PHYSICAL REVIEW A VOLUME 51, NUMBER 3 MARCH 1995

Semiclassical theory of Rydberg-wave-packet interferometry

Mark Mallalieu* and C. R. Stroud, Jr.The Institute of Optics, University of Rochester, Rochester, Nevi York 14627

(Received 28 June 1994)

A semiclassical approximation is derived for the autocorrelation function describing the excitationand detection of Rydberg wave packets by pairs of phase-locked laser pulses. The resulting expressionis in terms of a sum over Kepler trajectories, and provides a direct explanation of the periodicityand spreading of the wave packet at short times when the evolution is classical. More surprisingly,this solution also provides an accurate description of the complex revival behavior of the wavepackets, which is nonclassical. The phases of the wave packets, which are detected by the pulse-locking scheme, are also accurately given by analytical expressions derived from the semiclassicalsum. The phases of the wave packets at short times have a simple interpretation as the phases ofBohr-Sommerfeld, and are related to Berry's phase. The semiclassical expression for the phases ofthe nonclassical revival wave packets is simpler than the corresponding quantal solution.

PACS number(s): 03.65.Sq, 32.80.Rm

I. INTRODUCTION

Wave-packet states in Rydberg atoms are a matter ofcurrent experimental and theoretical study, due to theirusefulness for studying the classical limit. Such statesare deemed quasiclassical because the uncertainty in oneor more coordinates of the electron can be reduced toclose to the minimum value, leading to subsequent be-havior that has many characteristics in common with theclassical system. The simplest of these states is the ra-dial wave packet excited with a short laser pulse with noother external fields present [1—7]. These states are welllocalized in the radial coordinate due to the superposi-tion of many eigenstates with different principal quantumnumbers. However, they are not localized in the angularvariables because the states in the superposition all havethe same angular momentum. This can be overcome byusing external fields to mix the angular states. An exam-ple of this is the angularly localized wave packet [8] thatis excited with a laser pulse in conjunction with a radiofrequency field. These wave packets are localized only inthe angular variables as the states in the superpositionall have the same principal quantum number. A methodto prepare a wave-packet state well localized in all threedimensions has been proposed by Gaeta et al. [9].

The primary issue regarding these quasiclassical statesis the relationship between their dynamics and some un-derlying set of classical trajectories. The ionization ofthe angularly localized wave packet studied by Yeazelland Stroud [8] was properly accounted for by modelingthe ionization of an ensemble of Kepler trajectories withthe same energy and angular distribution as the wave

Present address: Department of Chemistry, University ofKansas, Lawrence, KS 66045.

packet. Turning this approach on its head, Gaeta et al.[9] developed a method to create three-dimensionally lo-calized wave packets by considering the effects of a shortelectric pulse on an ensemble of circular orbits intendedto model the initial circular-orbit eigenstate. The com-mon feature in both of these models is that the dynam-ics of the ionization process and wave-packet formation,respectively, occurred on a relatively short time scale.The dynamics of the radial wave packets, as well as wavepackets localized about a circular [10] or elliptical [ll] or-bit, can also be understood on short time scales by usingan ensemble of classical trajectories with the appropri-ate initial conditions. For a number of orbital periods allof the trajectories remain close to one another as theytravel around their Kepler ellipses, except when they ap-proach close to the nucleus where small differences in ini-tial conditions are magnified. However, this model breaksdown at longer times because small initial uncertaintiesin position grow with time due to the nonlinearity of theCoulomb potential. After a number of classical periods,the wave packet spreads so much that it is delocalizedabout the classical orbit, and quantum mechanical in-terference effects become important [10—12]. These in-terference effects can be accounted for with the use ofsums over classical-path amplitudes [13,14], maintainingthe close connection between the wave-packet dynamicsand the corresponding classical ensemble.

