Coarse-grained controllability of wavepackets by free evolution and phase shiftsE. A. Shapiro, Misha Yu. Ivanov, and Yuly Billig Citation: The Journal of Chemical Physics 120, 9925 (2004); doi: 10.1063/1.1730156 View online: http://dx.doi.org/10.1063/1.1730156 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/120/21?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Communication: Constructing an implicit quantum mechanical/molecular mechanics solvent model by coarse-graining explicit solvent J. Chem. Phys. 139, 081103 (2013); 10.1063/1.4819774 Transition representations of quantum evolution with application to scattering resonances J. Math. Phys. 52, 032106 (2011); 10.1063/1.3559003 Vibrational solvatochromism and electrochromism: Coarse-grained models and their relationships J. Chem. Phys. 130, 094505 (2009); 10.1063/1.3079609 Approximate resonance states in the semigroup decomposition of resonance evolution J. Math. Phys. 47, 123505 (2006); 10.1063/1.2383069 Complex Hamiltonian evolution equations and field theory J. Math. Phys. 39, 5700 (1998); 10.1063/1.532587

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Coarse-grained controllability of wavepackets by free evolutionand phase shifts

E. A. ShapiroSteacie Institute for Molecular Science, National Research Council of Canada, Ottawa, Ontario K1A 0R6,Canada and General Physics Institute, Russian Academy of Sciences, Moscow 117942 Russia

Misha Yu. IvanovSteacie Institute for Molecular Science, National Research Council of Canada, Ottawa,Ontario K1A 0R6, Canada

Yuly BilligSchool of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa,Ontario K1S 5B6, Canada

~Received 15 January 2004; accepted 9 March 2004!

We describe an approach to controlling wavepacket dynamics and a criterion of wavepacketcontrollability based on discretized properties of the wavepacket’s localization on the orbit. Thenotion of ‘‘coarse-grained control’’ and the coarse-grained description of the controllability ininfinite-dimensional Hilbert spaces are introduced and studied using the mathematical apparatus ofloop groups. We prove that 2D rotational wavepackets are controllable by only free evolution andphase kicks by AC Stark shift implemented at fractional revivals. This scheme works even if the ACStark shifts can have only a smooth coordinate dependence, correspondent to the action of a linearlypolarized laser field. ©2004 American Institute of Physics.@DOI: 10.1063/1.1730156#

I. INTRODUCTION

Facing the problem of controlling a quantum system, itis important to know whether the set of tools at hand—mathematically, the set of different Hamiltonians one canapply to the system—is sufficient to implement all desiredtransformations. With this kind of information, one can ei-ther look for intuitive schemes to implement the neededtransformation1,2 or rely on numerical or experimental opti-mal control algorithms.3 If, however, the set of control knobsat hand is proven to be insufficient, one can analyze why it isso and then efficiently search for additional control tools.

We use the definition of controllability of a quantumsystem, which implies the ability to generate any unitarytransformation, and hence any state to state transformation,with a given set of control knobs. Here controllability alsoimplies that any such transformation can be generated in acertain timeTf that does not exceed some boundary valueT0 ; for discussion of different definitions of controllabilitysee, e.g., Ref. 4.

Geometric approach to controllability has been very suc-cessful for finite-dimensional Hilbert spaces, i.e., in analyz-ing the controllability of a system where the evolution isconfined to a finite, predetermined set of levels. It has beenused in the areas of atomic and molecular control~see Refs.2 and 4–8!, as well as in the field of quantumcomputation.5–12 The following result is used most com-monly: one can generate any unitary transformation in anN-level system, applying the set of time-independent Hamil-tonians H0 ,...,Hk , if the Lie algebra generated by theseHamiltonians—the space of linear combinations ofH0 ,...,Hk and their iterated commutators—is of real dimen-sion N2.8,13

In the physical examples,H0 often stands for the unper-turbed Hamiltonian of the system andH1 ,...,Hk include dif-ferent types of perturbations one can switch on and off, oneat a time, for intervals of time that can be arbitrarily short. Inthe context of atomic and molecular controlH1 ,...,Hk areusually meant to be implemented by switching on and offlaser pulses of various spectra, intensity, and polarization. Itis important to keep in mind that the ability to applyH1 ,...,Hk over arbitrary time intervals implies arbitrarilyflexible shaping of the external fields~laser pulses! that pro-vide the perturbations. As a rule, for a finite number of levelstwo noncommuting Hamiltonians are sufficient to generateLie algebra of real dimensionN2.10

Thus, for a finite-level atomic or molecular system,knowing the transition matrix elements for laser pulses athand, one has a fairly straightforward way of telling whetherthe system is controllable or not.

