 20–22 and

w xexperiment 24 is the control of wave packet focus-ing on the I B potential surface.2

The linear version of control formulation for atarget inaccessible via one-photon process, such asthe two-photon pump-dump or pump-pump control,may only be possible under certain experimental

w xconditions 13–16,25 . It has been shown by Dubovw x Žand Rabitz 16 for the single coherent or phase-

. w xlocked control field, and by Yan et al. 15 for thepair of phase-unlocked control fields, that the opti-mal two-photon control in the pure state case canalso be formulated in terms of eigen equations. Theglobally optimal pair of phase-unlocked control fields

w xhas been identified unambiguously 15 . In this pa-per, we shall further identify the globally optimalfield in the phase-locked, or the single coherent field,

w xtwo-photon control process. cf. Section 3.3We shall also in this paper reformulate the exist-

ing optimal control theories, in both the phase-lockedscheme and the phase-unlocked scheme, in a system-atic and unified manner. It is hoped that a unifiedformulation that combines the Brumer-Shapiro’s co-

( )Y. Yan et al.rChemical Physics 233 1998 191–205 193

herent control with the optimal control theory bedeveloped in near future, so that one can make aclose comparison among various control schemes.The remaining of this paper is organized as follows.In Section 2, we present the general theory of opti-mal control via a single coherent field in terms ofboth the Liouville-space density matrix dynamicsand the Hilbert-space wave function dynamics, andin both the strong response regime and the weaktwo-photon control response regime in which theeigen equation is formulated. The theory to be pre-sented is general in the sense it is applicable toarbitrary matter systems. Application of the generalsingle field control formulation to two-surfacemolecular systems under the resonant pump-dumpcontrol excitation configuration is made in Section 3.In Section 4, we present the optimal phase-unlockedpump-dump control theory, again, in terms of boththe Liouville-space density matrix dynamics and theHilbert-space wave function dynamics, and in boththe strong response regime and the weak two-photoncontrol response regime where a pair of eigen equa-tions exist. In Section 5, we numerically implementthe eigen equations for the pump-dump pure-statecontrol of two targets in I molecular system in both2

the phase-locked and the phase-unlocked controlschemes. One target is chosen to be a minimumuncertainty Gaussian wave packet, and the other is avibronic eigen state. Both targets are highly vibra-tionally excited on the I X surface. The globally2

optimal field in each of the cases is calculated.Finally, we summarize and conclude our results inSection 6.

2. General theory of optimal control via a singlecoherent field

2.1. LiouÕille-space formalism

Let us start with a single coherent field controlcase in which the matter–field interaction Hamilto-nian in the electric dipole approximation can begenerally expressed as

H t sH yDe t . 1Ž . Ž . Ž .M

The molecule is assumed to be in a stationary state

Ž .r t before it interacts with the control field, which0Ž .is described here by the real function e t . In opti-

ˆmal control theory, we consider a target operator AŽ .and search for an optimal field e t that drives the

molecular system to maximize the overlap with thetarget,

Â t sTr Ar t , 2Ž .Ž . Ž .f f

at the specific target time t under a certain con-f

straint. Using the variation principle and the Liou-ville-space Green function method, the general con-

w xtrol equation can be obtained as 12

K t ;t sle t . 3Ž . Ž .Ž .f

In this equation, l is the constant Lagrange multi-plier for the finite incident field energy constraintwith no bias to any temporal or spectral shapes. The

Ž .key quantity in Eq. 3 is the Liouville-space controlkernel K , which is real and given formally by

ˆK t ;t s2 Re ir" Tr A t ;t Dr t . 4Ž . Ž . Ž .Ž . Ž .½ 5f f

ˆ ˆŽ . Ž . Ž . Ž . Ž .Here, r t sGG t ,t r t and A t ;t sA GG t ,t0 0 f f

denotes the forward-propagated system density ma-trix and the backward-propagated target operator,respectively, in the presence of the optimal control

Ž .field e t . Physically, the Liouville-space controlw Ž .xkernel K Eq. 4 describes the matter response to

an infinitesimal functional variation around the givenŽ .real external field e t . Since the control kernel Eq.

Ž .4 depends implicitly on the control field throughˆŽ . Ž . Ž .r t and A t ;t , Eq. 3 should be solved in af

Ž .self-consistent iteration manner. However, Eq. 3 isgeneral and applicable to the optimal control of anarbitrary material system, whether it is a multi-levelatom or a multi-surface molecule in either a pure ormixed state, via a single coherent field with an

Ž . Ž .arbitrary strength. Eq. 3 together with Eq. 4 cantherefore be served as the common background forformulating simplified optimal control equations in

w xvarious specific control cases 11,12,26 . The appli-cation of the general Liouville-space control theoryto the pure state case will be presented in the follow-ing subsection, and to the two-surface molecularsystem in the next section.

2.2. Hilbert-space formalism

We shall in this subsection present the generalw xHilbert-space formalism 10–12,19 for the optimal


control of a pure state system via a single coherentfield. Both the initial molecular system and the finaltarget are assumed to be in pure states. Let us denotethem as

< : ² < < : ² <r t s c t c t s n n , 5Ž . Ž . Ž . Ž .0 0 0

ˆ < : ² <As f f . 6Ž .Ž .In Eq. 5 we denoted explicitly that the system is

< Ž .: < :initially in a molecular eigenstate c t s n . This0

is in consistence with that the molecular system isinitially in a stationary state before it interacts withthe external field. We shall also assume that thesystem is non-dissipative so that it will remain as apure state governed by the matter-field Hamiltonian

Ž .in Eq. 1 . In this case, the forward-propagated sys-tem density matrix and the backward-propagated tar-

Ž .get in Eq. 4 have the following forms:

< : ² <r t s c t c t , 7Ž . Ž . Ž . Ž .

ˆ < : ² <A t ;t s f t ;t f t ;t , 8Ž .Ž . Ž . Ž .f f f

< Ž .: Ž . < Ž .: ² Ž . <with c t s G t ,t c t and f t ;t s0 0 f² < Ž . Ž . w Ž .xf G t ,t . The control yield A t Eq. 2 in thisf f

case reduces to

< < 2A t s c t , 9Ž .Ž . Ž .f f

where

² < :c t s f c t , 10Ž .Ž . Ž .f f

denotes the complex control yield amplitude. ByŽ . Ž . Ž .using Eqs. 7 , 8 and 10 , and the identity of the

Ž .Green function i.e. the time propagatorŽ . Ž . Ž .G t ,t G t ,t sG t ,t , the Liouville-space con-f 0 f 0

Ž . Ž .trol kernel in Eq. 4 has the form K t ;t sfw ) Ž . Ž .x Ž . w x2 Re c t f t ;t , and Eq. 3 reduces to 12f f

)2 Re c t f t ;t sle t . 11Ž . Ž .Ž . Ž .f f

Here

² < < :f t ;t ' ir" f t ;t D c t , 12Ž . Ž . Ž .Ž . Ž .f f

carries all the t-dependent information and can there-fore be viewed as the Hilbert-space control kernel.

