PHYSICAL REVIEW A 83, 023423 (2011)

Controlling the sense of molecular rotation: Classical versus quantum analysis

Yuri Khodorkovsky,1 Kenta Kitano,2,* Hirokazu Hasegawa,2,† Yasuhiro Ohshima,2 and Ilya Sh. Averbukh1

1Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel2Institute for Molecular Science and SOKENDAI (The Graduate University for Advanced Studies), Myodaiji, Okazaki 444-8585, Japan

(Received 6 December 2010; published 28 February 2011)

Recently, it was predicted theoretically and verified experimentally that a pair of delayed and cross-polarizedshort laser pulses can create molecular ensembles with a well-defined sense of rotation (clockwise orcounterclockwise). Here we provide a comparative study of the classical and quantum aspects of the underlyingmechanism for linear molecules and for symmetric tops, like benzene molecules, that were used for the firstexperimental demonstration of the effect. Very good quantitative agreement is found between the classicaldescription of the process and the rigorous quantum-mechanical analysis at the relevant experimental conditions.Both approaches predict the same optimal values for the delay between pulses and the angle between them, anddeliver the same magnitude of the induced oriented angular momentum of the molecular ensemble. As expected,quantum and classical analyses substantially deviate when the delay between pulses is comparable with the periodof quantum rotational revivals. However, time-averaged characteristics of the excited molecular ensemble areequally well described by these two approaches. This is illustrated by calculating the anisotropic time-averagedangular distribution of the double-pulse excited molecules, which reflects persistent confinement of the molecularaxes to the rotation plane defined by two polarization vectors of the pulses.

DOI: 10.1103/PhysRevA.83.023423 PACS number(s): 37.10.Vz, 33.80.−b, 42.65.Re

I. INTRODUCTION

Laser control of molecular rotation, alignment, and orien-tation has received significant attention in recent years (fora review, see, e.g., [1,2]). Interest in the field has increased,mainly due to the improved capabilities to manipulate char-acteristics of the laser pulses (such as time duration andtemporal shape), which in turn leads to potential applicationsoffered by controlling the angular distribution of molecules.Since the typical rotational motion is “slow” (∼10 ps) withrespect to the typical short pulse (∼50 fs), effective rotationalcontrol and manipulation are in reach. During the last decade,temporal rotational dynamics of pulse-excited molecules wasstudied [3,4], and multiple pulse sequences giving rise to theenhanced alignment were suggested [5–7] and realized [8–11].Further manipulations such as optical molecular centrifugeand alignment-dependent strong-field ionization of moleculeswere demonstrated [12,13]. Selective rotational excitation inbimolecular mixtures was suggested and demonstrated inthe mixtures of molecular isotopes [14] and molecular spinisomers [15,16].

Recently, several groups suggested a method for excitingfield-free unidirectional molecular rotation, in which theangular momentum along the laser beam propagation directionis provided by two properly delayed ultrashort laser pulsesthat are linearly polarized at 45◦ to each other [17–21]. Anexperimental demonstration of the effect was done in [21] forbenzene molecules. The mechanism behind this double-pulsescheme is rather clear, and is most easily explained in thecase of linear molecules [17–19]. The first ultrashort laserpulse, linearly polarized along the z axis, impulsively induces

*Present address: Institute for Solid State Physics, The Universityof Tokyo, 5-1-5 Kashiwanoha, Chiba 277-8581, Japan.†Present address: Department of Basic Science, The University of

Tokyo, 3-8-1 Komaba, Tokyo 153-8902, Japan.

coherent molecular rotation that continues after the end ofthe pulse. The molecules rotate under field-free conditionsuntil they reach an aligned state, in which the molecularaxis with the highest polarizability is confined in a narrowcone around the polarization direction of the first pulse. Thesecond short laser pulse is applied at the moment of the bestalignment, and at an angle with respect to the first pulse. Asa result, the aligned molecular ensemble experiences a torquecausing molecular rotation in the plane defined by the twopolarization vectors. The torque acting on a linear molecule ismaximal when the laser pulse is polarized at 45◦ with respect tothe molecular axis of the highest polarizability. This definesthe optimal angle between the laser pulses. The direction of theexcited rotation (clockwise or counterclockwise) is determinedby the sign of the relative angle (±45◦) between the first andthe second pulse in the polarization plane. This double-pulsescheme (see Fig. 1) was termed “molecular propeller” [17–19],as it resembles the action (side kick) needed to ignite a rotationof a plane propeller.

The heuristic arguments presented above are, in fact,essentially classical, although it is clear that a detailed quantumanalysis may be needed to consider the long-time evolutionof the excited molecules, or the operation of this scheme atrelatively weak pulses when only low-lying rotational statesare involved. That is why all the previous treatments of theproblem [17–21] presented full-scale quantum-mechanicalstudies, combining both analytical and numerical methods.

In the present paper, we provide a detailed comparisonbetween the classical and the quantum-mechanical analysis ofthe double-pulse scheme for exciting unidirectional molecularrotation, both for linear molecules and for benzene (that wasthe object of the experiment in [21]). We will demonstratethe predictive power of the classical treatment that correctlyprescribes the optimal delay between the pulses, and the anglebetween them, and will find the advantages and limitationsof such an approach in quantitative description of the effectat experimental conditions. We believe that by comparing the

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YURI KHODORKOVSKY et al. PHYSICAL REVIEW A 83, 023423 (2011)

complementary classical and fully quantum-mechanical viewson the same phenomenon, the reader will get a comprehensiveidea about the underlying physics.

A recent paper [19] studied quantum mechanically the“molecular propeller” scheme in the case of linear molecules.Dependence of the induced angular momentum on the pulses’intensity, and the delay between them, was analyzed. More-over, the angular distribution of linear molecules subject tounidirectional rotation was considered, and it was shown tobe confined in the plane defined by the two polarizationvectors of the pulses. This anisotropic angular distribution wascharacterized by the observable 〈cos2 ϕ〉, which was referredto as the azimuthal factor.