However, most real systems are infinite dimensional. Asimple yet fundamental model involving an infinite numberof levels is a bound motion of a wavepacket in a system withdiscrete infinite-dimensional spectrum, where any superposi-tion of states can be excited. Obvious examples include aharmonic oscillator, Rydberg atom, or rotor. A wavepacket isboth one of the most basic quantum-mechanical conceptsand one of the typical control entities. Experiments attempt-ing to utilize and control wavepacket dynamics span suchareas as photochemistry, intense laser-matter interaction, andquantum computation~see, e.g., Refs. 1, 2 and 14–16!.

Unfortunately, for infinite-dimensional systems, verylittle is known regarding controllability. Application of geo-metric methods to describe infinite-dimensional control isdifficult; the theory of infinite-dimensional Lie groups is not

JOURNAL OF CHEMICAL PHYSICS VOLUME 120, NUMBER 21 1 JUNE 2004

99250021-9606/2004/120(21)/9925/9/$22.00 © 2004 American Institute of Physics

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yet well developed. In general, the prediction is negative inthe sense that, using piecewise-constant controls, global con-trollability cannot be achieved with a finite number ofoperations.17 Controllability is achieved over some part ofthe Hilbert space, if the initial condition and the dynamicsare restricted to a finite-dimensional manifold~thus, the Liealgebra generated by all Hamiltonians at hand is finite di-mensional! ~see Refs. 7, 8, and 17!. Only in a few casesdealing with special types of controlling Hamiltonians canone achieve control over the infinite-dimensionalstateof thesystem7,18 ~rather than the whole evolution matrix! or over aninfinite-dimensional subset of all the possible evolutionmatrices.19

Moreover, if the number of levels involved is finite butvery large, then the general picture also looks rather pessi-mistic. In order to explore all the dimensions needed forcontrollability, as the dimension of the corresponding Liealgebra scales asN2, one has to scale accordingly the num-ber of control knobs: the number of piecewise constant non-commuting Hamiltonians, or the number of different laserfrequencies with controlled amplitudes and phases, etc. Al-ternatively, one has to apply sequences of interactions ofgrowing length. Note that many interesting controlexperiments16 are done in the nonperturbative regime, wherethe laser field couples and modifies many eigenstates, so thatit is rarely possible to single out a small isolated subsystemcontrolled by the field.

We suggest and study one possible way to deal withthese difficulties. Dealing with a wavepacket, one is mostlyinterested in the features of its localization. It is the degreeand the region of wavepacket localization in the phase spacethat are reasonable objectives of control. In the spirit of abulk of experiments on femtosecond control, we reformulatethe question of controllability following the idea thatonlycoarse-grained information on the wavepacket structure andlocalization is of interest.

We focus on wavepackets which evolve in a system withinfinite-dimensional spectrum with energies depending qua-dratically on the quantum number, such as a rigid rotor.Given a desired degree of wavepacket localization, we usethe wavepacket spreading and the so-called wavepacketrevivals20 to map complex infinite-dimensional evolutiononto that of a few replicas of the same wavepacket, evenlydistributed along the generalized angle variable on the corre-sponding classical orbit.

The number of such replicas is determined by theamount of information one wants to track. Their evolutionand controllability is described using the apparatus of loopgroups. The latter allows us to make conclusions about theconditions of controllability in an infinite-dimensional sys-tem.

The most important aspect of our coarse-grained ap-proach and the loop groups-based analysis is that they allowus to deal with the infinite-dimensional system withoutspecifying ~or restricting! the number of levels involved inthe dynamics. The coarse-grained approach allows us to in-troduce a finite-dimensional control basis where each basisvector is a wavepacket and is, in principle, infinite-dimensional—but is addressed as a whole.

In our control approach, the wavepackets are controlledby coordinate-dependent phase-shifts applied to the basiswavepackets. These phase-shifts can be implemented vianonresonant ac Stark shift induced by short laser pulses. As aresult of such short ‘‘kicks,’’ the Stark shift changes thephase of the wavefunction in a coordinate-dependent way,mixing all eigenstates.

