optimal control simulation of field-free molecular orientation: alignment-enhanced molecular...

Optimal Control Simulation of Field-Free Molecular Orientation:Alignment-Enhanced Molecular OrientationKatsuhiro Nakajima, Hiroya Abe, and Yukiyoshi Ohtsuki*

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

ABSTRACT: Nonresonant optimal control simulation is applied to a COmolecule to design two-color phase-locked laser pulses (800 nm + 400 nm) withthe aim of orienting the molecule under the field-free condition. The optimalpulse consists of two subpulses: the first subpulse aligns the molecule and thesecond one orients it. The molecular alignment induced by the first subpulseconsiderably enhances the degree of orientation, the value of which is close to anideal value at temperature T = 0 K. To confirm the effectiveness of thisalignment-enhanced orientation mechanism, we adopt a set of model Gaussianpulses and calculate the maximum degrees of orientation as a function of thedelay time and the intensity. In finite-temperature (T = 3.0 K and T = 5.0 K)cases, although the alignment subpulse can improve the degree of orientation,the control achievement decreases with temperature rapidly; this decrease canbe attributed to the initial-state-dependent (phase-shifted) rotational wavepacket motion.

I. INTRODUCTIONThe control of molecular alignment/orientation has attractedconsiderable interest for the past few decades as it has enabledmolecular processes to be induced in a molecular-fixed frame.When a molecule is in its ground electronic state, a torquerequired for its alignment/orientation control is generatedthrough dipole and/or induced-dipole interactions.1,2 To avoidpossible side effects due to the presence of an electric field,molecular alignment/orientation under the field-free conditionis favorable. As regards alignment control, impulsive Ramanexcitation of rotational states with a nonresonant laser pulse isoften used for this purpose,3−16 in which an aligned stateappears regularly (rotational revival) after the pulse.17 Because asingle pulse excitation has some inherent limitations thathinders the improvement of the degree of alignment,7,14 variouseffective multipulse schemes have been reported. In addition toaligning a molecule in a specific direction (1D alignment), therehave been several studies on 3D alignment by using laser pulseswith several polarization conditions and by a set of linearlypolarized pulses.18−22 Today, alignment control studies haveshifted their focus from the elucidation of basic mechanisms tooptimization as well as the wide range of applications.23−26

Examples of the latter include molecular orbital tomogra-phy24,25 and time-resolved imaging of a chemical reaction.26

For asymmetric molecules, molecular alignment controlwould be insufficient to realize molecular-fixed-frame experi-ments because the aligned molecules are regarded as a mixtureof molecules with opposite head-versus-tail orders.27−41 Theprimary difference between alignment control and orientationcontrol is that the latter requires parity-violating interactions.From the viewpoint of parity-violating interactions, orientationcontrol can be classified into two categories. In the firstcategory, a dipole moment that leads to the lowest-order parity-

violating interaction is utilized.29−36 Examples include theoreti-cal proposals that use (half-cycle) terahertz pulses, some ofwhich combine terahertz excitation with nonresonant laserpulses.29−32 Although the use of the terahertz pulses showspromise, there still remain technical problems in the generationof intense terahertz pulses. Since the early study by Friedrichand Herschbach,33 the combination of static fields with theexcitation by nonresonant laser pulses is known to achievereasonably high degrees of orientation.14,34−36 The control inthe presence of a static field, however, does not result incompletely field-free molecular orientation.In the second category, an intense two-color phase-locked

laser pulse, which typically consists of 800 and 400 nmcomponents, is used.37−41 The 400 nm component is usuallygenerated by the sum frequency generation of the 800 nmcomponent. The parity-violating interaction is introduced bythe hyperpolarizability. Some experimental studies demon-strated the effectiveness of this scheme by employing simplemolecules, such as CO40 and OCS.41 However, the observeddegrees of orientation are not so high. This limited success ismainly attributed to weak hyperpolarizability interactions,which also explains the requisite intense laser pulses used inthe experiments. In light of advanced pulse shaping techniquesin the 800 nm region, we may expect to achieve a reasonablyhigh degree of orientation with a shaped laser pulse with mildlyhigh intensity. From this viewpoint, Lapert et al.42 numericallydesigned two-color laser pulses on the basis of their optimal

Special Issue: Jorn Manz Festschrift

Received: May 29, 2012Revised: July 26, 2012Published: July 27, 2012

Article

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© 2012 American Chemical Society 11219 dx.doi.org/10.1021/jp3052054 | J. Phys. Chem. A 2012, 116, 11219−11227

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control method. In one of their simulations, they found anonintuitive solution, i.e., the parity-violating (asymmetric)interaction followed by a parity-conserving (symmetric)interaction.The present study focuses on the second category, i.e.,

control with two-color phase-locked laser pulses. Through acase study of a CO molecule, we explore an effective way torealize a high degree of orientation. To this end, we apply ourown optimal control algorithm to the design of an orientationpulse.43−45 As it is known that there exist multiple solutions inoptimal pulse design,46,47 our algorithm can find an alternativesolution with effectiveness similar to that reported by Lapert etal.42 In fact, we will propose an alignment-enhanced molecularorientation mechanism that will be derived from our optimalcontrol simulations. In section II, we briefly summarize theprocedure of our optimal control simulation. The algorithmicdetails are given in the Appendix. In section III, numericalresults are presented.