Ž . Ž .Eqs. 11 and 12 constitute the final Hilbert-spaceformalism for the optimal control of an arbitrarypure state matter system via a single coherent field

with an arbitrary intensity. The key quantity here isthe Hilbert-space complex control kernel f of Eq.Ž .12 . It should however combine with the complex

Ž . w Ž .xcontrol yield amplitude c t Eq. 10 to determinefw Ž .xthe optimal field cf. Eq. 11 . One may argue that

Ž .the phase of c t is irrelevant to the optimal controlf

process since it depends on the choice of energy zeroand the phase of wave function of either the target or

Ž .the system. However, that setting the phase of c t f

to be zero would introduce a certain artifact phasefactor to the f function, and thus, results in a pseudooptimal field in which the effect of quantum coher-ence among various composition subfields on mate-rial dynamics might not be considered properly. Thesituation would be different if we only consider theoptimal control in the weak response regime inwhich there exists a simple scaling relation between

Ž .c and f , and the phase of c t becomes irrelevant tof

the control dynamics.Before we proceed, let us analyze the order de-

pendence of the Liouville-space control kernel Kw Ž .x wEq. 4 and the Hilbert-space control kernel f Eq.Ž .x12 in an n-photon control process. The term n-photon process used in this paper refers to the weakresponse regime in which the leading term in thedensity matrix r is proportional to the nth power ofthe external field intensity. In this case, the control

Ž . w Ž .xoutcome A t Eq. 2 depends on the external fieldfŽ .e to the 2n th order, while the Liouville-space

Ž . w Ž .xcontrol kernel K t ;t cf. Eq. 4 depends on itsfŽ .2ny1 th order. It concludes that the general lin-earization to the Liouville-space control can only be

wachieved in the one-photon control process 12,17–x20 . Let us turn to the n-photon control process in a

pure state system, in which the Hilbert-space controlkernel is lead by f Žny1., and the control yield ampli-tude by cŽn.. We can therefore conclude that thelinear versions of the pure-state control formulationcan be obtained not only in the one-photon processw x Ž12,17–20 , but also in the two-photon pump-dump

. w xor pump-pump control process 16 .

2.3. Two-photon pure state control as an eigen prob-lem

As we have just mentioned, the relevant quantitiesin a two-photon pure state control process are the

Ž1.Ž .first order of the pure state control kernel, f t ;t f


and the second order of the control yield amplitudeŽ2.Ž . w xc t . We have 16f

Ž1. Ž0. Ž1.² < < :f t ;t s ir" f t ;t D c tŽ . Ž .Ž . Ž .f f

Ž1. Ž0.² < < :q f t ;t D c tŽ .Ž .f

tX X X

' dt B t ,t e tŽ . Ž .Ht0

t f X X Xq dt B t ,t e t , 13Ž . Ž . Ž .Ht

tt f X X XŽ2.c t s dt dt e t B t ,t e t . 14Ž . Ž . Ž . Ž .Ž . H Hft t0 0

In these equations B can be viewed as the Hilbert-space two-photon control response function definedby the following equation:

° Ž0. X Ž0. Xˆ² < < :f t ;t T tyt c t ,Ž . Ž .Ž .fXX ~ fortGt ,B t ,t 'Ž .

0,¢ Xfort-t .15Ž .

Here,2 yi H tr "MT̂ t ' ir" D e D , 16Ž . Ž . Ž .

is the second-order dipole transition operator, and

< Ž0. : yi HM Žtyt0 .r " < :c t se c tŽ . Ž .0

yi ń Žtyt0 .r " < :se n , 17Ž .² Ž0. < ² < yi HM Ž t fyt .r "f t ;t s f e , 18Ž .Ž .f

are respectively the free-propagated wave functionsfor the system and the target in the absence of

Ž . Ž .control field. Note that Eq. 13 or Eq. 14 requiresŽ X. XB t ,t be only defined for tGt . We however

Ž . Xextended its definition as Eq. 15 to t-t by zero.Ž . Ž .By doing that, we can recast Eqs. 13 and 14 as

t f X X X XŽ1.f t ;t s dt B t ,t qB t ,t e t ,Ž . Ž . Ž .Ž . Hft0

19Ž .t f1Ž2. Ž1.c t s dt e t f t ;t . 20Ž . Ž .Ž . Ž .Hf f2

t0

Ž .Eq. 20 describes a simple scaling relation betweenŽ1.Ž . Ž .f t ;t and c t in the weak response limit. Duef f

to this scaling relation, the optimal field in the weak

response regime will not depend on the phase ofŽ2.Ž . Ž . w Ž1.Ž .xc t , and Eq. 11 can reduce to Re f t ;t Af fŽ .e t . The two-photon pure-state control equation can

therefore be expressed as

t f X X X XRe dt B t ,t qB t ,t e t sle t .Ž . Ž . Ž . Ž .Ht0

21Ž .Ž .Eq. 21 recovers a recent result obtained by Dubov

w xand Rabitz 16 . For the optimal fields obtained fromŽ . Ž2.Ž . < Ž4.Ž . <1r2 < Ž2.Ž . <Eq. 21 , we have c t s A t s c tf f f

that is real and positive at the target time. We canfurther identify precisely the physical meaning of l

Ž .in Eq. 21 for the two-photon pure state control.This can be done by multiplying both sides of Eq.Ž . Ž . w x21 with e t , integrating the result over tg t ,t ,0 f

Ž .and further making use of Eq. 20 . We have

< Ž2. <ls c t rI)0, 22Ž .Ž .f

where

t f1 2Is dt e t , 23Ž . Ž .H2t0

is the total incident intensity of control field to theŽ .target time. Eq. 22 concludes that the eigenvalue

2 Ž . 2square l sA t rI is the two-photon control yieldf

with respect to the square of the incident optimalfield intensity. This result is consistent with thescaling law in the two-photon weak control responseregime. An optimal field is thus the eigenfunction of

Ž .Eq. 21 with a positive eigenvalue. Furthermore, theglobally optimal field in the case of pure statecontrol in the two-photon weak response regime cantherefore be obtained as the eigenfunction associat-

Ž . Ž .ing with the largest eigenvalue. Eqs. 21 – 23 con-stitute one of the main new results of the paper. As atheory for the single optimal control field, the result-ing e , which is either globally or locally optimal,may consist of a sequence of subpulses whose phasesare optimal locked to each other.

