optimal control of molecular alignment in dissipative...

Optimal control of molecular alignment in dissipative mediaAdam Pelzer, S. Ramakrishna, and Tamar Seidemana�

Department of Chemistry, Northwestern University, Evanston, Illinois 60208

�Received 9 October 2006; accepted 17 November 2006; published online 16 January 2007�

We explore the controllability of nonadiabatic alignment in dissipative media, and the informationcontent of control experiments regarding the bath properties and the bath system interactions. Ourapproach is based on a solution of the quantum Liouville equation within the multilevel Blochformalism, assuming Markovian dynamics. We find that the time and energy characteristics of thelaser fields that produce desired alignment characteristics at a predetermined instant respond indistinct manners to decoherence and to population relaxation, and are sensitive to both time scales.In particular, the time-evolving spectral composition of the optimal pulse mirrors the time-evolvingrotational composition of the wave packet, and points to different mechanisms of rotationalexcitation in isolated systems, in systems subject to a decoherering bath, and in ones subject to apopulation relaxing bath. © 2007 American Institute of Physics. �DOI: 10.1063/1.2408423�

I. INTRODUCTION

Coherent control strategies1–3 have been applied duringthe past two decades to a large variety of problems in scienceand technology, ranging from population transfer in atomicsystems4 through branching ratios in chemical reactions5 todirection of current in semiconductor devices6 and guidanceof light through solid nanoconstructs.7 While most of thisresearch has focused on exploring the controllability ofquantum systems through theoretical, numerical, and experi-mental studies, recent work has recognized the value of co-herent control as a spectroscopic tool, to better understandmolecular systems. Underlying the concept of “coherentspectroscopy” is the anticipation that a spectroscopy that ex-ploits the phase properties of light would provide new in-sights into material properties, beyond what is available fromconventional spectroscopies, which utilize only the energyresolution of lasers. Particularly inviting is the possibility ofextracting information with regard to the phase properties ofmatter.

References 8–11 introduced the notion of coherencespectroscopy in the context of two-pathway excitation coher-ent control.12–15 Within this scheme, it was illustrated theo-retically and experimentally that the material phase � f �alsotermed channel phase or molecular phase in the previousliterature�, observed as a shift of the sinusoidal yield curve incontrol experiments, contains interesting information regard-ing molecular continua, which is not available from conven-tional �phase incoherent� spectroscopies.8–11 More recently,several publications have illustrated the possibility of utiliz-ing the outcome of optimal control experiments to unravelreaction pathways and mechanisms.16–19

Related to quantum control, as well as to its applicationas a coherence spectroscopy, is the problem of nonadiabaticmolecular alignment induced by short, intense laser pulses.20

Here a short pulse populates a broad rotational wave packet

through sequential one- or two-photon transitions in each ofwhich ��J�=1, 2 units of angular momentum are exchangedbetween the molecule and the field. The phase relationamong the rotational components guarantees that such rota-tionally broad wave packets are transiently aligned after thepulse turnoff. The early research on nonadiabatic, field-freealignment21 introduced a near-resonance scheme for rota-tional excitation, where the wave packet is populated viasequential �J= ±1, Rabi-type cycles between two electronicstates. Subsequent work has utilized mostly a nonresonantscheme, where rotational excitation takes place via sequen-tial two-photon Raman-type cycles within a single vibronicmanifold.22–25 More recently, the possibility of resonantly ex-citing rotationally broad wave packets at near rotational tran-sition frequencies was illustrated.26

Formally, the mode of excitation plays no role �it isshown elsewhere27 that the equations of motion correspond-ing to one case can be rigorously transformed into thosecorresponding to the other�. Rather, it is the relative timescales of the laser pulse and the rotational motions that de-termine the phase relation among the rotational components,and hence the alignment extent and time evolution. The es-sential physics underlying the last statement can be under-stood based on an analytical model of the post-pulse revivalstructure of rotational wave packets.28 In the impulse limit,corresponding to short, intense pulses, only the time integralof the field matter interaction matters and pulse shape effectsplay no role. In this situation, the alignment induced by theshort pulse can be enhanced by application of a subsequentpulse, synchronized with the revival pattern of the rotationalwave packet generated by the first.23,29–31 In the more generalcase, however, the spectral composition of the pulse plays acrucial role in the alignment dynamics, inviting applicationof optimal control theories to manipulate the alignmentdynamics.32 During the past few years, several theoreticalstudies have applied generic algorithms to control the align-ment �or orientation� of isolated molecules.33,34 This methodwas shown to yield useful control mechanisms that depend

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strongly on the nature of the target state, but was restricted tothe impulsive regime. It was further shown that differencesin the delay between sequential pulses at constant spectralcontent of the pulses can lead to significantly different out-comes.