II. OPTIMAL CONTROL SIMULATIONWe consider a CO molecule modeled by a rigid rotor, which isspecified by a rotational constant, B, and an angular momentumoperator, J. The molecule is assumed to interact with a linearlypolarized laser pulse, E(t), through a permanent dipolemoment, μ, polarizability components, α∥ and α⊥, andhyperpolarizability components, β∥ and β⊥, where indices ∥and ⊥ mean the components parallel and perpendicular to themolecular axis, respectively. Let θ be the angle between thepolarization vector of the laser pulse and the molecular axis.Then, the total Hamiltonian is expressed as

μ θ α θ

β θ

= − −

−

H BJ E t E t

E t

( ) cos12

( )[ ( )]

16

( )[ ( )]

t 2 2

3(1)

where

α θ α α θ α= − +⊥ ⊥( ) ( ) cos2(2)

and

β θ β β θ β θ= − +⊥ ⊥( ) ( 3 ) cos 3 cos3(3)

The first term of the right-hand-side of eq 1 is the rigid-rotorHamiltonian, the eigenstates of which are the sphericalharmonic wave functions denoted by {|J M⟩}. The magneticquantum number, M, is conserved during the optical transitionsbecause the laser field is linearly polarized.In the present study, we consider a two-color laser field

ε ω ε ω η= + +ω ωE t t t t t( ) ( ) cos( ) ( ) cos(2 )2 (4)

that contains the two frequency components specified by ω and2ω with the relative phase, η. In eq 4, εω(t) and ε2ω(t) are theenvelope functions. If we further assume that ω and 2ω aremuch higher than those associated with the rotationaltransitions, we can take the cycle average over the opticalfrequencies of the laser field. The cycle-averaged Hamiltonian isexpressed as

α θ ε ε

β θ ε ε η

= − +

−

ω ω

ω ω

H BJ t t

t t

14

( )[ ( ) ( )]

18

( ) ( ) ( ) cos

t 2 22

2

22 (5)

The phase, η, determines the degree of asymmetry of theoptical interactions. For example, η = π/2 provides noasymmetry, which results in no control of orientation.Throughout the present study, we use a fixed value of η = 0.The dynamical behavior of the molecular system is described

by the time-evolution operator, U(t,0), that obeys the equationof motion:

ℏ ∂∂

= t

U t H U ti ( ,0) ( ,0)t(6)

When the rotor is initially in the lowest state, |ψ0⟩ = |J = 0 M =0⟩, the wave function at time, t, is given by

ψ ψ| ⟩ = | ⟩t U t( ) ( ,0) 0 (7)

In the optimal control simulation, we first choose a targetoperator, W, to specify a physical objective. The optimal pulseis designed so that it gives the maximum expectation value at aspecified time, tf:

ψ ψ= ⟨ | | ⟩F t W t( ) ( )f f (8)

For the sake of molecular orientation, the natural choice wouldbe

∑ ∑θ= = | ⟩⟨ |= =−

W P P P JM JMcos withJ

J

M J

J

0

max

(9)

where the projector, P, is introduced to restrict the number ofrotational states to be optically controlled. This is becausecontrolling the highly excited rotational states would not beexperimentally realistic or feasible. By applying calculus ofvariations under the constraint of eq 7,

δδε

δδε

= =ω ω

Ft

Ft( ) ( )

02 (9)

we obtain the optimality conditions:

ξ α θ ε β θ ε ε ψ⟨ | + | ⟩ =ω ω ω⎧⎨⎩

⎡⎣⎢

⎤⎦⎥

⎫⎬⎭t t t t tIm ( )12

( ) ( )14

( ) ( ) ( ) ( ) 02

(10)

and

ξ α θ ε β θ ε ψ⟨ | + | ⟩ =ω ω⎧⎨⎩

⎡⎣⎢

⎤⎦⎥

⎫⎬⎭t t t tIm ( )12

( ) ( )18

( ) ( ) ( ) 022

(11)

where the Lagrange multiplier that represents the constraint isexpressed as

ξ ξ ψ| ⟩ = | ⟩ = | ⟩t U t t t U t t W t( ) ( , ) ( ) ( , ) ( )f f f f (12)

Equations 7 and 10−12 are composed of the pulse designequations. We solve the nonlinear equations by adopting theiteration algorithms that guarantee monotonic convergence(Appendix).In the numerical calculations, we adopt the rotational

constant, B = 1.92 cm−1,48 corresponding to the rotationalperiod, Trot = 1/(2cB) = 8.68 ps, with c being the velocity oflight. The values of the polarizability components are takenfrom ref 49 so that α∥ = 15.63 au, α⊥ = 11.97 au, β∥ = 30.0 au,and β⊥ = 8.4 au. All the wave functions and their associatedLagrange multipliers are expanded in terms of the rotationalstates {|J M⟩} up to J = 17. The time evolution of the expansioncoefficients is calculated by the fifth-order Runge−Kuttamethod, in which the temporal grid is set to 2Trot × 10−6.