3. Pump-dump control via a single coherent field

3.1. Pump-dump control: LiouÕille-space theory

The general formulation presented in the previoussection is applicable to any particular molecular or


atomic system. In this section, we shall apply it tothe resonant control of a molecular system involving

< : < :the ground g electronic state and an excited estate. The molecule is initially in a stationary state,such as the thermal equilibrium or a ro-vibrationaleigenstate on the ground electronic g surface. Thepure electronic transition frequency v is assumede g

to be much larger than the characteristic frequenciesof nuclear motion on the two surfaces governed bythe molecular adiabatic Hamiltonians H and H .g e

Ž .We can therefore denote the real control field e t inthe resonant excitation configuration as

e t sE t eyi v e g t qE) t e i v e g t . 24Ž . Ž . Ž . Ž .Ž .Here, E t , the slowly varying complex envelop of

the optimal field, is what we are going to formulatein this section. As it oscillates at the nuclear dynamic

Ž .time scale, E t can be evaluated much more effi-Ž .ciently than the complete field e t , which is real

and but oscillates at the electronic transition region.Ž .The derivation of the control equation for E t from

the general formulation in the previous section in-volves the interaction picture and the electronic rotat-

Ž .ing-wave-approximation RWA . The interaction pic-ture is defined by the rotating frame unitary operator

yi HR tr " < : ² <e with H s"v e e , and the RWA ne-R e g

glects the small contributions that contain the highlyoscillatory factor of e" 2 i v e g t. The Liouville-space

w Ž .xcontrol kernel K Eq. 4 can be recast in thew xinteraction picture as 23

ˆK t ;t s2 Re ir" Tr A t ;t D t r t ,Ž . Ž . Ž .Ž . Ž .½ 5f I f I I

25Ž .where

D t sD e i v e gt qD eyi v e gt , 26Ž . Ž .I q y

with

< : ² < < : ² < †D sm e g , and D sm g e sD . 27Ž .q y q

Ž .In Eq. 27 , m is the electronic transition dipolemoment that usually depends on the molecular nu-clear degrees of freedom. The time evolution in

ˆŽ . Ž . Ž .either r t or A t ;t in Eq. 25 is governed byI I fw xthe RWA Hamiltonian 23 :

H t sH yD E t yD E) t , 28Ž . Ž . Ž . Ž .0 q y

with

< : ² < < : ² <H sH g g qH e e . 29Ž .0 g e

Ž .Note that D and D defined in Eq. 27 areq yphysically the creation and annihilation operators,respectively, for the electronic excitation or exciton.

< : < :They satisfy the following relations: D g sm e ,q< : < : < : < :D e sm g , and D e sD g s0. In the casey q y

of quantum field, the classical field envelop E shouldbe replaced by the field annihilation operator a, andîts complex conjugate E) by the field creation

† Ž .operator a . This analysis confirms that H t of Eq.ˆŽ .28 is indeed a Hamiltonian in the RWA, in which

w Ž . ) Ž .xannihilation or creation of a photon E t or E tis associated with a creation or annihilation of an

Ž .exciton D or D , respectively.q yFor simplifying the notation, we shall hereafter

remove the subscript of the interaction picture in theŽ .density matrix r t and the propagated targetI

ˆ Ž . Ž .A t ;t . By using Eq. 26 , we can now recast theI fw Ž .xLiouville-space control kernel K Eq. 25 as

) yi v te gK t ;t s K t ;t qK t ;t e qc.c.Ž . Ž . Ž .f q f y f

30Ž .

Here, c.c. stands for the complex conjugate to itsprevious term, and

ˆK t ;t s ir" Tr A t ;t D r t . 31Ž . Ž . Ž .Ž . Ž ." f f "

ˆŽ . Ž .In this equation, r t and A t ;t are respectivelyf

the forward-propagated system density matrix andthe backward-propagated target operator governed

Ž . Ž .by the RWA Hamiltonian H t of Eq. 28 . As wementioned earlier that D represents the creationqoperator and D is the annihilation operator for theyelectronic excitation, K and K defined in Eq.q yŽ .31 can be considered as the pump and the dumpcontributions, respectively. This classification is moretransparent in the pump-dump control via a pair ofphase-unlocked fields that will be considered in the

Ž . Ž .following section. By substituting Eqs. 24 and 30Ž .into Eq. 3 and followed by invoking the RWA, we

obtain the final Liouville-space pump-dump controlequation in the strong response regime as

K ) t ;t qK t ;t slE t . 32Ž . Ž .Ž . Ž .q f y f

3.2. Pump-dump control: Hilbert-space theory

The Hilbert-space pump-dump control equation inthe RWA can be derived similarly from either the


Ž . Ž .general Hilbert-space kernel f t ;t of Eq. 12 orf

the Liouville-space pump-dump control kernelŽ . Ž .K t ;t of Eq. 31 . For the pure state control, we" f

w Ž .xhave cf. Eq. 30

f t ;t s f t ;t e i v e gt q f t ;t eyi v e gt , 33Ž .Ž . Ž . Ž .f q f y f

or

K t ;t sc) t f t ;t . 34Ž .Ž . Ž . Ž ." f f " f

In these equations,

² < < :f t ;t s ir" f t ;t D c t , 35Ž . Ž . Ž .Ž . Ž ." f f "

with both the forward-propagated system wave func-Ž .tion c t and the backward-propagated target wave

Ž . Ž .function f t governed by RWA Hamiltonian H tŽ .of Eq. 28 . The final Hilbert-space pump-dump

control equation can be obtained either by using Eqs.Ž . Ž . Ž .11 , 24 and 33 together with the RWA or by

Ž . Ž .using Eqs. 32 and 34 . We have

c t f ) t ;t qc) t f t ;t slE t , 36Ž . Ž .Ž . Ž . Ž . Ž .f q f f y f

Ž .where c t is the complex control yield amplitudefŽ .given by Eq. 10 .