The concept of nonadiabatic alignment was recently ex-tended from the isolated molecule limit to dissipative mediawithin a quantum density matrix approach.35,36 One of theintriguing findings of this study is the information content ofthe alignment observables regarding the dissipative proper-ties of the medium and the molecule-solvent interaction. Ref-erences 35 and 36 note also the strong dependence of thepopulation relaxation rate, hence the lifetime of alignment inmedia, on the rotational level content of the wave packet.These findings invite the application of quantum control bothas a coherence spectroscopy and as a means of establishingspecific alignment properties. In the first instance, the resultsof Ref. 35 suggest the possibility of applying the machineryof quantum control as a probe of the dissipative properties ofinteresting media and of system-bath interactions. In the sec-ond instance they suggest the possibility of constructinglong-lived alignment in dense media, as would be requiredfor applications such as solution chemistry and quantum in-formation processing.

In the present study we address both questions withinoptimal control theory. Our goal is thus twofold. On the onehand, we expect that the unique coherence properties of ro-tational wave packets would furnish further insight into thedecohering properties of the medium when more detailedobservables than the highly averaged expectation value con-sidered in Refs. 35 and 36 are explored. One such observ-able, which is particularly sensitive to coherence properties,is the time and energy spectrum of the optimal field thatdrives the system to a specific target state. On the other handwe recognize that in the presence of a dissipative environ-ment the establishment of alignment with specific propertieswould require more sophisticated laser pulses than could bedesigned based on intuition or analytical models, pulseswhose time-dependent spectral composition accounts for thesolvent, as well as for the solute, dynamics. Our findingssupport these anticipations but reveal also unexpected fea-tures, as outlined below.

In the next section we develop the theory and in Sec. IIIwe present and discuss our results. In order to place thepresent work in context with the published literature, and toprovide a reference for the following subsections, we beginin Sec. III A with a brief study of the isolated molecule case.In this domain, our results reduce in one limit the two-impulsive pulse scheme of Refs. 23, 29, and 30, providing anillustration of the advantage of the approach. In a secondlimit the results of Sec. III A reduce to the ladder climbingmechanism of Ref. 26, suggesting a domain in which thismechanism is efficient. In general, however, the pulses gen-erated by the optimization algorithm are more complex andrich in information content than those observed previously,as is expected, since coherence plays a dominant role innonadiabatic alignment. With the dissipation-free case serv-ing as a reference, we proceed in Sec. III B to explore theeffect of population relaxation, and in Sec. III C the effects

of a purely decohering bath. Finally, in Sec. III D, we discussthe role played by temperature. The concluding sectionbriefly outlines avenues for future research.

II. THEORY

A. System Hamiltonian and the density operator

The system Hamiltonian �Hs� consists of the molecularpart �Hmol� and its coupling to the external field �Hint�t�� suchthat

Hs = Hmol + Hint�t� . �1�

For a linear rigid molecule one can express the molecularHamiltonian in the basis of rigid rotor eigenstates as

Hmol = �JM

�J�JM��JM� , �2�

where �J are the energy eigenvalues, �J=BeJ�J+1� withinthe rigid rotor approximation, Be is the rotational constant atthe equilibrium configuration, �� , � �JM�=YJM�� , �� arespherical harmonics, J and M are the total angular momen-tum and its space-fixed z projection, and �� , �� are the polarand azimuthal Euler angles, respectively. The time-dependent interaction with a linearly polarized laser field thatleads to near-resonant excitation of rotational states is ex-pressed in terms of the rigid rotor eigenstates as

Hint�t� = − �0��t� �J, J�, M

�JM�cos ��J�M��JM��J�M� . �3�

In Eq. �3�, �� 0= Z�0 is the matrix element of the dipole op-erator in the electronic basis, assumed directed along thebody-fixed z axis, and ��t�= z��t� is the time-dependent elec-tric field of the laser, assumed linearly polarized along thespace-fixed z axis.

