Periodic-orbit quantization of chaotic systems.

We demonstrate the utility of the periodic orbit description of chaotic motion by computing from a few periodic orbits highly accurate estimates of a large number of quantum resonances for the classically chaotic 3-disk scattering problem. The symmetry decompositions of the eigenspectra are the same for the classical and the quantum problem, and good agreement between the periodic orbit estimates and the exact quantum poles is observed. 1 It is a characteristic feature of dynamical systems of few degrees of freedom that the motion is often organized around a few fundamental cycles. These short cycles capture the skeletal topology of the motion in the sense that any long orbit can approximately be pieced together from the fundamental cycles. Moreover, many quantities of interest can be computed as averages over periodic orbits. In ref. [1] a highly convergent expansion around short cycles has been introduced and applied to evaluation of classical chaotic averages. The goal of this letter is to demonstrate that the curvature expansions[1] of periodic orbit sums[2, 3, 4] are an equally powerful tool for evaluation of quantum resonances of classically chaotic systems. In this approach, the averages over chaotic dynamical systems are determined from the zeros of dynamical zeta functions[5], de ned as expansions of in nite products of form 1= =Yp (1 tp) = 1 Xf tf Xp cp: (1) with weight tp associated to every primitive (non-repeating) periodic orbit (or cycle) p. The key observation is that the expanded product allows a regrouping of terms into dominant fundamental contributions and decreasing curvature corrections. Computations with zeta functions are rather straightforward; typically one determines lengths and stabilities of a nite number of shortest periodic orbits, substitutes them into (1), and estimates the zeros of (1) from such polynomial approximations. We shall apply here the expansion (1) to evaluation of repeller escape rates. The classical repeller escape rate is determined[6, 1, 7] by the largest zero of 1= (s) (s 2 real) with each prime cycle weighted by tp(s) = 1 p e sTp: (2) Here Tp is the period of the prime cycle p and p = ln( p) is its stability exponent, where p is the leading eigenvalue of the cycle Jacobian. The associated quantum amplitude is essentially the square root of the classical weight. This follows from the stationary phase formula[2, 3, 4] for determining the poles of the scattering matrix in terms of cycles, rewritten[8, 9] as the logarithmic derivative of the in nite product of functions Z(k) = Q1j=0 1 j (k), where the weights of the prime cycles for the di erent j's are t(j) p = e p(1=2+j)+ i hSp(k)+i p=2; (3) where Sp(k) is the action and p the Maslov index (in the 3-disk example considered below, k is the quantum wavenumber). The zeros of Z(k) in the complex k plane determine the eigenvalues or resonances of the quantum system; here we shall compute only those closest to the real energy axis, which are given by the zeros of 1= 0(k). As it stands, the Euler product (1) is a product over an in nity of prime cycles of arbitrary length, and its utility as a computational tool is far from obvious. It is one of many formally equivalent cycle averaging expressions, and its connection to the Gutzwiller periodic orbit sum [2] has been known for some time [10, 8, 9]. What has not been recognized is that the intuition derived from classical chaotic dynamics[1, 7] singles out one particular expansion, exploiting the fact that long periodic orbits can be approximated by short ones. 3 More precisely, there are two types of contributions to the curvature expansion (1): a set tf of fundamental periodic orbits and an in nite series of curvature corrections cp. The fundamental cycles have no shorter approximants; they are the \simplest" cycles in the sense that all longer orbits can be pieced together from the fundamental cycles as fundamental building blocks. The curvatures are di erences of long orbits and their estimates based on shorter orbits (see eq.(5) below). They account for corrections as one resolves the dynamics on longer and longer times, ie: ner and ner resolution in phase space. In averages dominated by positive entropy of unstable orbits, these di erences decay[1, 7] exponentially and the curvature expansions are expected to be highly convergent. Here we shall illustrate the convergence of curvature expansions by computing the classical escape rates and quantum resonances for scattering o three disks [11] (we refer the reader to refs. [9, 12, 13] for detailed discussions). In this model the classical motion can be visualized as a pinball bouncing in a plane between three equally spaced disks of equal radius, and the quantum dynamics is described by