A most interesting phenomenon occurs at times whenthe delocalized wave-packet state revives back into a well-defined wave packet, or into a set of symmetrically dis-tributed wave-packet replicas [10,12]. The revivals intomultiple copies of the initial wave-packet state occur atcertain &actions of the period for a full revival, and thusare called fractional revivals. During these &actional re-vivals, the wave function consists of a macroscopic su-perposition of identical wave-packet states which are dis-tinguishable. The coherence between these &actionalrevival wave packets can be expressed by the superpo-sition coefBcients which have been calculated by Aver-bukh and Perelman [12]. The phases of these amplitudes

1050-2947/95/51{3)/1827{9)/$06.00 51 1827 1995 The American Physical Society

1828 MARK MALLALIEU AND C. R. STROUD, JR. 51

can be measured using sequences of phase-locked pulses[15]. This technique involves a wave-packet interferome-try [16,17]. The various laser pulses excite identical wavepackets at different times, and the interference among thewave packets determines the final signal. The nature ofthe interference depends upon the evolution of the firstwave packet during the time before the second pulse ar-rives, and upon the phase relationship between the twopulses. It is this dependence on the optical phase whichcan be used to measure the fractional wave-packet phases[18,19].

One of the purposes of this paper is to relate thesefractional wave-packet phases to the properties of the rel-evant classical trajectories. This is accomplished with asemiclassical theory based on the classical-path represen-tation (CPR) of Alber and Zoller [1,3,20]. This method isuseful for studies of Rydberg wave packets excited &omthe ground state by short laser pulses, because it includesthe effects of laser polarization and the nonhydrogenicquantum defects in alkali-metal atoms in a natural way.In Sec. II, we derive the CPR of the phase-locked pulseprocess. This is used in Sec. III to show that the phaseof the wave packet as it evolves at short times can berelated to the classical action of the mean-energy trajec-tory while the wave packet remains well localized. A so-lution for the semiclassical sum which is accurate for longtimes is next obtained in Sec. IV, providing a basis forinvestigation of the fractional revivals. This solution issimilar to that obtained for the autocorrelation functionof one-dimensional [13] and three-dimensional circular-orbit Rydberg wave packets [14], which use the semiclas-sical theory of Tomsovic and Heller [21]. Analysis of theaccurate sum in Sec. V reveals that the fractional re-vivals are correlated with discrete sets of Kepler orbits.This relationship between the various sets of orbits andthe different wave-packet &actions includes the locationsof the peaks and their phases. We show how the probelaser pulse can be adjusted to selectively emphasize oneor another of the &actional wave packets.

II. CPR FOR PHASE-LOCKED PULSEMEASUREMENTS

The classical path representation is discussed exten-sively in the review of Alber and Zoller [1]. Their gen-eral theory handles more complex alkali atoms as wellas hydrogen, and the extension of the following anal-ysis to those cases is straightforward. This semiclassi-cal approach has been used previously to treat Ramantransition pump-probe processes. In that case, the re-sponse is proportional to the magnitude of a semiclassicalsum. We apply the method here to a pump-probe processwhich uses identical pump and probe pulses. This phase-sensitive pump-probe process has some advantages withregards to the signal obtainable in experiments, providedthat the relative phase between the two pulses is prop-erly controlled. When this is implemented by using pairsof phase-locked pulses, the evolution of the phase of thewave packet can be measured. In the sections following

this one, we show that this phase has an interesting con-nection to the semiclassical dynamics of the wave packet.First, we give the results of this measurement technique,then we derive its classical-path representation.

We consider excited states created by laser pulseswhich are tuned from the atomic ground state ~g) to anenergy e in the Rydberg series. If the laser pulse is muchshorter than the classical period of a classical electronwith energy e, a number of Rydberg states ~n) are ex-cited. This can been seen from the fact that the energylevel spacing of the Rydberg levels is inversely related tothe Kepler period of a classical electron at that energy.For a laser pulse which arrives at time ti, the excitedwave packet is accurately represented by

@wp(r, t, ti) = i exp(iu„ti) exp( —iet) ) 0 E(b )

x rp„(r) exp [—ib„(t —t, )], (1)

provided that the pulse is weak in the sense that most ofthe population remains in the ground state. The Rydbergwave functions are given by p„(r) = (r~n). The carrier&equency of the laser pulse is denoted by ~„, and peakfield strength and polarization are given by the vectorgo. The matrix elements of the field interaction betweenthe ground and excited states are designated by the Rabifrequencies 0 = (n~d. ge~g), where d is the dipole oper-ator. The Fourier transform of the laser pulse envelopef(t) is given by E(b ), and b„= e'„—e is the difFerencebetween the mean energy c = w„+ r~ and the energylevels e = —1/(2n2) (atomic units).