Within the Markov approximation, the reduced densityoperator ��t� evolves as

�

�t� = −

i

�Ls��t� − D��t� , �4�

where L and D represent the system Liouville and the dissi-pative superoperators, respectively. Thus

Ls��t� �Hmol + Hint�t�, ��t��−, �5�

and the dissipative superoperator includes all the effects thatarise due to coupling of the system Hamiltonian to a bath.The action of the dissipative superoperator on the reduceddensity operator is approximated within the multilevel Blochmodel as37

D��t� = − �J,M,J�,M�

1

2�KJMJ�M��JM��JM�, ��t��+

− KJMJ�M��J�M��J�M��JM��t��JM�

+ JMJ�M��pd� �JM��JM��t��J�M��J�M�� , �6�

where KJMJ�M� is the rate of population transfer from state�JM� to state �J�M�� and

JMJ�M��pd� is the pure decoherence38

034503-2 Pelzar, Ramakrishna, and Seideman J. Chem. Phys. 126, 034503 �2007�


rate between the states �JM� and �J�M��. Equation �6� ex-cludes terms that �i� transfer coherence from one pair ofstates to another pair, and �ii� transform population into co-herence and vice versa, both of which are accounted for inthe more general Redfield theory. These terms were shown tobe small for anharmonic systems,37 and are expected to benegligible for the highly anharmonic rotation spectra studiedhere.

The above formulation is applicable to both the gas celland the solution environment, but the mechanism of relax-ation and decoherence,38 and hence also the form andmethod of computation of the rates in Eq. �6�, differ qualita-tively in the former and the latter environments. In the nu-merical calculations of Secs. III B–III D, we focus on thecase of a gaseous medium, where the population relaxationand phase decoherence are dominated by binary collisions.Such two-body scattering events in the gas phase are typi-cally inelastic, involving both relaxation and decoherence,whereas the long range interactions dominating in solutionsare often dominated by elastic collisions that give rise topure decoherence. In most of the calculations below we con-sider the case of CO molecules interacting with an Ar-atombath, and apply the modified exponential gap �MEG� law39 tocompute the rotational relaxation rates.

B. Control fields for alignment optimization

Optimal control theory of molecular dynamics deter-mines the spectral composition of the laser pulse that willrealize a desired goal �e.g., optimize a specific observable� atthe end of the laser pulse, t= tf.

40 In the context of molecularalignment, the observable most often used to quantify thedegree of alignment and study its time evolution is the ex-pectation value of cos2 �, given, within a density matrix for-malism, by Tr�cos2 ��t� . Although this measure falls shortof providing a complete description of the alignment �sincehigher moments than the second play a role in intense laseralignment�, it is a common and transferable observable andhence applied also here. In order to maximize the expecta-tion value of interest at a time tf we seek the extremum of thefunctional A,

A = Tr�cos2 ��tf� −1

2�t0

tf

dt �t�E2�t� , �7�

where the second term on the right-hand side guarantees anupper limit on the field intensity in the time interval t0 to tf.The penalty function is taken to be time dependent to avoidsudden switch on and switch off of the control field.41 Theoptimal control field obtained from the functional of Eq. �7�is designed to ensure that the alignment, as characterized by�cos2 ��, will be maximized at the end of the control pulset= tf. Various other objectives can be envisioned, and can berealized within the same formulation by introducing differenttime dependencies for the penalty function. We have applieddifferent forms of �t� to realize alignment over a desiredwindow of time and to transfer the population to specificrotational states, for instance. For the purpose of the presentstudy, however, it is sufficient to focus on the objective of

maximizing the alignment at a given instance of time.The optimal control field E�t� thus obtained can be ex-

pressed in terms of the time-evolution superoperatorU�t , t0 ; E� as37

E�t� =i

�t��Tr�cos2 � U�tf, t; E�

��0 cos �, U�t, t0; E��t0��− , �8�

where the action of the time-evolution superoperator is givenexplicitly as

��t� = U�t, t0; E��t0� . �9�

Equation �8� provides a compact picture of the alignmentoptimization procedure. The density operator is first propa-gated from the initial time t0 to a time t, the commutation ofthe density operator with the dipole operator is next com-puted at time t, and the result is propagated until time tf bythe action of the second time-evolution superoperator. At thattime the result is acted on by the observable cos2 � and atrace is performed to obtain the control field E as a functionof time t. As the time-evolution superoperators, which deter-mine the control field, themselves depend on the electricfield, Eq. �8� calls for an iterative solution. A rapidly con-verging iterative scheme is proposed in Ref. 42, where anauxiliary density operator � that is propagated back in timeis introduced. In terms of the backward propagated densityoperator, the control field is recast compactly as

E�t� =i

�t��Tr��t; E��0 cos �, ��t��− , �10�

where ��t ; E� is defined as

��t; E� = U�t, tf ; E�cos2 � . �11�

Equation �11� can be shown to obey the equation of motion

�

�t��t; E� = −

i

�Ls��t; E� + D†��t; E� . �12�

By propagating separately the density operator ��t� forwardfrom the initial time, and the auxiliary operator ��t� back-ward from the final time, one obtains the control field for theintermediate time t, as envisaged in Eq. �10�. This separatepropagation results in an efficient iterative procedure.