In the present paper, we describe the same schemeusing a classical description of the double-pulse molecu-lar excitation, followed by a Monte Carlo averaging overthe thermal ensemble. We compare the time dependence ofthe azimuthal factor according to the classical analysis, to thequantum-mechanical one, and show that both results coincidefor initial times. In addition, we introduce a long-time-averaged angular distribution function, which convenientlydepicts the molecular orientation following the excitation.This distribution function seems to behave similarly in theclassical and quantum cases. Finally, we analyze classically theexcitation of the unidirectional rotation for more complicatedoblate symmetric top molecules, like benzene, and comparethis analysis with the full quantum-mechanical treatment ofthe problem.

The outline of the paper is as following. In Sec. II we studyclassically the rotational motion of a linear molecule kickedby a pair of short nonresonant laser pulses that are linearlypolarized in arbitrary directions. In Sec. III, we describethe response of a thermal ensemble of linear moleculesto such an excitation by using the Monte Carlo method.Angular distribution function and its time-averaged behaviorare studied in Sec. IV, and the results are compared to thefull quantum-mechanical treatment. In Sec. V, we analyze(both classically and quantum mechanically) the problemof inducing unidirectional rotation of benzene molecules.Section VI summarizes and concludes our study.

II. CLASSICAL DYNAMICS OF A KICKEDLINEAR MOLECULE

Here we consider classical dynamics of a model diatomic(or, more generally, linear) polarizable molecule that interactswith an ultrashort laser pulse. The molecule is treated as alinear rigid rotor. We assume that the central frequency ofthe pulse is far detuned from any molecular transition. Thepotential energy of the pulse interaction with the inducedmolecular polarization is given by [22]

V (θ,ϕ,t) = − 14E2(t)(�α cos2 β + α⊥), (1)

where �α = α‖ − α⊥ is the difference between the polariz-ability along the molecular axis and the one perpendicularto it, E(t) is the envelope of the electric field of the linearlypolarized laser pulse, and β = β(θ,ϕ) is the angle between themolecular axis and the direction of polarization of the pulse.Here θ,ϕ are the polar and the azimuthal angles characterizingthe direction of the molecular axis, respectively. We assume

the pulse duration is very short compared to the rotationalperiod, so that the pulse can be described in the impulsive(δ-kick) approximation. We define the dimensionless interac-tion strength P , which characterizes the pulse, as

P = �α

4h

∫ ∞

−∞E2(t)dt. (2)

Let us remind again the pulse sequence producing uni-directionally rotating molecules [17–19]: (1) at time t = 0molecules are kicked by a pulse linearly polarized along the z

axis; (2) at the time of the maximal molecular alignment (t =tal, which is the moment when the alignment factor 〈cos2 θ〉reaches the maximum value), a second pulse is applied,polarized at 45◦ to the z axis (we choose the polarizationvector to be in the x-z plane). We represent the molecule bya unit vector pointing from the center of mass toward one ofthe atoms (the center of mass is defined as the coordinates’origin). The tip of the vector defines a point on the surface ofthe unit sphere. The orientation of the unit vector is given by

r = (x,y,z) = (sin θ cos ϕ, sin θ sin ϕ, cos θ ) . (3)

The rotational velocity of the unit vector r can be describedby two orthogonal angular velocities vθ = θ and vϕ = ϕ sin θ

(the dot denotes a derivative with respect to time), or, in vectornotation,

v = vθ �eθ + vϕ�eϕ, (4)

where the spherical unit vectors are

�eθ = (cos θ cos ϕ, cos θ sin ϕ,−sin θ ),(5)

�eϕ = (−sin ϕ, cos ϕ,0).

This rotational velocity is referred to as simply “velocity” inthe following. The rotational kinetic energy in this notation isgiven by

T = 12I

(v2

θ + v2ϕ

), (6)

where I is the moment of inertia of the molecule.As the first step, we find the change in the molecular velocity

due to interaction with a single short laser pulse. For a z-polarized pulse, angle β is equal to the polar angle θ , and wecan use the potential energy from Eq. (1) to find the torque �

applied to the molecule in spherical coordinates:

�θ = −∂V

∂θ= −�α

4E2(t) sin 2θ, �ϕ = −∂V

∂ϕ= 0. (7)

Newton’s second law gives �θ = I vθ = I θ and �ϕ = I vϕ =I d

dt(ϕ sin θ ). We assume that the molecule does not change its

orientation during the short pulse (impulsive approximation).By integrating the Newton’s equations of motion over the shortduration of the pulse, we obtain the change of the velocity dueto the laser pulse action:

�vθ = −h

IP sin 2θ0; �vϕ = 0. (8)

Integrating again, and assuming that the molecule was initiallyat rest, we get

θ (t) = θ0 − h

IP t sin 2θ0, ϕ = ϕ0. (9)

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CONTROLLING THE SENSE OF MOLECULAR ROTATION: . . . PHYSICAL REVIEW A 83, 023423 (2011)

FIG. 1. (Color online) “Molecular propeller” (figure taken fromRef. [19]).

Here θ0 and ϕ0 describe the initial orientation of the molecule.Every molecule starts to rotate with a constant velocity(depending on θ0 and ϕ0), and r is tracing a circle passingthrough the north and the south poles of the unit sphere.

As the next step, we consider the action of a pulse linearlypolarized along some arbitrary unit vector p, and determinethe vector of the resulting velocity change �v for a moleculeoriented along some direction r0. The norm |�v| can be foundsimilarly to the previous case of a z-polarized pulse, and itis equal to h

I|P sin 2β0|, where β0 is the angle between the

polarization direction of the pulse p and the initial orientationdirection of the molecule r0. The direction of �v can be definedfrom the vector diagram in Fig. 2. Notice that �v is alwaysperpendicular to r0. Furthermore, �v is directed parallel orantiparallel to the vector component of p perpendicular to r0,which is equal to p − cos β0r0. As a result, we arrive at

�v = 2hP

Icos β0 (p − cos β0r0) . (10)

We can easily check that (10) gives the correct result for z-polarized pulse, for which p = (0,0,1), by transforming (8) toCartesian coordinates using Eq. (4).