The rest of the paper is outlined as follows. First, weintroduce a model of a 2D rotor and describe the wavepacketdynamics on the spreading-revival timescale.20 Using argu-ments about physical purposes of the control, we introducethe concept of the coarse-grained control and the controlmethod based on applying coordinate-dependent phase kicksto the wavepacket. The phase kicks are applied at the wave-packet’s fractional revivals. We introduce the mathematicaldescription of the control procedure using the apparatus ofloop groups and show that rotational wavepackets are con-trollable by the phase shifts which smoothly depend on thecoordinate. The analysis does not restrict the number of lev-els involved in the dynamics. Then, we discuss the condi-tions for limiting the amount of noise in the generated uni-tary transforms and possible generalization of our method toan arbitrary 1D wavepacket with weakly nonlinear spectrum.The last section presents our conclusions.

II. PHYSICAL REASONINGAND THE CONTROL MODEL

As a main example, we consider the control of a 2Drigid rotor confined to a plane. This system allows for theeasiest explanation of our ideas; generalization to other sys-tems with nonlinear spectrum will be discussed in Sec. V.

The 2D rotor’s spectrum and eigenstates are given by

EM5BM2, M52`,... , , ~1!

cM~f!51

A2peiM f, ~2!

where M is the quantum number of the planar rotationaleigenstate andf is the orientation angle of the rotor~Fig. 1!.

Figure 2~a! shows the so-called quantum carpet21—the2D map showing the evolution of the coordinate-dependentprobability distributionuC(f,t)u2 as a function of time. Forthis particular case the initial wavepacket isC(f,t50)5(MCMcM(f) with a Gaussian distribution CM

5exp@2M2/DM2#, DM56.This state corresponds to an initially oriented rotor. In

practice, an oriented state of a heteronuclear molecule can becreated by a strong ‘‘half-cycle’’ terahertz pulse22 or a com-bination of a dc electric field and an aligning laser pulse.23

At t50 the wavepacket is localized atf50 @Fig. 2~b!#.The initial localization is quickly lost: the wavepacketspreads along the rotational orbitf52p,...,p. However,the quantized nature of the spectrum leads to periodic relo-calization of the wavepacket—the so-called revivals.20 At therevival time

Trev52p/B, ~3!

the wavepacket revives exactly in its original form. AtTrev/2it revives as an exact replica placed at the opposite side of

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the trajectory,f5p. For any oddN, at any Trev /N therevival pattern consists ofN equally spaced replicas withwell-defined relative phases, such as those shown in Fig.2~c!. For evenN52K, atTrev /N the revival pattern consistsof K5N/2 equally spaced replicas. These moments and theirmultiples are called ‘‘fractional revivals.’’20

The controllability criterium which we shall use below isbased on the fact that in the wavepacket experiments one isusually interested in coarse-grained measures ofwavepacketlocalization, rather than in a complete fine evolution of allquantum amplitudes. For example, for a rotational wavepacket the commonly asked questions are whether the mo-lecular wave packet is oriented in some direction or not,whether it is aligned parallel to some direction or not, howhigh is the degree of alignment or orientation, etc. In thesame way, one can ask whether a vibrational molecular wavepacket is in the Frank–Condon region or not, whether or nota Rydberg electron is near the core where it can be ionized,etc. One is also interested in the degree of wavepacket local-ization in a region of interest. The approach explored belowcontrols such integral properties of the wavepacket localiza-tion on progressively finer scales.

Figure 3 shows evolution of the commonly used mea-sures of rotational orientation and alignment,^C(t)ucosfuC(t)& and ^C(t)ucos2 fuC(t)&. They illustratethe fact that the characteristic features in the wavepacketlocalization are most distinct at the moments of fractionalrevivals, where the wavepacket is relocalized into severalreplicas.

Consider for definiteveness the case of oddN. Since thepattern on the quantum carpet att.Trev /N is determined bythe relative phases of the wavepacket replicas at the momentof fractional revivalt5Trev /N, the intuitive idea is to con-trol the wavepacket evolution by inducing relative phaseshifts between these replicas att5Trev /N.24,25 Applyingphase shifts at fractional revivals creates superpositions ofNequally spaced replicas of the original wavepacket, and thuscontrols the evolution with the accuracy of 1/N-th of theorbit.

By ‘‘ coarse-grained controllability,’’ for a given N, weshall mean that one can implement arbitrary unitary trans-form in the basis ofN equally spaced replicas of the initialspatially localized wavepacket. Thus, arbitrary superpositionof these replicas can be created. Spatial structure at the scalefiner than 1/N is not resolved and is of no interest. The totalnumber of levels involved in the dynamics is notspecified—it is only relevant for structures with the scalefiner than 1/N. The coarse-grained basis will remain thesame even if the dynamics involves increasing number oflevels, but each basis wavepacket will change its shape at thesub-1/N scale.