The Journal of Physical Chemistry A Article

dx.doi.org/10.1021/jp3052054 | J. Phys. Chem. A 2012, 116, 11219−1122711220

Throughout the present study, the degree of orientation andthat of alignment are evaluated by the expectation values of cosθ and cos2 θ, respectively.

III. RESULTS AND DISCUSSIONA. Optimal Control Simulation in the Zero-Temper-

ature Case. In the zero-temperature case, the initial state isgiven by |ψ0⟩ = |J = 0 M = 0⟩. As the magnetic quantumnumber is conserved, the wave function is expressed in terms ofthe set of eigenstates, {|J M = 0⟩}. We choose Jmax = 6 to specifythe control target in eq 9. We then diagonalize the operator, cosθ, by using the set of rotational states, {|J M = 0⟩; J = 0, ..., Jmax= 6}, to obtain the eigenstate, |χ⟩, with the maximumeigenvalue, 0.949. The optimal pulse is designed to maximizethe expectation value, F = ⟨ψ(tf)|W|ψ(tf)⟩, with W = |χ⟩⟨χ| at aspecified final time tf = Trot. Figure 1 shows the convergence

behavior as a function of the number of iteration steps whenthe search parameters (Appendix) are constant, λ1(t) = λ2(t) =2.5 × 1038. The initial trial field is chosen as

ε ε εσ

π ζ ζ

ζ ζ

π ζ ζ

= = −−

×

≤ ≤

≤ ≤ −

− − ≤ ≤

ω ω

⎡⎣⎢

⎤⎦⎥

⎧⎨⎪⎪

⎩⎪⎪

t tt t

t t

t t

t t t t t

( ) ( ) exp( /2)

2

sin( /2 ) (0 )

1 ( )

sin[ ( )/2 ] ( )

(0)2(0)

0f

2

2

f

f f f (13)

with ε0 = 4.0 GV/m, σ = 1.5 ps, and ζ = 0.15 Trot. It clearlyshows the monotonic convergence of the present algorithm aswell as its numerical accuracy. In fact, the converged value of F= 0.978 is quite high (the ideal value Fideal = 1.00), which leadsto the degree of orientation, ⟨cos θ⟩(tf) = ⟨ψ(tf)|cos θ|ψ(tf)⟩ =0.930 (the ideal value is 0.949).Figure 2 summarizes the results of the optimal control

simulation. The optimal pulse in Figure 2a is composed of twointense subpulses together with several weak subpulses. We seefrom Figure 2b that the optimal pulse creates a highly orientedwave packet at the specified final time. If we remove the weaksubpulses, e.g., after t ≃ 0.8 Trot in Figure 2a, the degree oforientation slightly decreases in value from 0.930 to 0.922. Aswe have confirmed that the weak subpulses contribute little to

the orientation, we consider only the two major subpulses toexamine the control mechanisms below. The role of the firstsubpulse is to create an aligned wave packet. As shown inFigure 2c, the average value of the rotational quantum numberrapidly increases although no excitation of the rotational stateswith odd quantum numbers (i.e., odd-J states) can be seen. Thesecond subpulse excites the odd-J states that are essential torealize the orientation. This result suggests a “two-step” controlmechanism in which the control is achieved by a combinationof an “alignment” pulse and an “orientation” pulse. Accordingto this simplified picture, first, an alignment pulse creates analigned wave packet that equally contains the componentsoriented along one direction and those along the oppositedirection. When an orientation pulse is applied to this wavepacket, one of the components would be stabilized whereas theother components would become energetically destabilized andstart moving toward the opposite direction. These twocomponents would meet and cause constructive interferencesto realize a high degree of orientation. In this sense, the two-step mechanism could be referred to as an alignment-enhancedorientation and, thus, we will use this phrase in the following. Ifwe take a closer look at the structure of the second subpulse, wefind that it is characterized by a dip, the timing of whichcorresponds to the maximum expectation value of ⟨J⟩(t)(Figure 2c); that is, the first half of the second subpulse (i.e.,before the dip) excites the rotational states and the second half(i.e., after the dip) induces stimulated emission to excite theodd-J states. Through the excitation and/or redistribution ofthe population, the population in the even-J states and that inthe odd-J states are adjusted to have almost the same value(Figure 2c).If the above-mentioned two-step mechanism could achieve a

high degree of orientation, the set of optimal subpulses wouldbe mimicked by a set of Gaussian pulses, in which the

Figure 1. Convergence behavior as a function of the number ofiteration steps is shown by (a) the values of the kth objectivefunctional, F(k) = | ⟨χ|ψ(k)(tf)⟩|

2 (see text) and (b) the values of thedifference in objective functional between adjacent iteration steps,δF(k) = F(k) − F(k−1).