III. RESULTS AND DISCUSSION

In this section we apply the method outlined in Sec. II tostudy the controllability of linear rotors subjected to a dissi-pative bath. In Sec. III A we briefly discuss the dissipation-free case, which serves as a reference in the subsequent sec-tions. Here the system parameters are expressed only throughthe strength of the field-matter interaction, which scales asthe projection of the dipole moment onto the field polariza-tion vector �0 ·��t� and the system time scale rot=� /Be,which is determined by the reduced mass. The results of thissection are thus entirely general, applicable to any rigid lin-ear rotor, and it is pertinent to use reduced, system indepen-dent parameters for the field and matter attributes. FollowingRefs. 20 and 35 we introduce dimensionless time, intensity,

034503-3 Molecular alignment in dissipative media J. Chem. Phys. 126, 034503 �2007�


and temperature variables as t= t / rot, �R�t�=�� 0 ·��t� /Be,and T=kT /Be, respectively. In Secs. III B–III D, we studyseparately three aspects of controllability of alignment in dis-sipative media. Section III B explores the effect of popula-tion relaxation, induced by rotationally inelastic collisions,Sec. III C explores the effects of pure decoherence,38 inducedby elastic collisions, and Sec. III D studies the role of tem-perature. For these purposes it is important to consider arealistic molecular problem �and relevant to use dimensionedunits� and we use the CO/Ar problem as a well-studied testcase.

The shape and the fluence of the optimal pulse in this, asin other optimal control studies, is determined to a largeextent by the form of the time-dependent penaltyfunction.41,43 In the present study, we use a flat-top functionwith variable plateau amplitude m. Whereas the precisefunctional form of the flat top function is immaterial �herewe use the form

�t� = � 1

L�2t − tf

tf�100

+ m

−1

L�−1

with L=108�, its general structure plays a significant role in adissipative environment, as discussed in Sec. III B, where therational for the flat-top form is explained.

In all calculations discussed here we have chosen thefield parameters such that the rotational excitation will notexceed Jmax�10.

A. Time scales

Molecular alignment via creation of a broad rotationalwave packet is determined by the coherences among rota-tional �JM� levels. The maximum value of an alignment peak

depends strongly on the number of angular momentum statesthat make up the density matrix and hence on its diagonalelements. Control over the timing of alignment, however,depends solely on the coherences among the rotational states,and therefore on the time evolution of the off-diagonal ele-ments.

We begin this section by exploring the time range withinwhich the alignment maximum can be timed with precision.Figures 1�a�–1�d� consider the dissipation-free case �zeropressure� at a temperature of T=9.4�10−2 �corresponding to10 K for a CO molecule�, and illustrate the averaged align-ment �cos2 �� for four values of the time tf at which thealignment is to be maximized: tf small with respect to �Fig.1�a��, comparable to �Fig. 1�b�� and large with respect to�Figs. 1�c� and 1�d�� the natural system time scale �here therotational period rot�. Figure 1�a� shows that for times shortwith respect to the rotational period, tf � rot /2, the rapidrotational excitation in the course of the short pulse leads toa maximum of �cos2 �� at the target time. Nonetheless, ahigher peak is attained at a later time; it is clear that therotational components did not have sufficient time to opti-mally rephase. The controllability is dramatically improvedfor tf approaching and exceeding rot, as seen in Figs.1�b�–1�d�.

The results of Fig. 1�a�–1�d� have been anticipated. Inorder to manipulate the wave packet to attain a specific phaserelation at a given time, the individual constituents of thewave packet must have enough time for their phases exp�−iHmolt /�� to differ significantly from unity. Because thespectrum of Hmol involves level spacings of order Be, pulsesshort with respect to rot are unable to control these interac-tions.

The ability of optimal control theory to precisely timethe occurrence of field-free alignment is completely general,

FIG. 1. Optimal control of alignmentdynamics in isolated molecules. Thetemperature is 9.4�10−2 and time isgiven in units of the rotational period�see the text for the dimensionlessunits used�. �a�–�d� Average alignmentas characterized by the expectationvalue of cos2 �; �e�–�h� optimal fieldsgenerating the alignment in �a�–�d�;�i�–�l� Fourier transforms of the fieldsin �e�–�h�. The dashed lines in �a�–�d�mark tf, the point where the pulse endsand maximum alignment is targeted.



applicable to molecules of any symmetry, and is of bothfundamental and practical interest, since with unshaped shortpulses the peak of the alignment is not known ahead of ex-periment or accurate calculation. The possibility of preciselytiming the �field-free� alignment by application of a slowlyturning on, rapidly switching-off pulse � on� rot , off� rot� has been illustrated in the theoretical study of Ref.44 and was recently realized in several laboratories.45 In thiscase, however, the nature of the alignment and its rate ofdephasing are not subject to control, but are determined bythe laser intensity alone. We remark also that in linear andsymmetric top molecules �where the rotational motion isclassically stable20�, the optimal alignment available at tf re-curs indefinitely at each multiple of the rotational revivaltime, as long as coherence is maintained.

