Quantum Chaos Y2K Proceedings of Nobel Symposium 116

cod kr+

^

4

+ a(0)j

(8)

(6)

for kr large compared with v. This function is an asymptotic Bessel function of order v, [a superposition of Bessel 2 2 2 V{x) « -en K x/2aR , (7) and 2 Neumann functions] and i//(0) satisfies tp"{9)+ v(6) ^{9) = 0. The billiard boundary is ra(0) = R2 + £ c (0) for \x\ < a. The condition that the slope be large enough to and

\/a2. If this inequality is strong, 9 < —9a and (9 + 9a)~ << 1, with a similar formula near the small m states \jjm are approximately Airy functions con- 9 = n + 9a. It can be shown that a = \k{c,c(9+ n)— centrated near x = a. In Fig. 2 we show the state f 10,2 for £ c (0)) ± Jt/4 and v2 = Em - V(9) with " V{9) = k2R2 (Cc(0 + n) + Cc($))- F ° r large enough k there is an approxi£ = 0.05, a = R. One motivation for considering this case is that the mate solution i/r(0) = sin(m7t(0 + 0 a )/20 a ), |0| < 9a, I/J(9) nonisolated periodic orbits mathematically disappear for vanishes outside this region for |0| < n, and \p{9) — any finite £ no matter how small. However, if £ is small, there ±i^(0 + 7t). For this solution Em = (mn/29a)~. The energy is a region of phase space which remains close to the parameter k solves kR2 + Em/kR2 = (« T \)n- For large n bouncing ball orbits of the original stadium. Thus, there this has the approximate solution k2nm = ((n^fyn/Ri)' — are still special states related to the bouncing ball orbits EmIR2.. There are In — \ =p ^ radial nodes. of the standard stadium. However, it is complicated and We show this state for n — 10, m = 1, in Fig. 3. Notice not very enlightening to describe these states in terms of some changes of sign as compared with the bouncing ball the periodic orbits of the slightly sloped stadium itself. between the straight sides. In particular, a deviation of There is a parametrically different dependence on E of the the billiard sides in the direction of narrowing the channel states as compared with the energy correlations. We see gives a repulsive effective potential V, thus containing the below that if kRe << 1, the leading terms of the trace wave function. For a billiard which is a deviation from a formula, or in other words, the energy correlations due circle, an outward deviation gives the repulsive effective to the bouncing ball states, are little modified [15]. This potential. This gives for V(x) a steep sided well with a sloping bottom,

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3. Finding and approximating special states

Fig. 3. Special mode of a"baseball" stadium./?2 = 1.5a, R\ = 0.5a. Quantum numbers are n = 10, m = l.The theoretical wave function in this and the next figure is inaccurate near the center of the circle.

Fig. 4. Special low angular momentum mode in a Bunimovich stadium with a short side, a = 0.1/?. Quantum numbers are n = 10, m = 2.

2.4. Almost circular stadium The above Eq. (8) can be applied to find the low angular momentum states of any almost circular billiard. Energy levels and wavefunction statistics were extensively studied in this system [24]. Take r» = R + CD(0) to describe the billiard, with C0(rJ) small. For the Bunimovich stadium with side a << R , £D « a|cos0|. Polar coordinates at the center of the billiard are used and approximations neglecting (a/R)~ have been made. In this case the potential V(6) = 2k2aR\cos0\, an attractive triangular well near 6 — ±\K. It is periodic with period n. For large k2aR, the lowest eigenstates will be concentrated near 0 = ±\n. We show the state n = 10, m = 2 in Fig. 4. All of these variations on the Bunimovich stadium have been considered in the literature, except for the baseball stadium. The only special states previously pointed out, however, were for the Bunimovich stadium itself. © Physica Scripta 2001

3.1. History The whispering gallery modes and the bouncing ball modes corresponding to stable periodic orbits were first quantized by Keller and Rubinow by what is sometimes called the ray method. This technique starts with the classical mechanics or rays, and exploits caustics and adiabatic invariants. However, the assumptions usually made are unnecessarily strong, and these special states still exist even if the caustics are only a short time approximation. Another group of methods starts from the partial differential equation, the Helmholtz equation in the case of billiards. The parabolic equation method invented independently by Leontovich [18] and by Fock [19] chooses coordinates astutely, and finds appropriate scale factors, allowing an approximation to the partial differential equation. It often relies on the ray method to motivate the manipulation of the PDE. The etalon method of Babich and Buldyrev [20] is an improvement of this which involves choosing a characteristic example, or "etalon", which captures the essential features of the given problem. The extended Born-Oppenheimer method [EBO] has been used in this context only for bouncing ball states between parallel sides of a billiard. We showed above a couple of generalizations of this technique. Other generalizations are given elsewhere [4]. The last three methods are closely related. They use different language and motivation and make approximations in a different order, but the main result is the same. Systematic corrections to the leading result have been extensively studied for certain examples, and these corrections are apparently of a different form for the different methods. We prefer the EBO when it applies since it is simpler and better known than the other techniques. 3.2. Bogomolny operator We also have introduced a technique based on Bogomolny's surface of section transfer operator [21] which in certain ways is more general than the above methods. It also has the advantage that the transfer operator T is closely related to the trace formulas and to important "resummations" of the trace formulas. In addition, it makes somewhat more precise the notion of a region of phase space which is nearly integrable for short times. The operator T(s,s'\E) introduced by Bogomolny is a generalization of the boundary integral method used to obtain numerical solutions for billiard problems. The main equation of the boundary integral method is derived using Green's theorem. It takes the form H(s)-

jds'K(s,s'\E)n(s'),

(9)

a Fredholm integral equation. The kernel or operator K depends parametrically on the energy E. Eq. (9) has a nontrivial solution only when E is on the spectrum. For Dirichlet conditions, ix{s) = 9f(r)/3«, the normal derivative of the wavefunction evaluated at the position on the boundary labelled by s, the distance along the perimeter. The integral in Eq. (9) is over this boundary. The boundary coordinate s together with its conjugate momentum, the momentum parallel to the boundary ps, Physica Scripta T90

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R. E. Prange, R. Narevich and Oleg Zaitsev

gives the Birkhoff coordinates for a surface of section of a billiard. Bogomolny's method essentially approximates K(s,s'\E) by its asymptotic form, which we call T(s,s'\E). It generalizes to a broad class of surfaces of section, not restricted to billiards, for which there are often also exact kernels. For a billiard, in Birkhoff coordinates,

the special states, as well as the integrable case, however, only r — 1 integrals are well approximated by that method.

d(E) = J2d(E-

3.4. Nearly integrable phase space region

3.3. Special states and periodic orbits We take as an example the slightly sloped billiard described above. Introduce a reference billiard system consisting of a channel of width 2R, i.e. with boundaries at y = ±R. K l kL{sJ)+tl) The actual orbits are compared with the bouncing ball orbits T(s,s'\E) = M -j-.-\ ^ , (10) of this channel. We want to choose a surface of section such that the nomwhere L is the distance between boundary points, inal periodic orbits are described by a diagonal element, k = ~j2mEjh and fi = % for Dirichlet conditions. We shall T(x, x). Such a surface of section is the upper half of the suppress \i and write the prefactor as (.) for simplicity. billiard. It is convenient to label position along it by the corThe essential variation of T is given by the exponential responding position x on the upper side of the reference dependence on kL — hkL/h = S(s,s'\E)/h,where S is the billiard. [Note that a simple change of coordinates, e.g. action of the classical straight line orbit between boundary s -> s(x) does not affect the result.] 1 points s, s . We now approximate the actual T(x, x!) in a way valid for Because h is supposed to be small compared with typical \x — x*\ << R. Thus, we may expand classical actions the exponential is rapidly varying, and the method of stationary phase usually applies to integrals 2 (16) in which T appears. We deal always with billiard examples L(x, x) *4R + ± ( x - X') -QS(X) - QS(x') for which it is convenient to take n = 2m = 1, and kL is where Cs, given by Eq. (6) above, is small except in the typically large. endcaps. The first two terms in this formula are an approxiThe action S generates the classical surface of section map mation to the perfect channel, and QS approximates the diffrom ference between the sloped billiard and the channel. Assuming for the moment that kc,s{x) << 1, we can ps = dS/ds, ps> = —dS/ds'. (11) approximate Composition of powers of T have intermediate points of stationary phase determined by this map so in effect, longer T(x,x) « (.)e iA: *( 4+ ^) ! ) e -Ws(x)+U*r» (1 7 ) and longer orbits can be built up by iteration of the T operator. * ( .)e**( 4 + K^) 2 )(l - ik(£s(x) + Cs(x'))). (18) Corresponding to Eq. (9) is the fundamental equation This T operator has a form that allows direct solution of Eq. tj/(s) = fds'T(s,s'\E)\l/(s'). (12) (12). Next assume that the solution {//(x) of Eq. (12) is slowly varying and expand i/'(x/) « i/^(x) + (x7 — x)ip'(x)+ 1 in T Our new method, for the cases corresponding to special ^{x — x)~\j/"{x). The rapidly varying exponential 1 ensures that the important contributions to the x integral states, solves this equation directly and quasiclassically, that is, it exploits the stationary phase approximation to do the lie close to x, according to the estimate (xf — x)~~ R/k. One may carry out the integral of Eq. (12) and obtain the integral. equation i//(x) = e\p{i(4kR — EmR/2k))ip(x) with the conEq. (12) has solutions only for E values on the quasidition that ip satisfies the Schrodinger equation (3) with classical spectrum such that potential V of Eq. (7), where the replacement n% -> 2kR D(E) = det(l - T(E)) = 0. (13) is made. The energy eigenvalues k\ m are found by insisting that the .exponential is unity. Thus, This Fredholm determinant is a well denned version of the dynamical zeta function [22], a resummation of the trace for- Ak„,mR - EmR/2kn,m = 2nn. (19) mula [23]. The trace formula itself is given by Since kc,s becomes large as x goes into the endcap region, the approximation of Eq. (18) seems to break down. However, , ,„ - 1 T d\r\D(E) dosc{E) = — I m —— the wave function becomes small in that region. As long n dE as the approximate wave function continues to be small, -1 dlx\n{\ - T(E)) = —Im \,r (14) the approximation is satisfactory. v It is also possible to extend this result to a parameter n dE regime such that ki;s becomes of order unity or larger [1]. n dE'—'r It is only necessary that kqs be slowly varying compared r with the leading term. Here the density of states, d(E), is expressed as Ea) = dWeyl{E) + d0SC{E)

(15)

a

The above formulation can be regarded as defining a phase where dwevi is the smoothed density of states. The traces of space region where the system is nearly integrable. The powers of T are expressed in terms of periodic orbits when two arguments of T together with the surface of section the integrals are done in stationary phase. In the case of map give an area on the 2-dimensional phase space such that Physica Scripta T90

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> \d > 1. In the with vertical sides, say, x = 0, x = 1, and horizontal sides case of the opposite strong inequality, eka << 1 the x inte- y = 1, y = ex, where 0 < £ < < 1. The Born-Oppenheimer gral gives just a factor of 2a, to leading approximation. However, we saw that if *fzka > 1, there are strong effects on the wave functions. Thus, although the wave functions are sensitive to the slope, the energy level correlations are not in this parameter regime. This is a fairly general result. Namely, special states are quite sensitive to perturbations. The energy level correlations are much less sensitive to the same perturbation.

T is well approximated by Eq. (18). This, in turn, is the T operator for an integrable system, characterized by being a function of the coordinate difference, together with a leading correction. For the case discussed, the surface of section phase space is approximately \x\ S a, \p\ < Hk\x — x"\/R *» fiy/k/R. The two-dimensional surface of section phase space region defines a four dimensional phase space region, from which the orbits only slowly escape. A classical orbit at a typical point in this phase space region is not too different from a nearby classical orbit of the ideal integrable region. Of course, continuing this orbit for long times allows it to escape from the region. In the case at hand, almost all such orbits are chaotic with positive Lyapunov exponents. However, if the fundamental equation T\\i = \j/ has a solution ijj(x) which is large only in the nearly integrable region, there is no need to consider the very long orbits. That gives another condition on the perturbation. In terms of the effective potential, V(x), it means that V must become repulsive outside the region. The sign of V depends on the sign of the perturbation, and on the sign of the quadratic term in the integrable part of T. Thus, for bouncing ball states between parallel sides, a perturbation shortening the bounce path is repulsive, while for radial bounces between circular sides, a perturbation lengthening the path is repulsive.

4. More examples We briefly give a few more examples. The problems already mentioned can be solved by the Born-Oppenheimer method. In the examples of this section, the Born-Oppenheimer method is more limited or more cumbersome, or impossible to use, but the Bogomolny operator method succeeds. First, the whispering gallery modes can easily be studied. These modes are concentrated near the boundary of a sufficiently smooth and sufficiently convex billiard. The case in which the corresponding classical orbits are also confined near the boundary is the only one studied in detail in the literature. Assuming a smooth enough convex billiard with everywhere positive curvature on the boundary, the appropriate coordinates are s, the distance along the perimeter, and p, the distance from the perimeter toward the center of curvature at s. Let R(s) be the radius of curvature at point s. The Born-Oppenheimer ansatz is *P(r\s) = elfa$(p|^)i/'(j), © Physica Scripta 2001

Fig. 5. Contour plot of a numerically obtained whispering gallery mode in a stadium billiard, a = 0.05/?. There are no caustics in this case, and almost all classical orbits circulating near the boundary eventually escape. The distance between the short parallel lines is the length of the straight side. The wavenumber is k = 242.7611//?. Physica Scripta T90

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R. E. Prange, R. Narevich and Oleg Zaitsev

---:,

u,jiiwppww,'m-t

Fig. 6. State along the long side of a nearly square trapezoid. Slope of bottom ise = 1/16. Quantum numbers are n = 22, m = 2. In this figure and the next, the dashed line is the x—axis.

Fig. 8. A contour plot of a numerically obtained state near the (1,4) periodic orbits in a Bunimovich stadium billiard with a short straight side, a/R = 0.01. The wavelength of the state is X = 2.23969a. The parallel lines show the length of the short side, la.

and slow, so the EBO does not work. The T— operator approach is straightforward, but a little complicated [1]. In the T— operator approximation, there are two nearly degenerate states. That is, the T operator has inequivalent solutions of the same energy corresponding to orbits moving clockwise or counterclockwise. These states do not couple in the T operator approximation, and the appropriate symmetric combinations have nearly the same energy. From this figure it is seen that the states are made up of waves along the rays of classical angular momentum / « hk/\/2, i.e. along straight lines whose closest approach to the center is 1/V2- A perfect circle would have a caustic of this radius. Because the stadium is chaotic, such a caustic does not exist, except as a short time approximation. 5. Summary Fig. 7. State along the diagonal of the nearly square trapezoid of Fig. 6. Quantum numbers are n = 27, m = 1.

method may be used to find states concentrated near x = 0, similar to the states of the slightly sloped stadium. However, there are states associated with the (1,1) period orbits of the unperturbed square. These orbits are rectangles making a 45° angle with the x-axis. This may be solved using the Bogomolny technique by extending the billiard antiperiodically in the x direction, and using as surface of section the upper boundary, y=\. The effective potential is V(x) oc sk2\x\, - 1 < x < 1, repeated with period 2. States of this extended problem are superimposed to find a solution. A typical case is shown in Fig. 7. A last example is again the almost circular stadium. We now study states near a higher period orbit. In Fig. 8 we show a numerical state near the (1,4) almost square orbits in a stadium with side a/R = 0.01. We have not found any convenient coordinates which we can classify as fast Physica Scripta T90

We have reviewed some of the salient features of special classes of states which often occur in the systems of interest to quantum chaologists. Perhaps these cases are common because there is a tendency to construct billiards of straight lines and circles. The system as a whole may be chaotic, in other words, almost all orbits of the system may have positive Lyapunov exponents. However, for some shorter time, a subset of phase space may be close to that of a set of nonisolated periodic orbits of some integrable reference system, and this leads to the special states. These special states have rather striking wavefunctions. As a result, there are phenomena and even possible applications associated with them. In particular, the whispering gallery modes have long been known to give rise to interesting effects. One standard "application" is that special states often appear in weakly perturbed integrable systems. If these states are to be avoided, a regular resonant cavity must be constructed much more precisely than the condition dx << X, where Sx is the deviation from the ideal. Also, because there are a sequence of special states regularly © Physica Scripta 2001

Quantum Spectra and Wave Functions in Terms of Periodic Orbits for Weakly Chaotic Systems spaced in energy, the special states often numerically dominate the trace formula. Typically, there are two or more scales of variation in connection with the special states which can be identified. If coordinates can be found such that one coordinate is fast and the other is slow, standard adiabatic approximations, such as the Born-Oppenheimer approximation, can be used. Bogomolny introduced a surface of section transfer operator some time ago as a means of studying the spectrum, i.e. the trace formula. We have shown how this operator can also used to find the special eigenstates. The technique works if the operator can be approximated as a rapidly varying integrable part, and a more slowly varying correction to it. It is in some ways more general than the other methods. The special states are typically rather rare, in the sense that they are a small fraction of all the states in a given (high) energy range. In leading approximation, they do not couple to the other states. The energies of the states are predicted to good approximation, relative to the energy spacing of the given class of states, and even better absolutely. However, the accuracy is not necessarily good compared with the mean level spacing of all the levels. Further, it may happen that "accidentally" there is a nonspecial state with energy very close to that of the special state. Then terms neglected in our approximation can mix these states. Many phenomena are independent of such mixing, however. In this article we gave a number of examples of such special states, several of which appear for the first time in print. We hope our pictures will tempt an experimentalist to find some of these states in one of the several systems, water trays, acoustic plates, microwave and laser cavities, optical fibers, quantum dots, . . . , to which the theory applies. Acknowledgements Supported in part by the United States NSF grant DMR-9625549 and United States-Israel Binational Science Foundation, grant 99800319. R.N. was partially supported by the NSF grant DMR98-70681 and the University of Kentucky.

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References 1. Prange, R. E.. Narevich, R. and Zaitsev, 0., Phys. Rev. E 59. 1694 (1999). 2. Narevich, R.. Prange, R. E. and Zaitsev. O., Phys. Rev. E 62. 2046 (2000). 3. Narevich, R., Prange, R. E. and Zaitsev, O.. Physica E (to be published). 4. Zaitsev, O., Narevich, R. and Prange, R. E., Foundations of Physics (to be published). 5. Zaitsev, O., Ph. D. Thesis, University of Maryland, (2000). 6. Lord Rayleigh, "The Theory of Sound", (MacMillan, London, 1894), Vol. 2; Phil. Mag. 27. 100 (1914). 7. Agam, O. and Altshuler, B. L., cond-mat/0004190 (2000). 8. Kudrolli. A., Abraham, M. C. and Gollub. J. P., nonlin.CD/0002045 (2000). 9. Hackenbroich, G. and Mendez, R. A., cond-mat/0002430(2000). 10. Keller, J. B. and Rubinow, S. J.. Ann. Phys. (New York) 9, 24 (1960). 11. McDonald, S. W. and Kaufmann, A. N., Phys. Rev. Lett. 42, 1182 (1979). 12. Backer, A. et at., J. Phys. A 30, 6783 (1997). 13. Biswas. D. and Jain, S. R., Phys. Rev. A 42, 3170 (1990). 14. Tanner. G., J. Phys. A. Math. Gen. 30, 2863 (1997). 15. Primak, H. and Smilansky. U., J. Phys. A: Math. Gen. 27, 4439 (1994). 16. Heller. E. J.. Phys. Rev. Lett. 53, 1515 (1984). 17. Bai, Y. Y.. Hose, G., Stefanski, K. and Taylor, H. S., Phys. Rev. A 31, 2821 (1985). 18. Leontovich, M. A., Izv. Akad. Nauk SSSR, Ser. Fiz. 8, 16 (1944). 19. Fock, V. A., Izv. Akad. Nauk SSSR, Ser. Fiz. 10(2), 171 (1946) [J. Phys. USSR 10, 399 (1946)]; "Electromagnetic Diffraction and Propagation Problems", 2nd ed. (Sovetskoe Radio, Moscow, 1970) [1st ed. (Pergamon, Oxford, 1965)]. 20. Babich, V. M. and Buldyrev, V. S., "Short Wavelength Diffraction Theory" (Springer-Verlag. Berlin, 1991). 21. Bogomolny. E. B., Nonlinearity 5, 805 (1992), and references therein. 22. Voros, A., J. Phys. A 21, 685 (1988); Cvitanovic, P. and Eckhardt, B., Phys. Rev. Lett. 63. 823 (1989); Berry, M. V. and Keating, J. P., Proc. R. Soc. London, Ser. A 437, 151 (1992); Doron, E. and Smilansky, U.. Phys. Rev. Lett. 68. 1255 (1992). 23. Gutzwiller, M. C . J. Math. Phys. 12, 343 (1971). 24. Borgonovi, F. et at.. Phys. Rev. Lett. 77, 4744 (1996); Nockel, J. U. and Stone, A. D., Nature 385, 45 (1997); Frahm, K. M. and Shepelyansky, D. M., Phys. Rev. Lett. 78, 1440 (1997); ibid. 79, 1833 (1997); Borgonovi, F„ Phys. Rev. Lett. 80, 4653 (1998). 25. Morse, P. M. and Feshbach, H., "Methods of Theoretical Physics", (McGraw-Hill. New York, 1953), Vol. II. 26. Kaplan, L. and Heller, E. J., Physica D 121, 1 (1998).

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Bifurcation of Periodic Orbit as Semiclassical Origin of Superdeformed Shell Structure K. Matsuyanagi* Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan Received July 27, 2000

PACS Ref: 21.60.-n

Abstract Classical periodic orbits responsible for emergence of the superdeformed shell structure of single-particle motion in spheroidal cavities are identified and their relative contributions to the shell structure are evaluated. Fourier transforms of quantum spectra clearly show that three-dimensional periodic orbits born out of bifurcations of planar orbits in the equatorial plane become predominant at large prolate deformations. A new semiclassical method capable of describing the shell structure formation associated with these bifurcations is briefly discussed.

1. Introduction Regular oscillation in the single-particle level density (coarse-grained to a certain energy resolution) is called shell structure, and plays a decisive role in determining shapes of a finite Fermion system [1,2]. According to the periodicorbit theory [3-7] based on the semiclassical approximation to the path integral, shell structure is determined by classical periodic orbits with short periods. A finite Fermion system (like a nucleus and a metallic cluster) favors such shapes at which prominent shell structures are formed and its Fermi surface lies in a valley of oscillating level density, increasing its binding energy in this way. In this contribution, I would like to point out that (1) there is a unique application of the periodic orbit theory to a modern nuclear structure problem; i.e. to understand the mechanism of emergence of the superdeformed shell structure, and (2) bifurcation of periodic orbits is responsible for the formation of this new shell structure.

symmetry). The reason why superdeformed states can maintain their identities against compound nuclear states (whose level density is high) is that there is a barrier preventing the mixing between these different kinds of quantum states (associated with two local minima in the Hartree-Fock potential-energy surface). Therefore, in order to understand why superdeformed nuclei exist, we need to investigate the mechanism of producing the second minimum in the potential energy. It is certainly connected to an extra binding-energy gained by the formation of a new shell structure called the superdeformed shell structure. Our major subject is thus to understand the mechanism how and the reason why such a new shell structure emerges. The semiclassical periodic orbit theory is useful to gain an insight into the dynamical origin of it.

3. Spheroidal cavity model Let us consider the spheroidal cavity model as a simplified model for single-particle motions in heavy nuclei, and try to find the correspondence between quantum shell structure and classical periodic orbits. As emphasized in [6], the shell structure obtained for this model contains, apart from shifts of deformed magic numbers due to the spin-orbit potential (although they are important for realistic calculation of nuclear structure), the basic features similar to those obtained by the Woods-Saxon potential for heavy nuclei. Of course, it is necessary to examine the dependence on surface diffuseness. In fact, periodic orbits with small angular rotations between two successive reflections at the surface

It is my impression that bifurcations are often discussed in connection with "routes to chaos", but emergence of new ordered structure (in quantum spectra) through bifurcations is rarely discussed. 2. Nuclear superdeformation Superdeformed states are cold quantum states embedded in the highly excited warm region consisting of a huge number of compound nuclear states (see Fig. 1). Their shapes are similar to the spheroid with the major to minor axes ratio about 2:1. Of course, when we talk about "shapes" of a finite quantum system like a nucleus, we means intrinsic shapes associated with selfconsistent mean fields. Thus the superdeformation is a striking example of spontaneous symmetry breakdown. The mean field is rapidly rotating and generates a beautiful rotational band spectrum (to restore the broken * e-mail: [email protected] Physica Scripta T90

O

0.6

Deformation /S

Fig. 1. Illustration of the rapidly rotating superdeformed nucleus. Here T denotes the temperature, and the values of angular momentum / and parity it are those appropriate to the superdeformed band in 152Dy, which was discovered in 1985 by Twin et al. [8]. © Physica Scripta 2001

Bifurcation of Periodic Orbit as Semiclassical Origin of Superdeformed Shell Structure (like pentagons) disappear with increasing diffuseness parameter [9]. But, periodic orbits with larger angular rotations (like star-shaped orbits) survive at the realistic value of the diffuseness parameter for the nucleus [10]. Needless to say, the spheroidal cavity is a special integrable system. But, we have obtained similar results also for other parametrizations of prolate cavities (for which the Hamiltonian is non-integrable) [11]. Thus I believe that the spheroidal cavity model contains the basic features to get an insight into the problem of our primary concern; i.e. what is a proper semiclassical interpretation of the superdeformed shell structure. In the cavity model, single-particle equations of motion are invariant with respect to the scaling transformation (x,p, t) -*• (x, ap, a~[t) and the action integral Sr for a periodic orbit r is proportional to its length Lr: Sr(E = p2/2M) =
g(E) = !>(£-£«)= j^r £ ^ ~ ^) mir/2),

r

where g(E) denotes the smooth part corresponding to the contribution of the zero-length orbit and fxr is the Maslov phase of the periodic orbit r. The Fourier transform F(L) of the level density g(E) with respect to the wave number k is written as F(L)=

fdke-'kLg(E

=

h2k2/2M)

^

and bifurcations occur when the condition a b

sin(nt/p) sm(nq/p)

is satisfied [4,12], where a and b denote the lengths of the major and the minor axes, respectively. With increasing rj, planar orbits (4:2:1) bifurcate from the linear orbit that repeats twice along the minor axis. With further increase of rj, 3D orbits (p:t:q) = (p:2:1) with p = 5, 6, 7, • • • successively bifurcate from the planar orbits that turns twice (/ = 2) about the symmetry axis. These new-born orbits resemble the Lissajous figures of the superdeformed harmonic oscillator with frequency ratio a>±_: wz = 2:1. Every bifurcated orbit forms a continuous family of degeneracy two, which implies that we need two parameters to specify a single orbit among a continuous set of orbits belonging to a family having a common value of the action integral (1) (or equivalently, the length).

