VOLUME 74, NUMBER 9 PHYSICAL REVIEW LETTERS 27 FEBRUARY 1995

Semiclassical Spectra without Periodic Orbits for a Kicked Top

Peter Gerwinski, Fritz Haake, and Harald WiedemannFachbereich Physik, Universitat-Gesamthochschule Essen, 54117 Essen, Germany

Marek KusCentrum Fizyki Teoretycznej, Polska Akademia Nauk, PL-02-668 Warsaw, Poland

Karol ZyczkowskiInstytut Fizyki, Uniwersytet Jagiellonski, ul. Reymonta 4, PL-30-059 Krakow, Poland

(Received 24 June 1994)

We present an 6 0 approximation for the quasienergy spectrum of a periodically kicked top, validunder conditions of both regular and chaotic classical motion. In contrast to conventional periodic-orbittheory we implement the semiclassical limit for each matrix element of the Floquet operator rather thanfor the trace of the propagator. Even though a classical looking action is involved, the approximatematrix elements are specified in terms of complex ghost trajectories instead of real classical orbits. Ourmean error for the quasienergies is a surprisingly small 3% of the mean spacing.

PACS numbers: 05.45.+b, 03.65.Sq

Semiclassical approximations for quantum systemswhose classical limit is fully chaotic are mostly basedon Gutzwiller's periodic-orbit theory [1]. Even thoughexistence problems due to the exponential proliferationof periodic orbits with increasing period seem to havebeen overcome recently, it remains notoriously difficultto extract more than a few energy or quasienergy levels.Apart from exceptional systems like the cat map [2,3] andsome billiards on spaces of constant negative curvature[4—6] for which periodic-orbit theory happens to be exact,we know of no case where a large number of levels hasbeen determined semiclassically. The main obstacle thusfar lies in the practical impossibility to gather sufficientknowledge about very long periodic orbits.

In applying periodic-orbit theory to the baker' s

map, Dittes, Doron, and Smilansky [7] have found aquasienergy uncertainty scaling with 6 as b, @ ~ 1'i2;the ratio of b, @ and the mean quasienergy spacing thusdiverges with 6 ~ 0 as h '/ which means impossibilityin calculating the baker-map spectrum semiclassically.

Alternative semiclassical methods deserve exploration.For instance, Heller's Gaussian wave packet dynamicshas yielded surprisingly accurate long-time results whichmight eventually be employed to determine spectra [g].

We propose here a different scheme for implementingthe limit h 0 for periodically driven systems. We rep-resent the Floquet operator (i.e. , the unitary time evolutionoperator over a period of the driving) as a matrix in a ba-sis of coherent states. Each such state may be associatedwith a classical phase space cell which shrinks to a pointin the limit 6 0. After employing a suitable integralrepresentation for each matrix element, for instance a laFeynman, the semiclassical limit is implemented througha saddle-point approximation. Upon diagonalizing the re-sulting matrix we obtain our semiclassical quasienergyspectrum.

Systems with compact phase spaces and thus finite-dimensional Hilbert spaces should be the first to testour strategy. Any error possibly due to cutting infinitespaces to finite size is thus excluded. We have chosen aperiodically kicked top.

The effective size of 6 is here controlled by an angular-momentum quantum number j as h ~ 1/j. With j fixedwe are facing a Hilbert space of 2j + 1 dimensions.The spectrum then consists of 2j + 1 eigenphases aliasquasienergies of the Floquet operator. The specific Flo-quet operator we work with is expressed in terms of thecomponents of an angular-momentum operator J obeying[J,J~] = i J„etc., as

—tkJ /2J —iver J,. /2

This accounts for a rotation by an angle ~/2 about the

y axis followed by a nonlinear torsion about the z axis.We have chosen the torsion strength as k = 8 in orderto make sure that in the classical limit the phase spaceis dominated by chaos [9]; moreover, as a nearly regularcase, we have treated k = 1.