We proceed by applying the optimal control scheme as acoherence spectroscopy, to unravel the alignment mecha-nism. To that end we study, in Figs. 1�e�–1�l�, the time evo-lution of the optimal electromagnetic fields �Figs. 1�e�–1�h��and the spectral compositions of these fields �Figs. 1�i�–1�l��.Figures 1�e� and 1�i� analyze the electric field that maximizesthe alignment at tf, where tf � rot /2, cf. Fig. 1�a�. In this�tf � rot� domain, it is difficult to map the features of thecontrol field onto the dynamics of the alignment observable,as the short time scale of the pulse allows only for populationtransfer, not manipulation of phases between J-states. Weremark �Fig. 1�e�� that the optimal field is of the half-cycleform, and hence produces orientation �not shown here�. Fortf values not remote from 2 rot, e.g., Figs. 1�c�, 1�g�, and1�k�, one observes a clear “double-kick” structure, where thepulse cleanly separates into two subpulses, the second timedto a revival feature of the wave packet created by the first. Itis evident that the scheme proposed in Refs. 29, 30, and 23,of imparting to a rotational wave packet a short pulse syn-chronized with its rotational revival structure, is efficient.The clear separation into a double peak observed in Fig. 1�g�is only rarely found in the time-resolved fields �where energyis undefined�. It generalizes, however, into a robust multi-peak pattern with rich information content in partially time-resolved Fourier spectra, as discussed below. To the best ofour knowledge, the emergence of a two-pulse mechanism outof a feedback control study of alignment �orientation� wasnot observed in previous studies, likely since the time dura-tion allowed was too short to that end.33,34 The multiple kickmechanism, in the generalized sense discussed in the follow-ing sections, is restricted, however, to observation times tf

small with respect to the dissipation time scale, where thebath effect is negligible.

These results incorporate previous results of optimalalignment studies as special cases. References 30 and 46–48use ultra-short pulses acting at points of global or localmaxima to guide the alignment toward a predetermined ki-nematics limit. For target times somewhat longer than rot,features in our optimal fields always correspond to pulses atglobal maxima in the alignment, but also contain much moreintense pulses at times corresponding to minima �antialign-ment�. For very long pulses �tf �20 rot�, the low intensityregime of our results is reminiscent of the results of Ref. 26where a “ladder climbing” mechanism was observed. Our

optimal fields are somewhat more complex, however, as ourmodel takes into account the phase relationships whereasRef. 26 deals only with level populations. For the purpose ofthe present work, this section serves as a reference for thedissipative case, which we proceed to address in the follow-ing sections.

B. Population relaxation

In this section we discuss the response of the optimalcontrol algorithm to population relaxation �J-changing, in-elastic collisions� using the CO/Ar problem as an example.To that end we set the pure decoherence38 rate,

JMJ�M��pd� in

Eq. �6� to zero, determining the population transfer ratesKJMJ�M� using the procedure summarized in Sec. II. As in theprevious section, it is the time scales of the processes in-volved that determine the form of the optimal pulses. In theisolated molecule limit, Figs. 1, the system time scales aloneare imprinted onto the time evolution of the optimal field. Inthe presence of a dissipative bath, both the diagonal and theoff-diagonal elements of the density matrix evolve subject tothe combination of the molecular Hamiltonian and the dissi-pative operator, following phase factors of the forme�it/��Hmol+i��, where � is the relaxation rate. It follows thatwhen the target time is short with respect to the dissipationtime scale tf��1, the results of the previous section remainapplicable and the system time scales alone determine theoptimal field. In the general case, however, the field probesboth the system and the bath properties.