Thus the semiclassical trace formula for the level density is written as

~ g(E) + J2 Mk) cos(kLr -

143

(3)

5. Constant-action lines and Fourier transform Figure 2 displays the oscillating part of the smoothed level density in the form of a contour map with respect to the energy and deformation parameter. Regular patterns consisting of several valley-ridge structures are clearly seen. As emphasized by Strutinsky et al. [6], if few families of orbits having almost the same values of action integral Sr dominate in the sum in Eq. (2), the valleys in the contour map may follow such lines along which Sr stay approximately constant. In this figure, tick solid lines running through the spherical closed shells indicate the constant-action lines for tetragonal orbits in the meridian plane. It is clear that the valleys run along these lines. A detailed discussion on this point is made in Ref. [14]. On the other

m /2

~ F(L) + nJ2 e- "' Ar(idL) d(L - Lr). r

By virtue of the scaling property of the cavity model, the Fourier transform exhibits peaks at lengths of classical periodic orbits, so that it may be regarded as the "length spectrum" [4]. In the following, we shall make use of the Fourier transforms in order to identify the most important periodic orbits that determine the major pattern of oscillations in the coarse-grained quantum spectrum.

4. Bifurcation of periodic orbit As is well known, only linear and planar orbits exist in a spherical cavity. When spheroidal deformations occur, the linear (diameter) orbits bifurcate into those along the major axis and along the minor axis. Likewise, the planar orbits bifurcate into orbits in the meridian plane and those in the equatorial plane. Since the spheroidal cavity is integrable, periodic orbits are characterized by three positive integers (p, t, q), which represent numbers of vibrations or rotations with respect to three spheroidal coordinates. When the axis ratio t] of the prolate spheroid increases, hyperbolic orbits in the meridian plane and three-dimensional (3D) orbits successively appear through bifurcations of (repeated) linear and planar orbits in the equatorial plane. Bifurcation points are determined by stability of equatorial-plane orbits against small displacements in the longitudinal direction, © Physica Scripta 2001

Fig. 2. Oscillating part of the smoothed level density displayed as a function of the energy (in unit of h2/2MR\) and deformation parameter d. Here M and Ro denote the mass of the particle and the radius at the spherical shape, respectively The deformation parameter d is related to the axis ratio i] = a/b by <5 =~S(Y\- l)/(2f/ + 1) in the prolate case discussed in the text. Solid, dashed and dotted contour curves correspond to negative, zero and positive values, respectively. Constant-action lines for important periodic orbits are indicated by thick solid and broken lines (see text). This figure is taken from [13]. Physica Scripta T90

144

K. Matsuyanagi

Table I. Bifurcation points of short periodic orbits. orbit

(p.f.q)

(4:2:1) (5:2:1) (6:2:1) (7:2:1) (8:2:1)

axis ratio (a/b)

deformation <5

orbit length in Ra

V2

0.32 0.44 0.49 0.52 0.54

7.1 8.1 8.7 9.0 9.2

1.62

V3 1.80 1.85

z.

/9\ \f)\ \V7 y

. _/-aL«L. '••WJ

<3-'

5=0.6

^ \ •• '•

^ '

y

7 i 1

> (8:2:1)

Ml

Fig. 3. Length spectrum (Fourier transform of quantum level density) for the spheroidal cavity with S = 0.6 (axis ratio 2:1). At the bottom, the lengths (in unit of RQ) of classical periodic orbits are indicated by vertical lines. Long, middle and short vertical lines are used for 3D orbits, planar orbits in the meridian, and planar orbits in the equatorial planes, respectively. This figure is taken from [13].

98-

/

(7:2:1)

A

- -

76-

J

/'\f

5-

i \\/~~\ /*^

y^^\ J "*--, 1

4-

a

3- /

/

\

/

\ /

A -^r-VJ \ \

\

W

m--.

increasing deformation and reach maximal values. Then, they start to decline. Thus we conclude that the bifurcations of equatorial-plane orbits play essential roles in the formation of the superdeformed shell structure, and this shell structure is characterized by the 3D orbits (p:2:\). Some of these 3D orbits are displayed in Fig. 5. They possess similarities with the figure-eight shaped orbits in the axially symmetric harmonic-oscillator, that appear when the frequency ratio becomes exactly 2:1 [15]. It is important, however, to note a difference in that they exist in the cavity model for all deformation parameters S larger than the bifurcation points (not restricted to the special point of axis ratio 2:1). In view of the fact that more than 200 superdeformed rotational bands have been systematically observed and they have varying deformations in the range S = 0.4 ~ 0.6 [16-18], it seems more appropriate and general to define the concept of superdeformation in terms of the shell structure generated by these 3D orbits {p:t:q)=(p:2:\) (rather than geometrical shapes alone).

\ V A\

2-

"

•

-

'

6. Semiclassical method capable of treating the bifurcation

10-

X

Fig. 5. Three-dimensional orbits (5:2:1) and (6:2:1) in the superdeformed prolate cavity (axis ratio t] = 2). Their projections on the (x, v), (y, z) and (", x) planes are displayed. This figure is taken from [13].

(6:2:1) (5:2:11

X

•"•'

I

""l"

•

i

1

0.6

0?

We have evaluated the amplitudes A in the trace formula (2) by means of the Fourier transforms of quantum spectra. Now we attempt to calculate them by semiclassical method. Fig. 4. Deformation dependence of the Fourier amplitudes defined in Eq. (3), As is well known, however, the amplitude A evaluated by at lengths L = Lr of the butterfly-shaped hyperbolic orbit (4:2:1) in the merthe conventional stationary-phase approximation diverges idian plane and of 3D orbits (p:2:l). Solid curves correspond to those for at the bifurcation point. Thus, for describing the bifurcation equatorial-plane orbits from which these orbits are bifurcated. This figure phenomena under consideration, Magner et al. have develis taken from [13]. oped a periodic-orbit theory free from the divergence [19]. Here I would like to briefly discuss basic ideas of this work. hand, tick broken and solid lines in the region <5 = 0.3 ~ 0.8 Since the spheroidal cavity is integrable, we can develop indicate those for the five-point star-shaped orbits in the semiclassical method along the line initiated by Berry and equatorial plane and for the 3D orbits (5:2:1) bifurcated Tabor [5]. As usual, we start from the trace integral for from them, respectively. Good correspondence is found the level density in action-angle variables. In the convenbetween these lines and the valley structure seen in the tional scheme, however, one considers families of orbits with superdeformed region. Constant-action lines for the other the highest degeneracies (like 3D orbits) but those with lower 3D orbits listed in Table I also behave in the same fashion. degeneracies (like equatorial-plane orbits) are not necessThe magnitudes of contributions of individual orbits are arily taken into account. Hence we need to extend the found to exhibit a remarkable deformation dependence. Berry-Tabor approach in order to treat the bifurcation of Figure 3 shows the Fourier transform of the quantum interest. We thus consider all kinds of stationary points, spectrum at S = 0.6 (axis ratio 2:1). We see that these 3D and calculate (for the lower degeneracy orbits) the integrals orbits form prominent peaks in the range L = 8 ~ 9. over angles, too, by an improved stationary-phase method. Figure 4 displays the deformation dependence of the Fourier "Improved" here means that the trace integrals over both amplitudes \F(L)\ defined in Eq. (3) at lengths L = Lr of action and angle variables are calculated, as usual, by these orbits. We see that the Fourier peak heights associated expanding the exponent of the integrand about the stationwith new orbits created by bifurcations quickly increase with ary point up to the second order, but the integrations are 0.3

Physica Scripta T90

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Bifurcation of Periodic Orbit as Semiclassical Origin of Superdeformed Shell Structure

145

deformations. Fourier transforms of quantum spectra clearly indicate that 3D periodic orbits born out of bifurcations of the planar orbits in the equatorial plane generate the superdeformed shell structure. A new semiclassical method capable of describing the shell-structure formation associated with these periodic-orbit bifurcations is briefly discussed. Acknowledgements

Fig. 6. Deformation dependence of the amplitudes A for the second repetition of the short diameter 2(1,2) and the butterfly orbits 1 (1,4) in the elliptic billiard, calculated at kR0 = 50 by the improved stationary-phase method of Ref. [19]. Their absolute values are drawn by solid curves as functions of the axis ratio r\. For comparison, standard results of the extended Gutzwiller trace formula are plotted by short-dashed curves. This figure is taken from [19].

I would like to thank the organizing committee of the Nobel Symposium "Quantum Chaos Y2K" for giving me the opportunity to write my view as a discussion leader of one of the sessions on "Description of Quantal Spectra in Terms of Periodic Orbits." The symposium was very stimulating and useful; for instance, it was totally unexpected for me to learn that the periodic-orbit bifurcation in deformed cavities, of the type discussed here, can be used to get high power directional emission from lasers with chaotic resonators. This article is written on the basis of collaborations with Arita et al. [13] and Magner et al. [19],

References done within the finite physical region as in [5]. If the integration ranges are extended, as usual, to ±oo, singularity arises in the amplitude, since, at the bifurcation, a stationary point lies just on the edge of the physical integration range. The stationary points need not necessarily lie inside the physical region of integration over the action-angle variables, but they are assumed to be close to the integration limits. In fact,#a bifurcation occurs when one of the stationary points crosses the border from the unphysical region (negative values of the action variable) and enter the physical region. As we move away from the bifurcation points, thus obtained contributions from the lower degeneracy orbit families asymptotically approach the results of the conventional stationary-phase approximation. This approach has been successfully applied to the elliptic billiard model, and it is shown that the bifurcation of the (butterfly-shaped) hyperbolic orbit family from the repeated short-diameter orbit is responsible for emergence of shell structure at large deformations. For instance, it is clearly seen in Fig. 6 that the amplitude of the bifurcating orbit is significantly enhanced in the vicinity of the bifurcation point. Namely, we obtain the maximum instead of the divergence at the bifurcation point. We are now applying this approach to the spheroidal cavity model, and the result will be published in the very near future [19]. 7. Concluding remarks We have discussed quantum manifestations of short periodic orbits and of their bifurcations in the spheroidal cavity, and identified the classical periodic orbits responsible for the emergence of the quantum shell structure at large prolate

© Physica Scripta 2001

1. Bohr, A and Mottelson, B. R., "Nuclear Structure", (Benjamin, 1975), vol. 2. 2. Aberg, S., Flocard, H. and Nazarewicz, W., Ann. Rev. Nucl. Part. Sci. 40, 439 (1990). 3. Gutzwiller, M. C , J. Math. Phys. 12, 343 (1971). 4. Balian, R. and Bloch, C . Ann. Phys. 69, 76 (1972). 5. Berry, M. V. and Tabor, M., Proc. R. Soc. London A 349, 101 (1976). 6. Strutinsky, V. M., Magner, A. G., Ofengenden, S. R. and Dossing, T., Z. Phys. A 283, 269 (1977). 7. Brack. M. and Bhaduri, R. K„ "Semiclassical Physics", (AddisonWesley. Reading, 1997). 8. Twin, P. J. et at, Phys. Rev. Lett. 57, 811 (1986). 9. Lerme, J. et al., Phys. Rev. B 48, 9028 (1993). 10. Arita, K. et al.. to be published. 11. Misu, T. et al., to be published. 12. Nishioka, H., Ohta, M. andOkai, S., Mem. Konan Univ., Sci. Ser. 38, 1 (1991). 13. Arita, K., Sugita, A. and Matsuyanagi, K., Prog. Theor. Phys. 100, 1223 (1998). 14. Frisk, H., Nucl. Phys. A 511, 309 (1990). 15. I would like to point out that they appear through bifurcations also for irrational ratios, if anharmonic terms like octupole deformations are added; see [20-22]. I also like to mention that there is an analogous open problem concerning the semiclassical origin of the left-right asymmetry in nuclear shapes [23]. 16. Nolan, P. J. and Twin, P. J., Ann. Rev. Nucl. Part. Sci. 38, 533 (1988). 17. Janssens, R. V. F. and Khoo. T. L., Ann. Rev. Nucl. Part. Sci. 41, 321 (1991). 18. Berkeley Isotope Project Data File, http://isotopes.lbl.gov/isotopes/ sd.html. 19. Magner, A. G. et al.. Prog. Theor. Phys. 102. 551 (1999), and to be published. 20. Arita, K. and Matsuyanagi, K., Prog. Theor. Phys. 91, 723 (1994). 21. Arita, K. and Matsuyanagi. K„ Nucl. Phys. A 592, 9 (1995). 22. Heiss, W. D„ Nazmitdinov, R. G. and Radu, S., Phys. Rev. B 51, 1874 (1995). 23. Sugita, A., Arita, K. and Matsuyanagi, K., Prog. Theor. Phys. 100, 597 (1998).

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Physica Scripta. T90, 146-149, 2001

Wavefunction Localization and its Semiclassical Description in a 3-Dimensional System with Mixed Classical Dynamics M. Brack1, M. Sieber and S. M. Reimann1'3 1

Institute for Theoretical Physics, Regensburg University, D-93040 Rcgensburg, Germany Max-Planck-Institute for the Physics of Complex Systems, Nothnitzer Str. 38, D-01187 Dresden, Germany 3 Lund Institute of Technology, P. O. Box 118, S-22100 Lund, Sweden

2

Received August 30, 2000

PACS Ref: 24.75. +i, 03.65.Sq, 21.10.Dr, 47.20.Ky

Abstract

In the quantum-mechanical calculations for the energy shell correction SE, derived from the quantum spectrum of realistic nuclear shell-model potentials using Strutinsky's shell-correction method [7], it was found that the outer fission barrier of many actinide nuclei is unstable against octupole-type deformations [8]. In our present parametrization, this barrier is located at c = 1.53; by varying a from zero to ~ 0 . 1 3 , the energy SE is lowered by about ~ 2.5 — 3 MeV. (For the present qualitative discussion, we neglect the smooth liquid-drop model part of the total energy which varies much less in this restricted region of deformations.) This result could be well reproduced in our semiclassical calculations [3,4], Thereby, it turned out to be sufficient to include the two shortest periodic orbits (diameters and triangles) in each of the planes z, into the trace formula; the contributions of longer orbits were shown to be negligible [4]. In the upper parts of Figs. 1 and 2, the shape boundaries We have recently applied [3,4] the periodic orbit theory [5,6] p(z) are shown for two deformations with fixed elongation to a three-dimensional cavity model with axially symmetric c = 1.53. The first corresponds to the symmetric outer bardeformations that typically occur near the isomer minimum rier with a = 0, and the second to the asymmetric saddle with and the second maximum of the characteristic douple- a. = 0.13. The vertical lines show the positions z, of the planes humped fission barrier [1,2] of many actinide nuclei. The containing the periodic orbits. In the symmetric case (Fig. 1), boundary of the cavity in cylindrical coordinates (z, p) is the shortest orbits at zo = 0 are slightly unstable, wheras given by a shape function p(z); the deformations are denned those at — z2 = z\ = 0.414 are stable. In the asymmetric case by an elongation parameter c and an octupole-type left-right (Fig. 2), there is only one plane at zo = 0.671 containing asymmetry parameter a (see Ref. [1] for the detailed stable orbits. In the lower parts of the figures, we show definitions; the neck parameter h is fixed to be zero in the Poincare surfaces of section (p, tj>) for trajectories with present work). The shortest periodic orbits are found in (conserved) angular momentum component Lz = 0, where planes perpendicular to the symmetry (z) axis at locations p is the component of the momentum parallel to the tangent z,- given [6] by p'(z,-) = 0; they are just the shortest polygons plane at a reflection point and <j> is the polar angle in the (diameter, triangle, square, etc.) inscribed into the circular (z, p) plane of a reflection point at the boundary. (Note that cross sections of the billiard system with radii p(zt). The the Poincare mapping in these variables is not area oscillating part 8E of the total energy of the system con- preserving; this is, however, immaterial for the qualitative taining N fermions (we do not distinguish neutrons from interpretation of the figures.) In both cases we find two main protons and neglect the Coulomb and spin-orbit inter- islands of stability containing the fixed points corresponding actions) was calculated by the appropriate semiclassical to the shortest stable and unstable orbits mentioned above. trace formula. A uniform approximation was introduced Apart from some KAM chains of smaller islands correto describe the bifurcation of a single orbit plane (at sponding to higher resonances, the remainder part of the phase space is mainly chaotic. Note that the overall z0 = 0) into three orbit planes z, (i — 0, 1, 2) at the moment chaoticity is much larger in the energetically more stable of the neck formation where the shape function p(z) becomes asymmetric case. We thus find that the shell effect which convex (see Ref. [3] for details). The gross-shell features in leads to the energetic instability of the symmetric outer SE were emphasized by convolution of the trace formula barrier, at the same time pulls the system into a more chaotic over the wave number k by a Gaussian of width 0.6//? (R transition state. is the radius of the spherical nucleus), producing a smearing In Ref. [9], the microscopic origin of the instability against of the shell structure similar to that caused by the residual asymmetric deformations was linked to specific quantum pairing interaction in realistic nuclear models [1].

We discuss the localization of wavefunctions along planes containing the shortest periodic orbits in a three-dimensional billiard system with axial symmetry. This model mimics the self-consistent mean field of a heavy nucleus at deformations that occur characteristically during the fission process [M. Brack et al.. Rev. Mod. Phys. 44, 320 (1972); S. Bjornholm et al. Rev. Mod. Phys. 52, 725 (1980)]. Many actinide nuclei become unstable against left-right asymmetric deformations, which results in asymmetric fragment mass distributions. Recently we have shown [M. Brack et al., Phys. Rev. Lett. 79, 1817 (1997); M. Brack et al, in "Similarities and Differences Between Atomic Nuclei and Clusters", (AIR New York, 1998), p. 17] that the onset of this asymmetry can be explained in the semiclassical periodic orbit theory by a few short periodic orbits lying in planes perpendicular to the symmetry axis. Presently we show that these orbits are surrounded by small islands of stability in an otherwise chaotic phase space, and that the wavefunctions of the diabatic quantum states that are most sensitive to the left-right asymmetry have their extrema in the same planes. An EBK. quantization of the classical motion near these planes reproduces the exact eigenenergies of the diabatic quantum states surprisingly well.

Physica Scripta T90

© Physica Scripta 2001

Wavefunction Localization

P(z)

and its Semiclassical

Description in a 3-Dimensional

0.5

17

0.0

16

System with Mixed Classical Dynamics

147

:

•

11

-0.5

W

15

-1.5 -1.0 -0.5 0.0

0.5

z

1.0

•*«««H^9

1.5

9*

=3;

14

ji&t ** '~\*L£&K*Z

'life

13

12 11

:

:

°-° u4 v®^®/ &sfe& l©x©/ *& M# v T ^ S &€% i ^ y #te

— 10 0.0

fe,m

"- 4 a s s *

m

0.05

0.1 a

0.15

0.2

F(£. i. Quantum levels kjR for the lowest states i with L r = 4 at elongation c = 1.53. plotted versus asymmetry a The emphasized portions connected by avoided crossings denote some of the diabatic states (see text).

6 states having their probability maxima in the central equatorial plane or in two planes parallel to it. These states were Fig. 1. (Top): Cavity shape p(z) at elongation c = 1.53 and asymmetry a = 0.Q energetically most sensitive to the asymmetric deformations, The vertical lines indicate the planes containing the shortest periodic orbits. (Bottom): Poincare surface of section (/>, ) for L~ = 0. See the text for details. whereas all others were practically not. In order to check this behaviour in our present cavity model a n d to investigate the relation of these states to the leading periodic orbits, we have 1.0 calculated its q u a n t u m spectrum (with Dirichlet boundary conditions) at the relevant deformations. In Fig. 3 we show 0.5 the wave number spectrum kt = *J2mEj/h (in units of P(z) \/R) for the lowest states / with angular m o m e n t u m 0.0 L, = 4 obtained at c = 1.53 versus the asymmetry parameter •0.5 a. Whereas most levels vary only little with a, we clearly recognize a sequence of states whose energies decrease m o r e •1.0 and more steeply with increasing number /'. These states -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 are diabatic in that they connect different portions of the level spectrum through avoided crossings; in the upper part 1.0 &*&&"" of the spectrum we have emphasized them by heavy grey lines and labeled them by the numbers / of the adiabatic levels counted from the bottom. We shall show below that the energies of these diabatic states can be obtained from 0.5 an EBK quantization of the classical motion near the equatorial planes.

0.0 &

*

CL«

^ -0.5 ^~. J. * V

-1.0

r

h'tfSa.^r.'T...^...

0

1

2

3 , 4

Fig. 2. The same as in Fig. 1 for asymmetry a = 0.13. © Physica Scripta 2001

5

*g?

.V-H^K.!

The mechanism of the energy instability against a. now is this [9]: if one or two of the diabatic states are located just underneath the Fermi energy (and thus occupied), the system gains energy when oc is increased from zero. Since all the occupied " i n e r t " states slightly increase at larger values of a, the sum of occupied levels (and hence SE, see Ref. [7]) will exhibit a minimum at some finite value ao- In our model calculation [3,4] for Pu 2 4 0 with a fixed Fermi energy kf = 12.1/R, we obtained a0 ~ 0.13. In Fig. 4 we show probability distributions by contour plots of the squares of wave functions \\jjt(z,p)\2 in the (z, p) plane. The centre and b o t t o m panels correspond to ( = 10 at a = 0 a n d to / = 7 at a = 0.2, respectively, representing one diabatic state at the two ends of the a. Physica Scripta T90

148

M. Brack, M. Sieber and S. M. Reimann the same curvature radii of the equatorial planes. For an ellipsoid, the classical action variables are known [12]. In the vicinity of the equatorial plane they are given by the z component of the angular momentum, Lz/h = 0, ± 1 , ± 2 , . . . , by the action Ip for the radial motion in the equatorial plane,

K

-1.5 -1 -0.5

0

0.5

1

1.5

z

Jp2R\

- 12, - \L:\ arccos \L,J(pR{)\

2

0)

-L2/(p2RlR2)

K arcsin JR\/R2

y/p'-R^Rn-R^+Ti

and by the action /- for the motion out of this plane. (2)

-1.5

-1 -0.5

0

0.5

1

1.5

z

-1.5 -1 -0.5

0

0.5

1

1.5

z

Fig. 4. Contour plots of the probability distributions |i/<,(-, p)| 2 for selected states with L: = 4 at the elongation c = 1.53. The heavy black lines show the shape p(z) of the boundary. The grey vertical lines indicate the planes Zj containing the shortest periodic orbits. (Top): state ( = 1 1 at asymmetry a = 0.0. (Centre): state i = 10 at asymmetry a = 0.0 (Bottom): state i = 7 at asymmetry oc = 0.2.

interval. The top panel represents the beginning of the next higher diabatic state, starting as i = 11 at a. = 0. The probability maxima are clearly located precisely in the planes zi of the shortest periodic orbits, indicated by the heavy vertical lines. This pattern was found to be consistent: the probability maxima of all diabatic states are located in the planes of the shortest periodic orbits, whereas the maxima of all other "inert" states could not be correlated to any of the leading periodic orbits. In order to quantify the correspondence between the diabatic quantum states and the leading periodic orbits, we apply a method analogous to the quasimode construction for eigenfunctions located in the vicinity of stable periodic orbits (see, e.g., [10]). Consider the classical motion in the vicinity of a stable equatorial plane. It consists of small oscillations around this plane. In a linearized approximation, the motion is integrable and restricted to tori in phase space. The quasimode approximation consists in applying the EBK quantization conditions to this torus structure [11]. To derive the quantization conditions, we use the fact that the quasimode approximation depends only on the two curvature radii at the boundary of the stable orbit plane: the radius R\ in the equatorial plane itself, and the radius R2 perpendicular to it. We therefore replace our billiard system locally by an axially symmetric ellipsoid with Physica Scripta T90

These actions depend on the three constants of motion: the energy E (or the momentum squaredp 1 = h~k2), the angular momentum L-, and a conserved quantity K which vanishes for the motion in the equatorial plane and is the analogue of the second constant of motion in a two-dimensional elliptic billiard (given by the product of angular momenta with respect to the two foci). The actions (1), (2) are already given here in the linearized approximation obtained from the exact actions (given by elliptic integrals) through Taylor expansion up to first order in K. Eliminating K from Eqs. (1), (2), we arrive at

/„=;

Jp2R\

- 12. - \LZ\ arccos \L:/(pRy)\ (3)

I, arcsin-jRi/Ri -

l2/(p2RiR2)

to be used with the EBK quantization conditions L=h(n-\/2), H(m-\/A).