Figure 2. (a) Envelope functions, εω(t) (red) and ε2ω(t) (blue), of theoptimal orientation pulse (see eq 4) as a function of time when therelative phase η = 0. Time is measured in units of the rotational period,Trot. (b) Degree of orientation, ⟨cos θ⟩(t) = ⟨ψ(t)|cos θ|ψ(t)⟩, and thatof orientation, ⟨cos2 θ⟩(t) = ⟨ψ(t)|cos2 θ|ψ(t)⟩, are presented by thered-solid and blue-dashed lines, respectively. (c) The population in theeven-J states and that in the odd-J states are shown by black dashedand solid lines, respectively. The purple line shows the time-dependentaverage quantum number, the values of which are provided on theright ordinate. The two vertical dotted lines serve as a guide for theeyes.



alignment (first) and orientation (second) subpulses areapproximated by

εσ

ω= −⎡⎣⎢

⎤⎦⎥E t

tt( ) exp

2cos( )a

aalignment

2

2(14)

and

ε τσ

ω τ

ε τσ

ω τ

= − − −

+ − − −

ω

ω

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

E tt

t

tt

( ) exp( )

2cos[ ( )]

exp( )

2cos[2 ( )]

orientation

2

o2

2

2

o2

(15)

respectively. Here, we employ the fixed values of εa = 13.8 GV/m, ε2ω = (11.3/14.0)εω, σa = 110 fs, and σo = 240 fs, which areestimated from the optimal pulse in Figure 2a. The peakintensity of the alignment pulse (eq 14) is 2.5 × 1013 W/cm2,which is about one-fifth of that used in the orientationexperiment.40 The wavelength associated with the opticalfrequency, ω, is set to 800 nm. Note that the followingnumerical results do not depend on the choice of thefrequencies as long as they are much higher than those of therotational transitions. By using these Gaussian alignment andorientation pulses, we calculate the maximum degrees oforientation, ⟨cos θ⟩max, as a function of parameters τ and εω. Toensure that we find the maximum value for a given set ofGaussian pulses, we numerically integrate the equation ofmotion at least for one rotational period after the second pulsebecause the rotational wave packet of a rotor is known to showthe same degree of orientation at every rotational period, Trot(revival).Figure 3 shows the contour plot of the maximum degrees of

orientation, ⟨cos θ⟩max, as a function of the delay time, τ, andthe amplitude, εω, in which the position corresponding to theoptimal solution is marked by a cross. In Figure 3, we seeseveral regions with high values that appear around εω = 14.0GV/m and periodically along the delay time. The highest valueof 0.888 in Figure 3, which is located close to the cross,

confirms the effectiveness of the alignment-enhanced orienta-tion mechanism. For reference, in the absence of the alignmentpulse, the orientation pulse leads to the maximum degree oforientation, 0.417. We also note that for negative values of thedelay time, τ, i.e., when the orientation pulse appears before thealignment pulse, we could not find the regions with highdegrees of orientation (not shown). As this is in contrast withthe mechanism proposed by Lapert et al.,42 we can say that wehave found a different optimal solution from theirs.To see the details, we show the cuts along the dotted lines

(a) and (b) in Figures 4 and 5, respectively. Figure 4 shows the

maximum degrees of orientation as a function of the amplitude,εω, for a fixed value of τ = 0.49Trot(4.25 ps), in which thepopulation in the even-J states (dotted line) and that in theodd-J states (solid line) are also presented. All the lines inFigure 4 show oscillating behavior as a function of εω. Theiroscillation periods become shorter as εω increases because alarger value of εω leads to a larger Rabi frequency. Figure 4suggests that a larger amplitude of the orientation pulse doesnot necessarily lead to a higher degree of orientation. Theamplitudes associated with the high degrees of orientation arean indication that the even-numbered-state population is equalto the odd-numbered-state population. This could be under-stood by the fact that the interference between the even-J andodd-J states are essential in the orientation control. For morequantitative discussion, we expand the wave function in eq 7 in

Figure 3. Contour plot of the maximum degrees of orientation, ⟨cosθ⟩max, as a function of the delay time, τ, and the amplitude, εω, of theset of Gaussian pulses in eqs 14 and 15. For each set of Gaussianpulses, we calculate the time evolution at least for one rotationalperiod after the second pulse to ensure that we find ⟨cos θ⟩max (alsosee text). Time is measured in units of Trot. The two dotted linesorthogonal to each other are introduced for the discussion in Figures 4and 5.

Figure 4. Red line: maximum degrees of orientation as a function ofthe amplitude, εω, with a fixed value of the delay time, τ = 0.49 Trot(4.25 ps), which corresponds to the cut along the dotted line (a) inFigure 3. The population in the even-J states and that in the odd-Jstates are represented by black dotted and solid lines, respectively.

Figure 5. Red line: maximum degrees of orientation as a function ofthe delay time (measured in units of Trot) with a fixed value of theamplitude, εω = 14.0 GV/m, which corresponds to the cut along thedotted line (b) in Figure 3. Blue dashed line: degree of alignment atthe timing of the peak of each orientation pulse.



terms of the rotational states {|J M = 0⟩} and calculate theexpectation value of cos θ, i.e., the degree of orientation

∑ψ θ ψ

θ

⟨ | | ⟩

= * ⟨ + = | | = ⟩ +=

+

t t

C J M J M C

( ) cos ( )