Finally, we provide a formula for the time dependence of theposition r and velocity v of the point representing molecularorientation on the unit sphere during the field-free rotation,given the initial orientation r0 and initial velocity v0. As v isalways perpendicular to r, the tip of the r vector traces a circle

∆v

r0

β0

p

FIG. 2. A linear molecule, represented by r0, is kicked by a laserpulse linearly polarized along an arbitrary direction p. As a result, itreceives a velocity �v.

on the sphere. The orientation of the molecule at time t is

r(t) = r0 cos (v0t) + v0

v0sin (v0t), (11)

where v0 = |v0|. Taking the derivative, we find the molecularvelocity:

v(t) = −v0r0 sin (v0t) + v0 cos (v0t). (12)

III. ENSEMBLE AVERAGING AND MONTECARLO SIMULATION

We are interested in the response of a thermal molecularensemble to the double-pulse excitation, and in the calculationof ensemble averaged values of the alignment factor andother observable quantities. One way to calculate the timedependence of an observable averaged over the ensemble is tofind its value at time t using (9), or (11), and to integrate itover the initial angular distribution function [6]. For example,at zero temperature, the molecules are initially at rest andisotropically distributed in all directions. Therefore, for anyobservable f (θ,ϕ), we have to substitute the time-dependentexpressions (9), (11) for θ and ϕ to obtain f (θ0,ϕ0,t), and tointegrate over the isotropic distribution of θ0 and ϕ0:

〈f (τ )〉 = 1

4π

∫ 2π

0dϕ0

∫ π

0dθ0 sin θ0 f (θ0,ϕ0,τ ). (13)

Here the brackets denote averaging over the ensemble. For anonzero temperature, an averaging over the initial velocitydistribution should be performed as well. In a thermalensemble at temperature T , the velocities vθ,0 and vϕ,0 aredistributed according to the Boltzmann distribution:

P(v′θ,0,v

′ϕ,0)dv′

θ,0dv′ϕ,0 = 1√

2πσ 2th

exp

[−

(v′

θ,0

)2

2σ 2th

]dv′

θ,0

× 1√2πσ 2

th

exp

[−

(v′

ϕ,0

)2

2σ 2th

]dv′

ϕ,0,

(14)

where the dimensionless width of the thermal distribution σth

is given by σ 2th = IkBT /h2 = (2hBc/kBT )−1. Here B is the

rotational constant of the molecule, and v′ is the dimensionlessvelocity Iv/h. Instead of (13), we have

〈f (τ )〉 = 1

4π

∫ 2π

0dϕ0

∫ π

0dθ0 sin θ0

∫ ∞

−∞dv′

θ,0,

(15)∫ ∞

−∞dv′

ϕ,0 P(v′θ,0,v

′ϕ,0)f (θ0,ϕ0,v

′θ,0,v

′ϕ,0,t).

These integrals have fast oscillating integrands and are difficult(although possible) to calculate numerically.

A somewhat simpler way of finding the time dependenceof observables averaged over the molecular ensemble is bymeans of a Monte Carlo simulation. We start with moleculesisotropically oriented in space, with rotational velocitiesdistributed according to Eq. (14). These initial values areobtained using a numerical random number generator. In orderto randomly distribute points on the surface of the sphere, one

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YURI KHODORKOVSKY et al. PHYSICAL REVIEW A 83, 023423 (2011)

uses the probability function,

P(θ,ϕ)dθdϕ = P(θ )dθP(ϕ)dϕ

= 1

2sin θ dθ

1

2πdϕ. (16)

The realization of P(ϕ) is straightforward if one has a uniformrandom number generator. If wun denotes a random numberfrom a uniform distribution in the interval [0,1], then ϕrand

0 =2πwun. On the other hand, in order to realize P(θ ) one shoulduse the transformation method [23], Sec. 7.2. Starting with theequality of probabilities dw = 1

2 | sin θdθ |, we obtain θ rand0 =

2 arcsin√

wun.In order to realize the velocity distribution (14), one uses

the normal random number generator. Using the transforma-tion method again, we obtain vrand

θ,0 = σthwnorm and vrandϕ,0 =

σthwnorm, where wnorm denotes a random number from a normaldistribution with zero mean and standard deviation of unity.

Now we have all the ingredients needed to simulate thescheme for exciting unidirectionally rotating molecules, asexplained at the beginning of Sec. II. Randomly orientedmolecules are “kicked” by a pulse polarized along the z axis,and get velocity additions according to Eq. (10), which shouldbe added to their initial thermal velocities. The molecules thenevolve freely according to Eqs. (11) and (12). At each time step,we calculate the ensemble averaged alignment factor 〈cos2 θ〉.At the time of the maximal alignment, the 45◦ rotated pulseis applied, velocities are changed again according to Eq. (10),and another free propagation follows.

As the first application of this approach, we investigate theazimuthal factor 〈cos2 ϕ〉 that characterizes confinement ofthe molecular angular distribution to the x-z plane. This factortakes the unity value when all the molecules are confined to thisplane, and it is equal to 0.5 for the isotropic ensemble. In Fig. 3,

0 0.2 0.4 0.6 0.8 10.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.6

t/Trev

⟨cos

2 φ⟩

FIG. 3. (Color online) The ensemble-averaged azimuthal factor〈cos2 ϕ〉 as a function of time after the second pulse excitingunidirectional molecular rotation. It is calculated by the classicalMonte Carlo method with 5 × 105 molecules (dashed black), and bythe exact quantum (finite-difference time-domain) simulation (solidred). The polarization vectors are p1 = (0,0,1) and p2 = (1,0,1)/

√2,

and P = 5 for both pulses. The calculation was done for nitrogenmolecules at the rotational temperature of 50 K. The time is measuredin units of the quantum revival time Trev = 2πI/h.