Phase shifts required in our control scheme can beimplemented by coordinate-dependent interaction with a per-manent or induced dipole moment triggered by a short pulse.Indeed, letH0 be the field-free Hamiltonian andV(f,t)5V0(t)v(f) be the interaction with the external field, ap-plied over a very short timeT. If the field-free motion duringT is negligible,H0T!1, then the propagator is reduced to thecoordinate-dependent phase shift:

U~T!5e2 i *0T~H01V~f,t !! dt'e2 iv~f!*0

TV0~ t ! dt5e2 iAv~f!.~4!

Physically, each of the laser pulses must be shorter than theperiod of linear motion of the wavepacket. As shown byexamples in Refs. 24 and 25, this condition can be easilysatisfied within the current femtosecond technology. Notethat for induced dipoleV0(t)}E2(t) and thus the integral in

FIG. 1. Schematic view of a planar rotor.

FIG. 2. ~a! Density plot of evolution of the probability distribution for theangular wavepacket initially placed atf50. ~b,c! Probability distribution att50 andt5Trev/3 @dashed line on the panel~a!#, in arbitrary units.

FIG. 3. Time evolution of the measures of orientation and alignment for thewavepacket shown on Fig. 2.

9927J. Chem. Phys., Vol. 120, No. 21, 1 June 2004 Controllability of wavepackets

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Eq. ~4! is always nonzero. In the case of a permanent dipoleone needs the so-called ‘‘half-cycle pulses.’’26

For the interaction with a permanent dipole momentv(f)}cosfMR and for the nonresonant ac Stark shiftsv(f)}cos2 fMR, wherefMR is the angle between the mo-lecular~rotor! axis and the polarization of the radiation. Us-ing two orthogonal polarizations alongX and Y axis, oneobtainsv(f)}cosf andv(f)}sinf for the interaction withthe permanent dipole moment andv(f)}cos2 f and v(f)}sin2 f for the interaction with the induced dipole moment.The relationship 2 cos2 f215cos 2f shows that rescalingthe rotational constantB and a simple variable substitutionreduce the second case to the first one. Thus, without loss ofgenerality, we can limit our discussion to the case of a per-manent dipole moment. In the following, we will allow twomutually orthogonal polarizations, thus having sinf andcosf phase profiles at our disposal.

Several examples implementing the suggested controlidea in molecular vibrations and rotations are discussed inRefs. 24 and 25. Figure 4 presents a characteristic example.Let us start with the same oriented superposition of the rotoreigenstates as the one given in Figs. 2 and 3. AtTrev/4 thewavefunction consists of two replicas of the original wave-packet with the relative phase between them equal top/2. Atthis moment we apply to the wavefunction the angle-dependent phase shift, equal top/4 cosf, mimicking the ac-tion of a vertically polarized terahertz pulse. The single con-trol pulse changes the relative phases between thewavepacket replicas in such a way that the future dynamicsshows no signal for the wavepacket orientation^C(t)ucosfuC(t)&, while the alignment signal^C(t)ucos2 fuC(t)& acquires additional minima.

Intuitively, it is clear that the suggested control mecha-nism should work if the phase is~i! imprinted onto eachreplica of the original wavepacketas a whole, and~ii ! eachwavepacket replica can acquire arbitrary phase relative toothers. Such an ideal situation would be approached if eitherthe Stark shift had a stepwise rather than smooth sinf or

cosf shape, or if the wavepackets inf-representation wereinfinitely narrow. This ideal case was considered in Refs. 24and 25. Here, we focus on the realistic case of a smoothphase profile and make no assumptions about how narrowthe wavepacket replicas are.

Coarse-grained control up to 1/N-th of the orbit impliesthe ability to generate all spatial modulations from sinf andcosf to sinNf and cosNf ~or to sinN f and cosN f). Intu-itively, it is clear that having sinf and cosf phase profilesinthe exponentand the ability to adjust the strength of theinteractionA in Eq. ~4! should enable us to generate all re-quired spatial modulations. Indeed, exp(2iA sinf)5SJM(A)exp(2iMf) contains all required harmonics withamplitudesJM(A) significantly different from zero forM&A (JM are the Bessel functions!. The next two sectionsdescribe rigorous mathematical analysis that proves this in-tuitive statement. For the sake of simplicity we consider thecase of oddN; the case of evenN is slightly more cumber-some.

III. MATHEMATICAL FORMULATIONOF THE PROBLEM

Suppose that the initial state in the energy basis is givenby some vectorC(M ). Then in thef-representation the ini-tial state of the system is

C0~f!51

&(

M52`

`

C~M !eiM f. ~5!

