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Controlling Quantum Rotation With Light I. Sh. Averbukh*, R. Arvieu , and M. Leibscher* † ∗ Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel † Institut des Sciences Nucléaires, F 38026 Grenoble Cedex, France e-mail: [email protected] alignment (orientation) by laser fields to heavy-ioncolli- Semiclassical catastrophes in the dynamics of a quantum sions,andthe trappingofcoldatoms bya standinglight rotor(molecule)drivenbyastrongtime-varyingfieldarecon- wave. The paper is organized as following. Section II sidered. Weshow that for strong enough fields, asharp peak discusses semiclassical catastrophes in the dynamics of in the rotor angular distribution can be achieved via time- a driven quantum rotor with the help of a simple two- domain focusing phenomenon, followed by the formation of 2 dimensional model [18]. Section III applies these generic 0 angular rainbows and glory-like angular structures. Several 0 scenarios leading to theenhanced angular squeezing are pro- results to a thermal system of cold atoms driven by a 2 posed that use specially designed and optimized sequences pulsed optical lattice [19]. Section IV focuses on 3D ef- of pulses. The predicted effects can be observed in many fects important in the processes of molecular alignment n processes, ranging from molecularalignment (orientation) by and orientation by strong laser pulses [20]. a J laser fields to heavy-ion collisions, and the squeezing of cold atoms in a pulsed optical lattice. 3 1 II. TWO-DIMENSIONAL ROTOR PACS numbers: 42.50.-p, 42.50.Vk, 32.80.Pj, 33.80.-b 1 For the sake of clarity, we start with the simplest v model of a two-dimensional rotor described by the time- 1 5 I. INTRODUCTION dependent Hamiltonian 0 1 L2 Drivenrotorisastandardmodelinclassicalandquan- H = +V(θ,t) (1) 0 2I tum nonlinear dynamics studies [1]. Increasing interest 2 b 0 in the problem has arisen because of recent atom op- whereListhe angbularmomentumoperator,andI isthe / tics realization of the quantum δ-kicked rotor [2,3], and h momentum of inertia of the rotor. This model contains p novelexperimentsonmoleculeorientation(alignment)by mostofbthephysicswewanttopresenthere. Someimpor- - strong laser fields [4–7]. A strong enough laser field cre- tantadditionaleffectsthatappearin3Dwillbediscussed t n ates the so-called pendular molecular states [8–10] that below in Section IV. For a ”linear 2D molecule” having a are hybrids of field-free rotor eigenstates. By adiabati- a permanent dipole moment µ, and driven by a linearly u cally turning on the laser field, it is possible to trap a q polarized field, the interaction potential is molecule in the ground pendular state, thus leading to : v molecular alignment. The only way to reach a consider- V(θ,t)= µ (t)cos(θ) (2) i − E X able degree of alignment in this approach is by increasing the intensity of the field. However, many applica- where (t) is the field amplitude (i. e., of a half-cycle r E a tions may require only a transient molecular alignment pulse), and θ is the polar angle between the molecular (orientation), where the molecular angular distribution axis and the field direction. In the absence of interac- becomes extremely squeezedat some predeterminedmo- tion with a permanent dipole moment, the external field ment of time. It is well known that a physically related maycouple with induced anisotropicmolecularpolariza- problemofsqueezedstatesgenerationinaharmonicsys- tion. This interaction (being averaged over fast optical tem may be solved by a proper time-modulation of the oscillations) may result in the interaction potential pro- driving force (parametric resonance excitation). Behav- portional to cos2(θ) (see Eq.(17) in Section IV) [21]. Al- ior of a rotor in general, strong, time-varying fields is though these two forms of V(θ,t) may lead to different a much less-studied problem, although it is understood physical consequences (i. e., orientation vs alignment), that the long-persisting beats in the molecular angular the effects we will present are more or less insensitive to distributionmaybeinducedbyshortlaserpulses[11–17]. the choice of interaction. Therefore, we prefer to start In the present paper, we analyze generic features in with the more simple situation described by Eq.