{ 1 0 cos 0 cc}J

J J0

1

(16)

where CJ ≡ ⟨J M = 0|ψ(t)⟩. The conditions for {CJ} to give themaximum degree of orientation are derived by differentialcalculus subject to the normalization condition. Because all theequations involved in this maximization problem are symmetricwith respect to the even-J and odd-J states, a high degree oforientation is associated with the equal population of the even-Jand odd-J states. It explains the reason why the maximumdegree of orientation semiquantitatively coincides with theequal population as shown in Figure 4.Figure 5 shows the maximum degrees of orientation as a

function of the delay time, τ, for a fixed value of εω = 14.0 GV/m. For reference, we also plot the degree of alignment at thetiming of the peak of each orientation pulse. In general, thehigh degrees of orientation well coincide with the high degreesof alignment; that is, when the highly aligned rotational wavepacket is excited by the orientation pulse, a high degree oforientation tends to appear. It is worth noting that at τ ≃0.07Trot, τ ≃ 0.35Trot, and τ ≃ 0.63Trot, the high degrees oforientation approximately coincide with the timing thatcorresponds to the decrease of the degrees of alignment withlarge velocity (in the sense of classical mechanics). Thissuggests that there may be two kinds of alignment-enhancedorientation mechanisms.So far, we have assumed a fixed ratio of the amplitudes, ε2ω =

(11.3/14.0)εω, in the orientation (two-color) pulse. In Figure 6,

we show the maximum degree of orientation as a function ofthe ratio, εω/ε2ω, in which the total energy of the orientationpulse is set to the same value as that used in Figure 3. As for thedelay time, we assume a fixed value of τ = 0.49 Trot. In Figure 6,we also show the population in the even-J states (black dashedline), that in the odd-J states (black solid line), and the productof the amplitudes associated with the hyperpolarizabilityinteraction, εω

2ε2ω (thin red dashed line), the maximum valueof which is adjusted to 1.0 for illustrative purposes. Within theparameter values considered here, the highest degree oforientation is achieved when the ratio is adjusted for εω

2ε2ωto have the maximum value and the even-J and odd-J states are

(approximately) equally populated; that is, the εω:ε2ω = 1:1mixture of the frequency components is not an optimalorientation pulse. At the same time, we could say that thedegree of orientation is not so sensitive to the ratio comparedto the other parameters. such as amplitude (Figure 4) and delaytime (Figure 5).In the above discussion, the pulse parameters are chosen so

that they are consistent with the optimal pulse in Figure 2a. Tofurther illustrate the effectiveness of the alignment-enhancedorientation, we calculate the degree of orientation with anotherset of pulse parameters. In this example, the Gaussian pulses arecharacterized by narrower temporal widths and slightly higherintensities, i.e., the temporal widths and amplitudes are set to σa= σo = 50 fs, εa = 15.0 GV/m, and εω = ε2ω = 20.0 GV/m,respectively (see eqs 14 and 15). Figure 7 shows the maximum

degrees of orientation as a function of the delay time and thedegrees of alignment at the timing when the orientation pulse isapplied. Similarly to the results in Figure 5, we see a goodcoincidence of the two degrees of control, which can confirmthe effectiveness of the combination of the alignment andorientation pulses. In Figure 7, the alignment pulse creates asuperposition state that mainly consists of J = 0 and J = 2 states,resulting in a simpler behavior of the degree of alignment thanthat in Figure 5. Actually, in Figure 5, the aligned wave packet iscomposed of the rotational states with quantum numbers of upto ca. J = 4.

B. Alignment-Enhanced Molecular Orientation inFinite-Temperature Cases. In this subsection, we extendthe alignment-enhanced orientation scheme to finite-temper-ature cases. To evaluate the control performance, we firstestimate the “ideal” highest degree of orientation that would berealized by a specified maximum rotational state, Jmax, at aspecified temperature. For this purpose, we first solve theeigenvalue problem for each magnetic quantum number, M,

∑

θ χ χ

χ χ

| ⟩ = Θ | ⟩

| ⟩ = | ⟩⟨ | ⟩=

J M J M

cos withnM

nM

nM

nM

J

J

nM

( ) ( ) ( )

( )

0

( )max

(17)

with the eigenvalues, Θ0(M) ≥ Θ1

(M) ≥ Θ2(M) ≥ .... If we take into

account the restriction due to entropy, the ideal value attemperature, T, is calculated by

Figure 6. Red solid line: maximum degrees of orientation as a functionof the amplitude ratio, εω/ε2ω. The delay time is set to τ = 0.49 Trot(4.25 ps). Black solid and dashed lines: population in the even-J statesand that in the odd-J states, respectively. The normalized value ofεω

2ε2ω is represented by the red dotted line.

Figure 7. Red line: maximum degrees of orientation as a function ofthe delay time (measured in units of Trot) with fixed values of theamplitudes (see the text). Blue dashed line: degree of alignment at thetiming of the peak of each orientation pulse.