3.3. Pump-dump control of pure state systems in theweak response regime as an eigen problem

We shall now implement the RWA to the purew Ž .xstate two-photon control cf. Eq. 21 of the two-

surface molecular system. The two-photon process inthis case can also be referred to the resonant pump-dump control in the weak response regime, in which

Ž .the target is a nuclear wave function f t in theg f

ground electronic state. As discussed in Section 2.3,the key quantity in a general two-photon pure state

Ž .control is the function B of Eq. 15 . Using theŽ .interaction picture, we can recast B of Eq. 15 for

the two surface system as:

B t ,t X sB t ,t X eyi v e gŽtytX . , 37Ž . Ž . Ž .g

where

° Ž0. X Ž0. Xˆ² < < :f t ;t T tyt c t ,Ž . Ž .Ž .g f e gXX ~ fortGt ,B t ,t sŽ .g

0,¢ Xfort-t ,38Ž .

ˆwith the second order dipole transition operator Tew Ž .xdefined by cf. Eq. 16

2 yi H tr "eT̂ t ' ir" m e m. 39Ž . Ž . Ž .e

Ž .In deriving Eq. 37 , we also used the condition thatboth the initial system and the final target are in the

< Ž0.Ž .:ground electronic state, so that D c t s0 andy² Ž0.Ž . <f t ;t D s0. For completeness, we presentf q

Ž1. Ž2.Ž .also the relevant quantities, f and c t , in" f

terms of the Hilbert-space pump-dump correlationw Ž .function B . They are given as cf. Eqs. 19 andg

Ž .x20

t f X X XŽ1. )f t ;t s dt B t ,t E t , 40Ž . Ž . Ž .Ž . Hq f gt

tX X XŽ1.f t ;t s dt B t ,t E t , 41Ž . Ž . Ž .Ž . Hy f g

t0

and

tt f X X XŽ2. )c t s dt dt E t B t ,t E t . 42Ž . Ž . Ž . Ž .Ž . H Hf gt t0 0

Ž . Ž . Ž .Substituting Eqs. 24 and 37 into Eq. 21 ,Žfollowed by implementing the RWA, we have l)

.0

t f X X X X1 )dt B t ,t qB t ,t E t slE t .Ž . Ž . Ž . Ž .H g g2t0

43Ž .Ž .As we mentioned in Eq. 22 , the square of the

Ž .eigenvalue l in Eq. 43 corresponds to the controlŽ . < Ž . < 2yield A t s c t with respect to the square off f

the incident field energy,

l2 sA t rI 2 , 44Ž .Ž .f

where

t f 2< <Is dt E t . 45Ž . Ž .Ht0

Ž . Ž .Eqs. 43 – 45 therefore constitute the eigenequationsolution to both the globally and locally optimalpump-dump control fields in the weak responseregime under the RWA. Note that the resultingsecond order optimal pump-dump control amplitudeŽ2.Ž . w Ž . Ž .xc t Eq. 42 with Eq. 43 is real and positive atf

the target time t . As the nature of a single coherentf

control field theory, the resulting optimal pump-dumpE field may consist of a sequence of subpulses


whose phases are optimally locked to each other. Incontrast to the optimal control via a pair of phase-un-locked coherent fields considered in the next section,we shall hereafter also refer the control via a singlecoherent field considered in this section as thephase-locked pump-dump control scheme.

4. Optimal pump-dump control via a pair ofphase-unlocked coherent fields

4.1. Pump-dump control: LiouÕille-space theory

w xWe have recently 14,15 developed systemati-cally the theory of optimal pump-dump control via a

� 4pair of phase-unlocked coherent fields E , E in1 2

both the strong and weak response regimes. Here, wedistinguish the physical role played by each of theexternal fields in their interaction with the materialsystem. That is E denotes pump and E denotes1 2

dump. This distinction however may only be madeexactly in the weak response regime, and only whenthe two fields do not overlap spectrally. In the strongresponse regime, each field can act as both the pumpand the dump, and thus Rabi oscillations result. Weshall therefore use the condition of distinctness as aworking constraint in which the functional variation

Ž .dE t at the vicinity of the optimal field affects only1Ž .the pump process, while dE t only the dump2

process. This assumption is physically reasonable,and also crucial to the theoretical development of thephase-unlocked pump-dump control in the strongresponse regime considered at present and in thefollowing subsection.

In the phase-unlocked pump-dump controlscheme, the control target expectation value in Eq.Ž .2 should include an average over the random phase.

Ž . Ž . yi u Ž .Let us denote E t;u sE t qe E t as the co-1 2

herent superposition of E and E fields that are1 2

locked to each other at a specific relative phase² :usu yu , and PPP as the statistic average over2 1

the static random distribution of the relative phase u .The goal of control here is to find a pair of phase-un-

� 4locked pump-dump fields E , E that optimize the1 2

target expectation value,

ˆ² :A t s Tr Ar t ;u , 46Ž .Ž . Ž .f f

at the target time t . The RWA Hamiltonian whichf

Ž .governs the system density matrix r t ;u in Eq.fŽ .46 and other dynamic quantities in this subsection

w Ž .xis given by cf. Eq. 28

H t ;u sH yD E t ;u yD E) t ;u . 47Ž . Ž . Ž . Ž .0 q y

The final Liouville-space formulation for thephase-unlocked pump-dump control is given by the

w xfollowing two coupled equations 14,15 :

K t ;t sl E t , with js1and 2. 48Ž . Ž .Ž .j f j j

Here, l is a Lagrange multiplier for the finitej

incident energy constrain for the field E . The inclu-j

sion of two independent Lagrange multipliers, l1

and l , is consistent with the condition of the com-2

plete random phase between the two fields. The keyŽ .quantities in Eq. 48 are the pump control kernel K1

and the dump control kernel K . They formally2

relate to the two kernel contributions in the phase-Ž .locked control scheme, K and K of Eq. 31 , asq y

² ) :K t ;t s K t ;t ,u , 49Ž .Ž . Ž .1 f q f

² iu :K t ;t s K t ;t ,u e . 50Ž .Ž . Ž .2 f y f

Ž . Ž . Ž .Eqs. 48 – 50 together with Eq. 31 constitute thefinal Liouville-space formulation for the coupled andself-consistent solution to the optimal phase-un-locked pump-dump fields E , E in the strong re-1 2

sponse regime.

4.2. Pump-dump control: Hilbert-space theory

We shall now turn to the Hilbert-space formula-tion of the phase-unlocked pump-dump control in thestrong response regime. We assume that the initial

Ž . < Ž .: ² Ž . <system r t s c t c t and the final target0 0 0ˆ < : ² <As f f are both in the pure states. The only

incoherence that may arise is from the external driv-ing fields which are phase-unlocked to each other.