« = 1,2,3,...

(4)

m= 1,2,3,

(5)

Note that if one puts L = 0 above, Eqs. (3) and (5) yield precisely the radial EBK quantization condition (without further approximation) for a two-dimensional circular billiard with radius R\. Equations (3)-(5) represent our quasimode approximation. They implicitly yield the energies Enm (or wave numbers knm) of quantum states with angular momentum L, located near the planes of stable periodic orbits, which we expect to represent the diabatic states discussed above. The quantum numbers n and m count the wave function extrema in the z and p directions, respectively. (Correspondingly, n — \ and m — 1 count the numbers of nodes in the respective directions.) For example, the squared wavefunctions shown in the centre and bottom panels of Fig. 4 should represent the state (1,2), and the one shown in the top panel should belong to the state (2, 2). As a test of our interpretation of the diabatic quantum states and their semiclassical quantization, we compare in Fig. 5 the exact quantum spectrum kt (shown by solid lines) to the approximate EBK levels k„m, calculated here for the lowest states with angular momentum L: = 0. Due to our linearization of the actions (1), (2), the agreement should be best for small values of the quantum number n. It is, indeed, perfect for all states with n = 1 (shown in Fig. 5 © Physica Scripta 2001

Wavefunction Localization and its Semiclassical Description in a 3'-Dimensional System with Mixed Classical Dynamics

149

earlier been reproduced by a semiclassical trace formula [3,4]; thereby the fission path through the deformation space was shown to be determined by the constancy of the actions of the shortest periodic orbits. In the present paper we have demonstrated that these orbits lie at the centres of small islands of stability in an otherwise chaotic phase space. Its degree of chaoticity is increased by the shell effect causing the asymmetric deformations. We have also shown that the diabatic quantum states which energetically favor the asymmetry have their probability maxima precisely in the planes where the shortest periodic orbits are located. A quasimode approximation based on EBK. quantization of the linearized classical motion around the planes of the shortest stable orbits allowed us to uniquely assign quantum numbers (L:,n,m) to the diabatic quantum states and to semiclassically reproduce their energies rather well.

(23)

12

10

(2,2)

Pi (1,2)

(2,1)

Acknowledgements

0,1)

0.0

0.05

0.1

0.15

0.2

We acknowledge the help of P. Meier in constructing the energy level plots. This work has been supported by the Deutsche Forschungsgemeinschaft (M.B.) and a Habilitationsstipendium des Frcistaates Bayern (S.M.R.).

a Fig. 5. Quantum levels ktR of lowest states ;' with L. = 0 at elongation c = 1.53 versus asymmetry a (solid lines). Labels (n, m) indicate diabatic states; their levels knmR obtained by the quasimode approximation, Eqs. (3)-(5), are shown by dotted lines for « = 1 and dashed lines for n = 2.

by the dotted lines), which agree exactly with the quantum levels of the corresponding diabatic states. The agreement is less good for the states with n = 2 (shown by the dashed lines), but our semiclassical approximation still reproduces their correct slopes for larger a and allows for their unique assignment. Similar results are also obtained for larger values of Lz (for which the numerical agreement actually improves). In summary, we have established a correspondence between the shortest classical periodic orbits and a set of diabatic quantum states responsible for the onset of the left-right asymmetry in a simple axially deformed cavity model describing schematically the shapes of an actinide nucleus on its adiabatic path to fission. The energy gain due to the asymmetric deformations near the outer barrier had

© Physica Scripta 2001

References 1. Brack, M. et at. Rev. Mod. Phys. 44, 320 (1972). 2. Bjornholm, S. and Lynn, J. E., Rev. Mod. Phys. 52, 725 (1980). 3. Brack, M., Reimann, S. M. and Sieber, M„ Phys. Rev. Lett. 79, 1817 (1997). 4. Brack, M.. Meier, P., Reimann, S. M. and Sieber. M., in "Similarities and Differences between Atomic Nuclei and Clusters", (eds. Y. Abe et at), (AIP, New York, 1998), p. 17. 5. Gutzwiller, M. C , J. Math. Phys. 12, 343 (1971). 6. Balian, R. and Bloch, C , Ann. Phys. (N.Y.) 69, 76 (1972). 7. Strutinsky, V. M., Nucl. Phys. A 95, 420 (1967); A 122. 1 (1968). 8. Moller, P. and Nilsson, S. G„ Phys. Lett. 31B, 283 (1970); Pauli, H. C , Ledergerber, T. and Brack, M.. Phys. Lett. 34B, 264 (1971). 9. Gustafsson, C , Moller, P. and Nilsson, S. G , Phys. Lett. 34B, 349 (1971). 10. Voros, A., Colloques Internationaux du CNRS no. 237 (Aix-enProvence, France, 24-28 June 1974), ed. J. M. Sourian (1975), p. 277. 11. The quasimode treatment is limited to stable orbits. Note, however, that some wavefunctions are also found along planes of unstable orbits (see center of Fig. 4). This is similar to the scar phenomenon of the localization of wavefunctions near isolated unstable periodic orbits. 12. Richter, P. H.. Wittek, A., Kharlamov, M. P. and Kharlamov, A. P.. Z. Naturforsch. 50a, 693 (1995).

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Physica Scripta. T90 150-153, 2001

Neutron Stars and Quantum Billiards Aurel Bulgac1'2 and Piotr Magierski1'3 'Department of Physics, University of Washington. Seattle. WA 98195-1560, USA Max-Planck-Insitut fur Kernphysik, Postfach 10 39 80, 69029 Heidelberg, Germany Institute of Physics, Warsaw University of Technology, ul. Koszykowa 75, PL-00662, Warsaw. Poland

2

Received July 27, 2000

PACS Ref: 21.10.Dr, 21.65.+f, 97.60.Jd, 05.45+b

Abstract Homogeneous neutron matter at subnuclear densities becomes unstable towards the formation of inhomogeneities. Depending on the average value of the neutron density one can observe the appearance of either bubbles, rods, tubes or plates embeded in a neutron gas. We estimate the quantum corrections to the ground state energy (which could be termed either shell correction or Casimir energy) of such phases of neutron matter. The calculations are performed by evaluating the contribution of the shortest periodic orbits in the Gutzwiller trace formula for the density of states. The magnitude of the quantum corrections to the ground state energy of neutron matter are of the same order as the energy differences between various phases.

It is somewhat surprising to find out that the problem we are going to present here, which could have naturally emerged as a typical study case in quantum chaos, has somehow eluded the attention of this community. Various billiard problems, in particular the Sinai billiard in 2D, are paradigmatic in quantum chaos studies. The structures believed to appear in the crust of neutron stars, namely bubbles, tubes, rods and plates, known also as the "pasta phase" [1], are nothing else but one of several natural realizations of Sinai billiards. The energetics of these and similar phases at higher densities involving quarks has been studied for many years [1] (this reference list is not meant to be exhaustive). All the studies mentioned above have estimated the ground state energy of neutron matter only in the liquid drop or Thomas-Fermi approximation. In this approximation applied to finite fermi systems - such as an atom, a nucleus, a metallic cluster or a quantum dot - one fails to put in evidence the fact that in the case of closed shells these systems show an enhanced stability. Apparently, an agreement has been reached in literature concerning the existence of the following chain of phase changes as the density is increasing: nuclei —*• rods -> plates —> tubes —> bubbles —*• uniform matter. There are a couple of studies, where the role of quantum corrections has been taken into account, one being the Hartree-Fock calculations of Refs. [2], In Ref. [3] the shell effects due to the bound nucleons in nuclei immersed in a neutron gas only (mainly protons) have been computed. There it was concluded that this type of quantum corrections to the ground state energy will not lead to any qualitative changes in the sequence of the nuclear shape transitions in the neutron star crust. However, no other type of quantum corrections was suggested. We have shown that the quantum corrections to the ground state energy of neutron matter arising from the unbounded motion of fermions outside the bubbles in particular, are more important quantitatively and thus can lead to qualitative changes in * e-mail: bulgac(ajphys.washington.edu and [email protected] Physica Scripta T90

the physics of neutron matter at subnuclear densities []. In particular, since the magnitude of the quantum corrections to the ground state energy is of the same order of magnitude as the energy differences between various "pasta phases", there is no reason to expect that the sequence "nuclei —> rods —>• plates —>- tubes —>• bubbles —> uniform matter" is occurring and instead strong disorder could dominate over various regular lattices. Let us start our analysis with reviewing a somewhat simpler case, that of systems with only one bubble [5,6]. About half a century ago it was predicted that if a large nucleus was ever to be observed, most likely it would have either the shape of a torus or of a sphere with an empty spherical cavity inside (this is what is typically referred to as a bubble nucleus) [7]. Such geometries lead to a minimum of the sum of Coulomb and surface energies. Recently it was suggested that many fermion systems with a similar geometry can be created in the case of highly charged atomic clusters [8]. In order to focus our attention on the quantum corrections, we shall discard here the role of long range Coulomb interaction, which is easy to account for, and we shall sidestep the issue of bubble stability as well and refer the interested reader to earlier literature. In the case of atomic clusters the Coulomb energy plays a secondary role anyway [6]. For the sake of simplicity, we shall consider only one type of fermions with no electric charge. It is known that in the case of a saturating N-fermion system the total energy has the general structure E(N) = evN + esN2/i + ecNl/i + ESC(N).

(1)

The first three terms represent the smooth, liquid drop part of the total energy. ESC(N) is the pure quantum contribution, known as shell correction energy, the amplitude of which grows in magnitude approximately as oc Nl/6, see Ref. [9]. The liquid drop part of the energy in the case of billiard systems is referred to as the Weyl energy [10,11]. One of the simplest questions one can raise is: How should one determine the position of a bubble or void? Considerations based on naive symmetry arguments have been used in the past, without any critical consideration, to simply position the bubble in the very center of the system. Recently we have shown that moving the bubble off-center can often lead to a greater stability of the system, due to shell correction energy effects alone [5,6]. It is worth noting at this point that the shell correction energy bears a remarkable similarity to the Casimir energy in quantum field theory and in critical phenomena [12-14]. The Casimir energy can be thought of as a specific measure of the magnitude and nature of the fluctuations induced in the energy spectrum by the © Physica Scripta 2001

Neutron Stars and Quantum Billiards presence of various "obstacles" and is computed with a formula very similar to the one used to estimate the shell correction energy [4]. There is a large number of studies of various fermion systems, which shows that the character of the shell corrections is strongly correlated with the existence of regular or chaotic motion in the classical limit [11,16-18]. Integrable systems have conspicuous large gaps in their spectra, while chaotic systems have a spectrum characterized by rather small fluctuations and essentially no large gaps. If a spherical bubble appears in a spherical system and if the bubble is positioned at the center, then for certain magic fermion numbers the shell correction energy ESC(N), and hence the total energy E(N), has a very deep minimum. However, if the number of particles is not magic, in order to become more stable the system will in general tend to deform. Real deformations lead to an increased surface area and thus to an increased liquid drop energy. In the case of systems with bubbles/voids there is a somewhat unexpected extra soft deformation mode bubble positioning. Merely shifting a bubble off-center deforms neither the bubble nor the external surface and therefore, the liquid drop part of the total energy of the system remains unchanged. Therefore, one would expect that classically this deformation mode has exactly zero energy. Quantum mechanics at first glance seems to lend some support to this idea. In recent years it was shown that in a 2-dimensional annular billiard, which is the 2-dimensional analog of spherical bubble nuclei, the motion becomes more chaotic as the bubble is moved further from the center [19]. One might anticipate then that the importance of the shell corrections diminishes when the bubble is off-center and thus becomes easier to move a bubble, the further from the center it is. One might also naively assume that the most favorable arrangement is the one with the bubble at the very center, when the system has the highest symmetry. It came as somewhat of a surprise to find out that this is not the case most of the time and that the profile of the energy surface has a very unexpected structure [5,6]. We shall refer the reader to our previous work for more details concerning the physics of systems like bubble nuclei and revert now to the case of bubbles, rods, tubes and plates floating in an infinite neutron gas. All calculations we present here for infinite systems were performed at constant chemical potential. In order to better appreciate the nature of the problem we present here, let us consider the following situation. Let us imagine that two spherical identical bubbles have been formed in an otherwise homogeneous neutron matter. One can then ask the following apparently innocuous question: "What determines the most energetically favorable arrangement of the two bubbles?" According to a liquid drop model approach the energy of the system should be insensitive to the relative positioning of the two bubbles. A similar question was raised in condensed matter studies, concerning the interaction between two impurities in an electron gas. In the case of two "weak" and point-like impurities the dependence of the energy of the system as a function of the relative distance between the two impurities a is given by

151

homogeneous Fermi gas and Vi(ri) and V2(r2) are the potentials describing the interaction between impurities and the surrounding electron gas. At large distances, when kva » 1, this expression leads to an interaction first derived by Ruderman and Kittel [20,21]: cos(2£ F a) E(a) oc

,

,,. (3)

as where kF is the Fermi wave vector and m is the fermion mass. This asymptotic behavior is valid only for point-like impurities, when kFR <JC 1, where R stands for the radius of the two impurities. This condition is typically violated for nuclei and bubbles immersed in a neutron gas, for which kfR 2> 1- We have shown in Ref. [4] that in the case of large "impurities" (kpR 3> 1) and at constant chemical potential the interaction energy at large separations a becomes _. _ h2k\ (R\22sm(2kpa)

...

The most striking aspect of this interaction is the fact that it is inversely proportional to the square of the separation, instead of 1/a3 as in Eq. (3). The bubble-bubble interaction has a surprinsingly long range. Apart from Coulomb and gravitational interactions, there are no other longer ranged interactions known between two objects. One can advance a simple qualitative explanation of the difference in the power law exponent of the impurity-impurity interaction (3) and the bubble-bubble interaction (3) in terms of waves. A large "obstacle" reflects back more of the incident wave than a point object, which acts like a pure s-wave scatterer. We shall see below that perfectly planar surfaces lead to an interaction which decays with distance even slower, inversely proportional to the separation between surfaces. Naturally, when the incident and reflected waves interfere constructively, one does expect a more energetically favorable geometric configuration. Obviously, the mere fact that the interaction (3) oscillates shows that for each value of a = n(n + 3/4)/kp, where n is an integer, the potential has a minimum and one thus expects that a bubble-bubble molecule with such a radius can be formed. Moreover, for each such size various vibrational and rotational states, corresponding to small fluctuations of the bubble-bubble distance and angular velocity are most likely to exist. Tunneling between various molecular sizes should also appear. However, the treatment of such molecular systems and tunneling should be deferred until the bubble inertia is also estimated.

The arguments of the cosine in Eq. (3) and of the sine in Eq. (3) are nothing else but the classical action in units of h of the bouncing periodic orbit between the two impurities and this suggests the most likely method of attack for this problem, the semiclassical approximation based on the Gutzwiller trace formula for the density of states. As a matter of fact this is the way Eq. (3) has been derived [4]. The formation of various inhomogeneities in an otherwise uniform Fermi gas can be characterized by several natural dimensionless parameters: kpa » 1, where as above a is a characteristic separation distance between two such E{a) = \ j ^ jdr2 Vx{r{)X{ri -r2- a)V2(r2), (2) inhomogeneities; kpR^> 1, where R is a characteristic size of such an inhomogeneity; and kps = 0(1), where s is a typir cal "skin" thickness of such objects. The fact that the first where x( i — r2 — a) is the Lindhard response function of a © Physica Scripta 2001

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Aurel Bulgac and Piotr Magierski

two parameters, kpa and k?R, are both very large makes the adoption of the semiclassical approach natural. Since the third parameter, kps, is never too large or too small, one might be tempted to discard the semiclassical treatment in the case of saturating fermion systems altogether as unreasonable. However, there is a large body of evidence pointing towards the fact that even though this last parameter in real systems is of order unity, the seemingly unreasonable approximation kPs <SC 1, which we shall adopt in this work, is surprisingly accurate [11]. The corrections arising from considering k¥s = 0{\) lead mainly to an overall energy shift, which is largely independent of the separation among various objects embedded in a Fermi gas. On one hand, this type of shift can be accounted for in principle in a suitably implemented liquid drop model or Thomas-Fermi approximation. On the other hand, the semiclassical corrections to the ground state energy arising from the relative arrangement of various inhomogeneities have to be computed separately, as they have a different physical nature. We are thus lead to the natural assumption that a simple hard-wall potential model for various types of inhomogeneities appearing in a neutron fermi gas is a reasonable starting point to estimate quantum corrections to the ground state energy, see Refs. [5,6,11] and earlier references therein. One might expect that such simplifications will result in an overestimation of the magnitude of quantum corrections to the ground state energy, but that the qualitative effect would be reproduced. One can compute the quantum corrections to the ground state energy analytically for plate-like nuclei or voids with the neutron gas filling the space between slabs. The quantum correction to the energy for such a system is given by: Weyl

D~

2

£\Veyl

L3

7T3

tl-

2

V> 2ma 2 x N

£ ] P Aigsheu(s, Li)de,

Z3

Pout = PWeyl +J)\

ft

2-, ^i&hellCe,

J

(L\

N(N + \)(2N + \)(3N2 + 120

.

.

r

mLj

^cos(2nkLj) >

2nn'k^

1 /kpa s{ n

(6) 3N-\)

frkl 40aLm

,

(12)

,

sinh"(«K,-)

where h2k2 e = 2m '

(13)

1 (kfd\

M+,MH]

= ln

(14)

and hence we get: •fcshell _ ^

(V)

H A\

i=l

^-r i

n=l

[2nkpL{Cos(2nkpLi) + {2rrk\L2 — l)sin(2«&F£i)] (15)

(8)

stands for the integer part of the argument in the square brackets, and a = L — 2R is the distance between slabs and R is half of the width of the slab. Here L3 is the volume of an elementary (cubic) cell and the factor " 2 " in front stands for the two spin states. Using these formulas, one can show that the shell correction energy behaves as Z.3

,

n 3 sinh"(«Kj)

'kpa n

-shell

(12)

i)de

« i=i

and where N = Int

L

1 (kfd

\a) 4\~

2 h 1 n* L (\ ~~ L? 2ma2 2 \a)

(11)

The contribution due to one periodic orbit to the fluctuating part of the level density reads:

(5)

where the exact and the Weyl (smooth) energy [10,11] per unit volume are given by E

1

-shell

gshell(£, U) =

E-E,

-shell

argument. For x > 1, G(x + 1) ~ G(x), with properties G(x = n) ^ - 1 and - 1 < G(x) < 0.5. Note that at small separations a the shell correction energy is attractive in character. In the case of rods or tubes and bubbles we resorted to the Gutzwiller trace formula to estimate the corrections to the density of states. We were interested in what in the nuclear physics lingo would be called the "gross shell structure" and we discarded the fine details. As this approach implies a certain amount of spectral averaging one needs to account only for the existence of the shortest periodic orbits. As we are dealing with an infinite system there are no discrete levels and correspondingly no shells. However, the presence of "obstacles" induces fluctuations in both spatial matter distribution and density of states, which thus lead to the appearance of what we term shell correction or Casimir energy. For the case of spherical voids there are 26 periodic orbits between the nearest neighbors of three different lengths 2Li = 2(L - 2R), 2L2 = 2{L-j2 - R) and 2Li = 2(LV3 - R) in the case of a simple cubic lattice. Thus the shell correction/Casimir energy and density are equal to:

„(kfa • ( j

\ n

(9)

where G{x) is an approximately periodic function of its Physica Scripta T90

1 1 ^ Pout — PWeyl + 7 T 7 r 2 ^ L-4n^

^ sin(2«^F^i) ' £—, ~~uT, \' ^«sinh-(«Ki)

(16)

where Ai = 6, Ai = 12, ^ 3 = 8 respectively. Above, the summation over n is over repetitions of a given orbit and such summation becomes superfluous when one performs a spectral averaging in order to extract the "gross shell structure". This shell energy can be regarded as the interaction energy between bubbles immersed in neutron gas. The bubble-bubble interaction (3) was obtained from the above formula for the case of two bubbles only in the asymp© Physica Scripta 2001

Neutron Stars and Quantum Billiards totic limit. One can derive similar formulas for the case of rods [4]. The fact that the interaction energy between various homogeneities is described in terms of periodic orbits, points to another nontrivial aspect: the interaction between three or more such objects cannot be reduced to simple pairwise interactions. Orbits which bounce among three or more objects would lead to three-body, four-body and so forth genuine interactions. Such orbits were not yet included in our analysis, even though they can be accounted for in a straight forward manner. Since periodic orbits of this kind are typically longer, one expects their contribution to be masked when one performs a spectral averaging in order to extract the "gross shell structure". Lack of space prevents us from presenting more detailed results and discussing many other aspects of the energetics of inhomogeneous fermi systems. Hopefully, the few results we have highlighted here will convince the reader of the richness of these systems and of the spectacular role played by geometry and the chaotic versus integrable character of the single-particle dynamics. Besides static properties, one should expect that the dynamics of such systems will be extremely rich and challenging to describe.

References 1. Baym et at., Nucl. Phys. A 175, 225 (1971); Ravenhall. D. G. et a!., Phys. Rev. Lett. 50. 2066 (1983): Hashimoto, M. et at. Prog. Theor. Phys. 71. 320 (1984); Oyamatsu, K. et at. Prog. Theor. Phys. 72, 373 (1984); Lattimer, J. M. et at, Nucl. Phys. A 432, 646 (1985); Wilson. R. D. and Koonin, S. E. Nucl. Phys. A 435, 844 (1985); Lassaut, M. et at, Astron. Astrophys. 183, L3 (1987); Oyamatsu, K., Nucl. Phys. A 561, 431 (1993); Lorenz, C. P. et at, Phys. Rev. Lett. 70, 379 (1993); Pcthick, C. J. and Ravenhall, D. G., Annu. Rev. Nucl. Part. Sci. 45, 429 (1995); Watanabe, G. et at, see Los Alamos e-printarchiveastro-ph/0001273; Heiselberg, H.etat, Phys. Rev. Lett. 70. 1355 (1992). 2. Negele. J. W. and Vautherin, D., Nucl. Phys. A 207, 298 (1973); Bonche. P. and Vautherin, D„ Nucl. Phys. A 372, 496 (1981); Astron. Astrophys. 112, 268 (1982).

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3. Oyamatsu, K. and Yamada, M., Nucl. Phys. A 578, 181 (1994). 4. Bulgac, A. and Magierski, P., Los Alamos e-print archive astroph/0002377. 5. Bulgac, A. et at, in "Proc. Intern. Work, on Collective excitations in Fermi and Bose systems", (Eds. C. A. Bertulani and M. S. Hussein), (World Scientific, Singapore, 1999), pp. 44-61 and Los Alamos e-preprint archive, nucl-th/9811028. 6. Yu, Y. et at. Phys. Rev. Lett. 84, 412 (2000). 7. Wilson.H. A., Phys. Rev. 69 538 (1946); Wheeler, J. A., unpublished notes; Siemens, P. J. and Bethe, H. A., Phys. Rev. Lett. 18, 704 (1967); Wong, C. Y„ Ann. Phys. 77, 279 (1973); Swiatecki, W. J., Physica Scripta 28. 349 (1983); Myers, W. D. and Swiatecki, W. J., Nucl. Phys. A 601, 141 (1996). 8. Pomorski, K. and Dietrich, K., Eur. Journ. Phys. D 4, 353 (1998). 9. Strutinsky, V. M. and Magner, A. G., Sov. J. Part. Nucl. Phys. 7, 138 (1976). 10. Waechter, R. T„ Proc. Camb. Phil. Soc. 72, 439 (1972); Baltes, H. P. and Hilf, E. R.. "Spectra of Finite Systems", (Wissenschaftsverlag, Mannheim, Wien, Zurich: Bibliographisches Institut, 1976). 11. Brack, M. and Bhaduri, R. K.., "Semiclassical Physics", (AddisonWesley. Reading. MA, 1997). 12. Casimir, H. B. G., Proc. K. Ned. Akad. Wet. 51, 793 (1948); Mostepanenko, V. M. and Trunov, N. N., Sov. Phys. Usp. 31, 965 (1988) and references therein. 13. Fisher, M. E. and de Gennes, P. C , C.R. Acad. Sci. Scr. B 287, 207 (1978); Hanke, A. et at, Phys. Rev. Lett. 81, 1885 (1998) and references therein. 14. Kardar, M. and Golestanian, R„ Rev. Mod. Phys. 71, 1233 (1999) and references therein. 15. Bohigas. O. et at. Phys. Rev. Lett. 52, 1 (1984). 16. Balian, R. and Bloch, C , Ann. Phys. 67, 229 (1972). 17. Nishioka, H. et at, Phys. Rev. B 42, 9377 (1990); Pedersen, J. et at. Nature 353, 733 (1991); Brack, M., Rev. Mod. Phys. 65, 677 (1993) and references therein. 18. Strutinsky. V. M.. Sov. J. Nucl. Phys. 3. 449 (1966); Nucl. Phys. A 95, 420 (1967); ibid A 122, 1 (1968); M. Brack et at. Rev. Mod. Phys. 44, 320 (1972). 19. Bohigas, O. et at, Phys. Rep. 223, 43 (1993); Bohigas, O. et at, Nucl. Phys. A 560, 197 (1993); Tomsovic, S. and Ullmo, D., Phys. Rev. E 50, 145 (1994); Frischat, S. D. and Doron, E., Phys. Rev. E 57, 1421 (1998). 20. Ruderman, M. A. and Kittel, C , Phys. Rev. 96, 99 (1954). 21. Fetter, A. L. and Walecka, J. D., "Quantum Theory of Many Particle Systems", (McGraw-Hill, New York, 1971).