∑ ∑θ = Θ= =−

−| |p Tcos ( )J

J

JM J

J

J MM

ideal0

( )max

(18)

where pJ(T) is the probability that the rotor is thermally excitedto the Jth state.50 Figure 8 shows the estimation. We consider

the temperature at which the quantum numbers of thermallyexcited states are well below the specified Jmax. In a low-temperature region, e.g., T ≤ 5 K, the degree of orientationwould have a value larger than 0.7 even in the Jmax = 4 case. Attemperature T ∼ 30 K, it can have a value of ∼0.7 if we fullycontrol the rotational states with quantum numbers of up toJmax = 10; that is, Figure 8 may suggest that a reasonably highdegree of orientation would be realized even at finitetemperature.To see whether this “optimistic” scenario works or not, we

numerically solve the quantum Liouville equation with the setof Gaussian pulses given in eqs 14 and 15. At temperature, T,the density operator is expressed as

∑ρ ρ ψ ψ= = | ⟩ ⟨ |†t U t U t t p T t( ) ( ,0) ( ,0) ( ) ( ) ( )JM

JM J JM0

(19)

with |ψJ M(t)⟩ = U(t,0)|J M⟩, where the time evolution operatoris defined by eq 6.In Figure 9, the maximum degrees of orientation as a

function of the delay time, τ, and the amplitude, εω, are shownas contour plots at (a) 3.0 K and (b) 5.0 K. The definitions ofthese properties are the same as those in Figure 3. At 3.0 K (5.0K), the initial population in the lowest state, |J = 0 M = 0⟩, isreduced to 0.669 (0.458) due to the multiplicities associatedwith the magnetic quantum numbers. Similarly to Figure 3, theregions with high values tend to appear around εω = 14.0 GV/m and periodically along the delay time. The highest values are0.626 and 0.516 in the (a) T = 3.0 K and (b) T = 5.0 K cases,respectively. These highest values are located in the regionsassociated with the optimal conditions (see the cross in Figure3). Within the temperature ranges considered here, the resultsin Figure 3 (T = 0 K) and Figure 9 (finite temperature) shareseveral common features. However, we can see severaldifferences as well. One of the discrepancies is the differencein the number of regions with high values. The number ofregions as well as their peak values decrease rapidly astemperature increases.To see the decrease in greater detail, we show the cuts along

the fixed amplitude, εω = 14.0 GV/m, i.e., the maximumdegrees of orientation as a function of τ in Figure 10 (a) T =

Figure 8. Ideal value of the degree of orientation as a function oftemperature for the given maximum rotational state (see eqs 17 and18).

Figure 9. Contour plots of the maximum degrees of orientation, ⟨cosθ⟩max (see the captions of Figure 3), as a function of the delay time, τ,and the amplitude, εω, of the set of Gaussian pulses in eqs 14 and 15 at(a) T = 3 K and (b) T = 5 K. Time is measured in units of Trot. Thedotted line in each figure is introduced for the discussion in Figure 10.

Figure 10. (a) Cut along the dotted line in Figure 9a and (b) that inFigure 9b. In each figure, the red line shows the maximum degrees oforientation as a function of the delay time (measured in units of Trot).The blue dashed line shows the degree of alignment at the timing ofthe peak of each orientation pulse. For reference, the degrees oforientation in the absence of the alignment pulse are indicated by theblack thin-solid lines, (a) 0.194 and (b) 0.122.



3.0 K and (b) T = 5.0 K. The degrees of alignment at thetiming of the orientation-pulse peaks are also plotted. Forconvenience, the degrees of orientation achieved without thealignment pulse are shown by the horizontal thin solid lines,i.e., 0.194 and 0.122 in Figure 10a,b, respectively. We see thatthe alignment pulse considerably improves the degree oforientation in both cases. Roughly speaking, high values areobtained when the orientation pulse is applied to the highlyaligned wave packet; that is, an increase in the degree ofalignment leads to an improvement of the degree oforientation, although this is not always the case. The mostprominent exceptional cases appear at the delay time, τ ∼ 0.5Trot, in which the alignment pulse makes virtually nocontribution to the improvement of the degree of orientation(Figure 10a) and even makes worse it (Figure 10b).The time evolution of the degree of orientation associated

with the latter case (T = 5 K and τ ∼ 0.5 Trot) is shown inFigure 11. The bold solid (dotted) line shows the component,

the initial condition of which is the J = 0 (J = 1) state. If wefocus on each component, we find that the alignment pulseimproves the degree of orientation. However, the relative phaseof the two components is opposite, which results in thecancellation of the degree of orientation between them andleads to a poor degree of orientation as a whole. This“cancellation” behavior is consistent with a recent numericalstudy51 in which the authors pointed out that the directions ofthe induced orientation of a molecule with small hyper-polarizability become opposite depending on the odd or evenquantum number of the initial rotational state. The initial phaseof the wave packet will be explained by the fact that the wavepacket keeps the isotropic distribution just after the pulsedexcitation because the rotational motion cannot respondquickly.

IV. SUMMARYWe have designed a two-color optimal pulse to orient a COmolecule at T = 0 K. The optimal pulse consists of twosubpulses. The first subpulse aligns the molecule and thesecond one is applied to the aligned wave packet to control theorientation. This “two-step” control mechanism, i.e., alignment-enhanced molecular orientation, has been confirmed by

numerical simulation with a set of Gaussian pulses. Ournumerical findings in the T = 0 K case are as follows: (1) Highdegrees of orientation appear periodically as a function of thedelay time, which coincide with the timings of the high degreeof alignment. (2) A more intense orientation pulse does notnecessarily lead to a higher degree of orientation in the presentcontrol scheme. (The optimal intensity results in almost thesame population in the even-J rotational states and the odd-Jstates.) (3) The degree of orientation is not sensitive to theamplitude ratio of the 800 nm component to the 400 nmcomponent in the orientation pulse. In the finite-temperature(T = 3.0 K and T = 5.0 K) cases, although the alignment pulsecan improve control achievement, the degree of orientationdecreases with temperature rapidly. The primary reason for thisdecrease is attributed to the cancellation due to oppositeorientation directions that depend on the initial states.