At the fractional revivalTrev /N, N odd,C0(f) breaks intoN equally spaced replicasC0(f22p l /N) with equal ampli-tudesFl

(0) :

C~f,Trev /N!

5 (l 50

N21

Fl~0!

* C0S f22p l

N D , where uFl~0!u251/N. ~6!

Our goal is to be able to implement arbitrary transform act-ing on such superpositions, producing superpositions of thetype

C transformed~f!

5 (l 50

N21

Fl* C0S f22p l

N D , where (l

uFl u251 ~7!

with arbitrary complex amplitudesFl restricted only by thenormalization condition. Then we will have achieved the de-sired coarse-grained controllability.

The transformed state in Eq.~7! is expressed in the en-ergy basis as

C~M ! transformed5C~M !* D~M !,

where D~M !5 (l 50

N21

Fle2p iM l /N. ~8!

Note that according to Eq.~8! D(M )5D(M1N). Such pe-riodicity in the energy basis is the consequence of ourcoarse-grained approach. Being able to implement an arbi-trary coarse-grained transform onC~f! is equivalent to be-

FIG. 4. ~a! Quantum carpet and~b! orientation and alignment dynamics forthe wavepacket evolution controlled by the single pulse. The moment of thepulse application is shown by the arrows.

9928 J. Chem. Phys., Vol. 120, No. 21, 1 June 2004 Shapiro, Ivanov, and Billig

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ing able to generate arbitrary, butN-periodic unitary trans-form acting onD(M ). Thus, in the energy basis, our goal isto be able to generate an infinite-dimensional block-diagonalmatrix of the type

S .

A0

A0

A0

.

D , ~9!

whereA0 is an arbitrary unitary matrix of the sizeN3N.To implement the control, we use the free propagation

Hamiltonian H05diag(...,EM21,EM ,EM11,...), where EM

5BM2, as well asH1 ,H2 that correspond to applying shortpulses of vertical and horizontal polarization, impartingcosf and sinf phase profiles as explained in the previoussection. In the energy basis, the HamiltoniansH1 andH2 aretridiagonal:

H151

2 S . . .

1 0 1

1 0 1

1 0 1

. . .

D ,

~10!

H251

2 S . . .

2 i 0 i

2 i 0 i

2 i 0 i

. . .

D .

We consider the evolution operator exp(iH0t), but onlyfor discrete set of times—the multiples oft05Trev /N52p/BN, whereN is a fixed odd positive integer. The evo-lution operatorg5exp(iH0t0) is represented by a diagonalmatrix g5diag(...,z(M21)2,zM2

,z(M11)2,...), wherez5e2p i /N isan Nth root of 1.

In the next section we show that the evolution operatorsfor H1 andH2 together withg are sufficient to generate aneven wider group of the infinite, unitary, periodic block ma-trices of the type

S . . . . .

. A0 A1 . .

. A21 A0 A1 .

. . A21 A0 .

. . . . .

D . ~11!

We shall also discuss the condition of remaining within thesubgroup represented by the block-diagonal matrices~9!.

Mathematically, all possibleN3N matrices in Eq.~9!belong to theSUN group—the group of unitary matrices withthe determinant equal to unity:ATA5I , det(A)51. HereAT

is the transpose ofA and A is its complex conjugate. In theinfinite-dimensional Hilbert space, matrices Eq.~9! describe

rotations in all directions within the limits imposed by theblock-diagonal nature of these matrices and theSUN-natureof the blocks.

In the following section we show that, having matricescorresponding to the free evolution over the 1/N fractionalrevival and the phase kicks, we can make~infinitesimal!steps in any direction in the infinite-dimensional Hilbertspace within the limits of the group of matrices given by Eq.~11!. These matrices form an infinite-dimensional matrix rep-resentation of the so-calledSUN@z,z21# loop group.27

The SUN@z,z21# loop group is the set of allN3N ma-tricesA whose entries are not numbers but complex polyno-

mials in z,z21, with the conditionsATA5I and det(A)51.For the formal parameterz, there is a conventionz5z21:taking the conjugate of a matrix whose elements are polyno-mials of z, we take complex conjugates of the numericalcoefficients and replacez by z21. If z is moving along a unitcircle in the complex plane, the conventionz5z21 is natural.For every fixed value ofz, the matrix fromSUN@z,z21# be-comes a conventional matrix fromSUN . When we letz runover the unit circle, the corresponding matrix makes a loopin SUN . This interpretation of loop groups is used in thestring theory.28

Alternatively, an elementA of SUN@z,z21# can beviewed as a polynomialA5(kAkz

k in powers ofz andz21.In the infinite-dimensional matrix representation ofSUN@z,z21# given by Eq.~11!, theA0 blocks go on the maindiagonal,A21 go just below it,A1 just above it, etc.