A second laser pulse, identical to the first, arrives attime t2 and excites another wave packet which is thesame as the first wave packet, except for the evolution ofthe first wave packet and a phase factor which accountsfor the optical phase difference between the pulses. Thefinal population in the Rydberg series after the secondpulse has passed depends upon the interference betweenthe two wave packets. This population can be measuredwith a swept electric field that ionizes all excited states.The measured population is P(tg) = P, + P~(tg),which depends only on the delay between the two pulses,tg ——t2 —ti, not the absolute times ti and t2. The averagepopulation P is the incoherent sum of the population ofthe two wave packets and the interference population P~depends on the interference between the two wave pack-ets. Details of the calculation will be published elsewhere[15], we give the important results here. Assuming thatthe two laser pulses are phase locked, the contributionto the population due to the interference of the two wavepackets is given by Pc (tg) = P „,CI, (tg, est, Pg), with thephase locked autocor-relation function defined by

cc(ta age,'4e) = «',(&(&a; &ge) exp( ice) ). (2)

Here &Pg is the relative phase difference at which the twolaser pulses are locked and up is the &equency at whichthe phase is locked. This &equency does not need tocoincide with co&, but is assumed to be within the laserbandwidth. The phase-locked autocorrelation functiondepends on the optical phase Pg and the autocorrelation

SEMICLASSICAL THEORY OF RYDBERG-WAVE-PACKET. . . 1829

function of the wave packet in the rotating frame at e~g ——

(dg + E'g)

(@wp(0; &ge) l@wp(tz , &g'r))

The wave packet in the rotated frame is given by

gwp(r, t; est) = i ) B„P(6„)p„(r)exp[ —i(s„—age)t] .n

(4)

It would seem natural to choose the rotated &arne tobe at c, but as we shall see in Sec. V there are otherappropriate rotated &ames. The peaks of the autocorre-lation function in Eq. (3) have the same form as the pulseenvelope f(t) so long as the wave packet remains well lo-calized, as we shall see in the next section. However, thephase of the autocorrelation function at its peaks is verysensitive to its mean energy. This phase has a simplesemiclassical interpretation, and will also be discussed inthe next section.

We now find the classical-path representation [1,3,20]of the phase-locked pulse excitation. Using the resolventG+(s) = [s —H + i0] in Eqs. (3) and (4), the wave-packet autocorrelation function becomes

trajectories starting at the surface r corresponding tothe-outgoing Coulomb waves at energy c is propagatedusing classical equations of motion [24,25]. The incom-ing Coulomb waves are then expressed in terms of thesemiclassical propagation as a sum of classical-path am-plitudes involving trajectories which return to the core.The amplitude of the incoming wave due to a particulartrajectory is a product of the amplitude of the initial out-going wave at the point of departure &om the surface rand a semiclassical amplitude. This semiclassical ampli-tude has a magnitude which depends upon the stabilityof the trajectory versus initial angle of departure fromr, and represents a density of nearby trajectories. Thephase of the semiclassical amplitude depends upon theclassical action and Maslov index of the trajectory. TheMaslov index is related to the phase accumulated by thewave when it travels thru a caustic point.

The total wave function is then found by matching thisincoming wave with the wave function generated by thelaser. The self-energy is calculated &om this wave func-tion, and is now represented as a sum of classical-pathamplitudes. For the case of hydrogen with no externalfields, the classical-path representation of the self-energyis

exp[iSi (s)]1 —exp iSi(s)

C(td eee) = e(eep „,) exp(ieeeee) f de ~p(e —e)~

xZ(s) exp( —istic). (5)

= —il'(s) ) exp ikSi(s),k=1

(6)

The self energy of the -initial state lg) is given by Z(s) =(gld. goG+(s)d (elg) and is a two-photon transition ma-trix element &om the perturbation theory of two-photonprocesses. Although the transition &om the ground stateto the Rydberg series under consideration is a single-photon transition, the pump-probe process itself is a two-photon process because there is interference between thepossibility of an excited state absorbing a photon fromthe first or second laser pulse. The self-energy is re-lated to the wave function lU, ) = G+(s)d. golg) which isseen to satisfy the inhomogeneous Schrodinger equation(s —H —i0) lU, ) = d.golg). Once this equation is solved,the self-energy is readily obtained.