In order to explore the response of the field to the bath,we compare, in Fig. 2, the alignment dynamics generated bya field that was optimized so as to maximize �cos2 ��tf� inthe absence of dissipation, with the alignment dynamics gen-erated by a field that was optimized in the presence of popu-lation relaxation. Figure 2�a� considers the case of slow re-laxation compared to the observation time tf ��−1. Here thefield calculated in the absence of relaxation achieves thesame value of alignment as the field that was optimized inthe presence of the fully interacting bath. In Figs. 2�b� and2�c�, however, we see that the fields optimized to account forthe bath give larger values of alignment at tf. The phaseinformation imprinted in the ensemble by the field calculatedwithout relaxation is degraded before the end of the obser-vation time, and hence the pulse produces poor results.

Figure 3 quantifies and generalizes these observations.The averaged alignment achieved at tf at finite pressures isnormalized against the alignment at zero pressure and plottedversus the pressure. Comparison of Figs. 3�a�–3�c� with Fig.3�d� illustrates that the information on the bath time scales isnicely encoded into the averaged alignment. In each of thethree cases examined—Figs. 3�a�–3�c�—the results of thetwo pulses—the one optimized in the absence of dissipationand the one optimized in its presence—start deviating at apressure where the relaxation rate equals the observationtime �tf �1. In Fig. 3�a� this occurs around P�300 Torr, inFig. 3�b� around P�175 Torr, and in Fig. 3�c� around P�100 Torr. We note in passing �see Fig. 3�d�� that the relax-ation is faster the lower the temperature. This, at first glance,counter intuitive effect, is consistent with our expectation



and with the results of Refs. 35 and 36. As the rotationaltemperature increases, the wave packet is centered about ahigher J level and the level spacing between pairs of rota-tional constituents increases �the rotational energy levelspacing scales as 2BeJ in the rigid rotor limit�. Consequently,the rate of J-changing collisions �hence the population relax-ation rate� decreases sharply with temperature. Other conse-quences are discussed in Sec. III D.

The short-time Fourier transforms of the fields respon-sible for the alignment traces of Fig. 3�b� are shown in Fig.4. In the dissipation-free �isolated molecule� limit, Fig. 4�a�,the optimal pulse consists of a five subpulse sequence, inwhich the subpulses gradually shift to higher energy pulsecenter as the rotational excitation progresses and the rota-tional level spacing increases. In the presence of dissipationthe multiple kick scheme with increasing frequency becomesinefficient, as the phase relation established by the early sub-pulses is quickly disrupted by inelastic collisions with the

bath atoms. As the pressure, and hence the relaxation rate,increase, the subpulses migrate to later times, sufficientlyclose to tf to avoid relaxation before that time, thereby skip-ping the low energy “prepulse” at early time42 �Fig. 4�b��. Inthe limit of fast relaxation, the rotational excitation mecha-nism switches to a late, single intensive pulse that is suffi-ciently broad in energy to span the range of rotational levelspacing that our penalty function allows. It is interesting tonote the analogy of this mechanism to the mechanism dis-cussed in Ref. 49, where optimal control theory was appliedto control a photoinduced electron transfer reaction. Like-wise related is the study of population transfer in a three-level system in Ref. 42.

The foregoing discussion illustrates the role played bythe temporal form of the penalty function, �t� in Eq. �7�, ina dissipative medium. Previous research into the use of feed-back control to optimize the alignment in isolated molecules,has either ignored the discontinuous turn on or turn off of the

FIG. 2. Population relaxation effects: alignment of COin Ar bath for tf =40 ps and T=5 K. The dashed curvesillustrate the averaged alignment generated by pulsesthat were optimized in the absence of dissipation andused to align CO molecules in the population relaxingAr bath. The solid curves show the alignment generatedby pulses optimized in the presence of relaxation. �a� 50Torr; �b� 200 Torr; and �c� 400 Torr. Essentially identi-cal results are generated by the two pulses at low pres-sures �where the dissipation time scale is short com-pared to tf�, but at higher pressures the pulses in eachpair are qualitatively different in spectral content andlead to different alignment.

FIG. 3. Population relaxation effects: normalized aver-aged alignment of CO/Ar vs pressure. Solid curves withsuperimposed circles provide the alignment generatedby fields that were optimized in the presence of relax-ation and dashed curves with superimposed squares cor-respond to fields that were generated in the absence ofrelaxation. �a� tf =23 ps, T=5 K; �b� tf =40 ps, T=5 K;�c� tf =40 ps, T=3 K. Panel �d� shows the relaxationrate constant of CO in Ar at 5 K �solid curve� and at 3K �dashed curve� vs pressure, and is provided to assistin the interpretation of �a�–�c�.



pulse, focusing on the physics of the solution, or has used apenalty function of the sin2 shape to assure that the fieldvanishes at the end points of the time interval. This choicehas proven successful in previous calculations and we con-firmed that it does not sacrifice the efficiency of the controlalgorithm in the case of isolated molecules. Since, however,it forces the pulse to peak at the center of the time interval, itunnecessarily constraints the pulse to act toward the centerrather than closer to the end of the time interval, leading tofailure of the algorithm to optimize the target state in thepresence of significant dissipation. The “flat top” penaltyfunction used here allows freedom to allocate the availablefluence at any point in the pulse, thus giving a useful result.