(2). thedynamicsofaquantumrotordrivenbystrongpulses, By introducing dimensionless time τ = t¯h/I, and in- 2 andpresentastrategyfor efficientsqueezingofthe rotor teraction strength ε = µ (t)I/¯h , the Hamiltonian can E angular distribution by a sequence of pulses of moder- be written as ate intensity. The results of our research are related to 1 ∂2 a number of physical processes, ranging from molecular H = ε(τ)cos(θ) −2∂θ2 − b 1 The wavefunction of the system canbe expanded in the the help of the following semiclassical arguments. Con- eigenfunctions of a free rotor sider an ensemble of randomly oriented classical rotors subjecttoakick. Theangularvelocityofarotorlocated +∞ 1 at the angle θ is Ψ(θ,τ)= c (τ)einθ n √2π n=−∞ ω(θ)= Psin(θ) (5) X − In the absence of the field, the wave function takes the just after the kick, assuming negligible initial velocity. form For rotors from the regionof small θ <<1, the acquired +∞ velocity is linearly proportional to their initial angle, so Ψ(θ,τ)= 1 c (0)e−in2τ/2+inθ (3) that all of them arrive at the focal point θ = 0 at the n √2π n=−∞ same time X Despite a simple form of Eq.(3), the wave function ex- τ =1/P. (6) f hibits extremely rich space-time dynamics. In particu- This phenomenonis quite similar tothe focusingoflight lar, it shows periodic behavior in time with the period rays by a thin optical lens. For P >> 1, the shape of T = 4π (full revival) and a number of fractional re- rev the distribution at the focusing time τ is dictated by vivals at τ =p/sT (p and s are mutually prime num- f rev the aberration mechanism (deviation of the cos(θ) po- bers)[22]. Ananalyticalsolutionvalidforageneraltime- tential fromthe parabolic one), and it is P-independent. dependent fieldis unknownevenfor this simplestmodel. We consider the orientation factor O =< 1 cos(θ) > Much effort has been devoted to the case of extremely − (where angular brackets mean averaging over the state short field pulses (δ kicks) (see, e. g. [1], and references − of rotor)as a measure of the rotor orientation. For large therein). In general, as a result of a single kick applied enoughP, the time-dependent orientationfactor(for the to the rotor at τ =τ , the coefficients c transform as k n initials-state)maybeeasilyestimatedbyaveragingover +∞ the initially uniform classical ensemble of rotors having cn(τk+0)= in−mJn−m(P)cm(τk−0), (4) the velocity distribution of Eq. (5): O(τ) =1−J1(Pτ). mX=−∞ HereJ1(x) isBesselfunctionofthe firstorder. Themin- imal value O 0.418 of the orientationfactor is, in fact, where ≈ achieved in the post-focusing regime, at τ 1.84τ . As f ≈ +∞ seeninFig. 1(c), anewphenomenoncanbeobservedin P = ε(τ)dτ, the angular probability distribution just after the focusing. Sharpsingular-likefeaturesareformedinthe distri- Z −∞ bution, which are moving with time. Each of these fea- andJ (P)istheBesselfunctionofthenthorder. There- tures has a typical asymmetric shape, with pronounced n sultofmultiplekicksappliedatdifferenttimescanbeob- oscillations on one side and an abrupt drop down on the tained by combining transformations (4) after each kick other side. Again, the origin of this effect can be traced with a free evolution according to Eq.(3) between the in the time evolution of a classical ensemble of initially kicks. If the kicks are applied periodically to the system motionless rotors. with the period T , the system does not show chaotic After a kick applied at τ =0, the motion of the rotors rev behavior,andtheenergyaccumulatesquadraticallywith is described by time (the so-called ”quantum resonance” [23,24]). It is, therefore, quite natural to examine potential accumula- θ =θ0−P sin(θ0)τ (mod 2π) (7) tionofangularsqueezingoftherotorwavefunctionunder whereθ0istheinitialangle. Forτ <τf Eq.