Ž .The control overlap A t in this case has the formfw Ž .xof cf. Eq. 9

² < < 2:A t s c t ;u , 51Ž .Ž . Ž .f f

Ž . ) Ž . Ž .and K t ;t ,u sc t ,u f t ;t ,u as given by" f f " fŽ .Eq. 34 . The final coupled control equations for the

� 4pair of phase-unlocked pump-dump fields, E , E ,1 2w xare then given by 14,15

² ) :c t ,u f t ;t ,u sl E t , 52Ž . Ž .Ž . Ž .f q f 1 1

² ) iu :c t ,u f t ;t ,u e sl E t . 53Ž . Ž .Ž . Ž .f y f 2 2


Ž .Note that the control equations in Eq. 48 withŽ . Ž .js1 and 2 or Eqs. 52 and 53 should be solved

jointly and iteratively. The molecular dynamics in-volved in these equations are governed by the RWA

Ž . Ž .Hamiltonian H t;u of Eq. 47 , in which the pair ofpump-dump fields are locked to each other at therelative phase of u . A procedure of phase averagingshould be explicitly carried out in each step of theiterative solution to the optimal pair of phase-un-locked pump-dump fields. In the next subsection, weshall consider the pure-state control in the weakpump-dump response regime, in which the phaseaveraging can be performed analytically and theresulting optimal control equations can further beconverted into eigen problems.

4.3. Phase-unlocked pump-dump control as an eigenproblem

We shall now consider the case when both thepump and the dump fields are in the weak responseregime. In this case, the target expectation valueŽ .A t scales linearly with the incident pump energyf

I and the dump energy I . In the case of pure matter1 2

state system, we have

2tt f X X X

)A t s dt dt E t B t ,t E t .Ž . Ž . Ž .Ž . H Hf 2 g 1t t0 0

54Ž .

Here, the random phase average has been explicitlycarried out. The second order pump-dump controlresponse function B in Hilbert space is defined byg

Ž .Eq. 38 , the same as that in the phase-locked controlscheme. However, an optimal pair of phase-unlocked

� 4pump-dump E , E fields in the weak response1 2

regime satisfy the following coupled eigenequationsw x w Ž . x15 : cf. Eq. 43 for the phase-locked counterpart

t f X X X)dt B t ,t E t sl E t , 55Ž . Ž . Ž . Ž .H g 2 1 1

t

tX X Xdt B t ,t E t sl E t . 56Ž . Ž . Ž . Ž .H g 1 2 2

t0

Their eigenvalues relates to the phase-unlockedw Ž .xpump-dump yield as cf. Eq. 44

2l l sl sA t r I I . 57Ž . Ž .Ž .1 2 f 1 2

Ž . Ž .Eqs. 55 and 56 also state that the optimal dumpfield E at time t correlates only with the optimal2

wpump field E occurring earlier than t cf. that1X Ž .xt -t in Eq. 56 , and vice versa. This is consistent

with the causality in the pump-dump process inwhich the initial ground state wave function shouldbe pumped prior the dump interaction can occur.Note that this sequential ordering holds only at thewavefunction amplitude level. At the amplitude

Ž .square level as shown in Eq. 54 , however, it alsocontains the coherent contribution in which the dumpfield E) occurs between two pump fields, E and2 1

) w xE , interacting with the system 27 . By including1Ž . Ž X .the step function as Eq. 38 , i.e. setting B t ,t s0

if t-tX, we can replace the integration region in Eq.

Ž . Ž . w x55 or Eq. 56 by t ,t . Furthermore, we can0 fŽ . Ž .recast Eqs. 55 and 56 as

t f X X X† 2dt B B t ,t E t sl E t , 58Ž . Ž . Ž . Ž .H g g 1 1t0

t f X X X† 2dt B B t ,t E t sl E t . 59Ž . Ž . Ž . Ž .H g g 2 2t0

Here,

t fX XX XX XX X† )B B t ,t ' dt B t ,t B t ,t ,Ž . Ž . Ž .Hg g g gt0

60Ž .

w † xŽ X.and B B t ,t can be defined similarly. Eqs.g gŽ . Ž .55 – 60 constitute the final formulation for theoptimal pair of the phase-unlocked pump-dump fieldsin the weak response regime. The globally optimalpair of the phase-unlocked pump-dump fields are the

� 4eigenfunction E , E that associates with the largest1 22 Ž . Ž .eigenvalue of l in Eq. 58 or Eq. 59 .

5. Numerical results and discussion

In this section, we shall apply the theory devel-oped in the previous sections to the pump-dumpcontrol of the vibrational motion in an iodinemolecule, involving the ground electronic X and theexcited B surfaces. The potential functions for thesetwo surfaces are chosen as the same as those used in

w xRef. 22 . Shown in Fig. 1 are these two potentialsurfaces together with the pump-dump control


Fig. 1. Schematic diagram of the pump-dump control process inan I molecule that involves the ground X and excited B surfaces.2

scheme. For simplicity, we assume that the iodinemolecule begins at the vibronic ground ns0 levelin its electronic ground X state. The target time ischosen as t s300 fs.f

We shall only present the numerical results in theweak response regime for the pure state pump-dumpcontrol in both the phase-locked and the phase-un-

Ž .locked schemes. Therefore only Eq. 43 , and Eq.Ž . Ž . w Ž . Ž .x58 with Eq. 56 or Eq. 59 with Eq. 55 will beimplemented. Note that in the phase-unlocked con-trol scheme, the relative intensities among the vari-ous subpulses or components in the single coherencefield are totally determined by the eigen equation,

Ž .Eq. 43 . However, the intensities of the pump fieldand the dump field in the phase-unlocked schemecan each be scaled independently. For comparison,we set the incident energies from the individualpump and dump fields in the phase-unlocked case tobe the same as a half of the total energy from thephase-locked pump-dump field, i.e. I s I s Ir2.1 2

Further, we choose two targets, both on the groundX surface, to elucidate the control mechanisms ineither the phase-locked or the phase-unlockedschemes.

Control of waÕe packet focusing. The first targetis chosen as a minimum uncertainty Gaussian wavepacket to represent molecular focusing in phase-space. A minimum uncertainty Gaussian wave packettarget can be generally described by

y1r4f s 2p wŽ .g r r

=1 ip2exp y ryr q ryr . 61Ž . Ž . Ž .