■ APPENDIX: SUMMARY OF MONOTONICALLYCONVERGENT ALGORITHM

Coupled pulse-design equations are solved with the iterativeprocedure developed in our previous study.43,45 The envelopefunctions of the laser field, εω(t) and ε2ω(t), are artificiallydivided into two components so that εω (1)(t) and εω (2)(t), andε2ω (1)(t) ε2ω (2)(t), respectively. Then, all the equations arerewritten as a symmetrical sum of products of these artificiallydivided components. We start the iteration with the initial trialfield given in eq 13. Because of the flexibility of the algorithm,each iteration step can be represented in several forms. Here,we show one of the examples we actually used in the presentstudy. Each iteration step consists of two auxiliary steps, inwhich the first (second) auxiliary step calculates the electricfield associated with εω(t) [ε2ω(t)]. The algorithm at the kthiteration step is summarized as follows.Ausiliary step 1:

ξ

α θ ε ε ε ε

β θ ε εε ε

ξ

ℏ ∂∂

| ⟩

= − +

−+

| ⟩

ω ω ω ω

ω ωω ω

− − −

−− − ⎪

⎪

⎫⎬⎭

{t

t

BJ t t t t

t tt t

t

i ( )

14

( )[ ( ) ( ) ( ) ( )]

18

( ) ( ) ( )( ) ( )

2( )

k

k k k k

k kk k

k

( )

2(1)

( )(2)

( 1)2 (1)( 1)

2 (2)( 1)

(1)( )

(2)( 1) 2 (1)

( 1)2 (2)( 1)

( )

(A1)

with the final condition |ξ (k)(tf)⟩ = W|ψ(k−1)(tf)⟩, and

ψ

α θ ε ε ε ε

β θ ε εε ε

ψ

ℏ ∂∂

| ⟩

= − +

−+

| ⟩

ω ω ω ω

ω ωω ω

− −

− − ⎪⎪

⎫⎬⎭

{t

t

BJ t t t t

t tt t

t

i ( )

14

( )[ ( ) ( ) ( ) ( )]

18

( ) ( ) ( )( ) ( )

2( )

k

k k k k

k kk k

k

( )

2(1)

( )(2)

( )2 (1)( 1)

2 (2)( 1)

(1)( )

(2)( ) 2 (1)

( 1)2 (2)( 1)

( )

(A2)

with the initial condition |ψ(k)(0)⟩ = |ψ0⟩. In eqs A1 and A2, thesymmetrically divided electric fields are expressed as

Figure 11. Outside the box, the envelope functions of the 800 and 400nm pulse components are illustrated by the red and blue lines,respectively. The first pulse (second) pulse is the alignment(orientation) pulse. Inside the box, the red line shows the timeevolution of the degree of orientation at T = 5.0 K when the amplitudeand the delay time are set to εω = 14.0 GV/m and τ = 0.5 Trot,respectively. The contribution, the initial state of which is the J = 0 (J =1) state, is represented by the black solid (dashed) line. Time ismeasured in units of Trot.



ε ε

λ ξ α θ ε

β θ εε ε

ψ

−

= − ⟨ |

++

| ⟩

ω ω

ω

ωω ω

−

−

−− −

−

⎪

⎪

⎪

⎪

⎧⎨⎩

⎫⎬⎭

t t

t t t

tt t

t

( ) ( )

( ) ( ) 2 ( ) ( )

( ) ( )( ) ( )

2( )

k k

k k

kk k

k

(1)( )

(1)( 1)

1( )

(2)( 1)

(2)( 1) 2 (1)

( 1)2 (2)( 1)

( 1)

(A3)

and

ε ε

λ ξ α θ ε

β θ εε ε

ψ

−

= − ⟨ |

++

| ⟩

ω ω

ω

ωω ω

−

− −

⎪

⎪

⎪

⎪

⎧⎨⎩

⎫⎬⎭

t t

t t t

tt t

t

( ) ( )

( ) ( ) 2 ( ) ( )

( ) ( )( ) ( )

2( )

k k

k k

kk k

k

(2)( )

(2)( 1)

1( )

(1)( )

(1)( ) 2 (1)

( 1)2 (2)( 1)

( )

(A4)

where the positive function, λ1(t), called a convergenceparameter, characterizes the convergence behavior (speed,accuracy, etc.).Auxiliary step 2:

ξ

α θ ε ε ε ε

β θ ε εε ε

ξ

ℏ ∂∂

| ⟩

= − +

−+

| ⟩

ω ω ω ω

ω ωω ω

−

− ⎪⎪

⎫⎬⎭

{t

t

BJ t t t t

t tt t

t

i ( )

14

( )[ ( ) ( ) ( ) ( )]

18

( ) ( ) ( )( ) ( )

2( )

k

k k k k

k kk k

k

( )

2(1)

( )(2)

( )2 (1)( )

2 (2)( 1)

(1)( )