The state lU, ) is determined in several steps. Atsmall values of the radius, the wave function is given byCoulomb functions. For atoms with a complex core, theboundary conditions are matched at r ~ 0 with quan-tum defect methods [1],whereas for hydrogen a standardGreen's function method sufEces. At this step the outerboundary condition is satisfied by an outgoing Coulombwave which takes into account the fact that the wavefunction is being excited by a laser. At some suitableintermediate radius (r 50ao), which is outside theatom-laser interaction region, but still small on the scaleof the classical orbits at the energy being considered, theWKB form of the Coulomb functions can be used.

At this point the theory of Maslov and Fedoriuk[22,23], which uses a semiclassical approximation to theGreen's function G(q, q', s'), can be applied to the prop-agation of the wave function. An initial manifold of

when we restrict the two laser pulses to be well separatedfrom each other. [1] The energy normalized Fermi GoldenRule (FGR) transition rate I'(s) is closely related to thedensity of states and the square of the Rabi &equency0„ for energies e = e . It is a smooth extrapolation ofthe FGR ionization rate above threshold, and is approxi-mately independent of energy [1]. The classical action ofa trajectory at energy e after one Kepler period is givenin atomic units by Si(s) = $p . dq = 27rv. The actionvariable v is related to the energy by s = —1/(2v ), andcorresponds to the principle quantum number n of thequantized hydrogen atom. This expression has poles forinteger v = n; thus, it can be directly related to Bohrquantization.

The solution for the self-energy in Eq. (6) can now beput into Eq. (5) to get the semiclassical approximationto the autocorrelation function, C(tg, sgr) = C„(tg', sgr).Taking the laser pulse to have the Gaussian form f (t) =(sr'. 2) i) 4 exp( —t2/2r2) and pulling the FGR transitionrate at the mean energy I'(s) out of the integral resultsin

Csc(td& ssr) — exp(issstd) ) exp I—(E —s) 7„

+iS), (s) —istic] ds, (7

where we used P, —21'(s'). All that remains to bedone is to evaluate the integral. This requires makingan approximation to the classical action Sg(c) = kSi(e').For short times an approximation about the mean energy

1830 51MARK MALLALIEU AND C R STROUD JR

of the wave packet is adequate, as we

Wong imes a different a

n a previous solution-h thd

ernative approach is sie e integral. This a

c is similar to the useGga e aussian wa

an leek —Gutz wilier rop

a wi er range of astationary-phase solution

parameters than th

III. MEA SUREMENT OF THEBOHR-SOMMERFELD PHASES

The EhrEhrenfest regime of the wave-dfi d h se imes for which the wav

1OIl is

lllo li d bo t th

b'

hrate

' nas e wave packet and

theb 1 de y etermined b a

b hou e trajector at

ha

ergy.

S1Inimple semiclassical ice e' '

nof t

in erpretation as thethe mean-energ tra

e classical action

dli tl 1 t dt th B hr-gy rajectory. This cl

o e o -Som

K 1 b h T y o serves to put thj' into solvable form. T

e inte-e orm. The expansion of the

some energy rp is

s,(.) = s,(„)+(,-„)""(')BG'

(s —so)' O'Sg(s)2 0 E'

E' = 8'p(8)+ o ~ ~

xt e mean-ener Kgy Kepler period, with a s ng

in Fie y & s). This resu

].— mechaa e integration of the

anical expression btained &om

The imimportance of Eq. 10 is t p

in th ld he o -Sommerfeld action

mec anics. T pr e action was shown to

o e

8 ' h [2728] bli 'o f Ah

Be ', y Little'ohn

nique provided that the 1e pu ses are

termin a measurement of the rea

th ' E . (lo).If t e laser is tuned. to an ener suc

q.gy

wi appear just asg

or noninteger v the r skabl ho

' F' ponses shown t, and 100.75 w

in egration of Schr-sul 0 a

(lO) Ho thsim f d t d th

d 100 75 th h d1ocked autocorrel te a ion unction ur

ac ion makes the ph ase-

first return to thn purely imaginary f th

sum of the interferenceresponse, which is the

r erence population and thee average pop-

Prom elelementary classical mechanics

) .()=The second dn erivative of the t'e ac ion is given

I I I

1I I I I I I ~

k BT(s) 3kv T, (s) —= („(s—),

and determines thsmall changes

s e spreadin ratein energy about e. So lo

g e of trajectories f

k t lllo loca ized, it is validho e action ~8~ ab

isyiedsasim leG ' a w icp g

,

1

4LI (I I I I I I I I

0 1 2 3 4 5 6 7

t/T, (K)

I I I

8 9 10

&-(t', ) = p['(, --)t '~~

—e td„ I~ —&4(&)/&,'l' '

[tg —kT, (E)j2 )4[7.2 —z',

(lo)a( )j)

This describes a series ofries o peaks centered at 1mu tiples of

FIC. 1. Accuccuracy of central-orb

l to f to fo th fir e rst ten Kev= 00witha us

or a aser

representation.