4w "r r

Here, w is the variance of the target in position andr r

relates to w , the momentum variance, as w w sp p r r p p2Ž ."r2 ; r and p denote the position center and the

momentum center, respectively, of the Gaussian tar-get wave packet. These parameters are chosen as

1r2˚ ˚rs3.35 A, ps0 and w s0.06 A; thus the meanr r² < < :vibronic energy of this target is f H f s7557g g g

cmy1, about the same as the eigen energy of ns40vibronic level. The optimal pump-dump fields in theweak response regime are calculated in both the

Ž .phase-locked scheme by using Eq. 43 , and theŽ . Ž .phase-unlocked scheme by using Eqs. 58 and 59 .

Figs. 2 and 3 present respectively the temporal andthe spectral profiles of the globally optimal fieldsevaluated as the eigenvectors that correspond to thelargest eigenvalues in both control schemes. Thepump and dump components do not overlap eachother in either the time or the frequency domain, andthe incident energies from each components areequal. As we can see from Fig. 2 or Fig. 3, it isindistinguishable in this case, whether the control ismade in the phase-locked or in the phase-unlockedscheme.

In order to examine the control efficiency, weŽ .introduce the time dependent control yield h t as

w x23

2 <² < Ž2. : < 2 2h t sA t rI s f c t rI . 62Ž . Ž . Ž . Ž .g g

ˆŽ . w Ž .xIn this equation, A t sTr Ar t whose value at

Fig. 2. The temporal profiles of the globally optimal control fieldsin the phase-locked and phase-unlocked schemes for a minimum

˚ ˚Gaussian wave packet centered at 3.351 A with a width of 0.06 Aw xon the X surface cf. Fig. 1 . The molecule is initially at the

< :ground vibronic 0 level. The optimal pump-dump field in thephase-locked scheme consists of two subpulses which are coinci-dent with the pump and dump fields in the phase-unlocked controlscheme. The target is t s300 fs.f


Fig. 3. The spectral profiles for the same fields in Fig. 2.

w Ž .xthe target time t cf. Eq. 2 is the objective offŽ .control. The second identity in Eq. 62 is only

explicitly written for the phase-locked pure statecontrol scheme. In the phase-unlocked controlscheme, we shall replace I 2 by 4 I I and include1 2

² : Ž .the phase average PPP in Eq. 62 . From theprevious sections, we also know that the square root

1r2Ž .of the control yield at the target time, h t ,fŽ .equals l, the eigenvalue in Eq. 43 for the phase-

locked case, or lr2 where l is the eigenvalue in Eq.Ž . Ž .58 or Eq. 59 for the phase-unlocked case.

An alternative useful function to indicate the de-gree to which the control field drive the wave packet

Ž .to reach the target is the so-called achievement a t ,defined as

A tŽ .2a t sŽ . Ž4.ˆTr A Trr tŽ .g g f

<² < Ž2. : < 2f c tŽ .g gs . 63Ž .Ž2. Ž2.² < : ² < :f f c t c tŽ . Ž .g g g f g f

The second identity in the above equation is validonly for the pure state control with the phase-lockedfield in the weak response regime. In the phase-un-locked control scheme, it should include the randomphase averages for the numerator and the denomina-tor individually.

Ž . w Ž .xFig. 4 presents the yield h t Eq. 62 for theGaussian target controlled in the previous figures ineither the phase-locked or the phase-unlocked

Ž . 2Ž .scheme. As we mentioned that h t and a t differonly by a scaling constant, Fig. 4 can also serve for

Ž .the plot of a t , in which the achievement at theŽ .target time a t reaches about 0.87 in either of thef

Ž . w Ž .xFig. 4. The control yield h t Eq. 62 for the same system asŽ .Fig. 2. Shown are the relative values with respect to h t ,locked f

the yield at the target time controlled by the globally optimalphase-locked pump-dump field.

control schemes. Fig. 5 shows the evolution of thewave function under the globally optimal fields ineither the phase-locked or the phase-unlocked con-trol schemes. It can be seen that the controlled wavefunction at the target time overlaps pretty well with

Ž .the target dashed line .Control of eigenstate target: Raman pump-dump

control. We shall now turn to the optimal pump-dump

Fig. 5. The wave packet evolution, controlled by the globallyŽ . Ž .optimal field s in Fig. 2, on the excited B surface upper panel

Ž .and ground X surface lower panel . The unexcited portion of thewwave packet is removed. To visualize the control achievement cf.

Ž .xEq. 63 at the target time of t s300 fs, the target wave packetfŽ .dashed line is also included in the lower panel.


control of the second target which is taken to be anvibronic eigenstate. We shall refer this type of targetas the Raman pump-dump control. Before we pro-ceed, let us present the intrinsic relation between thepump-dump eigenstate control and the resonant Ra-man process. For generality, let us denote the initial

< Ž .: < :vibronic level as c t s i , and the final vi-g 0< Ž .: < :bronic target level as f t s f . Their corre-g f

sponding eigenenergies are e and e , respectively.i f

In this case, the weak pump-dump control responseŽ .function B of Eq. 38 assumes the following formg

w x15 :

B t ,t X seyi e f Ž t fyt .r "T tytX eyi e iŽt

Xyt 0 .r " .Ž . Ž .g f i

64Ž .ˆŽ . ² < Ž . < : w Ž .xHere, T t s f T t i cf. Eq. 39 , is the well-f i g

wknown T-matrix elements for the Raman more pre-w x Ž .cisely the stimulated emission pumping 28 SEP or

w x Ž .the stimulated Raman pumping 29 STIRAP f§ iw xtransition amplitude 30,31 . Since both the pump-

dump control and the SEP or STIRAP spectroscopicprocesses describe the same two-photon dynamics

< : < :that excite the initial i level to the final f levelon the ground surface via the intermediate excitedelectronic state, it is not surprising that they share thesame dynamical quantity. It would be possible to usethe control formulation developed in this section todesign the optimal SEP or STIRAP spectroscopicmeasurements or to extract the dynamic informationfrom the spectroscopic signals for the control feed-back.