FIG. 4. (Color online) Time-averaged angular distribution func-tion for an ensemble of nitrogen molecules kicked by a laser pulsewith P = 10. The curve plots and inset (a) are for molecules initiallyat zero rotational temperature. The dashed green curve presentsthe exact quantum result, and the dash-dotted red line displaysthe analytical classical distribution according to Eq. (19). The bluesolid curve and inset (a) are calculated by the Monte Carlo methodaccording to Eq. (18), with σbelt = 0.1 and N = 104. Inset (b) showsMonte Carlo results for the initial rotational temperature of 50 K. Forthe three-dimensional distributions, the distance between the originand the plotted surface in the direction defined by θ and ϕ is equal toρta(θ,ϕ).

we plot the classically calculated azimuthal factor 〈cos2 ϕ〉 asa function of time following the second pulse, along with theexact quantum result (for details of the quantum calculation,see [19]). The calculation here (and also at Figs. 4 and 5 below)is done for nitrogen molecules with the rotational constant ofB = h/(8π2Ic) = 2.00 cm−1, and polarizability anisotropy of�α = 0.69 A3 [24]. The rotational temperature is 50 K, orσth = 2.94.

It is clearly seen that the classical calculation (Monte Carlorealization using 5 × 105 molecules) perfectly reproducesthe first oscillation of the azimuthal factor. Obviously, laterfractional and full revivals are quantum features, which arenot reproduced by the classical treatment. It is also evidentthat the time-averaged azimuthal factor (that is around 0.52here) coincides for the classical and the quantum calculation.This value is slightly higher than the isotropic value of 0.5,hence the molecular distribution is slightly squeezed towardthe plane defined by the polarization directions of the first andthe second pulses. We see that a relatively simple classicalmodel reproduces the important features of this problem.

IV. ANGULAR DISTRIBUTION FUNCTION

In this section we estimate classically the angular distribu-tion function of an ensemble of linear molecules excited bytwo pulses, and compare it with the results of full quantum-mechanical simulation [19]. To achieve this goal, we use thekernel estimator method [25] to reconstruct the distributionfunction from a set of N � 1 points on the unit sphere, whichare participating in the Monte Carlo simulation. According tothis method, each one of the N points is surrounded by a narrow

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CONTROLLING THE SENSE OF MOLECULAR ROTATION: . . . PHYSICAL REVIEW A 83, 023423 (2011)

FIG. 5. Time-averaged angular distribution function for unidirec-tionally rotating nitrogen molecules that were excited by two delayedand cross-polarized pulses. The initial rotational temperature is zeroin (a) and is 50 K in (b) and (c). The calculation was done using theclassical Monte Carlo method, Eq. (18), in (a) and (b), and usingthe quantum finite-difference time-domain simulation followed bythe averaging over a revival period in (c). Both pulses had the kickstrength of P = 5. The following parameters were used in (a) and(b): σbelt = 0.1 and N = 104. The distance between the origin and theplotted surface in the direction defined by θ and ϕ is equal to ρta(θ,ϕ).

symmetric Gaussian function on the surface of the sphere.The sum of all Gaussians represents the angular distributionfunction. Its time dependence is determined numerically bypropagating independently N rotating molecules, as explainedin the previous sections.

A simple way to build a symmetric Gaussian functionaround a point ri(t) on a sphere (i denotes the molecule numberin the Monte Carlo realization) is by referring to the angleα between the unit vector to this point, and the unit vectorto a general point r in its neighborhood: cos α = r · ri(t).The corresponding (unnormalized) distribution function isexp (−α2/2σ 2), where σ is the distribution width (a freeparameter). Because only small width values are used (σ � 1),the quantity α2/2 can be replaced by the expression 1 −cos α, which is equivalent to it for α � 1, while for α > 1the Gaussian distribution function is negligible any way.Normalizing the two-dimensional surface Gaussian function,we finally obtain the following estimate for the time-dependentdistribution function:

ρ(θ,ϕ,t) = 1

N

N∑i=1

1

2πσ 2exp

(−1 − r · ri(t)

σ 2

). (17)

An important object showing the symmetry and anisotropyof our system is the time-averaged angular distributionfunction. A straightforward, but somewhat cumbersome way togenerate it is by numerical averaging the distribution function(17) over a long time period. A much simpler approach totime averaging is described below. After the end of the pulses,each molecule moves with a constant speed on some circle onthe surface of the unit sphere. Within a sufficiently long timeperiod, each molecule spends the same amount of time at each

length element of the circle (apart from very rare cases in whicha molecule stays strictly at rest). Thus, the long-time-averagedangular distribution function for each molecule is a “belt”(a circle with a width) on the unit sphere. Its orientationdepends on the initial position r0 and velocity v0 on theunit sphere, and its width is a free parameter σbelt, which,for consistency, is taken equal to the parameter σ thatwas introduced before. The total time-averaged distributionfunction is obtained, as before, by adding the “belts” for allthe molecules participating in the Monte Carlo simulation.

The orientation of the “belt” for the ith molecule is givenby the angular momentum unit vector �eL,i = r0,i × v0,i/v0,i ,which is perpendicular to the plane defining the circle ofrotation. Thus, the Gaussian “belt” for a single molecule islocated close to the points r on the unit sphere, for whichthe dot product �eL · r is nearly zero. Inserting the correctnormalization factors, we finally obtain for the time-averagedangular distribution function, ρta:

ρta (θ,ϕ) = ρ (θ,ϕ,t)