Physica Scripta T90

Physica Scripta. T90, 154-161, 2001

Scars and Other Weak Localization Effects in Classically Chaotic Systems E. J. Heller* Department of Physics and Department of Chemistry, Harvard University, Cambridge, MA 02138, USA Received September 8, 2000

PACS Ref: 03.40.Kf, 03.80.+r, 03.65.Sq.

Abstract There is much latitude between the known requirements of Schnirelman's theorem regarding the ergodicity of individual high-energy eigenstates of classically chaotic systems, and the extremes of random matrix theory. Some eigenstate statistics and long-time transport behavior bear nonrandom imprints of the underlying classical dynamics while simultaneously obeying Schnirelman's theorem. Here we review the issues and give evidence for the Sinai billiard having non Random Matrix Theory behavior, even as h —>• 0.

1. Introduction The field of quantum chaos roughly divides into work on the "baseline" random matrix theory (RMT) and its implications on one hand, and modifications to RMT imposed by the underlying dynamics, on the other. Most of the work on the dynamical modifications to RMT has focussed on quantum eigenvalue spectra; however we concentrate here on the eigenfunctions. The rigorous results on the nature of quantum eigenstates impose far weaker requirements than Gaussian random statistics would require. Specifically, theorems by Schnirelman, Zelditch, and Colin de Verdiere (SZCdV) [1] state that for a classically defined operator, the expectation value over almost all wavefunctions converges to the microcanonical average of the classical version of the operator, in the h -*• 0 limit. Since the operator is kept fixed as the limit is taken, these theorems provide information only about the coarse-grained structure of the eigenstates, and not about the structure at quantum mechanical scales, or any scale that decreases as h does. Known non-RMT behavior of eigenstates in a classically chaotic system include scarring, the anomalous enhancement (or suppression) of intensity near an unstable periodic orbit. The distribution of wavefunction intensities on a fixed periodic orbit can be computed in the semiclassical limit using the linear and nonlinear theory of scars [2^1], and is found to be very different from the Porter-Thomas prediction of RMT. Furthermore, upon ensemble averaging, a power-law wavefunction intensity distribution tail is obtained (and numerically observed) in chaotic systems, in contrast with the exponential falloff prediction of RMT. The fraction of strongly scarred states remains finite in the h —> 0 limit. Nonetheless, scarring poses no threat to the SZCdV ergodicity condition, for several reasons. First, the size of the scarred phase-space region surrounding the orbit scales as /i, tending to zero in the semiclassical limit. Second, the scar strength decreases as the inverse of the Lyopunov exponent for the orbit [2,5], limiting the effect *e-mail: [email protected] Physica Scripta T90

to the shorter periodic orbits. An enhancement factor affecting an ever smaller region of phase space is entirely consistent with ergodicity on coarse-grained scales. However, the scarring phenomenon does have very significant effects on physical quantities that depend on fine-scale structure, such as conductances and decay rates through small (or tunneling) leads [6,7]. Another example of markedly non-RMT behavior still consistent with SZCdV coarse-grained ergodicity is found in the "slow ergodic" systems, such as the tilted wall billiard and the sawtooth potential kicked map [8], and the Sinai billiard [9]. In these systems, the classical rate of exploration in momentum space is slow in some regions. For large h~x, the number of channels occupied by a typical eigenstate for the tilted wall billiard scales only as h~l/2 log A -1 , constituting an ever decreasing fraction of the 0(h~v) total available number of channels. However, the "bright" channels occupied by a given wavefunction tend to be evenly distributed over the entire phase space, and thus coarse-grained ergodicity still holds in the limit, even though the wavefunctions are becoming less and less ergodic at the single channel scale as h —*• 0. The reasons for this are discussed below.

2. Measures of ergodicity We review some important concepts related to the quantitative measurement of quantum structure and transport at "microscopic" (i.e. single-wavelength or single-channel) scales. A more extensive discussion may be found in [8]. Consider a classically ergodic system with quantum eigenstates \E„) and a test state basis \a). The test basis can be chosen to be the set of position states, momentum states, phase-space Gaussians, or any other set of states motivated by the physics of the problem. In a scattering problem one often finds it useful to use momentum states or channels as the reference basis \a), and look for localization of the full eigenstates relative to this basis. The matrix elements (a\E„) contain information on the the degree of localization. It is important to know whether or not known conservation laws are affecting the dynamics. That is, if a priori we know that e.g. energy or angular momentum is conserved, then we incorporate that fact into our analysis and do not brand a system as being nonergodic when we find it does not mix regions of phase space associated with different values of the conserved quantities. The natural test states may however span a range of values of the conserved quantities. There are two reasons this is not a trivial issue: © Physica Scripta 2001

Scars and Other Weak Localization Effects in Classically Chaotic Systems first, experimental situations often produce such test states. Second, we can quickly paint ourselves into a corner if we insist on using test states which already obey some known conservation law. As an example, consider a system with a nondegenerate energy spectrum. If we insist that test states must have fixed energy, they can only be eigenstates, and we are put into a rather awkward situation. In a billiard with energy conserved we may conveniently choose as test states Gaussian wavepackets which are localized in phase space and (necessarily) not eigenstates. It is easy to compute the classical intersection of each such Gaussian with any given energy hypersurface, and the actual quantum intensities Kalis,,)|~ can be normalized by this classical result, which is some smooth envelope (usually itself nearly Gaussian). In this way one can easily identify the degree of eigenstate localization (or deviation from ergodicity) due to quantum effects, as opposed to purely classical constraints. The classical constraints arise from very short time dynamics, shorter than a first collision time with the walls of a billiard for example. See Ref. [10] for a fuller discussion. In other cases we shall be able to assume that no conservation laws prevent each of the eigenstates \E„) from having equal overlaps with all of the test states. Example are quantized maps and S-matrix iterations. We assume for the remainder of the paper that all states are equally eligible a priori, and that no envelope corrections are required. Consider the set of (properly normalized) overlap intensities p"n = \(a\E„)\

(1)

In RMT (a natural baseline assumption in the absence of dynamical information about our system), the (a\E„) are predicted to be given by uncorrelated random Gaussian variables, real or complex as appropriate. The intensities /7JJ then follow a x2 distribution, of one or two degrees of freedom, respectively. Quantum localization will produce an excess of very large and very small intensities, compared to this baseline result. Necessarily, ipttn)a={pan)n =

VN.

(2)

where N is the dimension of the Hilbert space. The averages (...)„ are taken over all eigenstates \E„):

w =NvErf

(3)

£•»=!

The averaging (.. .)a over basis states \a) is defined similarly. The inverse participation ratio (IPR's) is a convenient way to codify the degree of phase space flow; with proper care the IPR's can also be interpreted in the cases where constraints apply [10]. IPRfl = P(a\a) = £ > : ) 2

(4)

with

E^= 1-

(5)

n

This has the dynamical interpretation, generalized to include © Physica Scripta 2001

155

states \b) # \a). P(a\b) = lim -

\(b\a(t))\2&t

(6)

T—•oo 1

as is easily seen by inserting complete sets of eigenstates on the right hand side (for non-degenerate eigenvalues). Also P(a\b) = Y.PanPbn-

(7)

A slightly different convex "entropy" measure of eigenstate localization at a given test state is discussed in [11]. The IPR P(a\a) measures the inverse of the number of eigenstates which have significant intensity at |a). Thus, equal intensities of all the eigenstates at \a) would imply P(a\a) = \/N; this level of ergodicity is of course almost never achieved in a chaotic system. Gaussian random fluctuations (RMT) produce IPR's of 3/iV (for real overlaps (a\E„)) or 2/N (for complex overlaps). IPR's exceeding the appropriate baseline value signal the presence of a localization mechanism beyond RMT, since the inverse of P(a\a) is effectively the number of phase space cells accessed[10] In the extreme localization limit where one eigenstate has all its intensity at \a}, we obtain the maximum possible value, P(a\a) = 1. Imagine for large N that the pan are random (within the normalization constraint) but not Gaussian random; perhaps a finite fraction of the pan for fixed a vanish for example, but the missing intensity varies randomly from state to state. The transport measure P{a\b) = J^np" pb„ will still be 1/iV within small fluctuations; only the fluctuations will be affected by the nature of the random distribition from which the p"n, pbn are chosen. The IPR becomes P(a\a) = 3 / ( / N)

(8)

where / is the fraction of intensities which do not vanish. The SZCdV theorem implies only that the transport measure < P{a\b) >, which is the macroscopically averaged version of P{a\b) (over non-/i dependent domains for \a) and \b)) be \/N. Fluctuations in P(a\a) could be large, even as h —>- 0. In principle then, it is possible that P{a\a) could decrease much slower than \/N for most states |a). Figure 1 shows typical pan for the RMT and random missing cases, and the resulting implications for the transport measures. As was shown previously, there is necessarily a connection between the P(a\b) and P(a\a) [8]. Assuming the elements/^ are statistically independent (but not necessarily Gaussian) variables gives the following relationship between N2 < P{a\bf > and N < P(a\a) >: N2 < P(a\b)2 > = 1 + j^(N < P{a\a) >) 2 + (9) Ci < (/O 4 > • The second term arises ,from the a = b and n = ri contributions to the sum on the left hand side. N 2 < P(a\b)2 > becomes anomalous if N < P(a\a) > grows faster than N ' / 2 . On the other hand, if N < P(a\a) > < 0(N{'2\ then the anomalous nature of the individual eigenstates need not qualitatively affect the transport properties of the Physica Scripta T90

156

E. J. Heller 4. Sinai billiard

Gaussian random

^MJtAJrw

P(ala) = 3/N P(blb) = 3/N P(alb) = 1/N

non-Gaussian random:

1 • • • i - r •(• | -

'If "1

P(ala) = K/N; K>3 P(blb) = K/N P(alb) = 1/N

Fig. I. Typical spectra/)" for RMTand random missing intensity cases, and the resulting P(a\a) and P(a\b)'s. The spectrum for the state \a) is shown above the ticks for the eigenvalue spectrum, and for \b), below.

system, even at the single-state level. This is a key point of the work of Kaplan and Heller [8,9]. The question we pose is: are eigenstates of classical chaotic systems very nearly like RMT would suggest, or do they localize, within the confines of SZCdV? The localization, if it exists, may be peculiar to a certain basis. Systematic localization in any basis which is itself localized is a violation of RMT. 3. Examples of localization An example of such extreme behavior is the case of the "bouncing ball" states [12], associated with non-isolated, marginally stable classical periodic motion. Such motion can trap a quantum wavepacket \a) for a time comparable to (or even longer than) the time at which individual quantum states are resolved, causing the wavepacket to have 0(1) overlap with only one or a few eigenstates, and leading to IPRfl = 0{N). From the SZCdV theorems, we easily see that that the fraction of bouncing ball states must tend to zero in the h —»• 0 limit. This kind of localization is easily visible to the naked eye; other kinds of localization, where the number of eigenstates having intensity at the test state \a) is large compared to 1 but small compared to the total number of states N, may be less easy to detect visually but may also be a statistically more important correction to RMT predictions, surviving at arbitrarily small values of h. In the slow ergodic systems such as the tilted wall billiard and the sawtooth potential kicked map [8], a highly anomalous IPR measure was predicted and observed for small h, with the system-averaged IPR scaling (with the IPR as normalized here) as v ^ / l o g / T 1 . Semiclassically, the degree of localization in such systems is even stronger, with the IPR scaling only as 1/log/T 1 . The difference is caused by diffraction, which dominates the phase space exploration and increases by v /T 1 the fraction of phase space occupied by a typical eigenstate. These same diffractive effects lead to almost perfect long-time transport between channels. We will illustrate non-RMT localization with the case of the Sinai billiard, one of the paradigms of classical and quantum chaos. As was shown in Ref. [9], the system localizes in channel space when viewed as a scattering system (essentially the Lorentz gas). Physica Scripta T90

4.1. Definition of system and motivation The Sinai billiard [13] is a prototypical example of strong classical chaos: it consists of a point particle bouncing freely in a rectangular cavity with hard walls, with a hard disk obstruction placed in the center of the rectangle. The system has positive entropy classically for any disk size; of course, this fact becomes relevant to the quantum mechanics only in the limit where the quantum wavelength is small compared to the size of the disk. (The mixing time Tma after which a typical wavepacket spreads over the entire available phase space is then short compared to the Heisenberg time TH, defined as h over the mean level spacing, at which the quantum dynamics becomes quasi-periodic and individual eigenstates and eigenvalues begin to be resolved.) The statistics of energy levels in the (desymmetrized) Sinai billiard has been found to be in good agreement with the GOE predictions of random matrix theory [14]. On the other hand, the eigenstate structure of the Sinai billiard turns out to be very different from RMT expectations, and the inclusion of short-time dynamical effects is essential for understanding its quantum ergodic properties. 4.2. Localization in Sinai billiards The Sinai billiard was the first nontrivial dynamical system shown to be ergodic with positive Lyapunov exponent [9]. In this sense it is the paradigm of chaos. It is also a unit cell of the Lorenz gas, a periodic array of hard disk scatterers (see Fig. 2a), with Dirichlet boundary conditions at the walls. A scattering system closely connected with both the Lorenz gas and the Sinai billiard puts the Sinai disk at

Fig. 2. (a) The Lorenz gas and two choices for a fundamental domain, (b) A Sinai like billiard related to the Lorenz gas. © Physica Scripta 2001

Scars and Other Weak Localization Effects in Classically Chaotic Systems

157

1 4

\

> U U exp(- i if) ^

4

2 * U exp(- i <|>)

^

U 2 exp(-2 ii)))

•

Fig. 3. The modified Sinai system, with a partial disk occupying a variable fraction / of the right hand vertical wall.

the end of a corridor of length a (Fig. 3). For numerical reasons we investigate a modified Sinai system with the circular disk off center and jutting only part way into the billiard; this is still a chaotic system (see Fig. 2b). The scattering wavefunction can then be expanded as V (x, y) = — ^ -

e_i*"x sm(mty/b) 1

73k a k x

(10>

Yd-J^Sm/tr ^ e -' sin(n'ny/b)

where for later convenience we have factored out a phase exp(2iA:„-a) from the n'-th column of the S-matrix. (If there is no scatterer on the right hand wall, this makes S1 the diagonal unit matrix, assuming Dirichlet conditions there). Now suppose that we reflect the scattered wave from the left wall back towards the right hand side, in accordance with the closed billiard problem we wish to solve. This can be done by imposing a boundary condition at the left wall, which need not necessarily be Dirichlet. (We indicate this by using a dashed line to represent this wall in Fig. 3.) If the wave is reflected from the left wall at x — 0, it returns with a new phase exp(i) given by the boundary condition at the left wall. We define Unn> = S„„, exp(-2ik„,a + \<j>). Setting i//n = exp(—'\k„x)sm(nny/b)/^/k^, (right-moving) wave is then (i +

t/

+

[/2 + ... )l/ , n

=

_L_lAn

(11) the net incoming

(12)

(see Fig. 3). Evidently, a bound state can be built up in the billiard if U has an eigenvalue + 1 . We can diagonalize the (/-matrix and consider the properties of its eigenstates. Since U is a unitary matrix, its eigenvalues lie on the unit circle. As we change the phase shift 0 at the left wall, the eigenvalues will correspondingly rotate around the unit circle; each of the N eigenvalues of U (assuming there are N open channels) will pass through +1 for some 0, so that every eigenstate of U is an eigenstate of the closed billiard with some boundary condition at the left wall and Dirichlet boundary conditions elsewhere. If one is willing to search through ranges of energies or of box lengths a one can find a set of eigenstates satisfying a particular boundary condition; this is a way of finding eigenvalues and eigenstates of the billiard with Dirichlet boundary conditions; they are given by eigenstates of U with eigenvalue 1 [15,16]. However here we do not seek the © Physica Scripta 2001

Fig. 4. Two typical eigenstates of the S-matrix for the Sinai-like scattering system.

Dirichlet solutions, since they are not special as far as their localization in the channel space (this has been tested numerically). This is of great value in gathering the statistics needed here. Two typical eigenstates of the [/-matrix are shown in Fig. 4; these show fairly obvious non-statistical mixing of different directions of propagation in the billiard (nonmixing of channels in the scattering approach). 4.2.1. Patterns in the channel (momentum) transport. The density plot of the transport measure Pnri for a typical case (f = 0.1 with 280 open channels, side lengths equal) appears at the top of Fig. 5. A dramatic pattern arises in the channel transport measure Pnn>. We have been making the point that Gaussian random wavefunction statistics are much stronger than required for SZCdV ergodicity, and that much coarser randomness can still lead to ergodic transport classically. We now see that transport in momentum space may even be highly organized, but in a way that still permits coarse grained SZCdV ergodicity. Physica Scripta T90

158

E. J. Heller

The fringe pattern changes with the length of the billiard, and as we now show represents alternating constructive and destructive interference due to the phase factors exp[2iA:„a], where a is the box length and kn is the horizontal wavevector. The S-matrix itself shows none of this fringing, but it is strongly evident already for S2: We have ^nn'

=

2—/ ^nn" ^n""''

(1 "^

n"

and since S is diagonally dominated for small/, the major contribution to S2n, for n ^ ri is ^nn^nri + ^nn'^n'n'

=

i>nn'\^nn + J«'n')

(14)

Of course Snn and S„>„> can interfere; these diagonal elements have factors exp[2i&„a] and exp[2iAva], respectively. Subsequent iterations reinforce this interference and give very sharp preferred channels that one can end up in when starting from a given initial channel. A plot of

0

SO

100

150

200

250

Wnri = |exp[2i^„a] + exp[2i^a]|" appears at the bottom of Fig. 5, and is seen to bear a close resemblance to the fringe pattern in P„„, (the exponent 28 is of course arbitrary and only serves to set the contrast of the plot). It should be kept in mind that the fine detail of the intensity modulations present in P„„> are absent in the lower plot, but the overall modulation of the regions of large and small Pnn> are almost identical. Interestingly, the special channels which correspond to classical free motion (never hitting the obstruction) show up on the diagonal as hyperbolic points of high density. This may be shown by expanding in the channel index (at least in the lower n region where the Taylor series holds for An ~ 1), e.g. | exp[2i/c„a] + exp[2ik{n+An)a] \ « 11 + exp[2ia (dkn/dn) An]\, where a is the length of the rectangular box. Since k„ = k„x = TJ2{E — (n2n2/2b2)) and kny — nn/b, where b is the height of the box, we have Fig. 5. Top: long-time transport probability between channels n and n' for the a = b = In (square) billiard, f=0.1 (m = li = 1). Bottom: the fringe pattern from Eq. (14).

Then the interference in Eq. (15) is maximally constructive for

i.e. exactly for the free motion trajectories. The special channels correspond with the hyperbolic regions along the diagonal in Fig. 5. The near-bouncing ball channels near the free propagation channels preferentially diffract symmetrically about these special channels, as evidenced by the local hyperbolic structure. This is again a consequence of the interference structure in Eq. (14). Essentially, there is a preference to scatter by a multiple of a reciprocal "lattice" vector, (2aAk = Irrni), reminiscent of Bragg scattering from a periodic structure with lattice constant a. The dramatic interference pattern is another interesting quantum signature of a short time effect, already evident after one iteration as explained above. It illuminates another Physica Scripta T90

variation on the theme of this paper: on scales finer than SZCdV, non-Gaussian statistics may prevail. Here, we see a very structured and nonrandom fringe pattern, which however varies on a scale proportional to H, doing no harm to the Schnirelman limit. 4.3. Near bouncing ball trajectories The disk covers a fraction/ of the right hand wall. We take that fraction to be between 0.04 and 0.28. In analogy with the map discussed above, a fraction 1 — / of the incoming wave is not scatterered on the first bounce, approximately independent of the incoming channel. Classically there are a finite number of angles f), with (a/b) tan 6 = n/m for integer m and «, which never hit the disk. For channels corresponding to propagation near these angles there is a reduction in scattering out of the initial channel. These channels are not true bouncing ball modes, © Physica Scripta 2001

Scars and Other Weak Localization Effects in Classically Chaotic Systems

\VvC\N \v\\\ \\\\v

N = 848 2 J t s q u a r e billiard

0.6

typical

but near enough to have a strong effect on lifetimes. ("Time" is now the number of iterations of the (/-matrix.) Figure 6 show the general and special angles and the "shadowing" effect which leads to slow decay. Ref. [9] contains a full discussion of the effects and number of these near "bouncing ball" trajectories. 4.4. Transport measures There are many statistics which can be gathered and compared to RMT predictions. For example, we can collect the distributions of pan, P(a\b), moments of these, etc. The most complete analysis of these can be found in Refs. [9] and [8]. There are separate expressions for large and small tails of such distributions. To generate such theoretical expressions we first find the short time envelopes predicted by the known dynamics, then make an assumption of RMT "filling" of those envelopes. This corresponds to statistical behavior at long times, subject to the constraints imposed by the earlier time dynamics. The envelopes differ for different initial conditions (here we consider various scattering channels as initial states) and the distribution of channels has to be summed over. The shapes of the envelopes can strongly skew the expected distributions of the p°, P(a\b), etc. away from RMT predictions. Indeed, the realization of the importance of antiscarring [7] was based on the fact that the corresponding to periodic orbits can have very strongly decaying wings, which yield far more smallpan's (correspondig to antiscarred states) than RMT would suggest. We consider first the return probability (inverse participation ratio) measures. The scaling relation Pf(lPR„ = x)=fV(fx) was predicted in [9]; a plot of fVtfx) vs fx for various values of the disk s i z e / is shown in Fig. 7, confirming this scaling over the whole domain of IPR values. We see also from the plot that the typical IPR in the Sinai system is ~ 2 / / , which for the values of/ considered is much larger than the RMT-predicted value of 3. We also see the expected broad distribution of IPR's, with N—independent width, in contrast to the RMT prediction that the spread in the IPR distribution should go to zero as l/y/N. The tail of the IPR distribution is predicted [9] to have the power-law behavior P(IPR„ = x) ~ l/fx2. The power law tail together with the cutoff in the maximum IPR lead to the prediction [9] (IPR„)„ = (Pnn) ~ log N/f. A plot of the the dependence of the average IPR on N a n d / is given in Fig. 8, where the agreement with this estimate is seen © Physica Scripta 2001

-

I1

0.3

Fig. 6. The near "bouncing ball" trajectories experience slow scattering; this in turn leads to non-RMT behavior of the eigenmodes.

-

0.103 o 0.140 x 0.184 + 0.232 *

0.4

near bouncing ball

-

f

0.5

\\vw

159

-

0.2

-

0.1

-

°

O

5

10

15

y = f * IPR*N Fig. 7. The probability distribution for the [PR's is plotted for various values of / , showing the predicted scaling behavior.

0.7

-

•

0.6 •

O

u-i 0.5

9

«

f 0.4

o

» *

+

+

*

? t

• ° * ? » oS * 1 I * * *

* f= f= f= * f=

0

9: 0.3

+

0.2

0.140 0.183 0.232 0.285

0.1

-

0

60

80

100

120

140

160

180

200

N Fig. 8. The average IPR is plotted, showing the predicted dependence on N and/.

to be excellent. As predicted, the mean IPR diverges logarithmically away from its ergodic value of 3 in the classical limit. The distribution of small intensities P„E„ for our S-matrix was also given in Fig. 8. As Fig. 9 shows, this too agrees well with theory for the Sinai model. Finally we consider the tails of the intensity and transport measures. From Ref. [9] we expect a cubic fall off in the tail of the PnEn intensity distribution: P(P„E„ = x) ~ l/fx* for x » 1//. In Fig. 10 we display the predicted and numerical results, showing good agreement between the two. This behavior is controlled by the near-bouncing ball dynamics. The tail of the transport distribution measure V{Pnn) is given by [9], V(P„„> = x) ~ l/f2x4. Figure 11 again demonPhysica Scripta T90

160

E. J. Heller \

-

•

•

\

f= 0.072 N = 395

-

1

Tail of Pnn'

\

\ theory P " data

\

e -P/2f

\

-1/2

c •

j„.„

V

r

\ \

•

0.4

0.8

1.4 x10

Fig. 9. The distribution of P„g. is plotted for small values of P„E„ and compared with theory. In this case the bump size is / = 0.072 and the number of channels is TV = 395.