For the Raman pump-dump control in the phase-unlocked scheme, there is further a symmetry rela-tion between the optimal pump field and its counter-

w xpart of optimal dump field 15 , which can be equiva-lently represented in the time-domain as

e ie iŽ t0qt .r "E) t q t seyi e f Ž t fyt .r "E t y t ,Ž . Ž .1 0 2 f

65Ž .

or in the frequency-domain as

yi v Ž t0qt f . ˆ ˆ)e E v sE vqv , 66Ž . Ž . Ž .2 1 f i

ˆ Ž . Ž .with E v being the Fourier transform of E t .j j

The above two equations lead to the following con-w xclusions 15 for the optimal pair of pump-dump

� 4 Ž .fields, E , E : i their frequency chirps are of the1 2Ž .same value but opposite in signs; ii their temporal

< Ž . < 2spectra, E t , are symmetric about the center, i.e.jŽ . w x Ž .t q t r2, of the interaction region t ,t ; iii the0 f 0 f

ˆ 2< Ž . <power spectrum of the optimal dump field, E v ,2Ž . Ž .is red Stoke shifted by v s e ye r" comparedf i f i

ˆ 2< Ž . <with that of the pump field, E v . The above1

three properties of optimal pair of pump-dump fieldshold for the control of eigenstate in the absence ofdephasing and relaxation processes.

In the present calculation, we choose the target of< : < :the Raman pump-dump control as the f s 8 vi-

< : < :bronic level, and the initial state is the i s 0vibronic ground level. Fig. 6 shows the temporalprofile of the phase-locked globally optimal pump-

Ž .dump field thin-solid , and those for the pump fieldand the dump field in the phase-unlocked scheme.

< Ž . < 2As predicted in the phase-unlocked pair, E t and1< Ž . < 2E t , are mirror image to each other at t r2s1502 f

fs. Also can be seen clearly is a pattern of interfer-ence among the various components in the phase-

Ž .locked pump-dump field thin-solid . In Fig. 7, wepresent the spectral profiles of those fields in Fig. 6.

ŽIn the phase-locked scheme the upper panel of Fig..7 , four evenly spaced subcomponents in the pump-

dump field can be seen clearly. The common fre-quency spacing here equals to Raman frequency ofv , which in this case is 1676 cmy1. Obviously, thef i

middle two components serve as both pump anddump, while the one carrying with the highest fre-quency acts as pump and the one with the lowest

Fig. 6. The globally optimal fields for the target of the vibronic< : < :eigenstate of f s 8 . The target time is t s300 fs. Thef

< :molecule is initially in the ground vibronic 0 level. The thin-solidcurve is the globally optimal pump-dump field in the phase-lockedcontrol scheme. The two dark-solid curves are respectively theglobally optimal pump and dump fields in the phase-unlockedcontrol scheme as indicated.


Fig. 7. The spectral profiles for the same fields in Fig. 6. Theupper panel is for the phase-locked and the lower panel for thephase-unlocked cases. Included is also the Raman excitation pro-

Ž . w Ž .xfile dashed line Eq. 67 in each panel. See text for details.

frequency as dump only. All these four componentsare optimally phase-locked to each other to maxi-mize the control yield. In comparison, the phase-un-

� 4locked pair, E , E , are much simpler; each field1 2Žconsists of only a single subpulse the lower panel of

.Fig. 7 . As it is predicted by the symmetry relation,the power spectrum of the optimal dump field is redshifted by v with respect to that of the pump field,f i

which is similar to that in the phase-locked scheme.Shown in each panels of Fig. 7 is also the finite-time

Ž . ŽRaman excitation signal dashed defined by assum-.ing t s00

2t f iŽvqe r " . tiS v ;t s dt e T t . 67Ž . Ž .Ž . HR f f i

0

It is clear that the globally optimal single-subpulsepump field in the phase-unlocked scheme carries thefrequency around where the Raman excitation profilereaches its global maximum. However, the globallyoptimal pump-dump field in the phase-locked schemeŽ .the upper panel of Fig. 7 should adjust itself such

Žthat the overall pump components the three higher.frequency subpulses make the best use of the whole

Franck-Condon factors within the full range of theRaman excitation profile. Another important feature

Ž . w Ž .xFig. 8. Control yield h t Eq. 62 for the same system as Fig. 6.The thin-solid curve is for the phase-locked control scheme, whilethe dark-solid is for the phase-unlocked control scheme.

of the Raman pump-dump control is that each of thesubpulses, in either the phase-locked or the phase-un-locked schemes, is as near-cw field as possible in the

w xtime interval t ,t of the control interaction could0 f

support. This means that each subpulses is nearlytransform-limited field which centers sharply at aspecified frequency.

Ž . w Ž .xFig. 8 represents the control yield h t Eq. 62Ž .for the phase-locked thin-solid and phase-unlocked

Fig. 9. The wave packet evolution controlled by the globallyŽ .optimal field s in Fig. 6. The controlled wave packet on the

Žground surface is nearly coincident with the target dashed line in.the lower panel at the target time in either the phase-locked or

phase-unlocked control scheme. Near perfect control is obtainedŽ . w Ž .xin view of the achievement a t Eq. 63 at the target time.f


Ž .dark-solid control of the eigen target. In this case,Ž .the yield h t at the target time in the phase-un-f

locked scheme is about 0.88 of that in the phase-Ž .locked scheme. The relative small value of h t inf

the phase-unlocked scheme with respect to that inthe phase-locked scheme can be easily understood bythe constructive interference among the various sub-components in the optimal phase-locking conditionwhich is lacked in the phase-unlocked scheme. Fig. 9shows the evolution of the wave packets on both

Ž .surfaces created by the optimal pump-dump field sin either the phase-locked and the phase-unlockedschemes. It is interesting that although their yields

Ž . Ž .are different, their achievements, a t of Eq. 63are almost identical in these two schemes; both of

Ž .them achieve nearly perfect control with a t f1f

at the target time.