(7)represents the ”quantum resonance” excitation. In this case, be- a one-to-one mapping θ(θ0) [see Fig. 2 (a)]. At τ = τf causeofthe exactquantumrevivalsatthe free-evolution the curve θ(θ0) touches the horizontal axis [Fig. 2 (b)]. stages, the effect of N kicks of a magnitude P is equiva- Atτ >τf theangleθ0becomesamulti-valuedfunctionof lenttotheactionofasinglestrongpulseofstrengthNP θ [Figs. 2(c),(d)]. Theclassicaltime-dependentangular (see, e. g. [23]). In Figure 1, we show numerically calcu- distribution function of the ensemble is given by lated time evolution of the probability density Ψ(θ,τ)2 | | after a relatively strong kick of a magnitude P = 85 f(θa,τ =0) f(θ,τ)= 0 (8) applied at τ = 0. Initially the rotor was in the ground dθ/dθa a | 0| s-state (cn(0)=δn0). For the chosenvalues of τ, several X distinct phenomena can be seen in these plots. First of The summation in Eq. (8) is performed over all possible all,thewavefunctionshowsanextremenarrowinginthe branches of the function θ0(θ) defined by Eq.(7). It fol- regionofsmallθ aftersomedelayfollowingthekick[Fig. lowsimmediatelyfromEq.(8)thatevenforasmoothini- 1(b)]. Thephysicsofthiseffectmaybeunderstoodwith tial distribution, f(θ,τ) may exhibit a singular behavior 2 u near the angles where dθ/dθ0a →0. The quantum nature ∆τk = u +kw of the rotormotion replaces the classicalsingularities by k k sharp maxima in the probability distribution with the uk+1 =uk u2k (9) Airy-like shape typical to rainbow phenomena. Indeed, − uk+wk this effect is similar to the formation of caustics in the wk+1 =wk+uk wave optics [25], and rainbow-type scattering in optics For large k, the last two finite-difference equations may and quantum mechanics [26–28]. We should stress,how- bereplacedbyasystemofcoupleddifferentialequations. ever,thatthelong-timeasymptoticregimesareradically The latter has an exact solution providing the following different for the classical and truly quantum motion of asymptotics: < θ2 > 1/√k and ∆τ 1/k. This the rotor. Thus, contrary to the classical limit, in which k k ∝ ∝ resultisingoodagreementwiththenumericallyobserved thecausticsexistforever,theygraduallydisappearinthe power-laws behavior of the graphs 3 (a) and 3 (b), and quantum case because of the overall decay of the initial it describes correctly their slopes at k >> 1. We note, rotational wave packet. On even longer time scale, an- that in contrast to the wave optics (in which the size otherquantumphenomenoncanbeseen,namelyrevivals of the focal spot is diffraction limited), our system may andfractionalrevivalsof the initial classical-likemotion. be, in principle, ”unlimitedly” squeezed in angle. We Figs. 1(e)-(i)showseveralexamplesoffractionalfociand also note that a quasi-periodic sequence of kicks applied rainbows in the angular distribution, which is a purely quantum effect. at τk+1 = τk +∆τk +Trev provides the same squeezing scenario for a quantum rotor. The introduction of the As we have demonstrated, a mere application of T -shift between pulses may be useful in the practical δ kicks at the condition of ”quantum resonance” does rev − realizations of the scenario to avoid the overlap between notleadtoaccumulatedangularsqueezing,andtheorien- short excitation pulses of a finite duration. tation is saturated at some finite asymptotic level. Here we suggest an excitation scheme that exhibits the de- sired accumulation property. As previously mentioned, III. SQUEEZING OF ATOMS IN A PULSED the wave function of the rotor reaches the state of the OPTICAL LATTICE maximal orientation(i. e., minimal O value) after a certain delay ∆τ1 following the application of the first kick In the present Section we apply the above results to at τ = τ1 = 0. We suggest to apply the second kick at anotherwell-knownsystem: coldatomsinteractingwith τ2 = ∆τ1. Immediately after the second kick, the sys- a pulsed opticallattice [19]. Opticallattices are periodic tem will keep the same probability density distribution. potentials for neutral atoms induced by standing light On the other hand, τ = τ2 will no longer be a station- wavesformedbycounter-propagatinglaserbeams. When ary point for O(τ) =<1 cos(θ)>(τ). The orientation − these waves are detuned from any atomic resonance, the factor O(τ), and its derivative are continuous and peri- ac Stark shift of the ground atomic state leads to a con- odic functions of time in the course of a free evolution. servative periodic potential with spatial period λ/2, half Therefore,O(τ)willreachanewminimumatsomepoint the laser wavelength (for a review, see, e. g. [29]). Such τ2+∆τ2 in the interval [τ2,τ2+Trev]. Clearly, the new latticespresentaconvenientmodelsystemsforsolidstate minimal value of the orientation factor is smaller than physics