The classical phase space, incidentally, is the spherelim, J 2/j2 = l. A point on that sphere may bespecified by angles defined by J, = j sin& cos@, JY =j sin 0 sin @,J, = j cosO. We have found it convenient towork with the complex stereographic projection variable

y = e'~ tan 2.0

In terms of this coordinate the classical limit yields thestroboscopic map of a point y into its image l as

(I = expl —ik (3)1+ VV")

Our semiclassical goal suggests to represent the Floquetoperator in a coherent-state basis. An angular-momentum

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VOLUME 74, NUMBER 9 PHYSICAL REVIEW LETTERS 27 FEBRUARY 1995

coherent state [10], to be denoted by lO, P) or ly), as-signs a direction to the angular momentum as (yl J., ly) =j sinOcoscb, etc. , with an uncertainty sinOAOAQ ~ 1/j,i.e., the best directional precision permitted by the com-mutation relations. In particular, such a state is the jointeigenstate lj, j) of J and J, with eigenvalues j(j + 1)and j, respectively. All the other coherent states can begenerated from this particular one by applying the rota-tion operator which turns the positive g direction into thedirection (O, @).

We shall need the matrix element of the Floquetoperator F between two coherent states [11]

Im(t)—10

ILP~~P

Pjt.-„

(I IFly) = )2 ik/2—

2(1+ I I*)(1+yy*)lj

2~k Re(t)10 20

with the exponent

V= — t +2ln(1+ v),2k

ik+tv = I e'1 —7

(4) Fig. 1. Gray-scale coded contour plot of le' "l in the complext plane. White regions indicate le'~l && 1 and dark onesle'

l« 1. The thick black line shows the path of constant

phase from —~ to +~ through saddles (thick dots) andinterlacing zeros of e' . Small dots refer to saddles not onallowed paths of integration.

In the semiclassical limit j ~, the t integral arisinghere lends itself to a saddle-point approximation. Saddlepoints are determined by

dV i 2v= ——t +dt k 1+v =0.

Not too surprisingly, this saddle-point equation is aslightly disguised form of the classical map, and thesaddle-point value of V(t) is closely related to the actionof that map [11]. However, by specifying both the initialpoint y and the final one I one overdetermines theclassical orbit, unless I happens to be the image of yunder the classical map. For generic pairs (y, I ) we maythink of a solution of the saddle-point equation as of aghost trajectory which lives in a complexified phase spaceand has a complex action [12—14).

The saddle-point approximation begins to make sensefor j upwards of j = 1. However, it is only for j » 1

that it suffices to use the saddle with the largest real partof V (t) from which both +~ and —~ can be reached inthe complex t plane by going downhill on the surfacel exp[j ReV(t)]l above that plane. For j smaller than about30 we have found it necessary to establish a path in thet plane connecting —~ to +~ through a sequence ofsaddles and interlacing zeros of the integrand, keepingImV(t) constant, and thus collecting the contributions of,in general, several saddles (see Fig. I). Incidentally, thequality of the semiclassical approximation for each matrixelement can be checked quite easily by evaluating thematrix element rigorously according to (4).

In a first semiclassical attempt at determining the Flo-quet spectrum we have chosen 2j + 1 different coherentstates ly„) as a complete basis. We have calculated the(2j + 1)2 matrix elements (y„lFly„, ) both rigorously andsemiclassically. Figure 2 displays, as a function of j, the

modulus of the semiclassical error for a typical matrix ele-ment. As expected for a saddle-point approximation, thaterror is inversely proportional to j. Finally, the semiclas-sical eigenvalues z„= r„e px(i@„) of the Floquet operatorare determined from the generalized eigenvalue equation

I I I 1 ~ I 1 I l ~ ~ I 1 I I

0.01

0.00110 100

FIG. 2. Modulus of the semiclassical error of a typical matrixelement of the Floquet operator versus j. The optimal fit is0.429j ', i.e., very close to a proportionality to 1/j.