IV. GEOMETRIC ANALYSIS

Mathematically, our goal is to show that our physicaltools allow us to make infinitesimal steps in any directionstaying within the limits ofSUN@z,z21#, that is, to generateany element of the corresponding Lie algebrasuN@z,z21#.With the ability to move in an arbitrary direction we shouldbe able to approximate any element of the group. For finite-dimensional Lie groups the Lie correspondence theoremguarantees that the whole group of transformations can begenerated if all of its Lie algebra is available. In the infinite-dimensional theory no analog of such a theorem has been yetobtained. Nonetheless, we can expect that having the LiealgebrasuN@z,z21# will allow us to approximate any ele-ment of the infinite-dimensional groupSUN@z,z21#.

The proof includes two theorems. Theorem 1 identifies alimited set of matrices which can be used to generate anyelement fromSUN@z,z21#. Theorem 2 shows how we canmake~infinitesimal! steps towards any of the matrices fromthis set, identifying the required sequences of the free evolu-tion and phase kicks. Thus, relying on the Theorem 1, Theo-rem 2 proves that we can generate the Lie algebrasuN@z,z21#.

Theorem 1: The groupSUN@z,z21# is generated by thefollowing N subgroups, each isomorphic toSU2 :

9929J. Chem. Phys., Vol. 120, No. 21, 1 June 2004 Controllability of wavepackets

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S a b

c d

1

.

.

1

D ,S 1

a b

c d

1

.

1

D ,...,S 1

.

.

1

a b

c d

D ,

S a bz21

1

.

.

1

cz d

D , ~12!

with (ca

db)PSU2 .

The proof of this theorem is given in the Appendix.Let us return to the HamiltoniansH0 ,H1 ,H2 . We see that the infinite matrixg5exp(iH0Trev /N) is N-periodic, and can be

identified with an element ofSUN@z,z21#. The matricesiH 1 ,iH 2 are alsoN-periodic and belong to the Lie algebrasuN@z,z21#.

Theorem 2: Let N be an odd integer. The operators$gm( iH 1)g2m,gm( iH 2)g2m% where m51,...,N generate the LiealgebrasuN@z,z21#.

Proof: In Theorem 1 we showed that the Lie groupSUN@z,z21# is generated by the set of matrices Eq.~12!. This impliesthat the corresponding Lie algebrasuN@z,z21# is generated by their 2N infinitesimal generators

S 0 i

i 0

0

.

.

0

D ,S 0 1

21 0

0

.

.

0

D ,S 0

0 i

i 0

0

.

0

D ,

S 0

0 1

21 0

.

.

0

D ,...,S 0

0

.

.

0 i

i 0

D ,S 0

0

.

.

0 1

21 0

D ,

S 0 iz21

0

.

.

0

iz 0

D ,S 0 z21

0

.

.

0

2z 0

D . ~13!

9930 J. Chem. Phys., Vol. 120, No. 21, 1 June 2004 Shapiro, Ivanov, and Billig

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Below we show that the infinite matrices associated withthese particular generators fromsuN@z,z21# can be writtenas the linear combinations

(m51

N

amgm~ iH 1!g2m

1 (m51

N

bmgm~ iH 2!g2m, with am ,bmPR. ~14!

The matricesgm( iH 1,2)g2m are all tridiagonal, with ze-

roes on the main diagonal, and are determined by theirupper-triangular parts, because of the conditionAT52A.The upper-triangular parts ofgm( iH 1)g2m,gm( iH 2)g2m are

i

2 (j PZ

zm~22k11!Ej , j 11

and

21

2 (j PZ

zm~22 j 11!Ej , j 11 .

HereEj , j 11 is a matrix that has the entry$ j , j 11% equal to 1,and all other entries are equal to 0, andz5exp(2pi/N) is acomplexNth root of unity.

We use the identity that holds for oddN:

(m51

N

zm~2k21!zm~22 j 11!5 HN,k5 j mod N,0, otherwise ~15!

~a sum of all theN different powers a complex root of 1,z (2k21)z (22 j 11), is equal to zero unless this root itself isequal to one!.

Thus fork51,...,N, the expressions

2

N (m51

N

2Rezm~2k21!~gm~ iH 2!g2m!