and nonlinear dynamics studies. In contrast to the previous one. By continuing this way, we will apply traditional solid state objects, the parameters of optical short kicks at iterative time instants τk+1 = τk +∆τk. lattices (i. e. lattice constant, potential well depth, etc.) By construction of this pulse sequence, the squeezing ef- are easily controllable. Many fine phenomena that were fect willaccumulate with time, in contrastto the ”quan- long discussed in solid state physics, have been recently tum resonance” excitation. This is demonstrated by observed in corresponding atom optics systems. Bloch Fig. 3, which shows calculated sequences ∆τ and k { } oscillations [30], or the Wannier-Stark ladder [31,32] are < 1 cos(θ) > (τ ) for a rotor initially in the s-state k { − } only few examples to mention. Time-modulation of the and being kicked by pulses with P = 3. The logarithm frequency and intensity of the constituent laser beams oftheorientationfactorgraduallydecreases,withoutany provide tools for effective modeling of numerous time- sign of saturation. dependent nonlinear phenomena. Since the initial pro- At the stage of a well-developed squeezing (O << 1), posal [33], and first pioneering experiments on atom op- thecos(θ)-potentialmaybeapproximatedbyaparabolic tics realization of the δ-kicked quantum rotor [2], cold one. It can be easily shown (both classically and quan- atoms in optical lattices provide also new grounds for tum mechanically) that in this limit our strategy pro- experiments on quantum chaos. vides the following recurrent relationships for the time Wedescribeatomsastwo-levelsystemswithtransition intervals ∆τ , successive values of the angular variance k u =< θ2 > 2O and normalized variance of the an- frequency ω0, interacting with a standing optical wave k k k gular velocity≈w =<( id/dθ)2 > /P2 : that is linearly polarized and has the frequency of ωl. k k − If the detuning ∆l = ω0 ωl is large compared to the − 3 relaxation rate of the excited atomic state, the internal temperatures. Itcanbeseen,thatafterthefirstfewkicks structure of the atoms can be neglected and they can be thelocalizationfactordemonstratesanegativepowerde- regarded as point-like particles. In this approximation, pendenceasafunctionofthekicknumber(astraightline the Hamiltonian for the atomic motion is in the double logarithmic scale). Although, the system demonstrates a reduced squeezing for higher initial tem- p2 H(x,p,t)= V(t)cos(2kx), (10) peratures, the slope of all of the curves in Figure 5 is l 2m − the same after the first several kicks, in full agreement where m is the atomic mass, k =ω /c is the wave num- with the general arguments of the previous Section (see l l ber of the standing wave. The depth of the potential Eqs.(9). Therefore, the accumulative squeezing scenario produced by the standing wave is V(t) = h¯Ω(t)2/8∆, may be an effective and regular strategy for atomic lo- l where Ω(t)=2d E(t)/¯h is the Rabi frequency, d~is the calization even at finite temperatures. z | | However, this does not mean that the accumulative atomic dipole moment and E(t) is the time-dependent squeezingistheonlyone(orthemosteffective)squeezing amplitude of the light field. In the case of rather short strategy. We have studied the best localization that can laser pulses, this Hamiltonian corresponds to that of the be achieved with a given number of identical δ-kicks, by two-dimensionalδ-kickedrotorconsideredintheprevious minimizing the localization factor Eq.(11) with respect Section, with θ = 2kx. In accordance with the previous l to the delay times ∆τ between the kicks. For clarity, discussion, many aspects of the dynamics of this system n we present here only the results for zero initial tempera- can be explained with semiclassical arguments. Here we ture of the atoms. Table 1 shows the best values of the provide results for a classical description of atoms in a localization factor found for up to five kicks, and com- pulsed opticallattice, with thermaleffects takeninto ac- pares them with the results of the accumulative squeez- count. ing strategy with the same number of kicks. While the WeperformedaMonte-Carlosimulationofthedynam- maximal atomic localization that can be achieved with ics of an ensemble of δ-kicked