wavefunctions which vanish on the boundaries of the disks. For billiard motion the momentum vm = hk = q2E=m is constant, the action Sp(k) is given by hkLp, where Lp is the length of the cycle p, and the quantum amplitude (3) associated with the cycle p is simply tp = j pj 1=2eikLp+inp . Here np is the number of bounces, and comes from the phase loss at every re ection (the boundary condition is jdisk = 0). We have chosen here the 3-disk scattering system because it captures the essential topology, stability and phase space structure of cycles in 2-d non-integrable potentials without the complications typical of motions in generic smooth potentials 4 (pruning of the symbolic dynamics, intermittency e ects due to marginally stable orbits; we shall return to these elsewhere). The dynamics here is geometrical optics, so the cycles are faster to compute than for arbitrary smooth potentials, and we were aided much by the work of Gaspard and Rice[9] in checking our results. The prerequisite for e cient use of curvature expansions is rm control of the symbolic dynamics[7, 13]. For su ciently separated disks, the symbolic dynamics is a ternary dynamics with alphabet f1; 2; 3g (the label of the disk the pinball bounces o ) and a single pruning rule prohibiting consecutive repeats of the same symbol[11, 9]. The corresponding curvature expansion (1) is straightforward, and converges well[13]. However, the C3v point group invariance[14] of the 3-disks problem simpli es and improves the curvature expansions in a rather beautiful way, which we now brie y sketch. The prime cycles fall into three classes of distinct symmetry; those invariant under rotations by 2 =3 (multiplicity 2), those invariant under re ections on symmetry axes (multiplicity 3), and the rest (multiplicity 6). By use of the standard character tables[14] it can be shown[13] that the corresponding contributions to the Euler product (1) factorize as follows: (1 tp)2 = (1 t1=3 p )(1 t1=3 p )(1 + t1=3 p + t2=3 p )2 (1 tp)3 = (1 t1=2 p )(1 + t1=2 p )(1 (t1=2 p )2)2 (1 tp)6 = (1 tp)(1 tp)(1 tp)4: (4) The three factors in this product contribute to the C3v irreducible subclasses A1, A2 and E, respectively, and the 3-disk zeta function factorizes into = + 2 E. Due to the symmetry, any 3-disk cycle can be pieced together from segments passing through the fundamental domain (see Fig. 1). The t1=2 p , t1=3 p weights in (4) have 5 direct physical meaning: they are the weights of the corresponding cycles restricted to the fundamental domain. Restriction to the fundamental domain also simpli es the symbolic dynamics: it becomes binary, with no restrictions on allowed sequences[11, 15, 13]. The ternary 3-disk f1; 2; 3g labels are converted into the binary fundamental domain labels f0; 1g by marking the backscatter by 0 and scatter to the third disk by 1. For ex., 23 = : : : 232323 : : : maps into : : : 000 : : : = 0 (and so do 12, 13), 123 = : : : 12312 : : : maps into : : : 111 : : : = 1, and so forth. (see Fig. 1). The Euler product (1) on each irreducible subspace is easily evaluated using the factorization (4). On the symmetric A1 and the antisymmetric A2 subspaces, the + and are given by the standard curvature expansion for the binary dynamics[1, 7]: 1= = (1 t0)(1 t1)(1 t10)(1 t100)(1 t101)(1 t1000) : : : = 1 t0 t1 (t10 t1t0) (t100 t10t0) (t101 t10t1) (t1001 t1t001 t101t0 + t10t0t1) : : : : : : (5) while for the mixed-symmetry subspace E the curvature expansion is given by 1= E = (1 + t1 + t21)(1 t20)(1 + t100 + t2100)(1 t210) : : : = 1 + t1 + (t21 t20) + (t100 t1t20) + t1001 + (t100 t1t20)t1 t210 : : : (6) Given the curvature expansions (5) and (6), the calculation is straightforward. Following ref. [9], we set the disk radius a = 1, x the disk-disk center separation R = 6 (for the sample values listed here), compute the eigenvalues and lengths of prime cycles up to 5 bounces (total of 14 cycles), substitute them into the curvature expansions, and determine the complex zeros; some hundred quantum resonances 6 are easily determined[13], with accuracy as good as 7 signi cant digits for the resonances closest to the real axis. A detailed discussion of this spectrum will be presented elsewhere[12]; here we only wish to illustrate the quality of the curvature expansions. In Table 1 we list a few typical results, illustrate their convergence by computing them with di erent maximal length cycles, and compare them to the numerical solutions for poles of the exact quantum scattering matrix[12]. The convergence of the curvature expansions is striking; they are many orders of magnitude more accurate than the estimates by other methods, and the gain in e ciency is dramatic: while the quantum scattering matrix requires computation of large truncations (of order of (70 70)) of in nite matrices with Bessel