1Im zm~2k21!~gm~ iH 1!g2m! ~16!

will indeed give all the required generators. This completesthe proof of the theorem.

In the language of laser pulses, Equations~14! and ~16!imply that for every possiblem we wait until the fractionalrevival g2m, then apply the laser pulse of the horizontal orvertical polarization—H1 or H2—and then applygm bywaiting until the full revival. We can as well applyH1 andH2 simultaneously, so that Eq.~14! looks like

(m51

N

gm~ iamH11 ibmH2!g2m, am ,bmPR. ~17!

These correspond to applying linearly polarized pulses, withthe angle of polarization defined byam /bm . To generateeach of theN matrices in Eq.~13!, we can useN differentpulses applied atN fractional revivals, with the polarizationsdefined in Eq.~16!. SinceN is odd, we would useN differentorientations of the field polarization in the plane. EvenNwould lead to onlyN/2 different polarizations and make themathematical proof more complicated. In particular, phasekicks at multiples ofTrev/2N would be needed. However,even before any proof, in practice for anyN52K one would

be satisfied with controllability forN52K21 andN52K11. Of course, in practice one also does not have to use suchcomplex pulse sequences; we only rely on them here for thesake of the mathematical proof.

V. CONCLUSIONS AND DISCUSSION

The main result of this paper is that wavepacketsarecoarse-grained controllable by free evolution and phase kicksgiven even by smooth coordinate-dependent interactionssuch as nonresonant Stark shifts. The two polarizations of thelaser field, creatingH1;cosf andH2;sinf, suffice to ap-proximate, for any oddN, any evolution matrix of the type~11!, and, as a subset, any evolution matrix of the type~9!,which corresponds to controllability with the accuracy up to1/Nth of the orbit.

This statement about coarse-grained controllability doesnot answer the question about the final shape ofN basiswavepackets. Is it possible to avoid the~undesirable! changein the shape of the basis wavepackets and the uncontrollablespreading of the initial wapepacket distributionC(M ), whilestill being able to create arbitrary superposition of the wave-packet replicas such as given by Eq.~7!?

An intuitive prescription has been described in Sec. II:one would like to use the wavepackets which are as narrowas possible and/or phase shifts with profiles as sharp as pos-sible.

It turns out that there is also the possibility of a generalsolution of this problem. Within the groupSUN@z,z21# ofthe evolution matrices~11! that we are able to generate, theblocksAi , iÞ0, are responsible for changing the shapes ofthe basis wavepackets by mixing different blocks in the inputstate vector, whileA0 provides the desired control withoutchanging the shapes of the basis wavepackets. It is the lasttwo operators in the set of the generators~13! that lead to thegeneration of nondiagonal blocks in the evolution matrix. Ifone uses all the possible control generators in Eqs.~16! and~17! except for those withk5N21,N, then only the finitegroup SUN , represented by the block-diagonal evolutionmatrices~9!, will be generated. There is also another poten-tial possibility: if a control sequence has taken us to an un-desired dimension, it may be possible to find a proper se-quence of the type~14!, ~16!, and~17!, which will bring usback to the block-diagonal evolution matrix of the type~9!.

These geometric considerations do not give any usefulpractical prescription for the control sequence. Long se-quences of the type~14!, ~16!, and~17!, with ‘‘infinitesimal’’laser pulses, are not feasible. The theory only proves thepossibility of noiseless control. To find a practical solution,one could use a numerical or an experimental optimizationalgorithm.3

The above analysis can be extended to wavepackets inweakly nonlinear 1D systems. Consider evolution of a wave-packetC(t)5(nCn(t)Cn centered aroundn0 in a systemwith quadratic spectrum,E(n)5E(n0)1v0(n2n0)1V(n2n0)2. In the semiclassical approximation this wavepacketcan be viewed as a carrierCn0

modulated by the envelopef (f,t)5(nCn(t)eif(n2n0),29 where the anglef is the phaseof the corresponding classical motion. Linear motion of the

9931J. Chem. Phys., Vol. 120, No. 21, 1 June 2004 Controllability of wavepackets

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wavepacket is given byv0 , while spreading and revivals ofthe envelope are defined byV: Trev52p/V.20 To describethe distribution on the corresponding classical orbit, one cantrack the shape of the envelope shifted by linear motion,f (f,t)5(nCn(0)exp@2iV0t(n2n0)

2#exp@if(n2n0)#, f5f

2v0t. Evolution of f (f) is identical to that of the wave-function of a 2D rotor above.