SEMICLASSICAL THEORY OF RYDBERG-WAVE-PACKET. . . 1831

pU II

II

yl

)Ck)~ e

U p

I I

2 3

t/T (F)4 5

FIG. 2. Wave-packet recurrence peak phase dependenceon mean energy. Accurate numerical integration of theautocorrelation function for mean energies correspondingto (a) P = 100 (solid line), 100.5 (dashed line), and(b) P = 100.25 (solid line), 100.75 (dashed line). The depen-dence of the phase on the central-orbit energy is in completeagreement with that predicted by the semiclassical theory.

IV. MULTIPLE PATHS IN THE NONCLASSICALREGIME

The breakdown of the solution in Eq. (10) is due to theapproximation involved in evaluating the integral Eq. (7),

ulation, is identically zero. At this energy there will be asign change every period, which creates the appearancein the response of a wave packet with twice the periodthat it should have.

Solutions to the classical-path representation obtainedby linearizing the dynamics about the mean energy havebeen used previously to describe the pump-probe re-sponse for Rydberg atoms in external fields, using dif-ferent pump and probe pulses [1,20,26]. In those cases,the nonclassical nature of the semiclassical phases mani-fests itself when different classical-path amplitudes inter-fere with one another. This occurs because closed orbitsleaving the atomic core at different angles can return tothe core at the same time, but having different values ofclassical action. In light of the results of this section, thephase-locked pulse technique could provide more detailabout the properties of such systems. Even in the caseof no external Gelds, however, interferences between dif-ferent classical-path amplitudes occur. As we see in thenext section, this in fact defines the nonclassical regimeof the radial Rydberg wave-packet evolution. To investi-gate that requires a new approach to the solution of theintegral in Eq. (7).

not the semiclassical method itself. The form of this in-tegral suggests the use of the stationary-phase methodwhen td becomes large enough, and that is the approachused previously by Alber et aL [3] and Alber and Zoller[1]. However, they did not use this result to investigatethe fractional revivals of the wave packet, as we shall inthe next section. Furthermore, a little reHection on theresults of the last section and the relevant classical dy-namics leads to an improvement which is good for bothlong and short times.

The classical analog for the excitation of a wave packetby a laser pulse is obtained by launching a classical en-semble of Kepler electrons &om the nucleus, with thestarting times of the members of the ensemble distributedabout the arrival time of the pulse. The energy distribu-tion of the classical ensemble must also mimic that of theexcited wave packet. The classical ensemble spreads as itevolves, due to the variation of the Kepler period T, (e)with energy e. At long enough times, the faster trajecto-ries will lap the slower ones. At this point the ensemblewill be spread entirely about the classical orbit. It makessense to consider only the subset of trajectories whichare in the vicinity of the nucleus when the second pulsearrives, as these are the ones that should be relevant tothe response. This follows from the dependence of theresponse on the wave-packet location when it is still welllocalized.

The trajectories in the vicinity of the nucleus whenthe second pulse arrives will be very close to an integerrepeat of their periods, so it is useful to account for thecontributions of all the trajectories near a given repeat oftheir periods in terms of the one which exactly repeats itsperiod at time td, . The kth reference trajectory is, thus,determined by the condition k7g = t~, where 7g = T,(sI, )is the orbital period of this trajectory. The reference tra-jectory energies eA, satisfy the stationary-phase conditionfor the integral in Eq. (7). The action variable of the kthreference orbit is

q2~k jThe classical actions of the nearby trajectories are de-

termined by the Taylor series given in Eq. (8), evaluatedat energy s&. Using this in Eq. (7) and performing theGaussian integral as in the last section results in

ein/4 ( (&s &)2 yC,.(&g., sgr) = exp(isget~) ) exp ~—

)

x exp (i8~ (tq) —i sf, td),

where Q = 1/r„+ i/g& and g&~—(&~(e&). The clas-

sical action of the kth reference orbit is denoted bySA, (tg) = Sg(sl, ). We note that the phase in the right-

most exponential is the full action 7Zq(td) = I "dtCq =SI,(tg) eI,td, where Zl, is t—he 'Lagrangian of the kth refer-ence orbit. This action plays a role in the time-dependentsemiclassical theory used previously [13,14] to calculatethe autocorrelation functions of Gaussian wave packetsin the Coulomb potential. The result in Eq. (12) is of