functions entries, the curvature expansions require evaluation of some dozen complex exponentials and a couple of sums and di erences. The judicious use of symmetry helps considerably; for example, going to the fundamental domain often doubles the number of significant digits for a given cycle length. The estimates could be further improved by extrapolations and knowledge of the analyticity properties of 1= , e.g. the positions of poles[7, 13]. No poles of the in nite product can occur within the half-plane of absolute convergence[17]. This also leads to an upper bound on the resonance lifetimes. The abscissa of absolute convergence can be determined as the leading zero of (5) with tp replaced by jtpj; Im(k) for all zeros of 1= must lie below kc (for the present a:R=1:6 example, kc = 0:121557 : : :). The convergence of the curvature expansion is best near kc (the longest lived resonances) and deteriorates as one moves further in the imaginary k direction. As Re k grows, the density of resonances increases. 7 Implications for the curvature expansion are that a certain number of terms have to be included before two resonances are distinguished; thereafter one again observes rapid convergence. This is illustrated in table 1 for the resonances k2 and k3, which have the same n=1 approximation. The explicit curvature expansions like (5) and (6) perhaps make it easier to explain our introductory claims about the exponential convergence of curvature expansions. A typical curvature expansion term involves a long cycle fabg minus its shadowing approximation by shorter cycles fag and fbg: tab tatb = tab(1 tatb=tab) = tab(1 e ab=2+ik Sab) where ab = a + b ab and Sab = Sa + Sb Sab. The exponential fall o of curvatures is a consequence of the smallness of the term in the brackets; and S are exponentially small for long orbits[16], typically O(e ab). Therefore, to resolve some scale k, one has to keep all di erence actions > 1= k. Their number increases like ( k)h= , i.e. roughly linearly (if the topological entropy h equals the average Lyapunov exponent ), and not exponentially, as one might expect naively. To summarize, we have demonstrated that the curvature expansions are a very e cient way of evaluating the classical and quantum periodic orbit sums. The essential ingredient for this success has been the physical insight that the dynamical zeta functions expanded this way utilize the shadowing of arbitrarily long orbits by shorter cycles; the technical prerequisite for implementing this shadowing is a good understanding of the symbolic dynamics of the classical dynamical system. Exploiting the symmetries of the problem, we are able to compute accurately a large number of resonances, using as input the actions and eigenvalues of as few as 8 2{14 prime cycles. We conclude with three more general comments on the relation of classical and quantum chaotic dynamics: 1. The curvature expansion approach presented here applies to strongly chaotic (non-integrable) systems, and is thus a quantization scheme for a class of systems complementary to those amenable to torus quantization. 2. The symmetry factorization (4) of the dynamical zeta function is intrinsic to the classical dynamics, and not a special property of quantal spectra (in which context it was used before[9]). 3. For the strange sets studied in refs. [1, 7], the curvature expansion is believed to be the exact perturbation expansion for classical chaotic averages: even though each cycle carries with it only its linearized neighborhood (analogue of the stationary phase approximation in the derivation of the quantum Gutzwiller sum), the union of the periodic points captures the invariant content of the full underlying smoothly curved dynamics. We expect similarly the quantum curvature expansion to overcome the limitations due to the linearization around the individual trajectories. P.C. is grateful to the Carlsberg Fundation for support, and to I. Procaccia for the hospitality at the Weizmann Institute, where part of this work was done. B.E. was supported in part by the National Science Foundation under Grant No. PHY82-17853, supplemented by funds from the National Aeronautics and Space Administration. We thank Peter Scherer for the quantum calculations and acknowledge stimulating exchanges with F. Christiansen, P. Grassberger, I. Percival, H.H. Rugh, U. Smilansky and D. Wintgen. 9 Figure caption 1. The scattering geometry for the disk radius/separation ratio a : R = 1 : 2:5. (a) the three disks, with 12, 123 and 121232313 cycles indicated. (b) the fundamental domain, ie: a wedge consisting of a section of a disk, two segments of symmetry axes acting as straight mirror walls, and an escape gap. The above cycles restricted to the fundamental domain are now the two x points 0 and 1 and the 100 cycle.