10

\

\ 100

'nn Fig. 11. The tail region of the plot of the f V distribution shows good agreement with the predicted quartic power law, for a 2n x 2n billiard, / = 0.23, 226 channels.

scales become infinitesimal as H —>• 0, yet they may contain infinitely many wavelengths in this limit. The earliest work in this area is scar theory [2,5,8,17], which showed that \ \theory\ the effects of the least unstable periodic orbits survive the \ \ P"3 \ h —>• 0 limit. However this is only one possible type of a N RMT N non-RMT "anomaly" in classically chaotic systems. a ' data \ Our first investigation beyond scar theory, using the so of called tilted-wall billiard [8], examined a very slowly ergodic classical system with the expectation that its eigenstates \ would be maximally likely to show non-RMT behavior. Indeed the eigenstates did show increasing localization on small scales as H-*• 0, while still of course obeying the SZCdV ergodic theorem. In that system it turned out that the transport was dominated by diffraction, which holds true 100 even as h —*• 0. 10 We have shown that the eigenstates of the Sinai system are Pn$ ever more strongly localized in a certain basis as H —> 0. The basis the scattering channels of the straight wall of the conFig. 10. The tail region of the PnEn distribution shows good agreement with the predicted cubic power law, for a 2K x 2ir billiard,/ = 0.23, 226 channels. fining box: essentially momentum space. We showed that the mean inverse participation ratio in the Sinai-like systems diverges logarithmically with increasing energy (or strates very good agreement with this estimate. Notice that decreasing ft), implying that wavefunctions are becoming the RMT prediction is P„„> = 1 for all channels n^ri. less ergodic at the single-channel scale as the classical limit is approached. The situation here is more remarkable than in the tilted billiard [8], since in Sinai systems the Lyapunov 5. Conclusion exponent is positive and classical correlations decay exponentially. A major conclusion of this work is that We have seen that RMT is strongly violated in a system the logarithmically increasing mean IPR is not due to the which is a pardigm of quantum chaos: the sinai billiard. bouncing ball states but instead to the "near-bouncing ball" Nothing we have found contradicts the SZCdV ergodicity channels, whose decay time is large compared to the typical theorem, of course. decay time l/f but still small compared to the Heisenberg The SZCdV theory predicts only coarse grained ergodicity time N at which individual eigenstates are resolved. of individual eigenstates in the h -*• 0 limit, which is much Once again we have shown that short time quantum weaker than the requirements of random matrix ensembles. dynamics and correlation functions have a permanent effect This gap, between random matrix ensembles on the one hand on the localization properties of the eigenstates, as in the and SZCdV on the other, leaves open many questions about case of scar theory. the true nature of eigenstates of classically chaotic systems Undoubtedly there are many more non-RMT effects in in the h -> 0 limit. The SZCdV result does not address eigenstates yet be uncovered in other systems, including the fluctuations of eigenstates on scales that shrink as some some that could affect important physical properties. positive fractional power of Planck's constant. Since such

Tail of Pn^

V

•

Physica Scripta T90

V

© Physica Scripta 2001

Scars and Other Weak Localization Effects in Classically Chaotic Systems Acknowledgements The author owes a great debt to Lev Kaplan, who was responsible for much of the work described herein and in the references given, (especially Ref. [9], from which this paper is derived). This research was supported by the National Science Foundation under Grant CHE-9610501. This work was also supported by the National Science Foundation through a grant for the Institute for Theoretical Atomic and Molecular Physics at Harvard University and Smithsonian Astrophysical Observatory.

References 1. Schnirelman. A. I., Usp. Mat. Nauk. 29, 181 (1974); Colin de Verdiere, Y.. Comm. Math. Phys. 102, 497 (1985); Zelditch, S., Duke Math. J. 55, 919 (1987). 2. Heller, E. J„ Phys. Rev. Lett. 53, 1515 (1984). 3. Kaplan, L. and Heller, E. J., Ann. Phys. (N.Y.) 264. 171 (1998). 4. Kaplan, L., Phys. Rev. Lett. 80, 2582 (1998). 5. Kaplan, L. and Heller, E. J., Phys. Rev. E 59, 6609 (1999). 6. Creagh, S. C. and Whelan, N. D., Ann. Phys. (N.Y.) 272. 196 (1999); Phys. Rev. Lett. 77, 4975 (1996); Narimanov, E. E., Cerruti, N. R., Baranger. H. U. and Tomsovic, S., cond-mat/9812165. 7. Kaplan. L., Phys. Rev. E 59. 5325 (1999).

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8. Kaplan, L. and Heller. E. J., Physica D 121, 1 (1998). 9. Kaplan, L. and Heller, E. J., Phys. Rev. A 62, 409 (2000). 10. Heller, E. J . J. Chem. Phys. 72. 1337 (1980); Heller, E. J. and Davis, M. J., J. Phys. Chem. 86, 2118 (1982); Stechel, E. B. and Heller, E. J.. Ann. Rev. Phys. Chem. 35, 563 (1984); Heller, E. J., Phys. Rev. A 35. 1360 (1987). 11. Mirbach, B. and Korsch, H. J., Ann. Phys. (N.Y.) 265. 80 (1998); Phys. Rev. Lett. 75, 362 (1995). 12. Espinoza Ortiz. J. S. and Ozorio de Almeida, A. M., J. Phys. A 30, 7301 (1997); Backer, A., Schubert. R. and Stifter, P., J. Phys. A 30, 6783 (1997); Phys. Rev. E 57. 5425 (1998); Tanner, G., J. Phys. A 30. 2863 (1997); Sieber, M., Smilansky, U„ Creagh, S. C. and Littlejohn. R. G.. J. Phys. A 26. 6217 (1993); O'Connor, P. W. and Heller, E. J.. Phys. Rev. Lett. 61, 2288 (1988). 13. Sinai, Ya. G„ Funct. Anal. Appl. 2, 61 and 245 (1968); Russ. Math. Surv. 25, 137 (1970). 14. Bohigas, O., Giannoni, M.-J. and Schmit, C , Phys. Rev. Lett. 52, 1 (1984). 15. Schanz, H. and Smilansky, U., Chaos. Solitons. and Fractals 5, 1289 (1995); Doron. E. and Smilansky, U„ Phys. Rev. Lett. 68. 1255 (1992); Chaos 2, 117 (1992); Nonlinearity 5, 1055 (1992). 16. Bogomolny, E., Chaos 2. 5 (1992). 17. Kaplan, L., Phys. Rev. E 58, 2983 (1998).

Physica Scripta T90

Physica Scripta. T90, 162-165, 2001

Tunneling and Chaos S. Tomsovic* Department of Physics. Washington State University, Pullman, WA99164-2814, USA Received August 24, 2000

PACS Ref: 03.65.-w, 05.45.+b

Abstract Over the proceeding twenty years, the role of underlying classical dynamics in quantum mechanical tunneling has received considerable attention. A number of new tunneling phenomena have been uncovered that have been directly linked to the set of dynamical possibilities arising in simple systems that contain at least some chaotic motion. These tunneling phenomena can be identified by their novel /(-dependencies and/or statistical behaviors. We summarize a sampling of these phenomena and mention some applications.

1. Introduction Quantum mechanical tunneling encompasses a multitude of fascinating effects that violate classically forbidden processes. In spite of the opposing quantum and classical behaviors, it is possible, through the use of semiclassical methods, to construct a theory requiring only knowledge of the classical dynamics. For example, one of the most familiar problems is that of potential barrier penetration. For a one-dimensional barrier and supposing that coupling to the environment can be neglected, a number of techniques are available that analytically continue the classical motion either into a complexified phase space or through the use of complexified time. However, attempts to generalize these techniques run into fundamental problems for systems with more degrees of freedom, even without adding in the complication of environmental coupling. A theory of multidimensional tunneling has thus resisted any single, universal treatment. It turns out that the difficulties can be traced baqk to the broad variety of underlying classical dynamics that emerge from multidimensional systems, and which are absent from one-degree-of-freedom systems. With better identification of the difficulties has come a number of new and novel tunneling phenomena linked to the presence of at least some chaotic motion. An excellent and fairly recent overview of the subject has been published as a pedagogical course by Creagh [1]. As chaos is ubiquitous in the dynamics of simple systems, i.e. those possessing a few degrees of freedom, there has been a major shift in focus towards investigating non-integrable dynamics. Whereas integrability may be "pleasing" in that it is amenable to analytic analysis, i.e. action-angle variables exist that render the dynamics trivial, in some sense it is extremely rare or can be considered as special. A few other categories of dynamics need to be introduced. For systems regarded as near-integrable, the dynamical instability tends to be quite weak, and perturbation theories often apply. Systems possessing a globalized chaotic region in phase space * e-mail: [email protected] Physica Scripta T90

and some stable motion are regarded as having a mixed phase space with a moderate level of instability. The extreme limit of fully chaotic dynamical systems often comes with highly unstable motion. Each of these dynamical possibilities creates the opportunity for unique tunneling behaviors. After a short description of a couple essential concepts, the rest of the paper gives a cursory outline of several advances and organizes them loosely by dynamical system type. No attempt has been made to be complete. It is just intended to give the flavor of the new dynamical directions under study.

2. Preliminaries With a few exceptions, we restrict our discussion mainly to the spectral properties of simple, environmentally isolated, bounded, few degrees-of-freedom (D) systems. In fact, paradigms of 2-D and periodically kicked 1-D systems have been invoked in the bulk of the recent work. The nature of eigenstates or the time evolution of initially localized states is an extremely interesting subject which ought to be included in such a discussion as this, but shall not be due to space constraints. More complete discourse and references can be found in Ref. [1]. 2.1. Integrable systems An inherent signature of tunneling is the existence of almost degenerate multiplets in a spectrum. For systems with a single reflection symmetry, doublets would appear if some positive measure of the classical trajectories occupied distinct phase space regions from their reflection symmetric counterparts. A semiclassical quantization of the motion and its symmetric partner would lead to the expectation of a degeneracy. The energy splitting of the degeneracy, AE, would be a measure of the strength of the tunneling. For integrable systems, semiclassical formulae take the form A£ = /*(J)/zexp(-S(J)/7!)

(1)

where J denotes the action variables of the system, and [A(J), S(J)} are smooth scalar functions of their arguments. Typically, ^(J) is closely related to angular frequencies, and S(J) to the action accumulated over a particular cyclical, classically forbidden path defined within the complexified dynamics. Of particular interest is the ^-dependencies of both the amplitude and the argument of the exponential. Often it is possible to control a parameter, such as a wavevector, which effectively allows a tuning of the value of h. In a log plot versus h, the tunneling splitting should exhibit a © Physica Scripta 2001

Tunneling and Chaos smooth, linear slope. The prefactor shows up as a logarithmic correction, and may be rather difficult to establish, but leaves the ^-dependence smooth. It is the contrast to this behavior that sets the new phenomena apart. 2.2. Dynamical tunneling Many of the recently discovered tunneling phenomena involve a counterpart to potential barrier penetration dubbed dynamical tunneling by Davis and Heller [2]. It frequently arises in the models of the nuclear motions of small molecules [3]. In dynamical tunneling, there is no energetic or potential barrier. Rather, the classically forbidden motions are dynamical in nature. In other words, constants of the motion other than energy are responsible for the forbidden features. The view from the underlying phase space structure can look indistinguishable between the two tunneling mechanisms. Separatrices, to which a potential barrier would give rise in phase space, or resonances can be responsible for isolating the symmetrically related motions. Unstable periodic orbits with their respective homoclinic tangles can also provide the barrier for dynamical tunneling. Curiously though, despite the phase space similarities, the spectral behavior can appear to vary in opposing directions. The potential barrier case leads to increasing tunneling splittings as energy is increased, and beyond the peak barrier energy the tunneling splittings vanish in the spectrum. On the other hand, many dynamical systems retain dynamical tunneling multiplets to infinite excitation energy with many of the splittings approaching zero in the limit. For example, homogeneous potentials lead to situations in which a constant fraction of the levels are involved with tunneling at all energies and the local energy average splitting decreases with increasing energy. 2.3. Chaotic eigenstates For integrable systems, semiclassical theory gives a one-to-one correspondence between particular classical motion and the eigenstates. Through the D relations J, = 2nh(n, +

V

-^

(2)

each energy level and eigenstate is related to a specific set of classical actions, {J}. Topologically speaking, all the eigenstates are similar and can be considered to belong to a regular class. This association fundamentally breaks down once chaotic motion exists in the system [4]. For nearintegrable and mixed phase space systems a non-neglible fraction of the phase space continues to be covered by motion possessing good action variables locally. A roughly corresponding fraction of the eigenstates continues to satisfy the criteria for being regular [5,6]. The remainder of the eigenstates can be lumped into a single irregular or chaotic class defined as being other than regular, but this underemphasizes the range of eigenstate characteristics that become possible once released from the burden of having to reflect regular motion. We mention three important examples of distinct irregular behaviors; this is not an exhaustive scheme. Crudely speaking, the ergodic subclass behaves locally like random waves subject to the ergodic measure S(E — H(p, q)) as it applies to wavefunctions [7]. If a system is not fully chaotic, it is understood that the (p, q) coordinates are restricted to a subset of © Physica Scripta 2001

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points approaching arbitrarily closely to a single chaotic trajectory. For this class to exist, the classical exploration of the ergodic subset of phase points must take place on a time scale short compared to the Heisenberg time scale that derives from the mean quantum level spacing. A second possibility arises because it is often possible to identify a variable in which the dynamics is diffusive. In this case, the irregular class of eigenstates may be strongly localized in the Anderson sense [8]. The kicked rotor provides the classic example [9]. A third class has been dubbed as being hierarchical [6,10]. They are associated with the hierarchy of time scales existing in the dynamics found at the boundaries between regular and chaotic motion. The states exhibit fractal character over some range of scales cutoff by H at one end, and the nature of the regular-chaos boundary on the other.

3. Near-integrable systems 3.1. An integrable

approximation

Our first example of a tunneling phenomenon requiring a serious look at the underlying dynamics is provided by a work of Ozorio de Almeida [11], and is exceptional in that it is the only topic discussed here in which the chaos is explicitly removed before treating the problem. However, it gives a theoretical approach to understanding dynamical tunneling in the presence of resonances, as are created by generic perturbations, and so it has been included. For small perturbations of an integrable system, motion in the neighborhood of low order rational ratios of angular frequency are replaced by resonances. The resonances may be enveloped by narrowly confined chaotic zones, however if the perturbation is weak enough, it is possible to average the Hamiltonian over the rapidly evolving angle variable of its unperturbed form. Effectively, through the combination of averaging and canonical transformations, the perturbed system is replaced by an integrable system whose phase space structure appears locally as a distorted pendulum. Two different tunneling processes emerge. The more familiar process is the tunneling between motion above and below the resonance leading to avoided crossings as a system parameter is varied. The other is a sort of tunneling of the state unto itself. If the resonance is large enough to quantize, its energy will be shifted from the standard semiclassical quantization because the wavefunction has "tails" that communicate directly and weakly from any lobe of the resonance to its neighboring lobes. 3.2. Analytic continuation It is not always necessary nor correct to average away the weak chaos present in the dynamics, and a new tunneling dependence on h was derived in the work of Wilkinson [12]. He was able to evaluate, by saddle point methods, the tunneling splitting integrals with the help of an analytic continuation of the regular dynamical motion. The explicit expressions contain a different prefactor power of h AE=

MW?12 exp(-Sfc(J)/*)

12

(3)

k

than found in Eq. (1), and multiple contributions are possible. In this case, the saddle points are isolated in disPhysica Scripta T90

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tinction to the contour integrations necessary in integrable systems. Although, the original derivation required separation of the motions in configuration space, the canonical form suggested, and he conjectured, that the results apply to more general dynamical tunneling situations. A criterium was also given for the cross-over between the regimes of applicability of Eqs. (1) and (3). With increasing perturbation, the analytic continuations can become limited to such an extent that the saddle points cease to exist, and the procedure eventually breaks down. 4. Mixed phase space systems We turn now to tunneling phenomena where chaos is taking a direct role in the process. 4.1. Chaos-assisted tunneling Consider a mixed phase space system whose parameters are such that each of its eigenstates fall mainly into a regular or ergodic class. Suppose further that there exists reflection symmetry leading to duplicate copies of regular trajectories that are well separated. One anticipates semiclassically degenerate doublets in the spectrum in proportion to the measure of regular motion. It turns out that these doublets are split by tunneling mechanisms completely different than discussed in Section 3. Instead, the tunneling proceeds indirectly via a compound process of wave amplitude breaking off bit by bit from the initial state near one regular domain, transporting chaotically to the neighborhood of its symmetric regular domain, and there reassembling its reflection of the initial state [6,13]. The fundamental process, and unsolved theoretical problem, is how the "phase space" tails of regular and ergodic wavefunctions join smoothly. The involvement of the ergodic eigenstates and spectrum create tunneling splitting fluctuations over several orders of magnitude [6,13,14] which overshadow any mean behavior similar to Eq. (1). Random matrix models have been introduced and shown to predict the observed splitting fluctuations successfully [13]. The first experimental evidence for chaos-assisted tunneling in a microwave cavity version of the annular billiard introduced by Bohigas et al. [15] has recently been reported by Dembowski et al. [16]. Note that it had previously been argued that chaos-assisted tunneling is involved in the decay, to normal deformation, of superdeformed nuclear states [17], but the decay involves many-body physics that makes it a less straightforward interpretation. It nevertheless involves the coupling of regular and presumably ergodic states. Several other important works have appeared, especially with regards to kicked systems or quantum maps [18]. However, we briefly discuss just one more issue [23]. The question arises whether symmetry is necessary; suppose that in a particular experimental situation it cannot be exactly maintained. Is there a possibility of seeing and confirming that chaos-assisted tunneling is occuring? In fact, symmetry is not required at all if two parameters can be controlled and independently varied. One parameter must force crossings between regular states that the tunneling will be splitting. The other parameter shifts the irregular levels with respect to the crossing levels. In direct tunneling, the tunneling splitting at the crossing is not influenced by the Physica Scripta T90

presence of other levels. In chaos-assisted tunneling, the splitting is completely determined by the whereabouts of the irregular levels, and the magnitudes of their avoided crossings with the regular levels. In the avoided crossings' parametric dependence, these distinctions are unmistakable. 4.2. Chaos-assisted ionization Chaos-assisted tunneling also arises in systems that are coupled to a continuum [20,21]. In the first reference, the authors have investigated certain regimes of the hydrogen atom exposed to either linearly or circularly polarized microwaves where nondispersive electronic wave packets can be created. There regular classical motion exists embbedded in chaotic dynamical regions of phase space. Trajectories diffusively transport to ionization thresholds. It happens that the tails of the electronic wave packets are connected by quantum tunneling with ergodic states localized in the chaotic dynamical part of the phase space which are in turn connected to the continuum. The authors found peculiar fluctuations of energies and ionization rates and successfully modeled them with the same basic random matrix model approach as introduced in [13] extended to include continuum coupling. In the second above reference, the authors investigate a quite different physical system, micro-optical cavities. These slightly deformed circular cavities have long-lived whispering gallery resonances created by total internal reflection. Qualitatively, certain parameter regimes of the system's asymptotic ray limit are dynamically similar to the above nondispersing wave packets, i.e. mixed phase space dynamics. The authors [21] find a range of deformations where it appears that chaos-assisted tunneling must be invoked to explain larger than expected lifetimes for these meta-stable resonances.

5. Fully chaotic systems For fully chaotic systems, recent works address tunneling between two ergodic states or between two localized states. We also give one last example reminiscent of Ozorio de Almeida's tunneling of a state unto itself [11] summarized in Section 3.1; although instead of involving a regular state, it involves an ergodic state.

5.1. Tunneling between chaotic states In a series of works, Creagh and Whelan have studied the tunneling behavior of ergodic states separated by a potential barrier [22]. They began by introducing a 2D double well for which the motion within each well was chaotic, and by giving a new approach to generating a trace formula directly for the splittings. A mean and fluctuating behavior could be separated and indentified with specific complex orbits. The fluctuations were controlled by orbits homoclinic to the real orbit connected to the optimal tunneling path through the barrier in a very similar way as a homoclinic orbit sum controls the autocorrelation function of chaotic wave packets [23]. They also studied the statistical properties of the tunneling splittings which are given by a generalized Porter-Thomas distribution. © Physica Scripta 2001

Tunneling and Chaos 5.2. Tunneling between strongly localized states Strong localization provides a quantum dynamical barrier that separates states as opposed to the classical dynamical barriers originally invoked in the definition of dynamical tunneling. For systems possessing an appropriate symmetry, multiplets of states that are further than a localization length from the "symmetry lines" will exist whose degeneracy is broken by tunneling [24]. With a generalization of the kicked rotor [25], Casatie/a/. [24] found, not surprisingly, that pairs of "double-humped" states existed, and that their quasienergies were split similar to the expectation for Mott states, AO ~ exp(-c 0 «o/0

(4)

where / is the localization length, n0 the number of states away from the symmetry line, and CQ a constant. The pairs did not exist too close to the momentum origin, and the localization length for states localized near the origin appeared to be a factor two different in their localization lengths. 5.3. Tunneling influence on band structure The band structure of a periodic system that happens to be completely chaotic classically can be heavily influenced by tunneling. For strong enough kicking strength, the kicked Harper equation serves as an excellent example [26]. Ergodic trajectories diffuse through the lattice. However, classically forbidden complex trajectories may hasten the process. LeBoeuf and Mouchet [27] have treated precisely this problem. As a function of kicking strength near integer values, allowed trajectories pass from the real to complex domains. With the use of the Gutzwiller trace formula in relations deriving from the spectral determinant, they compared results with and without the inclusion of these complex trajectories. They found that in some cases the inclusion of the complex trajectories significantly altered the spectrum as a function of the Bloch angle (quasi-momentum), and thus had an important impact on the band structure. 6. Summary The past ten or fifteen years have taught us that there exists a vastly richer collection of manifestations of tunneling than provided by the one degree-of-freedom double well. Despite several beautiful works, many, very basic, theoretical problems remain such as developing a theory for how regular and ergodic wavefunctions couple. In addition, new physical applications continue to surface. Given that tunneling impacts all domains of wave physics, not just quantum mechanics, the future of the subject promises to be very rewarding. Acknowledgements We gratefully acknowledge the efforts of the organizers, S. Aberg, K.-F. Berggren, and P. Omling, of the Nobel Symposium entitled "Quantum Chaos

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Y2K." as well as the support of the Nobel Foundation. Our apologies to S. Adachi, K. Ikeda, and A. Shudo for not including their tunneling studies due to space constraints; see [1] for a summary and references to their works.

References 1. Creagh, S. C , "Tunneling in two dimensions", in "Tunneling in Complex Systems", (ed. S. Tomsovic), (World Scientific, Singapore, 1998). 2. Davis, M. J. and Heller, E. J., J. Chem. Phys. 75, 246 (1981); Heller, E.J. and Davis, M. J., J. Phys. Chem. 85, 307 (1981). 3. Two examples are the water molecule, Lawton, R. T. and Child, M. S., Mol. Phys. 37 (1979) 1799; and benzophenone. Heller, E. J., "Wavepacket Dynamics and Quantum Chaology" in Chaos et Physique Quantique/Chaos and Quantum Physics, Proc. Les Houches Summer School, Session LII 1989, (eds. M. J. Giannoni. A. Voros, and J. Zinn-Justin), (North-Holland, Amsterdam, 1991). 4. Einstein, A., Verh. Deutsch. Phys. Ges. Berlin 19, 82 (1917); English translation by Jaffe, C , JILA rep. No. 116. 5. Percival, I. C , J. Phys. B 6, L229 (1973). 6. Bohigas, O., Tomsovic, S. andUUmo. D., Phys. Rep. 223 (2), 43 (1993). 7. Berry. M. V., J. Phys. A: Math. Gen. 10, 2083 (1977). 8. Anderson. P. W., Phys. Rev. 109. 1492 (1957); Rev. Mod. Phys. 50, 191 (1978). 9 Fishman, S„ Grempel, D. R. and Prange, R. E., Phys. Rev. Lett. 49. 509 (1982); Phys. Rev. A 29, 1639 (1984). 10. Ketzmerick, R., Hufnagel, L., Steinbach, F. and Weiss, M., Phys. Rev. Lett. 85, 1214 (2000). 11. Ozorio de Almeida, A. M., J. Phys. Chem 88, 6139 (1984); see also the work for narrow crossings by Uzer, T., Noid, D. W. and Marcus, R. A., J. Chem. Phys. 79, 4412 (1983). 12. Wilkinson, M., Physica 21D, 341 (1986). 13. Tomsovic, S. and Ullmo, D., Phys. Rev. E 50, 145 (1994). 14. Leyvraz. F. and Ullmo, D„ J. Phys. A: Math. Gen. 29, 2529 (1996). 15. Bohigas. O., Boose, D., Egydio de Carvalho, R. and Marvulle, V.. Nucl. Phys. A 560, 197 (1993); see also the elegant theoretical work of Frischat, S. D. and Doron. E., Phys. Rev. E 57, 1421 (1998). 16. Dembowski, C. et al., Phys. Rev. Lett. 84, 867 (2000). 17. Aberg, S., Phys. Rev. Lett. 82, 299 (1999); Khoo. T. L., "Tunneling from super- to normal-deformed minima in nuclei", in Tunneling in Complex Systems, (ed. S. Tomsovic), (World Scientific, Singapore, 1998). 18. Lin. W. A. and Ballentine, L. E„ Phys. Rev. Lett. 65, 2927 (1990); Phys. Rev. A 45, 3637 (1992); Roncaglia, R., Bonci, L., Izrailev, F. M., West. B. J. and Grigolini, P., Phys. Rev. Lett. 73, 802 (1994); Latka, M., Grigolini, P. and West, B. J., Phys. Rev. E 50, 596 (1994); Phys. Rev. A 50, 1071 (1994); Grifoni, M. and Hanggi, P., Phys. Rep. 304, 229 (1998); Kohler, S., Utermann, R„ Hanggi, P. and Dittrich, T., Phys. Rev. E 58, 7219 (1998); Plata, J. and Gomez Llorente, J. M., J. Phys. A 25, L303 (1992); Gutierrez, E. M. and Gomez Llorente, J. M„ Phys. Rev. E 52, 4736 (1995). 19. Tomsovic, S„ J. Phys. A: Math. Gen. 31, 9469 (1998). 20. Zakzrewski, J., Delande, D. and Buckleitner, A., Phys. Rev. E 57, 1458 (1998). 21. Nockel, J. U. and Stone, A. D., Nature 385 no. (6611), 45 (1997). 22. Creagh, S. C. and Whelan, N. D., Phys. Rev. Lett. 77, 4975 (1996); Phys. Rev. Lett. 82, 5237 (1999); Phys. Rev. Lett. 84, 4084 (2000); Ann. Phys. (N.Y.) 272, 196 (1999); preprint (2000). 23. Tomsovic, S. and Heller, E. J., Phys. Rev. E 47, 282 (1993). 24. Casati, G.. Graham, R.. Guarneri, I, and Izrailev, F. M., Phys. Lett. A 190, 159 (1994). 25. Bliimel, R. and Smilansky, U„ Phys. Rev. Lett. 69, 217 (1992). 26. LeBoeuf, P., FCurchan, J.. Feingold, M. and Arovas, D. P., Phys. Rev. Lett. 65, 3076 (1990). 27. LeBoeuf. P. and Mouchet, A., Phys. Rev. Lett. 73, 1360 (1994).