6. Summary and discussion

In this paper, we investigated the optimal pump-dump control theory in both the phase-locked schemeand the phase-unlocked scheme in a systematic andunified manner. We showed that in the conventional

Žsingle coherent field which is termed in this paper.also as the phase-locked control scheme, there are

two components, K and K , which can physicallyq ybe classified as the pump and dump contributions to

Ž .the control kernel K. In fact, K or K relates toq yŽ .the pump K or dump K control kernel in the1 2

w Ž . Ž .xphase-unlocked scheme cf. Eq. 49 or Eq. 50 .Similar relations for the Hilbert-space control kernels

w Ž . Ž .are also presented Eq. 36 versus Eqs. 52 andŽ .x53 . The simplest relation between the phase-lockedand the phase-unlocked control schemes is in theweak response regime in which there is only a single

wcommon Hilbert-space response function B Eq.gŽ .x38 . In this case, the optimal field in the phase-locked scheme is determined as the eigenvector ofthe algebraically symmetrized Hermitian operator,Ž † . w Ž .xB qB r2 cf. Eq. 43 , while those in the phase-unlocked scheme is as the eigenvectors of the geo-metrically symmetrized Hermitian operators, B† Bg g

† w Ž . Ž .xand B B cf. Eqs. 58 and 59 . As a preliminaryg g

effort, we numerically implemented the eigenequa-tion problem to a one-dimension quantum system forthe globally optimal fields in both control schemes in

the weak response regime. To our knowledge, theglobally optimal pump-dump field in the phase-locked scheme was not identified before. The effectof the phase-locking was analyzed in detail for boththe wave packet focusing target and the eigenstate

wRaman-like target in the system of consideration cf.xSection 5 . Further study based on the present unified

formulation should be carried to see how much ofthe weak field property is retained as the field inten-sity rises, and of the controllability as the systemincreases in complexity via either more quantumdegrees of freedom or faster dissipation and relax-ation processes.

In concluding, this work constitutes a systematicand unified theory of pump-dump control via both asingle coherent field and a pair of phase-unlockedcoherent fields. The formulation of the optimalphase-unlocked control scheme generalizes the origi-

w xnal Tannor-Rice work 8 and further complementsthe conventional theory of optimal control via a

w xsingle coherent field 9–12 . The method developedin this paper may also be used in establishing aunified formulation that connects Brumer-Shapiro’scoherent control scheme to the optimal control the-ory. As we showed in Section 5, the optimal fieldcan consist of multiple components with their phasesand amplitudes optimally modulated. In the eigen-state target control of study, all components in theoptimal field are nearly cw in nature and their fre-

w xquencies relate to each other cf. Fig. 6 . The abovefeatures are the ingredients of Brumer-Shapiro’s co-

w xherent control method 1–3 .

Acknowledgements

The support of the Research Grants Councils ofthe Hong Kong Government is gratefully acknowl-edged.

References

w x Ž .1 P. Brumer, M. Shapiro, Faraday Disc. Chem. Soc. 82 1986177.

w x Ž .2 M. Shapiro, P. Brumer, J. Chem. Phys. 84 1986 4103.w x Ž .3 M. Shapiro, P. Brumer, Int. Rev. Phys. Chem. 13 1994 187.w x Ž .4 R.G. Gordon, S.A. Rice, Ann. Rev. Phys. Chem. 48 1997

595.


w x Ž .5 C. Chen, Y.Y. Yin, D.S. Elliott, Phys. Rev. Lett. 64 1990507.

w x Ž .6 C. Chen, D.S. Elliott, Phys. Rev. Lett. 65 1990 1737.w x Ž .7 D.J. Tannor, S.A. Rice, J. Chem. Phys. 83 1985 5013.w x Ž .8 D.J. Tannor, S.A. Rice, Adv. Chem. Phys. 70 1988 441.w x Ž .9 A.P. Peirce, M.A. Dahleh, H. Rabitz, Phys. Rev. A 37 1988

4950.w x Ž .10 H. Rabitz, S. Shi, Adv. Mol. Vibr. Collision Dyn. 1A 1991

187.w x11 R. Kosloff, S.A. Rice, P. Gaspard, S. Tersigni, D.J. Tannor,

Ž .Chem. Phys. 139 1989 201.w x12 Y.J. Yan, R.E. Gillilan, R.M. Whitnell, K.R. Wilson, S.

Ž .Mukamel, J. Phys. Chem. 97 1993 2320.w x Ž .13 Y.J. Yan, J. Chem. Phys. 100 1994 1094.w x Ž .14 Y.J. Yan, J. Che, J.L. Krause, Chem. Phys. 217 1997 297.w x Ž .15 Y.J. Yan, J.S. Cao, Z.W. Shen, J. Chem. Phys. 107 1997

3471.w x Ž .16 V. Dubov, H. Rabitz, Phys. Rev. A 54 1996 710.w x Ž .17 I. Averbukh, M. Shapiro, Phys. Rev. A. 47 1993 5086.w x Ž .18 L.Y. Shen, S.H. Shi, H. Rabitz, J. Phys. Chem. 97 1993

12114.w x Ž .19 V. Dubov, H. Rabitz, J. Chem. Phys. 103 1995 8412.w x20 J.L. Krause, R.M. Whitnell, K.R. Wilson, Y.J. Yan, S.

Ž .Mukamel, J. Chem. Phys. 99 1993 6562.w x21 J.L. Krause, R.M. Whitnell, K.R. Wilson, Y.J. Yan, in: J.

Ž .Manz, L. Woste Ed. , Femtosecond Chemistry, VCH, Wein-¨heim, 1995, p. 743.

w x22 J. Che, J.L. Krause, M. Messina, K.R. Wilson, Y.J. Yan, J.Ž .Phys. Chem. 99 1995 14949.

w x23 J.L. Krause, M. Messina, K.R. Wilson, Y.J. Yan, J. Phys.Ž .Chem. 99 1995 13736.

w x24 B. Kohler, V.V. Yakovlev, J. Che, J.L. Krause, M. Messina,K.R. Wilson, N. Schwentner, R.M. Whitnell, Y.J. Yan, Phys.

Ž .Rev. Lett. 74 1995 3360.w x25 Y.J. Yan, J.L. Krause, R.M. Whitnell, K.R. Wilson, in: N.Y.

Ž .Yu, X.Y. Li Eds. , XIVth Intern. Conf. Raman Spec-troscopy, Wiley, New York, 1994, pp. 76–77.

w x26 J. Che, M. Messina, K.R. Wilson, V.A. Apkarian, Z. Li, C.C.Ž .Martens, R. Zadoyan, Y.J. Yan, J. Phys. Chem. 100 1996

7873.w x Ž .27 Y.J. Yan, L.E. Fried, S. Mukamel, J. Phys. Chem. 93 1989

8149.w x28 C.E. Hamilton, J.L. Kinsey, R.W. Field, Annu. Rev. Phys.

Ž .Chem. 37 1986 493.w x29 S. Schiemann, A. Kuhn, S. Steuerwald, K. Bergmann, Phys.

Ž .Rev. Lett. 71 1993 3637.w x30 S. Mukamel, The Principles of Nonlinear Optical Spec-

troscopy, Oxford, London, 1995.w x Ž .31 Y.J. Yan, S. Mukamel, J. Chem. Phys. 85 1986 5908.

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Optimal pump-dump control: phase-locked versus phase-unlocked schemes