= 1

N

N∑i=1

1

2π

√2πσ 2

belt

exp

(−

(�eL,i · r)2

2σ 2belt

), (18)

where the overline denotes a long-time average.Before considering the double-pulse scheme, we first

analyze the time-averaged angular distribution function formolecules subject to a single pulse polarized along the z axis.The result for zero temperature (all molecules are initially atrest) is shown in Fig. 4(a) and duplicated at the plot by thesolid line. In Fig. 4(b), the distribution function is plotted forthe same system, but at finite temperature. As expected, thedistribution is “cigar” shaped, and azimuthally symmetric. Atzero temperature, an analytical result can be obtained for thetime-averaged distribution function in the classical limit. Inthis case, all the trajectories of the kicked molecules are circlespassing through the poles of the unit sphere. These circlesare uniformly distributed in the azimuthal direction, and thedistribution function is ϕ independent. Therefore, at any givenvalue of θ we find the same total number of molecules. Asa result, the time-averaged distribution function multiplied bythe geometrical factor sin θ should be a constant. It is seenfrom here, that the distribution function averaged over longtime should be proportional to 1/ sin θ , or, after normalization:

ρta (θ,ϕ) = 1

2π2 sin θ. (19)

This equation was verified also by evaluating numericallyEq. (9) of Ref. [26], including summation over all branches.The function of Eq. (19) is plotted in Fig. 4 by a dash-dottedline. Our numerical Monte Carlo result in Fig. 4 (solid line)agrees well with the analytical result, but it is not singular atthe north and the south poles, because a finite width σbelt wasgiven to every molecule trajectory, and the singularities aresmeared out.

Finally, we compare the classical result to the quantumangular distribution function averaged over a single revivalperiod. The quantum result is given in Fig. 4 by a dashedline. Two features should be stressed here. First, the singularclassical behavior disappears in the quantum description.

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YURI KHODORKOVSKY et al. PHYSICAL REVIEW A 83, 023423 (2011)

Second, there is a fine oscillatory structure in the quantumdistribution, which is absent in the classical distribution.Besides these features, the distributions are similar. We alsocompare the values of 〈x2〉, 〈y2〉, and 〈z2〉, calculated bothclassically and quantum mechanically at the conditions shownin Fig. 4(b). Here the brackets denote the ensemble averagingand the overline denotes long-time averaging. The quantitiesx,y,z are the Cartesian components of the unit vector r. In bothcases, the same values of 〈z2〉 = 0.42 and 〈x2〉 = 〈y2〉 = 0.29are obtained. This should be compared to the “isotropic” valuesof 〈x2〉 = 〈y2〉 = 〈z2〉 = 1/3, and to the zero temperaturevalues of 〈z2〉 = 0.50 and 〈x2〉 = 〈y2〉 = 0.25.

For the double-pulse scheme, leading to the unidirectionalrotation of the molecules, we present the classical time-averaged angular distribution in Fig. 5(b), which is verysimilar, except for fine details, to the corresponding quantumdistribution in Fig. 5(c), presented also in [19]. The distributionis slightly squeezed to the xz plane (the plane defined by thepolarization directions of the two pulses). We expected 〈y2〉to be smaller than 〈x2〉 and 〈z2〉, and we indeed obtain numer-ically 〈x2〉 = 0.31, 〈y2〉 = 0.30, and 〈z2〉 = 0.39 classically,as well as quantum mechanically. The zero temperature resultis, obviously, more dramatic and is presented in Fig. 5(a).Here we obtain numerically 〈x2〉 = 0.29, 〈y2〉 = 0.26, and〈z2〉 = 0.45. These results demonstrate very good agreementbetween the classical and quantum calculations even forrelatively small values of the interaction strength P .

V. UNIDIRECTIONAL ROTATION OFBENZENE MOLECULES

Now we address a more difficult problem of the unidi-rectional rotation of benzene molecules. These molecules,considered as rigid bodies, behave as oblate symmetric tops.The classical treatment of the free rotation of a symmetrictop is analytically simple (although more complicated than therotation of a linear rotor), and is well documented in literature(see, e.g., [27], Secs. 33 and 35).

A. Rotation of benzene molecules—classical treatment

There are two moments of inertia characterizing a sym-metric top, I1 = I2 and I3, which are defined using the bodyfixed coordinate system x1,x2,x3, shown in Fig. 6, where themolecular symmetry axis is directed along the x3 axis. Thefigure is taken from [27], Sec. 33; the x2 axis is perpendicularto the plane of the figure and is not shown. For benzenemolecule, which is an oblate symmetric top, and has the formof a planar ring, the x3 axis is perpendicular to the plane ofthe ring and passes through its center. Hence, for benzene,I1 + I2 = 2I1 = I3 [27], Sec. 32.

When the molecule is freely rotating, the angular momen-tum, denoted by L, is constant in magnitude and direction. Therotational kinetic energy is also a constant of motion and canbe written as

T = L21

2I1+ L2

2

2I1+ L2

3

2I3= L2

‖2I1

+ L23

2I3, (20)

FIG. 6. The free rotation of a symmetric top with notation usedin the text.

where L1, L2, and L3 are the components of angular mo-mentum along the body-fixed coordinate axes x1, x2, and x3,respectively, and L2

‖ = L21 + L2

2. The free rotation is such thatthe angular momentum L, the angular velocity �, and themolecular axis x3 all lie in the same plane. The molecular axisx3 (as well as the vector �) performs precession around theangular momentum vector L with an angular velocity �pr,which is shown in Fig. 6. Its magnitude is

pr = L

I1. (21)

The molecular axis x3 moves on the surface of a cone and formsan angle θpr with the cone axis (which is along the vector L).

In addition to the precession motion, the top rotates aroundits x3 axis, but this motion is not important in our problem. Thereason is that this last rotation does not change the molecularorientation (defined by the molecular axis), and that the laserpulse does not influence this motion, as will be explained lateron.

We describe the orientation of the molecular axis by a unitvector r in the direction of x3. Its value at t = 0 is r0. Thevelocity of this vector is given by

v = �pr × r. (22)

We now define two more unit vectors. One is �eL, a unit vectorin the direction of L. Another is in the plane of the circle tracedby the tip of r:

r0‖ = r0 − cos θpr�eL

sin θpr. (23)

Using the notation defined, it is easy to see that the timedependence of r is given by

r(t) = cos θpr�eL + sin θpr

(r0‖ cos prt + v0

v0sin prt

).