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Relaxation and Fluctuations in Quantum Chaos Giulio Casati International Center for the Study of dynamical Systems. Universita degli Studi dell' Insubria. Via Valleggio 11, 22100 Como, Italy Istituto Nazionale di Fisica della Materia, Unita di Milano, Via Celoria 16, 20133 Milano, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Via Celoria 16. 20133 Milano, Italy Received July 13, 2000

Abstract We discuss quantum relaxation and fluctuation phenomena in the light of the corresponding classical behaviour. Quantum exponential localization and tunneling effects lead to the universal algebraic decay of the survival probability P(t)<x \lt. This behaviour is observed in particular in the time dependence of the ionization probability of Rydberg atoms driven by a monochromatic field. At a fixed time the quantum survival probability strongly fluctuates as some parameter is varied. These fluctuations have a fractal structure which is quantitatively related to the time dependence of the survival probability.

1. Introduction The study of the general features of quantum systems in the light of the chaotic properties of the corresponding classical systems has attracted an increasing interest in the last three decades. Besides the fundamental motivation to better understand the quantum classical correspondence, a main impulse has been provided by important applications in solid state physics, nuclear physics, quantum optics, chemistry and in general in problems involving radiation matter interaction and many body systems. More recently, quantum decoherence and quantum computation are likely to open new areas of application of "Quantum chaos". An exciting aspect of this field is that the great improvement of experimental techniques allows to perform very accurate experiments at the microscopic level which may test theoretical predictions as well as provide a guide for new directions of investigation. In this connection it is remarkable that the quantum kicked rotator introduced 25 years ago as a very abstract model of quantum chaos [1] has actually been realized in real laboratory experiments and is currently investigated in different groups [2-5]. Basically one would like to build a theory which allows to have a better understanding of some unexpected behaviour in several quantum contexts, when the strength of interaction does not allow the use of perturbative methods, as in the excitation and ionization of atoms in electromagnetic fields, dissociation of polyatomic molecules, scattering problems etc. In addition a number of different fluctuations have been observed in atomic, nuclear and solid state systems. How can they be characterized? Are they connected to the properties of classical chaotic motion? From the physical viewpoint, the problem of the behaviour of highly excited atoms in microwave fields, first investigated in the influential experiments of Bayfield and Koch [6], has by now revealed a large variety of themes of great interest. Indeed this problem lies at the intersection of several lines of contemporary research, so that methods and ideas originally developed in quite different areas find Physica Scripta T90

here a common ground of application. As a matter of fact, even the simplest theoretical model- the hydrogen atom in a monochromatic electric field- shows a beautyful variety of unexpected properties which are typical of strongly perturbed quantum systems. A careful analysis of this simple model, both theoretical and experimental, can therefore illuminate different intricate phenomena and serve as a solid base for the understanding of more complex systems as well as quantum-classical correspondence. In the original experiments of Bayfield and Koch[6] hydrogen atoms were prepared in a highly excited state and the frequency of the external field was about 40% of the resonant frequency for a single photon excitation to the next level and about 1 % of the photon frequency for excitation to continuum. Nevertheless, when the field intensity exceeds a critical value, inization takes place. Conventional quantum theory failed to account for the behaviour of this strongly perturbed system. On the other hand it was found that onset of classical chaos gives an essential contribution to the ionization process. In other words, since many photons are required to reach the continuum, ionization proceeds here via a chaotic diffusive excitation and classical mechanics provides the best way to describe this quantum chaotic ionization process. In this connection, we would like to recall that, strange that it may seem, the most efficient ionization mechanism is not a single photon but a multiphoton one [8,9]. A crucial step in understanding the quantum properties of this system has been the discovery of quantum localization [7,9,10]. This deep phenomenon of quantum suppression of chaotic diffusion, originally described in [1], is a typical occurrence and is due to quantum interference effects that prevent any diffusive like excitation process from going on indefinitely and can be considered a dynamical version of the Anderson localization well known in solid state physics. To gain a better insigth, let us consider the onedimensional Hamiltonian (in atomic units). 2

i

H = P— - - + sz sin(cof),

(1)

where the motion is assumed to take place along the field direction (z-axis, with z > 0). In order to compare classical and quantum dynamics it is convenient to use the scaled field strength so = £«$ and frequency a>o = um\, which completely determine the classical dynamics. The classical limit corresponds to /ieff = h/riQ —• 0, at constant £o, COO(«O denotes the initially excited state). © Physica Scripta 2001

Relaxation and Fluctuations in Quantum Chaos Besides greatly simplifying the theoretical and numerical work, the Id (1) model was also found to correctly describe the excitation of true, i.e., three-dimensional H atoms, initially prepared in very extended states along the field direction [9,10]. This fact has been actually of crucial importance since the study of the Id model has led to a clear physical picture of the excitation process. Quite interestingly, a very simple theoretical and numerical approach, which exposes in a transparent way the connection between the H atom problem and the kicked rotator, has been given by the introduction of the so-called Kepler map. This map takes a simple analytical form thanks to an important peculiarity of the problem, that is, the external field fully develops its perturbing influence on the free Keplerian motion of the electron mostly when the electron itself is in the vicinity of the perihelion. This is a consequence of the Coulomb singularity and leads to a kick-like influence of the external perturbation. Indeed, for cu0 > 1, the main change of the electron energy E occurs when the electron is close to the nucleus. As a consequence it can be shown that the classical dynamics is approximately described by the Kepler map [10].

167

bution becomes comparable to the number of photons required for ionization. Unexpected that these predictions may have been at their first appearence [7-10], they were confirmed by experimental results on the microwave ionization of H atoms [11-13]. In particular, in Ref. [12] a comparison was made between experimental results, numerical results obtained from the solution of the Schrodinger equation and theoretical predictions from the dynamical localization theory. The resulting agreement provides experimental evidence for the quantum suppression of the classically chaotic diffusion due to the localization phenomenon.

More recently, new experimental data on the microwave ionization of alkali Rydberg atoms [14] have shown that the survival probability P(t) exhibits an algebraic decay law P(t) oc t~x, with a «s 0.5. In the same paper, numerical simulations of quantum dynamics have been made, giving a value of <x consistent with experimental data. The origin of the slow algebraic decay was attributed to the underlying structure of classical mixed phase space composed by integ r a t e islands surrounded by chaotic components. However, the investigations of classical chaotic systems with mixed phase space showed that the probability of Poincare recur~N = N + ksmcj>, 0 = (p + 2nco{-2wN)-ir-, (2) rences to the same region, or the survival probability up to time /, decays algebraically with power ot~ 1.5 — 3 where N = E/co, k = 2.6eco~s/i, tb = tot is the phase of the [15-18]. Moreover, since the integral f™ P(x)dz is microwave field when the electron passes through the per- proportional to the measure of the finite chaotic region ihelion and the bar denotes the new values of variables. where the trajectory is trapped, the value of a should be In the quantum case, the change of N gives the number of greater than one. According to the correspondence principle, absorbed photons while the number of photons required one expects that, in the semiclassical regime, classical and to ionize the atom is N[ = \/(2n\aj). The quantum dynamics quantum systems exhibit the same decay law. Therefore of the model (2) is described by the quantum Kepler map for the above exponent a « 0.5 found in the experiments and the wave function \jj(): in numerical computations requires, in this respect, an explanation. In particular the question arises whether this ~$ = exp(-iH0(N))P exp(-ik cos <j))\p, (3) obtained value is generic or corresponds to some initial transient time behavior in a regime where quantum effects play where H0(N) = 2n/J-2ioN, N = -\d/d, cl =

< +oo), and the operator P projects probability over 2 and 3. However this is not the whole story. Ionization the states with negative energy (N < 0) [10]. probability and survival probability, once observed at a fixed On the base of the map (2), it is easily seen [10] that in the time, display wild fluctuations as a function of some parclassical case, diffusive ionization takes place for fields ameter. What is their nature? Are they connected to the intensities above the chaos border: £o > KC ~ l/(49a>0' ). structure of classical phase space? Are they determined The quantum excitation instead results from a sort of comby the classical decay rate of survival probability />(?)? petition between diffusion that would be predicted on classiOr by its quantum counterpart? These questions, discussed cal grounds and the localizing effect of quantum in Section 4, are important for the understanding of conducinterference. The latter tends to arrest the diffusive tance properties and, more generally, of the typical broadening of the wave packet at a maximum spread mechanisms which control the quantum dynamics of (localization length) of the order of the classical diffusion complex systems. coefficient. The quantum Kepler map, which was introduced as a convenient approximation of the actual quantum dynamics of 2. Quantum Poincare recurrences the H-atom in a microwave field, is closely related to the quantum kicked rotor, i.e. to the very model where A well known Poincare theorem states that, for a condynamical localization has been first identified and servative Hamiltonian system with an energy surface of unambiguously related to Anderson localization. Therefore, finite measure, a trajectory always returns, infinitely many the quantum Kepler map is just the theoretical link between times, arbitrarily close to its initial position; however the Anderson localization and the suppression of chaotic statistical distribution of recurrence times depends on the diffusion which takes place in the H-atom. In particular dynamics. it leads to the prediction that the localized distribution For a strongly chaotic motion, without stability islands, should be exponential in the number of absorbed photons. like e.g. the Arnold cat map [19], the probability P(t) to This allows for a simple interpretation of the derealization return or to survive in a given region after a time t, decays phenomenon: derealization and ionization takes place exponentially with t (as for the probability to have head when the number of photons within the localized distri- for n consecutive times in the coin's flipping game). For sys© Physica Scripta 2001

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terns with divided phase space instead, the complicated hierarchical structure leads to an anomalous power law decay of Poincare recurrences P{t) and of correlations functions C(t) [17,18,20]. Such slow decay is caused by the decrease, down to zero, of the diffusion rate for a trajectory when it approaches the chaos border determined by some critical invariant curve [17,20-23]. We recall that for a large classs of functions, the correlation C(t) is connected to Poincare recurrences P(t) by the relation P{t) ~ C(t)/t [17,24]. The above anomalous properties have been studied in great detail for classical systems [15,17,20-23]. The investigations of the corresponding quantum systems show that the structure of the eigenstates is closely related to the properties of classical phase space. In the integrable regime, eigenstates are located on the invariant KAM curves while in the chaotic regime they spread over the whole chaotic component in agreement with the Shnirelman theorem [25]. The structure of eigenstates can be seen in a pictorial way with the help of Wigner function and the Husimi distribution [26]. This representation allows to see graphically the qualitative change of eigenstates during the transition from integrability to chaos. In the chaotic regime it also allows to see scarred eigenfunctions in which the probability is concentrated near short unstable periodic orbits [27,28]. However in the chaotic regime, the majority of eigenfunctions are ergodic on the energy surface and the energy level statistics is well described by the Random Matrix Theory [29]. While in the regime of Hamiltonian chaos the structure of quantum eigenstates is now well understood [27-31], chaotic objects with fractal structure, which appear in dissipative classical dynamics such as strange attractors and repellers, were not studied in quantum mechanics. The main problem is that in the quantum case, dissipation is always accompanied by noise and generally one should study the density matrix [32,33]. In this way the problem becomes much more complicated than the Hamiltonian case and the fractal structure of strange attractors/repellers in quantum eigenstates had not been seen until recently [34]. In addition, also the question of how the Poincare' recurrences P(t) are affected by quantum dynamics was not addressed until recently. This problem becomes more and more important not only due to its fundamental nature but also in the light of recent experiments with mesoscopic systems. Indeed, different types of ballistic quantum dots can now be studied in laboratory experiments [35] and the phase space in such systems generally has a mixed structure. A different type of systems in which such effects should be observable experimentally is given by cold atoms in external laser fields where the Kicked Rotator model of quantum chaos has been built experimentally [2,3]. Possibilities of experimental investigation of slow probability decay in such systems has been discussed recently [4]. Finally, as discussed below, the time dependence of ionization probability of Rydberg atoms under microwave fields provides a different instructive example. The experimental studies of slow power correlation decay in the regime of quantum chaos are also important from the fundamental point of view, since the typical scale of correlations decay is much larger than the Ehrenfest time scale tE ~ In \/h on which the minimal coherent wave packet spreads over the available phase space. To the best of Physica Scripta T90

our knowledge no comparison of classical and quantum correlations in that regime has been made so far. Only recently such a comparison has been made in the regime of hard chaos with exponential correlations decay [36]. In this section we consider the quantum relaxation process in a dynamical model of quantum chaos where diffusion is caused by the underlying classical chaotic dynamics. This model, introduced in [37], describes a kicked rotator with absorbing boundary conditions (when the momentum is larger than some critical value). We would like to stress that the kicked rotator model provides a satisfactory local description of a large class of dynamical systems. In particular it approximates the Kepler map (2) and therefore it describes the dynamics of hydrogen atoms under an external periodic field. It can also be considered as a model of light trapped in a small liquid droplet with a deformed boundary in which the rays, with orbital momentum less than some critical value, escape from the droplet because the refraction angle exceeds the critical value [38]. In this open chaotic system absorption leads to the appearance of a fractal set in the classical phase space (strange repeller) [39]. The underlying classical fractal set should affect the quantum dynamics and find its manifestations in the structure of eigenstates. Indeed it is natural to expect that long living eigenstates will be associated with the above strange set on which classical orbits live forever. Therefore the eigenstates associated with this set should strongly influence the scattering process, relaxation and ionization into continuum. In our model, the evolution operator over the period T of the perturbation is given by ij, = U\JJ = p e - ' r « 2 / 4 e - * c o S 0 e - , T n V 4 ^ ;

(4)

where n = —id/dd, V is a projection operator over quantum states n in the interval (—N/2, N/2). Here, we put h = 1 so that the commutator is [h, 8] = —i and the classical limit corresponds to k —• oo, T —• 0 while the classical chaos parameter K = kT remains constant. In the classical limit the dynamics is described by the Chirikov standard map: n = n + A: sin 0 + ^ 1 ;

6 = e + j{n + n)

(5)

In this model all trajectories (and quantum probabilities) leaving the interval [—N/2, N/2] are absorbed and never return. Therefore, contrary to the standard kicked rotator model [1,40] in which the matrix of the evolution operator is unitary, the absorption breaks the unitarity of the evolution matrix so that all eigenvalues move inside the unit circle. In other words, each eigenvalue can be written in the form X = e~'e = exp (—iE — F/2) where r characterizes the decay rate of the eigenstate. In this way absorption corresponds to ideal leads without reflections back to the sample. A similar approach, in which coupling to continuum was studied on the basis of non Hermitian Hamiltonians, has been developed by Weidenmuller et al. (see for example [41]). In order to study the classical quantum correspondence in different dynamical regimes we consider in the following, three different situations: © Physica Scripta 2001

Relaxation and Fluctuations in Quantum Chaos

169

(0

Completely chaotic classical phase space and no quantum localization. (ii) Classically mixed phase space. (Hi) Completely chaotic classical phase space and quantum localization.

turn dynamics (4) for different N. We have found that the quantum probability Pq(t) follows the classical one up to a time tq after which it starts to decay at a lower rate (Fig. 1). More precisely, for very large times, the decay of Pq(t) is determined by the eigenvalue e = E - i r / 2 with minimal r = r m m . We determined the quantum relaxation (0 Strong classical chaos and no quantum localization. Here time t — 0.1 which correq by the condition In [Pq(tq)/P(tq)] we fix N/k — 4 and K = 7 so that the classical phase space sponds to 10% deviation. The values of t q, obtained in this is completely chaotic with no visible islands [42]. Due to way, strongly fluctuate with changing the system size N absorption, in this regime of strong chaos with one chaotic as is typical for mesoscopic systems. component, the classical probability to stay inside the A qualitative explanation for the time tq is the following interval {-N/2, N/2) decays exponentially with time: [36]. The complex eigenvalues of U are distributed in a P(t) ~ exp(-y c /). The time scale tc = l/y c ~ tD is determined narrow ring of width Ec inside the unitary circle [37]. This by the diffusion time tD required to reach the absorbing is typical for diffusive samples coupled to strongly absorbing 2 boundary. Since the diffusion rate is D =< An > / leads. As a result, N complex eigenvalues are homogen2 2 At ~ k-/2 then yc = Ec ~ l/tD = k /N . Moreover, since eously distributed in a ring of total area A « Ec and the disthe classical chaos parameter K = 7 and the ratio tance between them, in the complex plane, is 5 ss */Ec/N. N/k = 4 are fixed, the diffusion time tD is constant when In the classical limit this spacing goes to zero and one obtains N ->• oo and this allows to investigate the semiclassical a continuous density of poles. However, for finite N, the sepbehavior. In addition fD > > 1 which justifies the diffusive aration of poles is finite and can be resolved after a time approximation. tq ~ l/S. According to this argument, which is independent For the classical case, we numerically iterated the map (5) of the symmetry and dimensionality of the problem, the for M = 9 • 109 different initial conditions homogeneously deviation between quantum and classical probabilities will distributed on the line n = 0. The results demonstrate [36] take place at a clear exponential decay of the Poincare recurrences P(t) = exp(-y c / - b) with yc = 0.101882(1), b = 0.17774(5) r kN = tcJg (see Fig. 1). Even with such a high number of orbits, the (6) classical computations allow to obtain directly the probability P(t) with 10 % accuracy only up to the level where g is the conductance of the sample with Thouless P m 5 • 10"8(7ss 165). This limitation is due to statistical energy Ec = \/tc. Notice that tq «; tH where tH — \/A « errors appearing for finite number of trajectories. In spite N{h = 1) is the Heisenberg time, and A is the level spacing of this, the decay rate yc can be found with very high pre- inside the sample. Also it is interesting to note that the Ehrenfest time scale cision which allows to extrapolate the probability behaviour tE is much smaller than the quantum relaxation time tq: to larger times. 5 In order to study the quantum evolution we choose the tE/tq ~ In N/yfN < < 1. For example for N = 1.3 • 10 we corresponding initial condition in which only the level have tq = 254 while tE = InN/A « 9.4 (A-& \nK/2 « 1.25). n = 0 is populated and we numerically integrated the quan- This shows that the agreement between quantum and classical relaxation continues for a time scale which is much larger than the time of wave packet spreading. However, for t > tE there is no exponential instability in the quantum motion [40,43]. As a result, correlation functions of the • type 1 ' ' ft ' 0 C(T) = (sin 6(t) sin 6(t + T)) which, in the regime of strong InP(t) chaos, decay exponentially in the classical case -10 (ln|C| ~ —Ax), in the quantum case decay only during the -500 -20 Ehrenfest time scale up to l n | C | ~ - l n A f (fE <JC T «; tc). This is similar to what happens in closed (unitary) systems -30 X^ '••. -looo such as the kicked rotator [43]. This example shows that ) 5000 10( 003 -40 exponential relaxation is not necessary related to expolnP(t) t nential local instability and positive Kolmogorov-Sinai JS -50 entropy. In this connection I would like to mention that -60 -10 exponential decay of survival probability (together with a -70 normal Gaussian diffusive process), has been recently dis"~covered in a classical area preserving map which is mixing -80 -20 but has zero Lyapounov exponent, namely it has no t -90 ( 50 100 150 200 exponential instability [44]. •

•

OH

' • • • •

' • • • . .

100

200

300

400

500

600

700

800

900

1000

t Fig. 1. Classical and quantum probability decay for K = 7 and N/k = 4. The full line shows the fit to the classical decay Dotted lines give the quantum probability P q (/) for AT = 5001, 20001, 130001 (upper, middle and lower curves respectively). The lower insert shows the classical probability actually computed from M = 9 • 109 orbits and the fit is shown by the dotted line. The upper insert shows the classical (full line) and the quantum (dotted line) asymptotic decay for N = 5001. © Physica Scripta 2001

In conclusion, the classical probability P(t) to stay inside the sample [-N/2, N/2] decays exponentially for t > tD: P(t) ~ exp(-y c ?) with yc = \/tc. The quantum probability instead follows closely the classical one up to the quantum relaxation time scale tq ~ y/tc/A. Let us now turn to the structure of eigenfunctions. As it is known, in the classical case, the orbits which are never ionized and which stay forever inside the sample form a Physica Scripta T90

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fractal set (strange repeller) [39], Similarly to the problem of diffusion in the Lorentz gas [39], the information dimension d\ of this set can be expressed via the Lyapunov exponent A^lnK/2 and the probability decay rate yc~D/N2: d\ = 2 — yJA. In our case with yc » 0.1, A « 1.25 this gives d{ « 1.92. To obtain this set we studied the evolution of M = 2.2 • 109 classical orbits up to t = 100 map iterations [34]. Initially the orbits are homogeneously distributed in the phase plane. To analyze the structure of eigenfunctions in the phase space we used the Husimi function obtained from the Wigner function smoothed in the intervals An and A8 (AnA0 = 1/2) [34]. The ratio 5 = AQ/An was fixed in a way to have optimal resolution. The Husimi functions were constructed from antisymmetric eigenstates since they had the minimal values of T. It turns out that the quantum Husimi function for the eigenstates with T «s yc reproduces very well the fractal structure of the classical strange repeller on very small scales [34]. It shows close agreement between the classical and quantum data for large N corresponding to a small effective Planck constant /jeff: we call such states quantum fractal eigenstates. It is natural to expect some analogue of the Shnirelman theorem [25] for these quantum fractal eigenstates, so that in the quasiclassical limit they will be distributed over the classical fractal set according to the classical measure. The case for the states with r «s yc confirms such expectations [34]. However, the situation for the states with r = rmm is rather different. Indeed, generally, we observe there the appearance of scars on the underlying classical strange repeller, the silhouette of which, in spite of scars, is still clearly seen [34]. The strength of scars grows with decreasing N (increasing nef(). However, even for the largest N = 59049, the probability distributions f(n) projected on the n-axis demonstrate an evident difference between the classical and quantum cases (Fig. 2). We qualitatively understand this phenomenon in the following way: the quantum

state with the minimal value of T should stay away as far as possible from the absorbing boundaries so that quantum interference should redistribute the probability on the classical set and lead to some scarring around classical trajectories which correspond to short unstable periodic orbits located near the center. Indeed Fig. 2 shows a pronounced peak near period two orbits. We would like to point out that the complex eigenvalues of the evolution operator U can be considered as some poles of the scattering matrix. Therefore, we can expect that similar quantum fractal eigenstates will appear in the problems of chaotic scattering in the quasiclassical regime. One of the possible models to study such effects is the three discs problem where the gap in the T-rates has been discussed in [45]. Such quantum fractal eigenstates can also be studied in experiments with chaotic light in micrometer-size droplets [38] where the classical dynamics is governed by a map analogous to the map (2). (ii) Mixed phase space. In this section we consider the case with fixed ratio N/k = 4 and take the classical chaos parameter K = 2.5, so that the classical phase space has a hierarchical structure of integrable islands and chaotic components. A typical example of classical and quantum survival probability decay for this case is shown in Fig. 3 [46]. The classical probability P(t) decays with a power law with exponent <x ~ 2 in a range of six orders of magnitude. Here the exponent is slightly different from the typical value 1.5. Indeed as discussed in [17,20-22] the exponent can vary from system to system (and even oscillates with In?) depending on the local structure of phase space in the vicinity of the critical boundary invariant curve whose rotation number can play an important role. The quantum survival probability Pq(t) is plotted for different values of N ( effective h is proportional to \/N). In agreement with the correspondence principle, the quantum probability follows the classical value during a rather long time scale which grows with N. For longer times instead, the quantum probability Pq{t) appears to approxi-

0.00005 X

X

J

i

0.00004

n n

II II

Jl

;i /1

11 i i

0.00003

X

ii

0.00002

t^3

t^Vk-

l.e-05

*^fc

0.0

-N/2

n

N/2

Fig. 2. Probability distribution f(n) over the unperturbed basis n for quantum eigenstates for the case N = 59049, T = 0.1080 ^ yc after 100 iterations of the map (dotted curve), r = r m j n after 60 iterations of the map (dashed curve). The full curve is the classical distribution. Crosses mark periodic orbits with period one at (0,0) and period two at (n = ±n/T for 8 = n/2,"in12). The asymmetry of/(«) for the classical case is due to round off computer errors. Physica Scripta T90

0

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10 12 14 16 18

lnt Fig. 3. Classical (thick continuous line) and quantum (thin continuous and dotted lines) probability decays for K = 2.5 and N/k = 4. The two dashed straight lines show slope 2 and 1. The quantum curves correspond to N = V with p increasing from 5 (upper continuous) to 10 (lower dotted). The starting conditions are two symmetric lines at n = ±/V/3, both for the classical and quantum evolution. © Physica Scripta 2001

Relaxation and Fluctuations in Quantum Chaos

171

t ~ exp (2n/l). Since n is proportional to the total measure fi ~ n oc I In /, it follows that the survival probability is Pq(t) * C/t (7) P (t) = d/j./dt ~ l/t in agreement with (7). The same estiq where C is some constant. This latter decay continues during mate can be obtained by expanding an initial state over evolution operator (4) with a rather long interval of time (4 orders of magnitudes in time the eigenstates of the 2 11 6 probabilities |c„|~~ e~ " /£ and ionization rates T„ ~ for the case N = 3 ). Of course (7) represents an intermedi2 2 ll |c„| ~ e~ " /D. Then the survival probability is given by ate asymptotic behavior since for / -> oo the decay will 2 r P (t) ~ J^°|c„| e»'d« ~ D/t. Our data at K = 7 for differq be exponential Pq(t) ~ exp(—rmjnf) where r m j n is deterent values of k (k = 5,6, ...10) give C = aDb, with mined by the minimal imaginary part of the eigenvalue X a = 0.27(5) and b = 0.92(13). of the evolution operator U (Ui//X = e~"t/^). The above derivation of the decay law (7) refers to the (Hi) Classical chaos and quantum localization. In order to obtain a much better numerical evidence and a qualitative regime of quantum localization of chaos. It is interesting understanding of the origin of quantum behavior (7), we to observe that for the integrable case one still has the rates r n ~ \c„\1~ e~"llc!r where £eff is an effective length deterstudied [46] the probability decay in a simpler case where the classical phase space is completely chaotic (K = 7) mined by tunneling in the classically forbidden region. Howbut the wavefunctions are localized inside the sample of size ever here the fluctuations in time are stronger since £eff N, due to the phenomenon of quantum localization. To this depends on the local structure of invariant curves and islands end we choose the sample size N = 500 and the quantum in the integrable domain. Nevertheless, as can be seen in parameter k = 5 so that the localization length /«» Fig. 4, for the quasi-integrable case K = 0.5 the global decay k2/2 N = 500. Initially the probability is concentrated slower way than the corresponding classical one. For to at n = 0. As it is seen, the classical probability decays, the above reasons, even in the case of mixed phase space, asymptotically, exponentially with time P(t) = 0.11 exp the decay follows the l/t law in accordance with numerical (—yt). The value of y can be found from the solution of data. The l/t behaviour can continue up to a time the Fokker-Planck equation with absorbing boundary con- fmax oc exp(N/leff) ~ l/r m in which is determined by the ditions which gives y «s Dn2/2n2, where D = fik2/2 is the minimal decay rate in the system. For t > ; max the quantum classical diffusion rate (see Eq. (4) in [37]). The asymptotic survival probability Pq(t) decays exponentially with time exponential decay starts after the diffusive time to ~ 1/y. Pq(t) oc exp (—///max). This corresponds to eigenstates which The value of /? depends on the classical chaos parameter are localized in the center of the island. We have been able to and for K = 7 is /? ta 2.8 [37]. Therefore the expected theor- observe [46] such localized asymptotic Husimi function for 4 5 etical value is y = 6.9 • 10~4 which is close to the numerical N = 3 . However, already for N — 3 the time / max is too 4 large for it to be reached it in our numerical simulations value y = 6.4 • 10" . where we followed the evolution up to 3 • 107. Apparently The quantum probability (Fig. 4) follows the classical one this time is already comparable with / max and this explains during some interval of time after which it exhibits a clear l/t the strong oscillations of Pq(t) for / > 106 (Fig. 3). We would decay, according to Eq. (7), for several orders of magnitude. like to stress that / max grows exponentially with N (N ~ l/h). This quantum decay can be understood in the following way. As a result, the quantum decay l/t in semi-classical regions The quantum eigenstates are localized with localization proceeds up to enormously long times. In this regime the length I « D/2 [47]. Therefore the absorption time for a correlation functions practically do not decay because state at a distance n from the absorbing boundary, is C(t) oc tP(t) ~ const. mately follow the decay law

PM

-8

>v

£ -io "12

In this section we have discussed a new universal law for probability and correlation decay in quantum systems both in the exponentially localized regime and in regimes in which classical dynamics has a mixed phase space [46]. It should be possible to observe this universal behavior in the kicked rotator model which has been studied in experiments with cold atoms [2,3]. Similar effects should be also observable in the experiments with micrometer droplet lasing discussed recently in [38].