1832 MARK MALLALIEU AND C. R. STRgUD, JR.

1.0

l4)

0.5

0.0

(b)-0.6

0.4

0.2I

1614 15 17

FIG. 3.&/v, (e)

. 3. Accuracy of reference-tation for a w

re erence-orbit classical- a

part of the axcie y a ion u

ca cu ated with an accu ca integra-

h f 'b' h lfe a revival.

C..(t„;e,) = C, t;,&I g ~ st dI egr)i~/4= e exp(ie'grtg)

)Xgk

exp —8' 7

x exp[iRI, (tg)],

whicic is identical to tto the i

o the stationar — ha [ ]~ g

the ne next section for st dwill prove useful

'

th & t'ac ional revivals.

r I I

0.25

OOO -4

0.50

thhe same form as thoem ployed is diferent H

~ though the meth das ose solution

reduced action i den . enceforth

me o

, in or er to kee the, we shall call 8 t

uan a ion in i s. 3accurate

g.a revivals the s

accurate. Th'e semiclassical solut'

nonclais result can be used to

so u ion is very

1 fh is

ri ntxon in Eq. (12 . T1 1 h

t b b'1 t d'

e interference bet's complete con-

Eq. 12 .e ween the terms i

un er ying

n elimitast m oo tallowin th

oo t e condition w

g e approxirnatE . (12) 1(i

V. CORRESPONDENCE BFRACTIONAL W P

DISCRETEWAVE P

ETE KEPLER ORBITS

050 -'

)~ 0.25

0.00

(b)Thee wave-packet 1

when it undevo ution '

1 10is especiall

a iona actions of thes o e revival period

VT,. = T(e)—

-0.25I

2218 20 21

&/v

FIG. 4.

'rc ( ~). 4. Accuracy of reference-

tation for a we erence-orbit classical-

1t

autocorrelation funct'pulse. The real

1- th(1 o h 1ca culated with an acc

s e ine com-ica sntegra-

f ' (b' near the two-thre a in the

o- xr s revival.

The full actions can be ap rox'

h '1 1 poe re ationshi

e y revealinip between the Ke le

e actional revivals of tep er trajec-

ld k ftreeatment here h h

s oo ar astrae. s

e w ic nonethely, we provide 1

tant features.e ess illustrates the '

a simp er

The auto ' ea s o t

es e impor-

e autocorrelation peaks of t ek fh

m ave maximum cmsint e

constructive interfer-

SEMICLASSICAL THEORY OF RYDBERG-WAVE-PACKET. . . 1833

ence. For simplicity, we consider &actions of the revivalperiod such that f = 1/my, where my is an integer. Tofacilitate the solution for the times of total constructiveinterference, it is useful to have an expression for the ac-tion of each reference trajectory relative to that of onewhich is close to the mean energy. The first step is to Gnda time t~ = kyT, (E ) at which the mean-energy orbit itselfis a reference orbit. To keep this time within a Keplerperiod of the &actional revival time, a convenient choicefor the integer kf is

Trev

T, (s) 3m'

where the Hoor brackets denote the largest integer smallerthan the quantity enclosed. For times td, g kyT, (s) themean-energy orbit is not a reference orbit, but the energyof the reference orbit given by ky7~ = t~ is close enoughto the mean energy that we can safely expand about itrather than the mean energy, for times within a classicalperiod of fT„„

The action variables v& of the nearby reference orbitscan be found &om that of the kf th reference orbit withthe help of 7i, = 2vrv&s = T~&[1+ 3(vi, —vi~)/vt ~]. Thisis used along with k7i, = kgb~ to get

n= [vJ . (20)

Across the interval kyT (s )( tg-( (kg+ 1)T,(e'-), thereare mf equally spaced times at which the reference orbitswill satisfy Eq. (19). This can be seen from the referenceorbit spacing given by Eq. (16) and a simple linear ex-trapolation of the orbits vA,

&across the interval. The

&actional revival autocorrelation peaks are centered at

4 =Ikf+ + IT(&=)

t' v. j &

2m' mf )j = 0, 1, 2, . . . )mf —

) (21)

corresponding to the reference orbits,

course the Bohr orbits. Although the role of the Bohr or-bits might be expected, the role of the half-integer orbitsis perhaps surprising.