particles which were ini- twopulsesisalmostthesameforaccumulativesqueezing tially uniformly distributed in space and had a thermal andfortheoptimalsequenceoftwopulses,theoptimized momentum distribution. Figure 4 shows the spatial dis- results for three and more pulses are much better. tribution of atoms at different times after a single kick. Forillustration,wechoosethesequenceoffouroptimal In the upper row, the initial temperature of the ensem- pulses to visualize the dynamics behind the localization ble is zero, and we can observe focusing [Fig. 4(a)] and process. Figure 6 (a) shows the spatial distribution of formation of caustics [Fig. 4(b)] as it was described in the ensemble of atoms at the time of arrival of the sec- Section II. In Fig. 4(c) and (d), the average thermal en- ond pulse. Note that the second pulse is not applied ergyoftheinitialensembleiscomparablewiththeenergy at the time of the maximal localization. On the con- supplied by a kick. As a result, instead of sharp peaks trary, the optimized procedure finds it favorable to wait in the spatial distribution (like in Figure 4 (a) and (b)), after the focusing event, until the distribution becomes we observe some broader spatial structures that are still rather broad. The optimal four-pulse sequence we found reminiscentofthefocusingphenomenonandtherainbow requires applying the third and the forth pulses simulta- effect. neously,thus producinganeffective ”doublepulse”. The We examined the accumulative squeezing approach in spatialdistributionatthetimeofthecombinedthirdand application to the above system. The angular localiza- fourth pulses can be seen in Figure 6 (b). The distribution factor O =<1 cos(2kx) > introduced in Section l − tionisonlyslightlymorelocalizedthaninFig. 6(a),and II describes now the spatial width of atomic groups lo- only the last pulse (with the double strength) squeezes calized in the minima of the light-induced potential. In the ensemble at the time of the maximal localization, the classical limit, the localization factor after applying thus bringing most of the atoms to the optical lattice n kicks is given by minima [see Fig. 6 (c)]. 1 ∞ π O(tn)=1 dω0 dθ0ρ(ω0)cosθn, (11) − 2π Z−∞ Z−π IV. ORIENTATION AND ALIGNMENT OF A 3D where θ is determined by ROTOR n n−1 Under certainconditions,the processofmolecularori- θn =θn−1+∆τn ω0− sinθi!. (12) entation (or alignment) by laser fields can be described i=0 X byastronglydriventhree-dimensionalrigidrotormodel. Here, ω0 = 2klv0, where v0 is the initial velocity of an Althoughmanyfeaturesinthedynamicsof3Drotorsare atom, and ρ(ω0) describes the thermal distribution of similartothosealreadydiscussedforthetwo-dimensional the velocities. Figure 5 displays the amount of spatial case, there are two principal differences that we want to squeezing for the series of 100 kicks for different initial emphasize. The first one may be traced in the evolution 4 ofaclassicalensembleof3D-rotorsbeinginitiallyatzero plotted on a sphere. Red color corresponds to the high temperature. The probability of finding a rotor (driven probability density while the blue color presents a low by a linearly polarized field) at a certain solid angle el- probability. Initial isotropic ensemble corresponds to an ement sinθdθ is determined by the initial distribution uniform solid angle distribution [see Fig. 7 (a)]. In Fig- function, f0(θ0) as follows: ure 7 (b), the distribution is plotted at the ”focal time” definedaccordingto Eq.(6). In the two-dimensionalcase f(θ)sinθdθ =f0(θ0)sinθ0dθ0. (13) considered in the previous Sections, the distribution at the focal time is characterized by a sharp peak at θ = 0 Therefore, the probability density of finding a rotor at and a broad background for larger values of θ. In the the angle θ at some latter time is three-dimensionalcase,thedistributionhasanadditional f(θ)= f0(θ0a)sinθ0a , (14) sharpdipatθ =0,asdiscussedabove. Figures7(c)and dθ/dθa sinθ (d) show the angular distribution at t > tf. In Fig. 7 Xa | 0| (c) the formation of the ”corona” around θ = 0 can be where the summation is done over all branches of the seen. Inaddition,wecanobservearingofrelativelyhigh θ0(θ) function (see Section II). The probability density probabilitymovingfromthenorthpoletothesouthpole