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det((y. lF ly. ) —~(y. ly. )) = o

The nonorthogonality of the coherent states notwithstand-ing this equation would yield the exact eigenvalues z„ ifwe were to use the exact matrix elements (y lFly„); inparticular, all moduli r„would come out as unity, due tothe unitarity of F.

Our interest is, of course, in the roots arising with thesemiclassically approximated matrix. Figure 3 displaysthe rms deviation 6 r of their moduli r„ from unity,again as a function of j', the mean is now taken overall 2j + 1 eigenvalues. Like for the matrix elements,the error scales roughly as 1/j. The same scaling isfound for the rms deviation AP of the phases @„ from

VOLUME 74, NUMBER 9 PH YSICAL REVIEW LETTERS 27 FEBRUARY 1995

2~/(2q + 1)0.1 =

~ k I ~ I I I f ~ k I ~ ~ k I II

only dominant saddleall allowed saddles

o o

0.01 =kkkkkkkkl

10

oook k k k k I I I I

100

FIG. 3. The radial error of the eigenvalues scaled by the meanspacing versus j. —0.5 0 0.5their exact counterparts, as shown in Fig. 4. While theproportionality of the semiclassical errors to 1/j should beexpected, it is, in fact, a most welcome surprise that eachapproximate eigenphase can be uniquely associated withits exact counterpart. This is illustrated in Fig. 5 for j =10. A more quantitative manifestation of the astoundingsuccess of our approximation can be read off from Fig. 4:The phase error is as small as 3% of the mean spacing.

The small ratio of the eigenvalue uncertainty to themean spacing here incurred for the kicked top demandstheoretical explanation. To that end we are presentlyworking out the next term in the asymptotic expansion ofthe matrix elements of which the saddle-point contributionis but the leading term.

In order to check on the reliability of our results, wehave explored a second method for establishing the Floquetspectrum from the semiclassically approximated matrixelements (I )F~y). %'e now work with an overcompletebasis of N coherent states with N ~ 2j + I for which itisconvenient to adopt a slightly modified normalization,

n = 1,2, . . . , %.

FIG. 5. Exact (+) and semiclassical (0) Floquet eigenvaluesfor j = Io. In all 21 cases a unique association is possible.

which may be related to the unitary (2j + 1) X (2j + 1)Floquet matrix F„with respect to an arbitrary completeorthonormal basis Op, ), p, = 1, 2, . . . , 2j + I}as

2j+1F „= g At„F~„A„„, A,„=(v)$„). (10)

Here we have introduced a rectangular (2j + 1) x Nmatrix A and its adjoint At.

A piece of linear algebra now comes in handy. Aslong as no approximation is made in evaluating thematrices involved, among the N eigenvalues of F =AtFA there must be N —(2j + 1) vanishing ones whilethe remaining 2j + 1 eigenvalues must coincide withthose of FAAt.

To appreciate this lemma we inspect the matrix

(AA') .=~ g(viz. ) &y. I ~) (11)(2j + 1)AS

This gives rise to an % X N "extended" Floquet matrix

+') o0.1 =

k k ~ k ~ kI

~ f ~ I t I I k

only dominant saddleall allowed saddles

3 0

0.01 =k k k I k k I I l

QQ0 0~

k k k k k ~ k I I

102

100

FIG. 4. The phase (quasienergy) error of the eigenvalues isroughly 3% of the mean spacing 2m/(2j + 1).

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We have sneaked in a factor AS/b. S and choose, in thedenominator, AS = 4m/N, i.e., the mean surface elementon the unit sphere around each of the N points (8„,P„).Upon expressing that surface element, in the numerator,in terms of the variable y, we obtain

AAt =

Indeed, for N ~ the foregoing sum can be written asthe integral defining the resolution of unity by coherentstates [10]. In that limit, therefore, the eigenvalues of thematrix (AtFA)„, include the 2j + I eigenvalues of theFloquet matrix F„„aswell as N —(2j + I) zeros.