In view of Eqs. (19) and (16), the fractional revivalpeak locations will be more clearly expressed in terms ofthe periods of a given Bohr orbit near the mean-energy,rather than the periods of the mean-energy itself. Theparticular Bohr orbit used for the time scale is chosen tobe

vA, —vI, = —lmf, (16)1.0 - '

where t = k —ky. The assumption (vy —vy&) (& vi, zwas used. The expression (16) can be used to get anapproximation to the reference trajectory energies e& byexpanding them about e&,f'

(vi —v~, )I'ev'

(17)

The product (s& —s& )t~ simplifies using the definitionky

tg/7i, ~= ky and the fact that the ratio t~/T„„= 1/my

does not change significantly for times within a classicalperiod or so of T„„/my. The action 81, = 2vrkvA, is easilyfound in terms of Eq. (16), and after a little algebra weAnd

0.5-CL)

00~ ~~ ~

I

27

0.8

IC

0.4-

0.0-

29.5

I

28

30

29I

30

30.5

mfRg( tg) ~ 27l kgb& —'Ei td + 27rl vA,

&+

(mod 2vr). (18)

The first two terms on the right hand side of Eq. (18) areindependent of L. For maximum constructive interferenceto occur between the contributions in the sum in Eq. (13),the terms which depend upon l must be a multiple of 2a.This leads to the condition

0.6-IC

0.3-

0.0-

39.83 40.17

&/'T, (8-„)

40.50

vi„+ = 0 (mod 1) .2

This condition on the reference-orbit action variables forthe &actional revival peaks is reminiscent of Bohr quanti-zation for the energy levels. If my is even (odd) then vgamust be integer (half integer). We call the orbits withaction variable an odd multiple of 1/2 the half-integerorbits, and the orbits with integer action variables are of

FIG. 5. Insensitivity of revivals to small variations in en-ergy. The real part of the autocorrelation function is shownnear the (a) first full revival, (b) one-half revival, (c) one-thirdrevival. In all cases the revival peaks are centered where ei-ther the Bohr or the half-integer orbits have a multiple oftheir orbital period. The mean energies of the curve corre-spond to (a) v = 90 (solid line), P = 90.33 (dashed line),v = 90.67 (dotted line); (b) P = 180 (solid line), v = 180.33(dashed line), v = 180.67 (dotted line); (c) v = 360 (solidline), v = 3600.33 (dashed line), v = 360.67 (dotted line).

1834 MARK MALLALIEU AND C. R. STROUD, JR.

Kv, ~

——n+ —+ j —lm~,2

j = 0, 1, 2, . . . , my —1.

(22)

The index r. = my (mod 2) indicates whether it it theBohr orbits or the half-integer orbits that the peaks arealigned with. For simplicity we have used the time scaleT,(s„)of-the mean Bohr orbit even for odd my, when thetime scale given by n+ 1/2 would be more appropriate.

The revival peaks for the first full revival, the half re-vival, and the third revival are shown in Fig. 5 for a rangeof mean energies. The plots show the wave-packet auto-correlation function as determined from an accurate nu-merical integration of Schrodinger s equation. Regardlessof the value of n, the peaks are centered at times wheneither the Bohr or the half-integer orbits have a multipleof their orbital periods, in accordance with Eqs. (21) and(22).

Just as the &actional revival peak locations are insen-sitive to small changes in the mean energy, the phasesof the fractional revivals should also be insensitive tosmall changes in the mean energy. It is best to workin the rotated kame at czar

——c- for the phase-lockedpulse measurement given in Eq. (2). To get the phaseof each &actional revival peak in the appropriate rotat-ing frame, it is necessary to calculate the rotated actionQz =

Rgz (t~) + et& at ea'-ch fractional peak time tz. It

has already been shown that all of the reference orbitswill have the same actions modulo 2' at these times, soit is simplest to consider the action of the reference orbitwith A: = ky. Proceeding as before, r& —e- is expandedas in Eq. (17). The use of Eqs. (21) and (22) results in

(23)

To get the total phase of each peak, the phase correc-tion of vr/4 which is present in the stationary-phase resultin Eq. (13) needs to be included. This factor comes &omthe contributions close in energy to the reference orbit.The kactional revival phases are

vr & v. 7r(&+ &)& .