f(θ)hasasingularityifoneofthefactorsinthe denomi- of the sphere. This ring is a three-dimensional analog natoriszero. Asinthe2Dcase,thezeroesofthefraction ofthe rainbowstructuremodifiedby the thermaleffects. dθ/dθa give rise to the formation of the rainbow-like After the ring arrives at the south pole, a singular fea- 0 |structur|es. But in addition, the geometrical factor sinθ ture appears around θ = π, with a hole in the center can get zero too, which leads to the additional singular- caused the repulsive centrifugal force [Fig. 7 (d)]. The ities at θ = 0 and θ = π. This kind of singularities are formation of this sharp and robust structure in the an- responsible for the formation of the corona and glory ef- gular distribution is analogous to the glory effect in the fects in the optical and quantum-mechanical scattering wave optics. [26,27]. All the described phenomena, namely, focusing, caus- The second difference appears at finite initial temper- ticscreation,andaccumulatingsqueezingarenotspecific ature of the ensemble of 3D rotors. Because of the con- for the simplest models consideredabove, but are rather servation of the angular momentum projection onto the commonfeaturesthatcanbeobservedundergeneralcon- fieldpolarizationdirection,aneffectiverepulsivecentrifu- ditions of a strong excitation of the quantum rotor. In- gal force appears that prevents the rotors from reaching deed,forstrongenoughdrivingfieldonemayneglectthe the exact θ =0 and θ =π orientations. As a result, two initial rotational energy stored in the rotor, and use the holes in the angular distribution of a driven 3D thermal above quasiclassical ideas to describe its dynamics. The ensemble should be always present at θ = 0 and θ = π role of the initial state of the rotor (even thermally av- [34]. eraged) is reduced to the formation of a frozen classical- We present below the results of Monte-Carlo simula- like initial angular distribution of the rotational ensem- tion of the dynamics of a classical ensemble of three- ble. For example, for the dipole-type interaction, Eq. dimensional rotors (linear molecules) driven by a lin- (2), the angular focusing may be analyzed by consider- early polarized time-dependent field. The corresponding ing classical dynamics at small angles (θ 1): ≪ Hamiltonian is d2 +ǫ(τ) θ =0. (16) H = 2m1r2 p2θ+ sipn2φ2θ!−µE(t)rcosθ, (15) Here,τ =t¯h/I,a(cid:20)nddτǫ2=µE(t(cid:21))I/¯h2,andtheinitialcondi- tionisθ(τ0)=θ0,dθ(τ0)/dτ =0(whereτ0isanymoment where m is the reduced mass of the molecule, r is in the past before the beginning of the pulse). Focusing the (fixed) distance between the atoms, µ is permanent time τ is defined by θ(τ ) = 0. Because of the linear- f f dipole moment, θ and φ are Euler angles, and p and ity of Eq.(16), the position of the focusing times, and θ pφ are related canonical momenta. For the driving field number of focusing events do not depend on θ0, but are described by a δ pulse, the equations of motion can be determinedonlybythepropertiesofthepulseǫ(τ). This − easily integrated (see [35]). boundary problem may be solved analytically in several At zero initial temperature, p = 0 and p = 0, and special cases only (for δ-pulse considered above, or for θ0 φ0 theangleφ(t)isanadditionalconstantofmotion. Inthis a step-like ǫ(τ) dependence). In general, determination case, the dynamics of the system is reduced to that of a of focusing times requires numerical solution of Eq.(16). two-dimensional rotor, beside the effect of the geomet- Based on the previous analysis, we expect that a new rical factor described above. Figure 7 demonstrates the angular rainbow appears immediately after each focus- timeevolutionoftheangulardistributionofanensemble ing event. with finite initial temperature. Here, the probability of Figure 8 shows numerically calculated time-evolution finding arotorinside the solidangleelementsinθdθdφ is of a three-dimensional quantum rotor (linear molecule) 5 coupledto the externallaserfieldvia the anisotropicpo- [4] D. 