(24)

For the velocity we obtain

v(t) = pr sin θpr

(−r0‖ sin prt + v0

v0cos prt

). (25)

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CONTROLLING THE SENSE OF MOLECULAR ROTATION: . . . PHYSICAL REVIEW A 83, 023423 (2011)

It is important to stress that the tip of the unit vector r does notnecessarily trace a circle of a unit radius on the surface of theunit sphere, as for the linear molecule. It can trace any circle,which can even shrink to a point, in the case of 1 = 2 = 0and 3 = 0. A unit circle is traced only if θpr = π/2, or 3 = 0and 1 = L/I1.

Although we are dealing here with a more complicatedmolecule, the interaction with an ultrashort laser pulse isdescribed by the same formula as in Eq. (1). The reason is thatthe polarizability tensor of the symmetric top has the samesymmetry as the one of the linear molecule. The interactiondepends, as before, on the electric field envelope squared,on �α, which is negative in the case of benzene, and onthe angle between the pulse polarization direction p and themolecular axis orientation r0. Notice that a linearly polarizedlaser pulse does not produce a torque parallel to the molecularaxis.

The torque exerted by the pulse is, therefore,

� = − 14E2(t)�α sin 2β0 �ep×r0 , (26)

where �ep×r0 is a unit vector in the direction of the cross productp × r0. This equation shows that the laser pulse “kicks” thesymmetry axis of the benzene molecule toward the plane,which is perpendicular to the polarization vector p. UsingNewton’s second law and integrating over the short time of thepulse, we obtain for the change in the angular momentum,

�Lx = −hP sin 2β0(�ep×r0

)x, (27)

and two similar equations for the y and the z compo-nents. Angle β0 can be found using p · r0 = cos β0. Now,after the new angular momentum vector is known, andassuming the molecule did not move during the interactionwith the pulse, the molecular rotation is completely definedby Eq. (24). Naturally, new values of θpr, �eL, r0‖, v0, and pr should be found using the new value of L. Specifically,θpr is found using the relation cos θpr = �eL · r0, r0‖ is thendefined using Eq. (23), pr – by Eq. (21), and v0 – usingEq. (22).

The Monte Carlo simulation for benzene molecules hasseveral differences compared with the one used for linearmolecules. Here, the kinetic energy is more easily separablewhen expressed using the angular momentum componentsalong the body-fixed coordinate system, as in Eq. (20).

We start with random generation of the angular momentumcomponents L‖ and L3, according to the thermal distribution,

P(L′‖)dL′

‖ × P(L′3)dL′

3

= 1

σ 2th,1

exp

[− (L′

‖)2

2σ 2th,1

]L′

‖ dL′‖

1√2πσ 2

th,3

exp

[− (L′

3)2

2σ 2th,3

]dL′

3,

(28)

where we introduce σ 2th,1 = I1kBT /h2 and σ 2

th,3 = I3kBT /h2,similar to Eq. (14), and where L′ represents the dimension-less angular momentum L/h. Employing the transformationmethod, and using the notation introduced after Eq. (16),we obtain explicitly for the random components: (L′

‖)rand =√2 σth,1

√ln [1/(1 − wun)] and (L′

3)rand = σth,3wnorm. Know-ing these components, we deduce the magnitude of the angular

momentum L = √L2‖ + L2

3, but its direction is yet to bedefined. We obtain also the precession velocity using Eq. (21),and θpr from the relation cos θpr = L3/L.

Next, we determine the direction of the angular momentumin the laboratory coordinate system, which is given by thevector �eL. We do this by randomly generating two sphericalangles θL and ϕL. These two angles are distributed the sameas θ and ϕ in Eq. (16), because the tip of the vector �eL isdistributed uniformly on the surface of the unit sphere.

Finally, by randomly generating an angle ϕ, uniformlydistributed between 0 and 2π , we obtain the initial orientationof the molecular axis on the cone around the vector L. Now,the coordinates of r0 are completely defined in the frame inwhich the z axis is fixed along the vector L. These coordinatesare given by (sin θpr cos ϕ, sin θpr sin ϕ, cos θpr). Using tworotational matrices, the coordinates of r0 in the laboratoryframe are then found.

The above Monte Carlo procedure was used to generatedata that are discussed later in Sec. V C.

B. Quantum-mechanical treatment

Here we outline the quantum-mechanical description ofunidirectional rotation of benzene molecules by two delayedultrashort laser pulses with crossed linear polarizations. Detailswill be presented in [28]. The molecular Hamiltonian, H0, inthe field-free conditions is obtained simply by replacing thecomponents, Li (i = 1,2,3), of the classical angular momen-tum by the corresponding quantum-mechanical operators, Ji ,in Eq. (20). The rotational eigenstates and eigenenergies aregiven by the time-independent Schrodinger equation,

H0|r〉 = Er |r〉, (29)

where r stands for an index to identify the eigenstates,explicitly expressed by a set {J,K,M}, with J being thetotal angular momentum, and K and M being its projectiononto the molecular symmetry axis and the space-fixed axis,respectively. At finite temperature, an initial ensemble is amixture of molecules in many different eigenstates, but wefirst take a single rotational level as an initial state, |ri〉.The nonadiabatic interaction with the nonresonant ultrafastlaser field, given by Eq. (1), converts the stationary state to arotational wave packet, |�(t)〉, which is expanded as

|�ri(t)〉 = U (t,0)|ri〉 =

∑r

Cri ,r (t) exp(−iωr t)|r〉. (30)

Here, U (t2,t1) is the time-evolution operator from time t1 tot2 and ωr = Er/h. The expansion coefficients, Cri,r (t), varyduring the interaction with the laser pulses, and subsequentlybecome constant after the laser field vanishes. They can bederived by solving the time-dependent Schrodinger equation(TDSE) with the initial condition, |�(t → −∞)〉 = |ri〉,

ih∂

∂t|�ri

(t)〉 = [H0 + V (t)]|�ri(t)〉, (31)

where V (t) is given in Eq. (1). Substituting Eq. (30), this TDSEis recast into the following coupled differential equations for

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YURI KHODORKOVSKY et al. PHYSICAL REVIEW A 83, 023423 (2011)

the expansion coefficients:

ihd

dtCri ,r (t) =

∑r ′

〈r|V (t)|r ′〉 exp(−i�ωr ′,r t)Cri,r ′ (t),

(32)

with �ωr ′,r = ωr ′ − ωr . These coupled equations can besolved numerically to determine the complex expansioncoefficients, once the matrix elements for V are specified.