C. Pure decoherence

The previous section illustrates the effect of populationrelaxation, arising from inelastic collisions with the bath at-oms. Here we complement this discussion by investigatingthe effects of elastic collisions. These disrupt only the phaseinformation and hence lead to decay of the off-diagonal ma-trix elements of the density matrix but do not exchange en-ergy between the system and the bath, leaving the popula-tions intact.

In order to isolate the effects of pure decoherence38 fromthose of population relaxation, calculations were performedwith zero population relaxation at various values of the puredecoherence rates in Eq. �6�. To compare the effects of popu-lation relaxation and phase decoherence of similar timescales, we contrast the alignment calculated for 40 ps deco-herence rate with that calculated at 150 Torr, where the decaytime is about 40 ps. Figure 5 compares the averaged align-ment obtained in the presence of a purely decohering bathwith that obtained in a bath that induces population relax-ation. Our goal of maximizing the alignment at tf =40 ps isclearly attained in both media. We note, however, the distinctdifference in the mechanism by which the peak alignment isattained. In the case of a population relaxing bath �solidcurve in Fig. 5�, the rotational excitation is rapidly built justbefore the final time, while the phase relation among the

rotational components is optimized to establish the peak atthe target time tf =40 ps. The baseline of �cos2 �� thereforeremains close to the isotropic value of 1/3 until just before tf.In the case of the purely decohering bath �dashed curve inFig. 5�, the algorithm points to a more efficient mechanism,where rotational population is built early in the pulse and thephase relation among the rotational components is estab-lished subsequently. As a consequence, the baseline of�cos2 �� reaches its final value of about 0.45 already at themidpoint of the pulse. We note also that subsequent to t= tf

the baseline of �cos2 �� gradually decays �on a 40 ps timescale� in the relaxing bath, whereas in the purely decoheringbath it remains constant.

The alignment strategy adapted to best handle puredecoherence38 is further clarified in Fig. 6, which illustratesthe short-time Fourier transform of the pulse optimized toestablish alignment at tf =40 ps for different pure decoher-ence rates. One clear distinction from the analogs analysis ofthe relaxing bath in Fig. 5 is that the multiple kick schemefavored in the isolated molecule limit is not replaced by a

FIG. 4. �Color� Population relaxation effects: short-time Fourier transform of optimal tf =40 ps pulses at 5K and �a� 0 Torr; �b� 100 Torr; and �c� 200 Torr.

FIG. 5. Pure decoherence effects: averaged alignment for tf =40 ps. Thedashed curve is computed assuming a purely decohering bath, whereas thesolid curve �corresponding to CO in a 150 Torr Ar bath�, accounts only forrotationally inelastic collisions. The dissipation time scale is 40 ps in bothcases.



single intense pulse just before tf, although its structureclearly mirrors the changing properties of the bath as thedecoherence rate grows. In particular, the alignment strategychanges from a five-pulse to a four-pulse and finally to athree-pulse sequence as the decoherence time scale increasesfrom �−1 � tf through �−1� tf to �−1� tf /2. Also, as all Jlevels decohere at the same rate, the isolated molecule strat-egy of increasing the frequency with each pulse remains ef-ficient. Finally, we note that the main peak is timed toroughly the center of the pulse, where much of the popula-tion excitation is established, as in the isolated molecule caseand in contrast to the relaxing bath case.

D. Role of temperature

The previous sections studied the effect of different dis-sipative mechanisms on the controllability of nonadiabaticalignment and examined the alignment and the optimal fieldsas probes of the dissipative properties of the bath. In thissection we complement these studies by examining the effectof temperature on the alignment dynamics and on its control-lability when subject to dissipation.

The effect of rotational temperature in the isolated mol-ecule, molecular beam limit �where, by contrast to the densemedium case considered here, the different modes are notequilibrated� is well understood.20,25 The revival features be-come sharper due to averaging over an increasingly largerensemble of initial conditions as the temperature increases.At the same time, the alignment decreases in magnitude, dueto the thermal population of progressively higher M sub-strates. In the limit where the rotational energy kBT is largerthan the field-matter interaction, the alignment is lost. Thesegeneral effects are observed also here, as illustrated in Fig.7�a�. Although the major characteristics of alignment andantialignment features are quite similar at all temperatures,the peak values are much reduced as temperature increases.The controllability of the alignment time, however, is notaffected by temperature.