We have diagonalized the semiclassical version of theextended matrix F, after approximating each of its Nelements as described above, for various values of thequantum number j and the extended dimension N. In

VOLUME 74, NUMBER 9 PH YS ICAL REVIEW LETTERS 27 FEBRUARY 1995

0.1 =

2~i(2j + 1)0.01 =

semiclassicalquantum

3y.VV VVV

0.001 =-

0.0001

FIG. 6. Phase error determined with an overcomplete basis(dimension N ) 2j + l) versus the scaling variable p, =N/(2j + 1), for j = 30. The semiclassical error (O) saturatesat 3% of the mean spacing. With exact matrix elements thephase error (+), due to the nonorthogonality of the coherentstates, decays roughly as p, 4. The data for other values of jyield the same two "curves. "

all cases we have found N —2j —1 eigenvalues withvery small moduli, r « 1, and 2j + 1 nearly unimodulareigenvalues which, in fact, lie close to the exact ones.Figure 6 displays the dependence of the phase error 5@of the nearly unimodular eigenvalues, in units of themean level spacing, on the two dimensions 2j + 1,N. Toour surprise the data suggest that Ap[2~/(2j + 1)]depends on 2j + 1 and N only through the single scalingvariable p, = N/(2j + 1). Roughly, /J, @[2vr/(2j + 1)]decreases like p, until p, has grown to about 2.5 andthen becomes independent of p, , staying close to 0.03.This means that all levels can be determined to within3% of the mean level spacing by increasing the extendeddimension to a little less than 3 times the "physical'dimension 2j + l. It is most gratifying to see that thelimiting accuracy obtained is practically identical to theone incurred in our first semiclassical strategy. Thisstrengthens our expectation that the error can be explainedby the next-order term of the asymptotic 1/j expansionfor the matrix elements of F.

Our strategy is not restricted to conditions of classicalchaos. We have carried out calculations like the ones de-scribed above for k = 1, a case close to integrable. Resultsof similar, actually even slightly better accuracy were ob-tained. We should not forget to mention our considerableefforts to check the insensitivity of our semiclassical spec-tra to the precise locations of the coherent states on the unitsphere, as long as these locations form a more or less reg-ular grid with no pair of points much closer together thana mean nearest-neighbor spacing.

In concluding, it may be well to highlight the differenceof our method to the more familiar one of Gutzwiller's interms of the number of trajectories employed. We workwith (2j + 1)z ghost orbits rather than with exponentiallymany real periodic orbits (for a Hilbert space of 2j + 1

dimensions all periodic orbits with periods from 1 to j

would be needed). Incidentally, our ghost-orbit approachalso differs from Ref. [15] where real orbits were used toapproximate matrix elements.

We are not spared the efficiency problem which stillso notoriously plagues semiclassical approximations oflarge spectra. In our case that problem is not due toany proliferation of classical orbits. The extra burden weheap on ourselves, compared to the diagonalization of theexact Floquet matrix, is the saddle-point approximationfor each matrix element. But efficiency, even though itwould be most welcome and should indeed be strivenfor, was not the primary goal for the moment. We arealready glad to see that a full spectrum with genericfluctuations of a dynamical system with global classicalchaos is indeed accessible by semiclassical methods, i.e.,

by approximations exploiting the effective smallness ofPlanck's constant.

We gratefully acknowledge support by the Sonder-forschungsbereich Unordnung und grooe Fluktuationenof the Deutsche Forschungsgemeinschaft. M. K. andK. Z. acknowledge partial support by KBN GrantNo. P302 10206. We have enjoyed fruitful discussionswith Eugene Bogomolny, Stephen Creagh, PatricioLeboeuf, and Nils Lehmann. We are especially indebtedto Jonathan Keating for his patient help with the intrica-cies of the saddle-point technique.

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