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Conf. on Stochastic Behavior cently, the accumulated squeezing scenario [18] has been in Classical and Quantum Hamiltonian Systems, Como, realized experimentally in this system [36]. Moreover, Italy (Springer-Verlag, Berlin, 1977), p.334 the relatedsqueezingapproachesmayfindapplicationin [24] F.M.IzrailevandD.L.Shepelyansky,Dokl.Akad.Nauk atom lithography of ultra-high resolution [37,38]. In the SSSR249, 1103(1979); Sov.Phys.Dokl.24,996 (1979) caseofmolecules,theconsideredeffectsmayrevealthem- [25] Yu. A. Kravtsov and Yu. I. Orlov, ”Caustics, Catas- trophes and Wave Fields” (Springer Series on Wave selves in the angular distribution of fragments produced Phenomena, 15), 2nd edition, (Springer-Verlag, Berlin, byintenselaser-fieldmolecularinteraction. Themostdi- 1999). rectevidence canbe achievedina two-pulseexperiment, [26] K.W.FordandJ.A.Wheeler,Ann.Phys.7,259(1959) in which the first strong non-resonant pulse attempts to [27] M. 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Goldstein, Classical Mechanics (Addison-Wesley, [3] H.Ammann,R,Gray,I.Shvarchuck,andN.Christensen, Reading, 1980) Phys.Rev.Lett. 80, 4111 (1998) [36] M. Raizen (privatecommunication) 6 [37] I.Sh. Averbukh,patent pending Fig. 7. Time evolution of the angular distribution [38] forarecentreviewofatomlithographysee,e.g.aspecial of a (classical) ensemble of δ-kicked 3D rotors at finite issue on nanomanipulation of atoms, Appl. Phys. B 70, temperature. The probability of finding a rotor inside issue 5 (2000) (D.Meschede and J. Mlynek eds) the solid angle element sinθdθdφ is plotted on a sphere. Picture (a) shows the distribution at τ = 0 (time of the Figure Captions: kick). Figures(b),(c),and(d)correspondtoτ =τ ,τ = f 3.3τ and τ =5τ , respectively. The initial temperature f f Table 1. The table shows the minimal value of the lo- corresponds to k T =(1/100)µ2r2[ ∞ (t)dt]2/I2. calization factor that can be achieved with a fixed num- B −∞E berofkicksusingaccumulativesqueezingscenario(O ) R acc Fig. 8. Contour plot for the time-dependent and optimized sequence of pulses (Oopt). angular distribution function 2πsin(θ)Ψ(θ,τ)2 of a | | three-dimensional rotor (molecule) subject to a strong Fig. 1. Angular distribution of a quantum rotor ex- ”polarization-type” interaction, Eq.(17) with a Gaus- cited by a strong δ-kick (P = 85). The graphs corre- sian pulsed laser field: 2(t)(α α )I/4h¯2 = 3 spond to (a) τ = 0.5τ , (b) τ = τ , (c) τ = 2τ , (d) E k − ⊥ × f f f 103exp[ (τ/0.01)2]. Here τ = t¯h/I. The molecule re- τ =τf+Trev/2,(e)τ =τf+Trev/3,(f) τ =τf+Trev/4, − sides initially in the isotropic ground angular state (J = (g) τ = 1.8τ +T /2, (h) τ = 1.8τ +T /3, and (i) f rev f rev 0,m=0).Angularfocusingandrainbowsemergingfrom τ =1.8τ +T /4, respectively. f rev each of the focal regions can be seen. The angular rainbows are made of moving rings located symmetrically Fig. 2. Classical map representing the final angle θ as with respect to the equatorial plane. afunctionofinitialangleθ0 for(a)τ =0.5τf,(b)τ =τf, (c) τ =3τ , and (d) τ =10τ . f f Fig. 3. Accumulative angular squeezing. Graphs are shown in double logarithmic scale. Fig. 4. Spatial distribution of a classical ensemble of atoms after a single δ-kick. In Figs. (a) and (b), the initialtemperature,T oftheensembleis zero. In(c)and (d), k T = (1/9)k2[ ∞ V(t)dt]2/m, where V(t) is the B l −∞ depth of the potential produced by the standing wave, R andk is Boltzmannconstant. Figures(a) and(c) show B the spatial distribution at focal time τ = τ , while in f figure (b) and (c) τ =2.5τ . f Fig. 5. Accumulative squeezing of atoms in a pulsed optical lattice at finite temperature (classical description). The minimal localization factor as a function of the kick number is shown in double logarithmic scale. The solid line corresponds to zero initial temperature. The dashed and dotted lines corre- spond to k T = (1/9)k2[ ∞ V(t)dt]2/m and k T = B l −∞ B (4/9)k2[ ∞ V(t)dt]2/m, respectively. l −∞ R R Fig. 6. Spatialdistributionfortheoptimizedsequence of four δ-pulses. The upper row shows the spatial distribution averagedover100 atomic ensembles (eachcon- taining 5000atoms), the lowerfigures show the distribution of atoms in one of the ensembles. In (a), the distribution is plotted at the time of the second pulse, that is delayed by τ =3.02τ with respect to the first pulse, f in (b) - at the time of the (combined) third and fourth pulsesdelayedbyτ =1.35τ afterthe secondpulse. 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