If the spreading of the wavepacket is much slower thanits linear motion, then one can utilize the existence of twovastly different time scales similar to using two differentpolarizations in the above analysis. Specifically, the samecoordinate-dependent Stark shift can be applied at those mo-ments of time when the original wavepacket is atf50 andwhen it is atf5p. This technique was used in Ref. 24 tomodel quantum logic gates in a vibrational molecular wave-packet.

Finally, let us comment on the connection of the controlregime studied here with the field of quantum chaos. A 2Drotor, kicked periodically with the same kind of the kick—H1 or H2 of the fixed amplitude—is the most fundamentalmodel system studied in that field.30 It is well known that ifthe kick period is not commensurate with the wavepacketrevival time, then the quantum kicked rotor follows the be-havior of its classical counterpart until saturation of the dif-fusion due to dynamic localization.30,31However, if the kicksare applied at fractional revivals, then the behavior of thequantum kicked rotor is drastically different. In particular,average energy of the system grows quadratically rather thanlinearly with time.

On the one hand, the main difference between the gen-eral analysis developed here and the case of a resonantlykicked rotor is that here the kicks are in general not periodicand not identical—the strengths, polarizations, and time-delay between the kicks can change.

On the other hand, to the best of our knowledge thetheory of a resonantly kicked rotor32 has only been devel-oped for kicks applied at full revivals. The role of interfer-ences at fractional revivals, the width of the resonances andtheir role—all very important issues for the control of quan-tum chaotic systems33—are not known. We hope that thegeometric analysis introduced here, i.e., the knowledge ofwhat kind of pulse sequences are responsible for the quickclimbing along the momentum ladder@Eq. ~16!#, could beuseful in addressing these problems.

ACKNOWLEDGMENTS

We acknowledge fruitful discussions with A. Stolow, P.Bucksbaum, C. R. Stroud, Jr., H. Rabitz and V. Ramakrishnaand financial support from NSERC.

APPENDIX: PROOF OF THEOREM 1

Let us outline the proof of Theorem 1. First of all, usingstandard algebraic manipulations, one can show that thegroup generated by theseSU2 subgroups also contains thesubgroups

S .

1

a bz2k

.

czk d

1

.

D , ~A1!

as well as the diagonal matrices diag(zs1,...,zsN) with s11¯

1sN50.Indeed, if in the firstN21 matrices in~12! we seta

5d50, b52c51, then we will be able to create an arbi-trary permutation of basis vectors. Using permutations andthe first matrix in~12! with a, b, c, dnot fixed, we can mixany two vectors, i.e., create any matrix of the form~A1! withk50. Adding the last matrix from the set~12!, we can gen-erate any matrix of the form~A1!.

Now we can prove the theorem by induction onN. Tomake the inductive step, we need to show that multiplying agiven matrix inSUN@z,z21# by matrices of the form~1!, wecan make the first column to be

~1,0,...,0!T. ~A2!

Since the matrix is unitary, this will force the first row to be~1, 0,..., 0!, and the problem is reduced to matrices of sizeN21. Again we shall use induction, now on the width of thecolumn.

Dealing with a given single column, we shall sequen-tially use matrices of the form~A1! to simplify it to the form~A2!.

Let us, first, define width of a polynomial f (z)5( j 5s

n ajzj as w( f )5n2s11 ~here we assumeanÞ0, as

Þ0). The widthw(C) of a columnC5( f 1(z),...,f N(z))T isdefined as the maximal width of its entriesf i(z).

If in the column we have two different lines with poly-nomials of the maximal width, then we shall be able to mixthem together so that the width in at least one of them will bereduced. The claim below guarantees that there could not bea situation when there is only one line with polynomial ofmaximal width.

Claim: If the width of a columnC of a matrix inSUN@z,z21# is greater than 1, then the column contains atleast two entries of the maximal width.

First, note that for a polynomialf (z)5( j 5sn ajz

j , thehighest power ofz in the expression

f ~z! f ~z!5S (j 5s

n

ajz2 j D S (

j 5s

n

ajzj D

is n2s5w( f )21. Note thatCTC51, which holds since thematrix is unitary. Hence, if the width ofC is greater than 1,the terms of the highest power ofz, w(C)21, in CTC

5( i 51N f i(z) f i(z) must cancel out. Thus there should be

more than one entry of the maximal width.Now it is easy to see that having two entries inC of

maximal width, we can apply an appropriate matrix of the

9932 J. Chem. Phys., Vol. 120, No. 21, 1 June 2004 Shapiro, Ivanov, and Billig

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form ~1! to reduce the number of entries of maximal width.Continuing by induction, we complete the proof of the theo-rem.

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