If another laser pulse is irradiated onto the molecule att = τ , the wave packet created by the first pulse, representedin Eq. (30), is further modified by the interaction with thesecond pulse. The resultant wave packet at t after the secondpulse is expanded as

|�ri(t)〉 = U ′(t,τ )|�ri

(τ )〉 =∑

r

Bri ,r (τ ) exp(−iωr t)|r〉,

(33)

where the prime on U indicates the properties of the secondpulse. The quantity Bri,r is the transition amplitude from theinitial |ri〉 state to |r〉 by the interaction with the two pulses,and is represented by

Bri,r (τ ) =∑r ′

Cri,r ′C ′r ′,r exp(−i�ωr ′,r τ ). (34)

To evaluate the transition amplitude given in Eq. (34), itis convenient to adopt the space-fixed axis system defineddifferently from the previous one. Here, we set the z axisalong the laser propagation direction and the polarization ofthe first pulse parallel to the x axis. Then, the interaction givenin Eq. (1) is recast as [29]

V (θ,ϕ,t) = 18E2(t){�α [1 + cos(2ϕ)] cos2 θ

−�α cos(2ϕ) − (α‖ + α⊥)}, (35)

or, re-expressed as

V = − 1

12E2(t)

{(α‖ + 2α⊥) − �αD

(2)∗0,0

+√

3

2�α

[D

(2)∗−2,0 + D

(2)∗2,0

]}, (36)

where D(j )p,q is the rotational matrix [30]. With this expression,

the evaluation of the matrix elements for a symmetric-top basisset is straightforward by applying the spherical tensor algebra,and the following selection rules are derived:

�K = 0,�M = 0 (then �J = 0,±2 for KM = 0 and �J =

0,±1,±2 for KM = 0), or�M = ±2 (then �J = 0,±2 for K = 0 and �J =

0,±1,±2 for K = 0).We set the polarization of the second pulse tilted against

that of the first one by angle �ϕ. When the new axis system isdefined so as the x ′ axis is set parallel to the polarizationof the second pulse, the new and old angular coordinatesare related as ϕ′ = ϕ − �ϕ. Consequently, the symmetric-topwave function transforms as

|J,K,M〉 �−→ |J,K,M〉 exp(iM�ϕ). (37)

By taking the transformation into account, Bri,r in Eq. (34) isrewritten, if the second pulse is a replica of the first one, as

Bri,r (τ ) =∑r ′

Cri,r ′Cr ′,r exp[−i(�ωr ′,r τ − M ′�ϕ)],

(38)

with r ′ standing for a set of {J ′,K ′,M ′}. The transitionamplitude for the two pulses can be evaluated for an arbitraryτ at once, if the set of the coefficients, Cri,r ′ , are derived bysolving the coupled equations in (32).

The expectation value of a given observable A for a certaininitial condition with finite temperature is derived as thefollowing ensemble average,

〈A〉(τ,t) =∑ri

Wri〈�ri

(t)|A|�ri(t)〉

=∑ri ,r,r ′

WriB∗

ri ,r ′ (τ )Bri,r (τ )〈r ′|A|r〉 exp(i�ωr ′,r t),

(39)

where Wriis the Boltzmann factor for |ri〉 with appropriate

degeneracy factor due to nuclear-spin statistics [31]. For thediscussion of the molecular alignment and orientation, we takecos2 θ , J 2, and Jz (corresponding to the classical Ly), as A.

C. Classical versus quantum treatment

In this section we analyze the process of exciting unidi-rectional rotation of benzene, following the setup of [21].The scheme operation is practically the same as describedafter Eq. (2), except of several details specific for the benzenemolecules. Shortly after the first ultrashort pulse, polarizedalong p1 = (0,0,1), the molecules experience a transientantialignment, as opposed to alignment in the case of linearmolecules. This happens because �α is negative for benzene,as opposed to linear molecules, such as nitrogen, and thedirection of the angular momentum transferred to the moleculeby the pulse is reversed. Therefore, shortly after the firstpulse the molecular axis angular distribution is squeezed tothe plane perpendicular to p1. Quantitatively, after the firstpulse, the ensemble average of 〈cos2 θ〉 decreases below theisotropic value of 1/3. The second delayed and tilted laserpulse brings this anisotropic ensemble to the rotation with apreferred sense. In the following, we shall consider the timedevelopment of the alignment factor 〈cos2 θ〉 after the firstpulse, and the dependence of the average value of the inducedangular momentum as the function of the delay of the secondpulse.

We show first the classical results. The top part of Fig. 7presents the alignment factor 〈cos2 θ〉 as a function of timepassed after the first pulse [polarized along p1 = (0,0,1)], forthree values of the pulse strength P . Here and below we use thefollowing parameters for the benzene molecule [32]: rotationalconstants B = h/(8π2I1c) = 0.190 cm−1, C = h/(8π2I3c) =B/2, and the polarizabilities α‖ = 6.67 A3, α⊥ = 12.4 A3. Allplots are for rotational temperature of 0.9 K (as in experiment[21]), which corresponds to the dimensionless parametersσth,1 = 1.29 and σth,3 = √

2σth,1 = 1.82.The plots for 〈cos2 θ〉 show that the antialignment becomes

more pronounced and occurs at earlier times as the pulse

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CONTROLLING THE SENSE OF MOLECULAR ROTATION: . . . PHYSICAL REVIEW A 83, 023423 (2011)

0.2

0.4

0.6⟨c

os2 θ⟩

0 1 2 3 4 5 6

−0.2

0

0.2

time (units of I1/h)

⟨Ly⟩ /

⟨L2 ⟩0.