4 ( mq) m~(24)

This is a much simpler expression for the fractional re-vival phases than the corresponding expression which isbased on an analysis of the eigenstate amplitudes [12,15].

These phases can be measured by the scheme given byEq. (2). If the optical phase Pe is equal to the phase ofthe peak P~, the measurement will produce a maximum.In [15] we show how this works for the my = 2 and 3kactional revivals. Here, we show the my ——4 revival inFig. 6 and the my = 5 revival in Fig. 7. From Eq. (24),

1.0

0.5

0.0

-0.5

-1.0I

1.0-~ e 0 5

0.0

-0.5

10

1.0-0.5

0.0--0.5

-1.0

1 0

0.54)

0.0

-05--1.0

1.0

0.5

0.0

-0.5

-1.0

29.75 30.00 30.25 30.50 30.75 31.00

'~T. ( .—)

1.0

0.5

0.0

-0.5

-1.0 I I I I

24.0 24.2 24.4 24.6 24.8 25.0

t/T, (C-„)

FIG. 6. Phase-locked autocorrelation function at I/4 re-vival. The laser pulse is tuned to u = 360.0 (solidline), v = 360.5 (dashed line), with a pulse length ofrz ——T (s-)/25. The response is shown for a phase-lock of(a) @e = 0, (b) Pe = vr/4, (c) Pe = —3s'/4. For this re-vival, the autocorrelatiou peak phases are given by Po ——7r/4,P& = Ps = 0, and Pg ———3s/4.

FIG. 7. Phase-locked autocorrelation function at 1/5 re-vival. The laser pulse is tuned to p = 3600 (solidline), u = 360.5 (dashed line), with a pulse length of7„=T, (e'„-)/25. The response is shown for a phase-lock of(a) Pe = 7r/5, (b) Pe = —7r/5, (c) Pe = —n. . For this revival,the autocorrelation peak phases are given by Pp = Pg = 7r/5,Pg ——Ps = —7r/5, and Pg = —s.

51 SEMICLASSICAL THEORY OF RYDBERG-WAVE-PACKET. . . 1835

the peak phases of the I/O revival are Po ——vr/4, Pq ——

Ps —0, and P2 ———3m. /4. This is in agreement with thephase-locked autocorrelation function shown in Fig. 6,with the locked-phase set to each of these values. For theI/5 revival the peak phases. are given by Po ——P4 ——vr/5,

P~ = Ps = —~/5, and P2 ———7r, which is in agreementwith Fig. 7.

VI. CONCLUSIONS

We have found accurate semiclassical solutions duringboth the classical and quantum regimes of wave-packetevolution. The solution for short times demonstratedthat even in the "classical" regime there is a quantummechanical phase of the wave packet which is a physicalobservable, as it can be measured by a simple phase-locked laser pulse scheme. This quantum mechanicalphase is related to the Bohr-Sommerfeld action of theKepler orbit at the mean energy of the wave packet. Thesemiclassical treatment for the wave-packet dynamics inthe nonclassical regime is surprisingly accurate. The &ac-tional revival behavior, although an inherently quantum

phenomena, is accurately reproduced from interferencesamong classical-path amplitudes. The various fractionalwave packets are seen to correspond to discrete sets oforbits, some of which are Bohr orbits and others whichhave half-odd values of their action variable. A simpleexpression for the phases of the fractional wave packetswas derived from the semiclassical solution. These phasesare directly related to the classical action of the discreteorbits of the fractional revival wave packets.

Even though the classical dynamics for the systemstudied here is integrable, the semiclassical sum exposeda rich and intriguing interplay between the classicaland quantum systems as the wave packet evolves. Theclassical-path representation examined here has been ap-plied previously to nonintegrable systems, such as thediamagnetic Kepler problem, for short times only. Thesesystems are currently of interest for the implications theycarry for the meaning of the correspondence principle. Areexamination of these problems in light of the resultspresented here promises to be fruitful for further clari6. —

cation of the classical limit of microscopic quantum me-chanical systems.

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