(2)( ) 2 (1)

( )2 (2)( 1)

( )

(A5)

with the final condition |ξ(k)(tf)⟩ = W|ψk)(tf)⟩, and

ψ

α θ ε ε ε ε

β θ ε εε ε

ψ

ℏ ∂∂

| ⟩

= − +

−+

| ⟩

ω ω ω ω

ω ωω ω ⎪

⎪

⎫⎬⎭

{t

t

BJ t t t t

t tt t

t

i ( )

14

( )[ ( ) ( ) ( ) ( )]

18

( ) ( ) ( )( ) ( )

2( )

k

k k k k

k kk k

k

( )

2(1)

( )(2)

( )2 (1)( )

2 (2)( )

(1)( )

(2)( ) 2 (1)

( )2 (2)( )

( )

(A6)

with the initial condition |ψ(k)(0)⟩ = |ψ0⟩. In eqs A5 and A6, thesymmetrically divided electric fields associated with the 2ω-component are expressed as

ε ε

λ ξ α θ ε

β θ ε ε ψ

−

= − ⟨ |

+ | ⟩

ω ω

ω

ω ω

−

−

t t

t t t

t t t

( ) ( )

( ) ( ) {4 ( ) ( )

( ) ( ) ( )} ( )

k k

k k

k k k

2 (1)( )

2 (1)( 1)

2( )

2 (2)( 1)

(1)( )

(2)( ) ( )

(A7)

and

ε ε

λ ξ α θ ε

β θ ε ε ψ

−

= − ⟨ |

+ | ⟩

ω ω

ω

ω ω

−t t

t t t

t t t

( ) ( )

( ) ( ) {4 ( ) ( )

( ) ( ) ( )} ( )

k k

k k

k k k

2 (2)( )

2 (2)( 1)

2( )

2 (1)( )

(1)( )

(2)( ) ( )

(A8)

where the positive function, λ2(t), is another convergenceparameter associated with the second auxiliary step.

To prove the monotonic convergence, we consider thedifference in the objective functional between the kth and (k −1)th iteration steps, F(k,k−1) = δf1

(k) + δf 2(k), where

δ ψ ψ ψ ψ= | | − | |− −f t W t t W t( ) ( ) ( ) ( )k k k k k1( ) ( )

f( )

f( 1)

f( 1)

f

(A9)

and

δ ψ ψ ψ ψ= | | − | | f t W t t W t( ) ( ) ( ) ( )k k k k k2( ) ( )

f( )

f( )

f( )

f

(A10)

For example, δf1(k) can be rewritten as

δ δψ δψ= | | +− −f t W t Re P t( ) ( ) 2 { ( )}k k k k1( ) ( , 1)

f( 1)

f 1 f

(A11)

where |δψ(k,k−1)(tf)⟩ = |ψ(k)(tf)⟩ − |ψ(k−1)(tf)⟩ and P1(t) =⟨ξk(t)|δψ

(k,k−1)(t)⟩. If we differentiate P1(t) with respect to time,we have

ξ α θ ε

β θ εε ε

ψ ε ε

ξ α θ ε

β θ εε ε

ψ ε ε

=ℏ

⟨ |

++

× | ⟩ −

+ℏ

⟨ |

++

× | ⟩ −

ω

ωω ω

ω ω

ω

ωω ω

ω ω

− −

−

−

−− −

− −

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎧⎨⎩

⎫⎬⎭

⎧⎨⎩

⎫⎬⎭

tP t t t

tt t

t t t

t t

tt t

t t t

dd

( )i

8( ) 2 ( ) ( )

( ) ( )( ) ( )

2

( ) [ ( ) ( )]

i8

( ) 2 ( ) ( )

( ) ( )( ) ( )

2

( ) [ ( ) ( )]

k

kk k

k k k

k

kk k

k k k

1(k)

(1)( )

(1)( ) 2 (1)

( 1)2 (2)( 1)

( )(2)

( )(2)

( 1) 2

(k)(2)

( 1)

(2)( 1) 2 (1)

( 1)2 (2)( 1)

( 1)(1)

( )(1)

( 1) 2(A12)

Substituting eqs A3 and A4 into eq A12 and then integratingboth sides of eq A12 over t ∈ [0, tf], we obtain

∫ λε ε

ε ε

=ℏ

−

+ −

ω ω

ω ω

−

−

P t tt

t t

t t

2Re{ ( )} d1

4 ( ){[ ( ) ( )]

[ ( ) ( )] }

tk k

k k

1 f0 1

(1)( )

(1)( 1) 2

(2)( )

(2)( 1) 2

f

(A13)

whereby δf1(k) ≥ 0 is proved. Similarly, we can show δf 2

(k) ≥ 0and therefore F(k,k−1) ≥ 0, which proves the monotonicconvergence of the iteration algorithm.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: [email protected].

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSWe thank Professor Jorn Manz for stimulating discussions inSendai and for helpful comments. We also acknowledge usefuldiscussions with Professor T. Nakajima. This work was partlysupported by a Grant-in-Aid for Scientific Research (C)(23550004) and also in part by the Joint Usage/ResearchProgram on Zero-Emission Energy Research, Institute ofAdvanced Energy, Kyoto University (B-22).



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