5

P=−3

P= −10

P= −10

P= −1

P= −3

P= −1

FIG. 7. (Color online) Classical treatment. The top panel showsthermally averaged alignment factor 〈cos2 θ〉 for benzene moleculeskicked by a single pulse, plotted as a function of time after the pulse.The pulse has a strength of P1 = −1,−3,−10 (black, blue, and redlines online, respectively) and polarization direction p1 = (0,0,1).The bottom panel displays the (normalized) value of the orientedangular momentum 〈Ly〉/

√〈L2〉 induced by a second ultrashort pulse,

shown as a function of the delay of this pulse. The polarizationdirection of the second pulse is p2 = (−1,0,1)/

√2 (at 45◦ to p1),

and the value of P2 is the same as that of P1, with the same colorcode used for the curves. The initial rotational temperature is 0.9 K.The calculations were done using the Monte Carlo method with 105

realizations.

intensity increases, as is naturally expected. Similar to the caseof linear diatomic molecules, we anticipate that the secondpulse should be fired at the moment of the maximal squeezingof the molecular angular distribution (i.e., when the alignmentfactor takes the minimal value). In this case, the maximalaverage torque is applied to the molecules, facilitating themost emphasized unidirectional rotation. This is indeed seenfrom the bottom part of Fig. 7. Here the quantity 〈Ly〉/

√〈L2〉

(calculated after the second pulse) is plotted as a functionof the time delay between the first pulse (with polarizationdirection p1) and the second pulse with polarization direction

0 1 2 3 4 5 6

−0.2

0

0.2

0.4

time (units of I1/h)

⟨Ly⟩ /

⟨L2 ⟩0.

5

0.2

0.4

0.6

⟨cos

2 θ⟩

P= −3

P= −3

P= −10

P=−10

P= −1

P= −1

FIG. 8. (Color online) The same quantities as in Fig. 7 are plottedhere. They were calculated using the quantum-mechanical treatment.Half and full rotational revivals are seen, which are not present inFig. 7.

0.2

0.4

0.6

⟨cos

2 θ⟩

0 0.2 0.4 0.6 0.8 1

−0.2

0

0.2

0.4

time (units of I1/h)

⟨Ly⟩ /

⟨L2 ⟩0.

5

P= −1

P= −10

P= −10

P= −1

P= −3

P= −3

FIG. 9. (Color online) Comparison of the classical results ofFig. 7 (solid lines), and quantum-mechanical results of Fig. 8 (dashedlines) at short times. The values of P are −1,−3,−10 for black, blue,and red lines, respectively.

p2 = (−1,0,1)/√

2, where the angle between p1 and p2 is 45◦.The maximal modulus of the transferred angular momentumhappens indeed when the second pulse is applied at themoment of the maximal antialignment. We also notice thatthe optimal delay becomes shorter with the strength of thepulses.

Figure 8 presents the same quantities calculated fullyquantum mechanically, following the procedure outlined inSec. V B. On the long-time scale, it demonstrates quantumrotational revivals, such as half and full revivals, that areabsent in Fig. 7 for obvious reasons. For a detailed comparisonwith the classical results at shorter times (corresponding tothe conditions of the experiment [21]), Fig. 9 displays theclassical and the quantum graphs overlapped on the same plot.The results seem to be practically identical at short times,and the agreement is improving with the increase of |P |, asmore rotational states are involved in the dynamics of the wavepacket. Parameters of the experiment [21] correspond to thecurves for P = −3 in Fig. 9, with the delay time chosen nearthe minima of the curves. As follows from Fig. 9, the rotationaldynamics in this range is adequately described by the classicaltreatment.

VI. CONCLUSIONS

We studied the problem of exciting unidirectional molecularrotation with the help of two delayed cross-polarized laserpulses using classical mechanics, and compared this approachwith the results of fully quantum-mechanical treatment. Theproblem was analyzed both for linear molecules and forsymmetric tops, like benzene. Although the comprehensive de-scription of the fine details of rotational dynamics undeniablyrequires the full-scale quantum formulation, we found a verygood agreement between the classical and quantum approachesin a wide range of experimentally relevant parameters. Theclassical model identifies the origin of the mechanism, definesthe optimal delay between the pulses, and the best anglebetween them. Moreover, it accurately predicts the magnitudeof the induced oriented angular momentum, and correctly

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YURI KHODORKOVSKY et al. PHYSICAL REVIEW A 83, 023423 (2011)

reproduces the behavior of the optimal delay with pulsestrength, as it was observed in [21]. At the same time, a fullyquantum treatment is required to describe quantitatively theoperation of the double-pulse scheme for relatively weak laserpulses, when only low rotational states are involved. In thiscase, considering quantum interference of excitation pathwaysby the first and the second pulses [21] provides an instructivecomplementary view on the origin of the phenomenon.Moreover, operation of the scheme at long delays betweenthe pulses (comparable with the period of rotational revivals)needs quantum description as well. At the same time, quantumand classical results agree well again if the long-time-averagedobservable quantities are considered. We illustrated this bycomparing classical and quantum time-averaged molecularangular distributions that show anisotropy and confinementto the plane spanned by two polarization vectors of the laserpulses.

We believe that this comparative study provides the readera comprehensive view on the physics behind the double-pulsescheme for ultrafast preparation of molecular ensembles withoriented rotational angular momentum and controlled sense ofrotation.

ACKNOWLEDGMENTS

The authors are thankful to Kenji Ohmori for stimulatingthis collaborative work. Y.K. and I.A. appreciate many fruitfuldiscussions with Yehiam Prior, Sharly Fleischer, and ErezGershnabel. Financial support for this research from the IsraelScience Foundation is gratefully acknowledged. Additionalfinancial support from a Grant-in-Aid from MEXT Japan,C-PhoST, and “Extreme Photonics” is also appreciated. Thisresearch is made possible in part by the historic generosity ofthe Harold Perlman Family.

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