Controlling Quantum Rotation With Light

16 Pages·0.23 MB·English

by  I. Sh. Averbukh

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 Nucl´eaires, F 38026 Grenoble Cedex, France
e-mail: [e-mail]
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 increas-
ing 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 focus-
ing. 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 cer-
tain 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 distribu-
tion 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. Normand, L. A. Lompre, and C. Cornaggia, J. Phys.
larization interaction. The time averaged value of this B 25, L497 (1992)
interaction is (see, e. g. [21,9]) [5] P. Dietrich, D. T. Strickland, M. Laberge, and P. B.
Corkum, Phys. Rev.A 47, 2305 (1993)
V(θ,t)= 1/4 2(t)[(αk α⊥)cos2(θ)+α⊥] (17) [6] G. R.Kumar et al.,Phys. Rev.A 53, 3098 (1996)
− E −
[7] J. Karczmarek, J. Wright, P. Corkum, and M. Ivanov,
Here α and α are the components of the polarizabil-
k ⊥ Phys. Rev. Lett. 82, 3420 (1999).
ity,parallelandperpendiculartothemolecularaxis,and
[8] B. A. Zon and B. G. Katsnelson, Sov. Phys. JETP 42,
(t) is the envelope of the laser pulse. In this case there
E 595 (1976)
are two opposite equilibrium directions θ = 0,π, so the [9] B. Friedrich and D. Herschbach, Phys. Rev. Lett. 74,
field aligns the molecules but not orients them. The ap- 4623 (1995); J. Phys. Chem. 99, 15686 (1995).
plied pulse has a Gaussian shape of a finite duration (i. [10] W. Kim and P. M. Felker, J. Chem. Phys. 104, 1147
e. it is not a δ-kick). As seen from Fig. 8, both focusing (1996)
andcausticsformationoccurinthesystem,similartothe [11] C. H. Lin, J. P. Heritage, and T. K. Gustafson, Appl.
previousconsideration,withthedifferencesattributedto Phys. Lett. 19, 397 (1971); J. P. Heritage, T. K.
another symmetry of the problem. For instance, the an- Gustafson, and C. H. Lin, Phys. Rev. Lett. 34, 1299
(1975).
gular rainbows are made of moving rings located sym-
[12] L. Fonda, N. Mankoˇc-Bor´stnik, and M. Rosina, Phys.
metrically with respect to the equatorial plane. Note,
Rep. 158, 159 (1988)
that the glory and corona are not seen because of the
[13] P. M. Felker,J. Phys.Chem. 96, 7844 (1992)
sin(θ) factor incorporated into the distribution function
[14] T.Seideman,J.Chem.Phys.103,7887(1995);ibid.106,
shown in Fig. 8.
2881 (1997)
[15] T. Seideman, Phys.Rev.Lett. 83, 4971 (1999)
[16] J. Ortigoso et al.,J. Chem. Phys.110, 3870 (1999)
V. CONCLUSIONS [17] L. Cai, J. Marango and B. Friedrich, Phys. Rev. Lett.
86, 775 (2001)
The predicted effects may be observedin awide range [18] I. Sh. Averbukh and R. Arvieu, Phys. Rev. Lett. 87,
ofsystemswithstronglydrivenrotationaldegreesoffree- 163601 (2001)
dom. Possible examples range from heavy-ion collisions [19] M. Leibscher and I. Sh.Averbukh,submitted,2001
[20] I.Sh.Averbukh,R.Arvieu,andM.Leibscher,tobepub-
(when highly excited wave packets of nuclear rotational
lished
states are produced [12]) to molecules subject to strong
[21] R. W. Boyd, Nonlinear Optics, Academic Press,
laser pulses, and cold atoms trapped by standing light
(Boston,1992)
waves. The spectacular features described in this pa-
[22] I.Sh.AverbukhandN.F.Perelman,Phys.Lett.A139,
per may be observed in the spatial distribution of an 449 (1989); Sov. Phys.JETP 69, 464 (1989)
atomic ensemble driven by pulsed optical lattices. Re- [23] F. M. Izrailev et al., Proc. 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. V.Berry, Adv.Phys. 25, 1 (1976)
orient (align) the molecular ensemble, while the second [28] S. D. Bosanac, J. Chem. Phys. 95, 5732 (1991)
short delayed pulse creates fragment ions. [29] P.S.JessenandI.H.Deutsch,Adv.AtmMol.Opt.Phys.
37, 95 (1996)
[30] M. B. Dahan, E. Peik, J. Reichl, Y. Castin, and C. Sa-
lomon, Phys. Rev.Lett. 76, 4508 (1996)
[31] S. R. Wilkinson, C. F. Bharucha, K. W. Madison, Qian
Niu,andM.G.Raizen,Phys.Rev.Lett.76,4512(1996)
[32] Qian Niu,Xian-GengZhao,G. A.Gerogakis, and M. G.
[1] F. Haake, ”Quantum Signatures of Chaos”, Springer- Raizen, Phys.Rev.Lett. 76, 4504 (1996)
Verlag, (Berlin, 1991) [33] R.Graham,M.Schlauutmann,andP.Zoller,Phys.Rev.
[2] F. L. Moore, J. C. Robinson, C. F. Bharucha, B. Sun- Lett. 45, R19 (1992)
daram, and M. G. Raizen, Phys. Rev. Lett. 75, 4598 [34] B. A.Zon, Eur. Phys. J. D8, 377 (2000)
(1995) [35] H. 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 rain-
bows 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 de-
scription). The minimal localization factor as a func-
tion of the kick number is shown in double logarith-
mic scale. The solid line corresponds to zero ini-
tial 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 dis-
tribution averagedover100 atomic ensembles (eachcon-
taining 5000atoms), the lowerfigures show the distribu-
tion of atoms in one of the ensembles. In (a), the dis-
tribution 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. Fig-
f
ure(c)showsthedistributionofatomsatthetimeofthe
maximal squeezing.
7
Noof kicks Oacc Oopt
2 0.33 0.31
3 0.26 0.20
4 0.21 0.11
5 0.18 0.07
TABLEI.
8
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T
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T
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