^ v

\\.

V\. N v \ ^^^V

3. Ionization rate in Rydberg atoms 0

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lnt Fig. 4. Classical (continuous lines) and quantum (dashed lines) probability decays for K = 0.5 (grey lines) and K = 1 (black lines); parameter k = 5. The evolution starts at n = 0 and the probability is absorbed for n < 0 and « > 500. The two thin straight lines have slope one, while the dotted grey curve shows the fit for the classical decay at K = l: P = 0.11 exp(—yt), y = 6.4 10- 4 . © Physica Scripta 2001

In the light of the results obtained in the previous section we can now turn to the investigation of the ionization probability for a hydrogen atom in a microwave field. To this end, we choose the initially excited state with principal quantum number n = no and numerically studied the survival probability P(t) in a linearly polarized monochromatic electric field e(/) = e sin(a>/). The quantum evolution is numerically simulated by the Kepler map (4), by the one-dimensional model of a hydrogen Physica Scripta T90

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atom (1), and by the 3d model for atoms initially prepared in states extended along the field direction and with magnetic quantum number m = 0. The comparison of these three models shows that the essential physics is captured by the \d model. We introduce an absorption border for levels with n > nc which, for the Kepler map, corresponds to N > Nc ^ — 1/'{Inlay) [48]. Such border occurs in real laboratory experiments, for example as a consequence of unavoidable static electric fields experienced by the Rydberg atoms during their interaction with the microwave field. The absorption border nc can be varied in a controlled way via a static electric field es, the static field ionization border being esn^. % 0.13. The results of quantum simulations [49] for a situation similar to the experimental one (Fig. 2(b) in Ref. [14]) show that the quantum data for the survival probability P(t) obtained from the quantum Kepler map and from the Id hydrogen atom model agree with each other and with the numerical computations of [14]. However all these data are strongly different from the classical probability decay, which displays a slope a « 2. The reason of this disagreement should be attributed to the fact that the initial state «o = 23 is not in the semiclassical regime. Our data for the Husimi distribution [49], show that a significant part of the probability is trapped inside the stable island at n ~ 20 (wn3 « 1). For this reason the probability decays slowly during a long time f « 105 after which it drops faster. If «o is increased significantly, the semiclassical regime is reached and the quantum probability decay becomes close to the classical one up to the time scale f# ~ 104. O u r data show [49]that tu is proportional to «o (at fixed £0, too), in agreement with previous estimates of Ref. [46], according to which in oc 1/^eff- After this time the quantum 1/f decay is clearly observed in agreement with the results of [46]. In Fig. 5 we show a more realistic case in which, initially,

Fig. 5. Survival probability for £o = 0.1, co0 = 2.6, «o = 60, «c = 64: quantum Kepler map (dotted line), quantum (solid line) and classical (dashed line, ensemble of 3 x 106 trajectories) Id hydrogen model. The straight lines have slopes 1 and 2.15, the latter coming from a fit of the classical decay for 5 x 102 < t < 2 x 104. Inset: quantum survival probability for the Id model (solid line) and the 3d model (dot-dashed line). Physica Scripta T90

classical and quantum probabilities decay in a very similar way and where only after a time ( # - 5 x 102, the quantum survival probability starts to decay more slowly (P(t) oc 1/f) than the classical one which decays as l/fa, with a ~ 2.15. This case corresponds to n0 = 60 and can be observed in experiments similar to those performed in [14]. Again, the quantum Kepler map gives a qualitatively correct description of the ionization process up to very long interaction times. The comparison of quantum simulations for the Id hydrogen atom model and the 3d dynamics is shown in the inset of Fig. 5. It demonstrates that both dynamics give very close results, confirming that the essential physics is captured by the Id model. We have placed the absorption border near the initial state (nc — 64) in order to have Pc = £/ANc w 3.5 > l, where 1$ = 3.3£21a>0~lo/3«21 is the localization length in number of photons [10] and ANC = («o/2tW())("o/wc — 1) is the number of photons required to reach the absorption border. In this way the probability can go out very easily and the 1/f probability decay is observed after a short transient time of the order of 20 microwave periods. On the contrary, when pc < 1, as in the case of «o = 23 (pc « 0.3), strong fluctuations around the 1/f decay take place. This is analogous to the huge (log-normally distributed) conductance fluctuations in a disorder solid with localization length smaller than the system size [50]. In conclusion, the universal decay law discussed in the previous section has been shown here to describe the ionization rate of Rydberg atoms. In particular, the theory is also consistent with previous numerical and experimental results. 4. Quantum fractal fluctuations As mentioned in the introduction, the theoretical predictions concerning the microwave ionization of hydrogen atoms were confirmed by numerical and experimental results [11-13]. Nevertheless, the agreement with localization theory holds only "in the average". Indeed, both the numerical and the experimental data exhibit more or less sensible deviations from the average prediction [12]. This is hardly surprising, because the dynamical localization theory was just meant to yield a gross description of the quantum dynamics. Actually, as it was pointed out in [10], one would expect even stronger fluctuations than were actually observed. Indeed, according to theory, the quantum distribution is strongly fluctuating around its average exponential shape, and "...these fluctuations will affect also the ionization rate which will have a very irregular fine structure" [10] p. 1429. Actually, ionization probability (at a fixed time and field intensity) turns out to depend in such a complicated way on the microwave frequency, that it could be called a chaotic function of the latter [51]. Localization theory provides a natural and very general setting for understanding these fluctuations. Indeed they represent a peculiar kind of sensitive dependence on parameters that should be expected to appear any time, when some classical diffusion is suppressed by quantum localization. In particular, in the Kepler map formulation, the excitation of a one-dimensional hydrogen atom in a microwave field is described by a quantum map which acts on suitable wave functions. In an appropriate representation, these wavefunctions are defined on a one-dimensional © Physica Scripta 2001

Relaxation and Fluctuations in Quantum Chaos lattice, where the nth site corresponds to the absorption of n photons by the atom [51]. The problem of finding eigenvectors and eigenvalues of the Kepler map has a definite resemblance to the equation for eigenfunctions of a particle in a one-dimensional disordered lattice. The disorder is here associated with a "pseudo-random potential" which depends on the microwave frequency co. It follows that different values of co for a given atom correspond to different samples in the solid-state model. In other words, even a sligth change in co will produce a completely different realization of the pseudo-random potential. Therefore, the problem of microwave excitation of Hatoms is essentially a localization problem (at least in the case where the frequency is larger than the level spacing). The essential feature here is that localization takes place in a. finite lattice, because just a limited number of photons can be absorbed before going into the continuum. We have, therefore, two characteristic lengths (in number of photons); one is just the maximum number of photons, and the other the photonic localization length. The theory we have developed rests on the basic assumption that the excitation process and the ionization probability are essentially determined by the ratio of these two characteristic lengths. When this ratio is large, the localization effect will preclude ionization even when the classical dynamics is totally chaotic; in the opposite case, one should expect strong ionization. When compared to the theory of Anderson localization, this assumption is just the basic ansatz of the scaling theory for localization in finite samples. That theory aims at providing a description of how the conductance of a finite sample depends on the size of the sample at very low temperatures, and it was found to give a satisfactory average picture. Nevertheless, the actual behaviour of a given sample was found to exhibit wild fluctuations around the average scaling behaviour. The point is, of course, that while eigenstates are individual and reproducible, they are, nevertheless, random in structure, in other words, they exhibit random fluctuations around their average exponentially localized shape. Since conductance is determined by the trasmission coefficient across the finite sample, it is also affected by analogous fluctuations. In the hydrogen atom problem, the ionization probability is determined by the rate of exponential decay of those quasi-energy eigenfunctions which have a significant overlap with the initially excited states, over a distance (in number of photons) detemined by che experimental conditions. This rate is determined by the localization length. Then the Kepler map formalism suggests that the ionization probability at fixed interaction time and field intensity should display the same kind of fluctuations as conductance does. This expectation has been confirmed [51] by numerical experiments in which the ionization probability was computed as a function of the microwave frequency at a fixed interaction time. On purely qualitative grounds the dependence looks indeed random. In more general terms, it has been recently proposed [52,53] that classical algebric decay of P{t) oc r - *, should lead to fractal conductance fluctuations with fractal exponent a directly related to the exponent a. Indeed correlation functions of quantum scattering fluctuations are semiclassically related to decay laws of classical survival ,C! Physica Scripta

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probabilities [55]: for instance, classical exponential decay, typical of pure hyperbolic situations, results in Lorentzian quantum correlations. In the framework of this general semiclassical theory it was pointed out in [56] that algebraic classical decay leads to quantum correlation functions (in energy) which follow a fractional power law, C(AE) ~ C(0) - const x {AE)* at values of AE which are semiclassically small but quantum mechanically large (such that the semiclassical approximation remains valid over time scales ~ h/AE). In particular, such a behavior should be detected in the conductance fluctuations of mesoscopic devices, which are observed on varying an external magnetic field, when the phase space of the corresponding classical system has a mixed structure. Ketzmerick [52] has pointed out that, in the presence of such fractional behavior, the graph of conductance versus magnetic field must display a fractional fractal dimension when observed on not too small scales. Using certain assumptions which open the way to analogy with fractional Brownian motion, he has proposed a relation between the fluctuations fractal dimension and the exponent of algebraic decay of classical survival probabilities. According to these results the fractal dimension of these fluctuations should be (j = 2 — a/2. Experimental results about fractality of magnetoconductance fluctuations in gold nanowires were first reported in Ref. [57]; more recently, Sachrajda et al. [53] have given experimental evidence of fractal fluctuations in mesoscopic realizations of soft billiards. However, Ketzmerick's relation could not be tested on such experimental findings, because the underlying classical dynamics is not so well controlled, as to allow for reliable estimates of classical decay. On the other hand, severe difficulties stand in the way of testing the relation by quantum numerical simulations of realistic models. However, in a recent paper [54], the quantum survival probability as a function of a magnetic field has been computed for a system with a classically mixed phase space. It was found that, on changing an external parameter which has the meaning of an applied magnetic field, such quantum probability exhibits fluctuations with a fractal structure in quantitative agreement with Ketzmerick's prediction. The model system is described by a map which allows for efficient numerical simulation of both classical and quantum decay. The system we study is a variant of the Kicked Rotor, known as the Separatrix Map [58]: 7 = I + ks'md

18)

0 = 0+7Tn(|7|/r),

The map dynamics is chaotic in a bounded region of size Al ~ 2kT around / = 0, and becomes regular at large j / | , so it has a mixed phase space, due to hierarchical stable islands near the "chaos border" at the crossover from chaotic to regular motion. The parameter r sets scale lengths in 7, as the dynamics is unchanged if k, r and / are varied keeping their ratios constant. We "open" our system by restricting the dynamics (8) to / > 0, and by taking the line 7 = 0 as an absorbing boundary. Due to long surviving orbits, which stick a long time close to stable islands, a population initially concenPhysica Scripta

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trated inside the chaotic region decays in time with exponent a * 4/3 [59], The corresponding quantum map is obtained by the quantization rule / -> Hh = \hd/d6.

0.00003

U = Pexp (-iT[(« + (j>) ln((« + <j))h/r) - (h + <£)]) exp(—\kh~x cost)),

(9)

where P is projection on states n > 0. The phase cj> describes a Aharonov-Bohm magnetic flux hep through the ring parametrized by 8, and does not affect classical trajectories. We consider quantum dynamics on the discrete lattice parametrized by the quantum number n, and define survival probability at time t inside a sample of length A" as

~

0.000022

0.000015

7.5e-06 -1.0

/vo = I>„(o|2,

(10)

0.0

Fig. 6. Survival probabilities vs. flux at fixed times t = 5 • 103 (upper) and t = 104 (lower), parameters are r = 400, k = 1 and T = 3.22.

with the wavepacket initially concentrated at n = n0. As we are dealing here with survival probability rather than conductance, Ketzmerick's original argument has to be slightly modified. Arguments presented in [54] show that P^t) in (10) describes a fractional Brownian motion, the graph of which has fractal dimension: 2--

(for a < 2).

(11)

This is Ketzmerick's relation [52] for conductance fluctuations, now shown to be valid also for survival probability at fixed time. The value of <x to be used in (11) is a = 4/3 given by the survival probability decay numerically computed [54]. Therefore (11) predicts fractal dimension a « 4/3. We also studied the quantum correlation functions C(5(f>) =< P+ifyPt+s+it)

(12)

which, in the semiclassical theory, are related to the decay laws of classical survival probabilities. If P(t) oc t~*, then at small 3<j) C ( ( t y ) = C ( 0 ) - c o n s t |
(13)

To check the above predictions, we have computed fluctuation patterns at 4 different times (two examples are given in Fig. 6, from which we have extracted fractal graph dimensions a and the exponents ac of correlation functions (formula (13)). Fractal dimensions have been computed by means of the modified box counting algorithm used in [53]: results are shown in Fig. 7 [54]. We tested the algorithm, by computing fractal dimensions of curves generated by Iterated Function Systems, for which a is analytically known [60]. The algorithm gives correct results, with but a slight overestimate of about 1%. We have thus found that quantum fluctuations of survival properties in dependence of magnetic flux exhibit a fractal structure if observed on appropriate scales, in quantitative agreement with Ketzmerick's formula. Notice that the model system we have considered is one widely used in investigating effects of hierarchical stable islands, and results drawn from it can be considered representative of generic hamiltonian systems with a mixed phase space. We would like to close this paper with the following remarks: Physica Scripta T90

Fig. 7. Fractal analysis for the data shown in Fig. 3. The graph of P^(t) vs. was covered by vertical strips of width d(j>, and the largest excursion of P^(t) in each strip was recorded. Summing over all strips, and dividing the result by S(j>, we obtained the quantity N(S(j>) plotted on the vertical axis. Diamonds correspond to t\ = 104, squares to h = 4 • 104. The corresponding dimensions are G\ = 1.38 ±0.02 and with formula (14) (full lines) yields exponents a c equal to =1.33 ±0.02 and to 1.39 ± 0.02 respectively.

(/) Relation (11) has been derived via a semiclassical argument and therefore a. is the power of the algebraic decay of the classical survival probability. The interesting question arises in cases where the classical and quantum decay are different. For example, in the case of kicked rotor (Fig. 4) with K = 1 and k —•• 5, the classical survival probability decays exponentially while the quantum survival probability decays as \/t. Recent computations [61] of the quantum survival probabilty fluctuations for this case have shown that the fractal dimension is a— 1.5 which show that the exponent a which enter in (11) is the one given by the quantum decay law. (ii) The results presented in this paper point at a peculiar type of quantum fluctuations, which seem intimately connected with the localization phenomenon. However, whereas in the Anderson case the randomness of the fluctuations can be ultimately traced back to the "external" © Physica Scripta 2001

Relaxation and Fluctuations in Quantum Chaos randomness of the potential, in the models discussed in this paper, the only possible source of such fluctuations can be just classical chaos. It is interesting to remark that we have here a kind of "structural" instability, i.e., one which shows up upon changing some external parameter; in the classical case, no such instability can appear (at least, on such small scales), because of the smoothing effect of phase averaging. In this respect, we recall that strong empirical evidence has been obtained [9,62] that the quantum evolution is very stable, in sharp contrast to the extreme sensitivity to initial conditions which is the very essence of classical chaos. This fact demonstrates that, at least insofar as the measurement process is not taken into account, quantum dynamics is much more predictable than classical dynamics. In conclusion, one may be tempted to say that the quantum suppression of chaotic diffusion which is produced by quantum interference is being paid at the price of such intrinsically quantum instabilities.

24. 25. 26. 27. 28. 29. 30. 31.. 32. 33. 34. 35. 36. 37. 38. 39. 40.

References 1. Casati, G., Chirikov, B. V., Ford, J. and Izrailev. F. M., in Lectures Notes in Physics. (Springer Verlag, Berlin. 1979), Vol. 93. p. 334. 2. Moore, F. L., Robinson, J. C , Bharucha, C. F., Sundaram, B. and Raizen, M. G„ Phys. Rev. Lett. 75. 4598 (1995). 3. Ammann, H„ Gray, R., Shvarchuck, I. and Christensen, N., Phys. Rev. Lett. 80, 4111 (1998). 4. Niu, Q. and Raizen, M. G„ Phys. Rev. Lett. 80, 3491 (1998). 5. Oberthaler, M. K., Godun, R. M., d'Arcy. M. B., Surnmy, G. S. and Burnett, K... Phys. Rev. Lett. 83. 4447 (1999). 6. Bayfield, J. E. and Koch. P. M., Phys. Rev. Lett. 33, 258 (1974). 7. Casati, G., Chirikov, B. V. and Shepelyansky, D. L., Phys. Rev. Lett. 53, 2525 (1984). 8. Casati, G., Chirikov. B. V., Guarneri, I. and Shepelyansky, D. L., Phys. Rev. Lett. 57, 823 (1986). 9. Casati, G., Chirikov, B. V., Guarneri, I. and Shepelyansky, D. L., Phys. Rep. 154, 77 (1987). 10. Casati, G., Guarneri, I. and Shepelyansky, D. L., IEEE J. Quant. Electron. 24, 1420 (1988). 11. Galvez, E. J.. Sauer, B. E., Moorman, L., Koch, P. M. and Richards, D., Phys. Rev. Lett. 61. 2011 (1988). 12. Bayfield, J. E., Casati. G., Guarneri, I. and Sokol. D. W., Phys. Rev. Lett. 63. 364 (1989). 13. Arndt. M., Buchleitner, A., Mantegna. R. N. and Walther, H., Phys Rev. Lett. 67, 2435 (1991). 14. Buchleitner, A. et at., Phys. Rev. Lett. 75, 3818 (1995). 15. Geisel, T„ Zacherl, A. and Radons, G., Phys. Rev. Lett. 59, 2503 (1987). 16. Lai, Y.-C, Ding, M., Grebogi, C. and Bliimel, R.. Phys. Rev. A 45. 8284 (1992). 17. Chirikov, B. V. and Shepelyansky, D. L., Proc. IX Int. Conf. on Nonlinear Oscillations (Kiev 1981), Naukova Dumka 2, 420 (1984) [Engl. Trans. Preprint PPPL-TRANS-133, Princeton (1983)]; Physica D 13. 395 (1984). 18. Chirikov, B. V. and Shepelyansky, D. L., Phys. Rev. Lett. 82, 528 (1999). 19. Lichtenberg, A. J. and Lieberman, M. A., "Regular and Chaotic Dynamics," Springer-Verlag, New York, 1992. 20. Karney. C. F. F., Physica D 8, 360 (1983). 21. Meiss. J. and Ott. E., Phys. Rev. Lett. 55, 2741 (1985); Physica D 20, 387 (1986). 22. Chirikov, B. V. and Shepelyansky, D. L., in "Renormalization Group", (Eds. D. V. Shirkov, D. I. Kazakov and A. A. Vladimirov), (World Sci. Publ., Singapore, 1988), p. 221. 23. Ruffo, S. and Shepelyansky, D. L., Phys. Rev. Lett. 76, 3300 1988.

© Physica Scripta 2001

41. 42.

43. 44. 45. 46. 47. 48.

49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59.

60. 61. 62.

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Artuso, R„ Physica D 131, 68 (1999). Shnirelman, A. I., Usp. Mat. Nauk. 29, 181 (1974). Chang, S. J. and Shi, K. J., Phys. Rev. A 34, 7 (1986). Heller, E. J., Phys. Rev. Lett. 53. 1515 (1984). Agam, O. and Fishman, S., Phys. Rev. Lett. 73, 806 (1994). Bohigas, O.. "Les Houches Lecture Series", 52, (Eds. M.-J. Giannoni, A. Voros, and J. Zinn-Justin), (North-Holland, Amsterdam, 1991). Casati, G. and Chirikov, B. V., "Quantum Chaos", (Cambridge University Press, Cambridge, 1995). Stein, J., Stockmann, H.-J. and Stoffregen, U., Phys. Rev. Lett. 75, 53 (1995). Haake, F., "Quantum Signatures of Chaos", (Springer, Berlin, 1992). R.Graham, in "Quantum Chaos", (Eds. G. Casati, I. Guarneri and U. Smilansky), (North-Holland, Amsterdam, 1993), p. 241. Casati. G., Maspero, G. and Shepelyansky, D. L.. Physica D 131, 311 (1999). Marcus, C. M., Rimberg, A. J., Westervelt, R. M., Hopkins, P. F. and Gossard. A. C , Phys. Rev. Lett. 69, 506 (1992). Casati, G., Maspero, G. and Shepelyansky, D. L., Phys. Rev. E 56, R6233 (1997). Borgonovi, F., Guarneri, I. and Shepelyansky, D. L., Phys. Rev. A 43, 4517 (1991). Nockel. J. U. and Stone, A. D.. Nature 385, 45 (1997). Gaspard. P. and Nicolis, G., Phys. Rev. Lett. 65, 1693 (1990). Chirikov, B. V., "Les Houches Lecture Series". 52. 443, (Eds. M.-J. Giannoni, A. Voros, and J. Zinn-Justin), (North-Holland, Amsterdam, 1991). Lewenkopf, C. H. and Weidenmuller, H. A., Ann. Phys. 212, 53 (1991). Actually, as recently shown by Chirikov (B.V. Chirikov. preprint Budker INP 99-7 Novosibirsk, 1999), for K = 1 there is a large stability island in the phase space. However this island is due to accelerator modes and does not influence the large time behaviour of survival probability. Shepelyansky, D. L„ Teor. Math. Fiz. 49, 117 (1981); Physica D 8, 208 (1983). Casati. G. and Prosen. T„ Phys. Rev. Lett. 85, 4261 (2000). Gaspard, P., in "Quantum Chaos", (Eds. G. Casati, I. Guarneri and U. Smilansky), (North-Holland, Amsterdam, 1993), p. 307. Casati, G., Maspero, G. and Shepelyansky, D. L.. Phys. Rev. Lett. 82, 524 (1999) Shepelyansky, D. L„ Phys. Rev. Lett. 56, 677 (1986) We introduce complete absorption for levels with n>nz after each microwave period (At = 1). We checked that our results do not change significantly for a wide range of variation of At around this value. Benenti, G.. Casati, G., Maspero, G. and Shepelyansky. D. L., Phys. Rev. Lett. 84, 4088 (2000) Pichard, J. L., Zanon. N.. Imry, Y. and Stone, A. D., J. Phys. France 51, 587 (1990). Casati, G., Guarneri, I. and Shepelyansky, D. L.. Physica A 163 205 (1990). Ketzmerick, R.. Phys. Rev. B 54, 10841 (1996). Sachrajda, A. S. et ai, Phys. Rev. Lett. 80, 1948 (1998). Casati, G., Guarneri, I. and Maspero, G. Phys. Rev. Lett. 84, 63 (2000). Bliimel, R. and Smilansky, U., Phys. Rev. Lett. 60, 477 (1998); Doron, E„ Smilansky, U. and Frenkel, A., Physica 50D, 367 (1991). Lai, Y., Bliimel, R., Ott, E. and Grebogi, C , Phys. Rev. Lett. 68 3491 (1992). Hegger, H. et ai, Phys. Rev. Lett. 77, 3885 (1996). Chirikov. B. V., Phys. Rep. 52, 263 (1979). According to recent results by Chirikov. B. V. and Shepelyansky, D. L. Phys. Rev. Lett. 82. 528 (1999), the actual asymptotic decay should follow the universal exponent —3, which should however show up after quite long times t ~ 106. Prior to that, the decay follows the non-universal exponent we have numerically found [54]. As our computations do not involve such long times, the latter exponent is the appropriate one to be used here. Barnsley, M. F., Fractals Everywhere, Academic Press (1993). Benenti, G. Casati, G., Guarneri, I. and Terraneo, M., in preparation. Casati. G.. Chirikov, B. V., Guarneri, I. and Shepelyansky, D. L., Phys. Rev. Lett. 56. 2437 (1986).