One new feature introduced in the dissipative environ-ment, which does not have an analog in the isolated moleculecase, has been briefly commented on in Sec. III B and is seen

also in Figs. 7 and 8, namely, with increasing temperature thecenter of the wave packet shifts to higher J and hence therelaxation rate decreases. Due to the increasing stability ofthe system toward population relaxation with increasing tem-perature, we find in Fig. 7�a� that the second revival peak at15 K surpasses the first revival peak at 10 K. Likewise, thefirst and second revival peaks at 30 K are essentially equal inmagnitude to the primary alignment peak at tf. As pointedout in our earlier study, similar resistance to inelastic colli-sion effects could be more easily attained by coherent meansof shifting the center of the wave packet to high J states.

Figure 7�b�, providing the Fourier transforms of the op-timal fields responsible for the alignment in Fig. 8�a�, illus-trates the adjustment of the optimal pulse to the increasingrotational population. With increasing temperature, higherfrequency components are introduced so as to handle theincreasing rotational level spacing in the initial ensemble. Atthe same time the superfluous low frequency field compo-nents are suppressed.

The short-time Fourier transforms shown in Fig. 8 pro-vide a complementary view of the diminishing effects ofdissipation with increasing temperature. The trend seen when

FIG. 6. �Color� Pure decoherence effects: short-timeFourier transform of optimal tf =40 ps pulses with no

population relaxation for �a� �pd�=0; �b� �pd�−1=40 ps;

and �c� �pd�−1=20 ps.

FIG. 7. Temperature effects: �a� averaged alignment for tf =23 ps at 200Torr. �b� Fourier transform of the optimal fields that generate the spectra in�a�. The temperature is 10 K�¯� , 15 K�- - -�, and 30 K �solid curve�.



comparing 10 K ensembles at various pressures is reversedhere for constant �200 Torr� pressure: the pulse moves to actearlier and earlier as the temperature is raised, because thelarger population in the high J levels becomes increasinglyimmune to the effects of population relaxation. At suffi-ciently high thermal rotational energy the regular multipulsescheme is recovered.

IV. CONCLUSIONS

Our goals in the research summarized in the previoussections have been to explore the controllability of nonadia-batic alignment in dissipative media, and to use optimal con-trol theory as a coherence spectroscopy of the dissipativebath and the bath system interactions. Our numerical ap-proach has been based on solution of the quantum Liouvilleequation within the multilevel Bloch formalism, assumingMarkovian dynamics. We applied a low frequency near-resonance field to establish alignment, but our findings areindependent of the mode of inducing rotational excitationand apply equally to alignment by nonresonant �IR� pulses orby near electronic resonance pulses.

On the one hand, our results illustrate that optimal con-trol theory is able to control the alignment dynamics in adissipative medium. On the other hand they point to a quali-tative difference in the mechanism through which the algo-rithm optimizes the rotational dynamics in the presence andin the absence of dissipation. As anticipated, the alignmentfeatures can be timed with precision only over time rangescomparable to �or long with respect to� the rotational period,and comparable to �or short with respect to� the dissipationtime scales. Our numerical results quantify this general state-ment for the example of the alignment dynamics of CO mol-ecules in an Ar bath.

More interesting, and potentially more useful, than thecontrollability question, is the information content of the op-timal fields generated by feedback control algorithms. In thecontext of the alignment problem, we find that the observ-able fields contain a wealth of new information about thealignment dynamics and the system-bath interaction, whichis available from neither optimal control studies in isolated

molecules nor studies of alignment in dense media with un-optimized pulses. In particular, we find that the time andenergy characteristics of the laser fields that maximize thealignment at a given instance respond in distinct manners todecoherence and to population relaxation, and are sensitiveto both time scales.

Rotational wave packets provide a fascinating probe ofthe coherence and dissipative properties of dense environ-ments, which expresses itself in the observables of alignmentexperiments. Our previous studies35,36 pointed out this prop-erty and explained its origin, but focused solely on the infor-mation content of �cos2 �� in �unoptimized� alignment ex-periments. The present work illustrates that other observablesthat track different facets of rotational wave packets provideadditional insights. We hope that an experimental study ofthe questions discussed here will become possible in the nearfuture.

ACKNOWLEDGMENT

The authors are grateful to the Department of Energy,Grant No. DAAD19–03-R0017, for support of their researchon the alignment problem.

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