Physica Scripta T90

Physica Scripta. T90, 176-184, 2001

Rydberg Electrons in Crossed Fields: A Paradigm for Nonlinear Dynamics Beyond Two Degrees of Freedom T. Uzer School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430, USA Received July 31. 2000

PACS Ref: 05.45.+b, 32.80. Rm, 32.60.+i

Abstract Systems with two degrees of freedom are comparatively well-understood in contrast to the other, more common, variety which requires more than two degrees of freedom for its description. These multidimensional systems will be the focus of this article, which deals with quantal problems in the Correspondence Principle limit. Our systems of choice are highly excited ("Rydberg") atoms placed in external fields the Hamiltonians of which show soft chaos.

1. Introduction The classical dynamics of multidimensional systems undergoes a fundamental change when the number of degrees of freedom exceeds two: Beyond that threshold, a wealth of new physics becomes possible [1]. Yet forays into this vast area used to be comparatively rare until recently partly because beyond this divide we lack diagnostic tools comparable in power to the Poincare surface of section, and newly developed computational methods such as Frequency Map Analysis offer the only hope of analyzing the new physics [2]. The stark qualitative contrast between systems with two degrees of freedom ('2dof) and those with more can perhaps be best illustrated in the space of actions of integrable and nearly-integrable systems [3]: In a conservative system with 2dof invariant tori form impenetrable obstacles to free movement and only diffusion across the stochastic layers is important. However, in more than 2dof, stochastic layers merge to form the Arnol'd web and surviving tori can no longer impede diffusion [4]. Diffusion becomes richer now, and, in particular, it allows for the possibility of large-scale diffusion along resonances, a diffusion process generally associated with Arnol'd. Stochastic motion might carry the system arbitrarily close to any point on the energy shell by travelling along the resonance layers and simply circumventing tori [5]. Of course, in realistic systems, both kinds of diffusion occur. This survey is devoted to quantal systems whose Hamiltonians show soft chaos [6]. Our problems of choice are highly excited ("Rydberg") atoms [7-9] which require more than 2dof for their description. The traditional view of highly excited atoms has been revolutionized in the last three decades by two classes of experiments, both of which are intimately connected with chaos. In particular, they form testing grounds for quantal manifestations of classical chaos, and most of our knowledge about this controversial subfield of chaos can be traced back to these two fundamental experiments. The first one concerns Quasi-Landau (QL) oscillations in Rydberg atoms placed in strong magnetic fields [10-13] (the Quadratic Physica Scripta T90

Zeeman Effect, QZE, also known as the Diamagnetic Kepler Problem, DKP) [14-16] and the second is multiphoton ionization of hydrogen in strong microwave fields [17] (see [18] for a review). The interpretation of the QL oscillations in terms of a particular periodic orbit of the Hamiltonian [19] - an elementary example of Gutzwiller's Formula [6] - ushered in a wide-scale application of classical mechanics to a wide variety problems, which until recently had been considered the exclusive domain of the quantum theory [20-24] - a very fruitful approach from which we are still benefitting. QZE is a 2dof problem because of the cylindrical symmetry imposed by the static magnetic field. The interpretation of the microwave ionization problem remained a puzzle to atomic theory until its stochastic, diffusional nature was uncovered through the then-new theory of chaos [25]. In the intervening decades the microwave problem has been researched so intensively that we can fairly claim to understand the various models in which the symmetries of the problem can be exploited to reduce its dimensionality, ionization by linearly and circularly polarized radiation (LP and CP respectively) being cases in point [26-29]. A number of recent experiments have broken new ground in the dynamics of multidimensional chaotic systems, in addition to being fascinating atomic physics in their own right. Among these are the high-resolution spectroscopy of Rydberg atoms in crossed static electric and magnetic fields [30-33] ("EXB"), and the microwave ionization of hydrogen using elliptically polarized radiation ("EP") [34,35], to name just two with 3 or more dof. The crossed fields experiments have revealed a rich and puzzling array of periodic orbits [36] which we are still trying to unravel, hoping that Poincare's conjecture about multidimensional systems, namely that "the periodic orbits are extremely precious because they are our best and almost our only means to penetrate this fortress, previously considered unapproachable" [37], is as practical as it is true. One look at the new class of microwave experiments shows that they do not lag far behind the crossed fields problem in complexity: The ionization yield curves interpolate between the limiting cases of LP and CP in an uneven way, and their rich features depend very sensitively on the polarization [35]. Direct descendants of the classical treatments which were so successful in the two limits [38] experience difficulties in accounting for key features of the experiment in strongly perturbed regimes away from these limits. This paper is organized as follows: In the next section we review a key diagnostic tool for multidimensional mechanics called Frequency Map Analysis [2]. After a (necessarily © Physica Scripta 2001

Rydberg Electrons in Crossed Fields: A Paradigm for Nonlinear Dynamics Beyond Two Degrees of Freedom brief) overview of the static EXB problem we consider time-dependent multidimensional problems: These concern hydrogen subjected to microwave fields. We first present research on ionization by a combination of static and microwave fields (2 1/2 dof, the 1/2 coming from the time dependence), and then move on to rotating fields: The construction of Trojan wavepackets using a CP field [39,40] is followed by ionization in an EP microwave field, which takes place in 3 1/2 dof. We conclude with chaotic scattering in the EXB problem, and our application of Transition State Theory to it to reveal its fractal structure. Unconserved angular momenta, which are the source of many surprises in physics, are a common theme running through these studies. We will argue that the crossed fields problem is a paradigm for a wide class of such multidimensional problems since so many of these and other physically significant problems can be mapped onto it. 2. A new diagnostic tool: frequency map analysis The most common tool for depicting the dynamics of a system with 2dof is the Poincare Surface of Section. It is, however, not applicable in a general way for multidimensional systems. We will restrict our subsequent remarks and demonstrations to 3dof systems, although the general notion is valid beyond 3dof. A complete overview of the dynamics for regular (integrable or near-integrable) 3dof systems can be obtained on the following dimensionality considerations: In an integrable 3dof system the motion is confined to threedimensional invariant tori [3,5,41,42], which consequently describe the "physics" of the motion. Invariant tori are characterized by a set of three actions, the motion along the tori being described by three conjugate angles [3]. As an alternative to the actions, invariant tori may also be described by their fundamental frequencies f\,f_,f of the motion. If we restrict ourselves to a fixed energy in a conservative system the action space of this system, i.e., the space of invariant tori, becomes two-dimensional. This space may be displayed easily, thus describing the physics. Two ingredients matter in the mapping: (/)

the choice of an appropriate two-dimensional set of initial conditions that capture all the invariant tori, and (ii) a method that characterizes the motion on the invariant tori. Before going into particulars, let us consider a system that is rendered non-integrable by a perturbation. According to the KAM theorem [3], if the perturbation is small enough, most of the invariant tori are preserved. Trajectories that are launched on one of these tori remain on it thereafter, executing quasiperiodic motion with a fixed frequency vector / = (f\,fi,fi) which depends only on the torus. The family of tori is parametrized over a Cantor set of frequency vectors, while chaotic behavior can - and generically does - occur in the gaps of the Cantor set. These slightly deformed tori are fixed structures of the system. It is possible to find them numerically, and to interpolate between them to form an action-angle coordinate system in which regular (quasiperiodic) motion appears uniformly circular, and weakly chaotic motion stands out as slight departure. © Physica Scripta 2001

Yll

Only tori on which the motion is resonant, i.e., where the frequencies fulfill a relation

mlf[+mzf2+mifi=0

(1)

(mi,m2,mi being integer numbers), and tori close to resonant tori are destroyed, leaving the overall structure of phase space intact. Small regions of chaotic motion form around the remnants of resonant tori that merge to form a network of resonances in action space [4], the Arnol'd web [3]. In principle, diffusion along the resonances, the so-called Arnol'd diffusion [3,43] might take the system arbitrarily close to any point on the energy shell. This is not possible in a 2dof freedom system where the action space at fixed energy is one-dimensional, with the result that invariant tori impede large-scale diffusion. A good method for the characterization of invariant tori should be capable of both providing accurate descriptions of invariant tori, and at the same time measuring the extent of local diffusion, in order to distinguish regular from chaotic regions. In principle, the actions themselves, which remain approximately conserved even on broken tori, could provide such a means. However, it is far from straightforward to compute the actions without knowing the shape of the tori. Instead we use a different method to characterize invariant tori, namely Frequency Map Analysis [2,44,45] (FMA), which was originally developed for celestial mechanical applications, i. e. to study the origin of chaotic planetary obliquities [46] and chaos and order in the solar system [47,48]. Frequency Map Analysis is a numerical method that is capable of providing a clear representation of the global dynamics of multi-dimensional systems. It relies on computing the frequency vectors associated with each of the invariant tori. Although these frequencies are, strictly speaking, only defined and fixed on these tori, the algorithm will compute a frequency vector over a finite time span for any initial condition. On the KAM tori, this vector is a very accurate approximation to the true frequencies, whereas at resonances and in chaotic regions it provides a natural interpolation between these fixed frequencies. Invariant tori are characterized by analyzing the dynamical properties of single trajectories to represent KAM tori or their remnants, and the change of the frequency vector with time measures the diffusion strength. On an invariant torus with fundamental frequencies / , a dynamical variable g(t) can be expressed as S(0 = l > e x p [ i ( i i i < * \ / ) f ] ,

(2)

with complex coefficients c& and integer vectors m(k\ Sampling at consecutive time steps, the FMA yields very accurate approximations to the most important frequency combinations Tk = {tn(k),f) and the associated coefficients cu- Even when the quasiperiodicity condition is fulfilled only approximately, the Frequency Map Analysis remains very accurate for the determination of local diffusion, revealing even the weakest resonances. It is thus clearly superior to, say, a numerical analysis of the maximum Lyapunov exponent as a chaos indicator. Physica Scripta T90

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Mathematically, Frequency Map Analysis relies heavily on the observation that when a quasiperiodic function f(t) in the complex domain C is given numerically, it is possible to recover a quasiperiodic approximation of f(t) in a very precise way over a finite time span [-T, T\, several orders of magnitude more precisely than what simple Fourier series could provide [2]. Indeed, when one computes the Fourier series off(t) over the finite interval [-T, T], one assumes that/(f) is periodic of period T, which is obviously not true. This restriction is not present in the algorithm for the numerical analysis of fundamental frequencies described by Laskar [45]. The extent of chaos can be quantified by the local diffusion strength D, which for our purposes is defined as D = y C A ^ + tA^)2

(3)

where A means the difference between the time intervals [0, T] and [T,2T] with a suitable chosen T for the two most important frequency combinations T\. and Ti computed using the FMA. In the two-dimensional action space, if one shades the resulting figures according to diffusion strength, the network of resonances appears as a maze of bright lines with enhanced diffusion. Chaotic regions, where diffusion is strongest, appear as extended bright areas. The computational effort for the Frequency Map Analysis is usually less than the effort required for the integration of the trajectories, but may be comparable to it. We have applied this diagnostic tool to order and chaos in crossed fields [49], mode-mode interactions in molecules [50], excitonic systems in external fields [51] and to visualize the Arnol'd web for an atom in crossed fields [52,53].

3. The Rydberg atom in static crossed fields While the problem of a Rydberg atom interacting with a strong magnetic field (the Diamagnetic Kepler Problem, DKP, also known as the Quadratic Zeeman Effect, QZE) has been fairly well understood as a result of sustained research in the past two decades [14,15], the superficially similar scenario resulting from the addition of a perpendicular electric field - the so-called crossed field arrangement [30,31,49,54-57] - remains the least understood of all Rydberg problems. This is all the more remarkable since the crossed fields problem is an experimental accessible paradigm for a wide variety of outstanding issues in atomic and molecular physics, solid-state physics [58,59], nuclear physics [60], astrophysics [61], and celestial mechanics [62,63], especially because of its analogy to the Restricted Three-Body Problem [40,63-65]. The deceptive simplicity of the Hamiltonian for a Rydberg atom placed in crossed electric and magnetic fields (in atomic units) [66] H=l-p2--r+jL:+^-(x2+y2)

+ Fx

(4)

is belied by the rich nonlinear dynamics it generates. Here, the magnetic field B is in units of 2.35 x 105 Tesla and points in the z direction, the electric field F is in units of 5.14 x 10"V/m and points in the x direction, and Lz is the r-component of the angular momentum L = r xp. Physica Scripta T90

This problem is so complex because no continuous symmetry survives the extensive symmetry breaking induced by the two fields. The symmetry breaking of the Coulomb potential induced by a single external field (in practice the magnetic field is of greater interest since the Stark problem is separable) affects the three quantum numbers n, I, m/ of the Rydberg electron differently: As long as a single field direction is present, the magnetic quantum number remains good, / breaks down extensively, whereas n breaks down only gradually with increasing magnetic field. In contrast, the extensive symmetry breaking introduced by two misaligned fields leaves no continuous symmetry intact and thereby allows the electron to move freely in three degrees of freedom. This extensive symmetry breaking makes it impossible for global constants of the motion to exist, as was shown recently [67]. One can imagine the wealth of complex behavior that can follow such a release. The experimental challenge has been taken up by Raithel etal. [31-33] who, in a landmark series of experiments, have identified a class of quasi-Landau (QL) resonances in the spectra of rubidium Rydberg atoms in crossed electric and magnetic fields. Similar to the original QL resonances observed by Garton and Tomkins [10], this set of resonances is associated with a rather small set of planar orbits of the crossed-fields Hamiltonian which is known to support an enormous number of mostly non-planar periodic motions [31]. The propensity of the these experiments towards planar orbits has been explained in terms of adiabatic constants [68,69]. In contrast with the DKP, and despite exploratory work [63,70,71], the systematics of periodic orbits in the crossed-fields problem has not been unravelled yet, leaving many open questions. This is especially true where quantum calculations are concerned [72,73]. It has also been found that the velocity-dependent, Coriolis-like Lorentz force in Newton's equations causes the ionization of the electron to exhibit chaotic scattering [74-78] with its characteristic signature of Ericsson fluctuations [79]. Velocity-dependent forces are common in all areas of physics ranging from atomic to astrophysics and are notorious for playing havoc with conventional physical notions: Most strikingly, it is not possible to define a potential energy in their presence and consequently counter intuitive equilibria can be established at maxima in phase space as in the case of Trojan asteroids, as we will see below. Atomic systems in magnetic and electric fields exhibit the complications of velocity-dependent forces with great clarity [30-33], especially during the ionization process [56,69,76] which has confounded conventional treatments. We will have more to say on this interesting subject at the end of this article.

4. Microwave ionization in more than two degrees of freedom Most of the early theoretical investigations of microwave ionization in LP fields used 1 dof+time models in order to simplify the calculations. This was a reasonable assumption at the time because many of the experiments themselves considered extended, quasi-one-dimensional hydrogen atoms [18] in which the angular momentum of the Rydberg electrons was so much smaller than their principal quantum © Physica Scripta 2001

Rydberg Electrons in Crossed Fields: A Paradigm for Nonlinear Dynamics Beyond Two Degrees of Freedom numbers - the atoms resembled needles in which the electron was bombarding the core with zero angular momentum. These models, further approximated as the so-called Kepler Map [80], proved to be perfectly adequate to explain most experimental observations [26]. More careful investigations show the map to be noncanonical [81]. However, new classes of experiments involve more general excitations, and multidimensional models are needed to understand them. Indeed, it was shown early on that the estimates of ionization thresholds for the 1-dof model can be improved by a multidimensional model [82] which maps the entire problem onto two interacting oscillators. 4.1. Microwave ionization in strong parallel fields: 2 1/2 degrees of freedom Recent experiments in which a strong static electric field is added parallel to a LP microwave field of comparable strength [83,84] have once again presented atomic physicists and nonlinear dynamicists with new challenges. The experiments of Spellmeyer et al. [83] on lithium have been analyzed semiclassically and have revealed fascinating bifurcations as the ratio of the two fields is varied. Newer and more detailed experiments [84] show ionization yield curves which are rich with regular oscillatory features and signatures of resonance transitions in the form of sharp dips. Since the strong fields in this 2 1/2 dof system cause standard time-dependent perturbation theory to fail, an approach is needed which sorts out the many transitions that arise from the interplay of the strong static fields and the dressed states created by the strong microwave field. We have recently shown [85] how these observations can be satisfactorily understood in terms of multifrequency transitions [86] driven by a single-frequency microwave field between Floquet (or Quasi-Energy) states (QES) [87]. Because the fields are strong, transitions between quasienergy states are possible at a very large number of frequencies simultaneously and coherently despite the fact that the incident field is monochromatic. We circumvented the special difficulties associated with this setup through an innovative mapping of the Hamiltonian [88] which clarifies the roles of the two fields. Moreover, the infinite summations of inter-shell matrix elements generic of higher-order perturbation treatments are reduced to simpler intra-shell computations. Remarkably enough, a satisfactory classicalmechanical treatment is still outstanding.

4.2. Rydberg electron motion in CP fields-atomic counterparts of the Lagrangian equilibrium points The first observations of ionization of Rydberg atoms by circularly and elliptically polarized microwave fields [27] showed a substantially higher threshold for field ionization for CP than that for LP or EP. This observation was interpreted by proposing that in a frame co-rotating with the CP field, the ionization proceeds essentially in the same way as in a static field. Soon afterwards, theoretical work [28,29] showed that matters were more complicated and that the effect of rotation on the ionization must be included explicitly (see also [89-93]). To understand why, let us examine the most general Hamiltonian for a hydrogen atom in a rotating microwave field.
179

We take as our Hamiltonian in the Schrodinger equation HV = [>P (in atomic units) as H = Ha + ze„ cos cot + C,XE„ sin cot

(5)

where Ha is the hydrogen Hamiltonian, c, is the ellipticity degree, e„ — E„(\ + c) 2 , and E„ is the field intensity [94]. One can attempt to view the excitation process from a rotating frame, but one that rotates with the variable frequency of the electric field: cp(t) = <;co(cos~ cot + c sin" cot) .

(6)

This leads to the conserved quantity K = Ha + zE{t) - cp(t)Ly

(7)

with E(t) = e„(l - k2 sin2 cot) ' , and k2 = 1 - c,2 , all to be used in K

where n is the principal quantum number. In the LP limit (c, = 0) the problem is simplified by transforming to the Kramers-Henneberger frame (for a review see [95]) which oscillates with the microwave field. In the opposite limit of CP, the two fields become static and a conserved Hamiltonian, which is the so-called Jacobi constant [96], results from viewing the problem in a frame rotating with the uniform speed of the CP field [97]. The lack of one-to-one correspondence between energies and Jacobi constants (note how every angular momentum Lv connects to a different Jacobi constant), the particular way the microwave field is switched on is liable to change the outcome. This introduces a strong dependence on the way the sample is prepared and has been a major cause of complication in interpreting CP data [29]. How do celestial mechanicians cope with dynamics in rotating frames? During his researches on the periodic motions of the moon [98], the astronomer Hill introduced [99,100] the concept of the Zero-Velocity Surface, "ZVS" (this commonly used term has been criticized in [101]). The ZVS amounts to calculating [96,99,102]

V = H_^±l±fl.

(9)

This concept can also be usefully applied to the ionization threshold problem, and the ZVS in this case shows clearly the conditions under which orbits ionize and why giant orbits circle the core with great stability [97]. In the CP case, the ZVS has a saddle and a maximum, and orbits with a K smaller than the saddle are kept from penetrating to the core, and thus from ionizing. The stability of orbits in this scenario led us directly to the recognition [97] that the Rydberg atom in a CP field shows the Brown or Trojan bifurcation familiar from the Restricted Three-Body Problem (RTBP) [96]. That led to research on nonspreading electronic wave packet construction, which has become an active field [39,40,103,104]. Wavepackets representing electrons are notoriously difficult to hold together in realistic potentials. Indeed, the quesPhysica Scripta T90

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tion of how to localize an atomic electron in all three space dimensions has been at the forefront of science ever since Schrodinger's attempt, in 1926, to synthesize coherent electronic wavepackets in the hydrogen atom [105]. Recently, we have discovered a practical (i.e., experimentally feasible) solution to this long-standing question: our recipe for field-stabilized coherent (i.e., non-spreading) atomic states, was inspired by celestial mechanics and arose first in the context of the hydrogen atom in a CP field [27,106]: Eberly and co-workers [64,107-109] and we [110] discovered the existence of an outer Lagrange equilibrium point in the hydrogenic CP problem that supports stable classical motion even though this point is a potential maximum. Lagrange equilibria are responsible for, among other things, the stability of Jupiter's Trojan asteroids in celestial mechanics [96]. Despite early excitement, the analogy between Rydberg atoms and planetary systems is imperfect since the finite size of Planck's constant imposes an absolute scale on the quantum problem [111,112]. The atomic analogs of Lagrange equilibria are stable over only a limited range of parameters, and placing a finite-size minimum uncertainty wavepacket at an equilibrium point becomes a delicate balancing act. We demonstrated in [103,110-113] that by combining magnetic and CP fields in a crossed fields arrangement it is possible to prepare stable, non-spreading "Trojan" wavepackets which are direct analogs of an electron in Bohr's model of the hydrogen atom. Radiative energy losses due to the acceleration of the electron - Schott's objection to the original Bohr theory based on classical electromagnetism are miniscule for such highly excited Rydberg electrons [114,115]. Further, instead of working at a maximum, by using a magnetic field it is possible to create a genuine outer potential well whose depth is much larger than kT in the atomic potential for a Rydberg atom [116,117]. The electron is localized both radially and angularly in a frame rotating with the microwave field, thereby creating an atom with a giant dipole moment [103]. Our time dependent quantum calculations confirm that the wavepacket travels around the nucleus on a periodic, circular Kepler orbit without significant dispersion or spreading [113]. The theory can best be explained in terms of the ZVS's of the problem [117]. In the dipole approximation and atomic units (H — me = e = 1) the energy for a hydrogen atom subjected to a circularly polarized microwave field and a magnetic field perpendicular to the plane of polarization is

p-

1

(o,.

H=^—r+^(xpy-ypx)

a>-

-,

i

+ -±(p?+r)

— F (x cos co( t + y sin cof t).

where the terms are as follows; t h e kinetic energy, the C o u l o m b potential, the paramagnetic energy, the diamagnetic energy a n d the interaction with the radiation field. The magnetic field is taken to lie along the positive z—direction and, coc = eB/mec is the cyclotron frequency, (Of is the microwave field frequency a n d F its strength. In a synodic frame rotating with the field frequency cof Physica Scripta T90

the Hamiltonian becomes w P~ 1 / c\ H = K=— ( t o f - —)(xpv en2 + ^-(x2+y2)-Fx

-ypx) (U)

where K is once again the Jacobi constant and the coordinates are now interpreted as being in the rotating frame. Note that K is again a variant of the crossed-fields Hamiltonian. A key point is that this configuration of fields allows the coefficient of the paramagnetic term in this equation to be varied or eliminated. By mixing coordinates and momenta the paramagnetic term prevents the normal separation of the Hamiltonian into potential and (positive definite) kinetic parts: nevertheless, a potential energy may still be defined if cof = coc/2—thereby eliminating the paramagnetic term, and leading to a saddle point and an outer harmonic minimum in the potential. In the laboratory frame the equilibrium at the minimum corresponds to a circular orbit in the plane and localization of the electron in this well produces a giant atomic dipole rotating at the microwave frequency in the x — y plane. Prior to examining the dynamics it is convenient to scale coordinates and momenta; '

2/3

-1/1

i

/i">\

r =(o-'>r,p =cac p. (12) Dropping the primes and assuming planar motion yields the Hamiltonian H = K. = -(p\+p2) + g (x2

r

\

[Q-~)(xpy-ypx) -/

(13)

+y2)-zx

where K. = K/cu2^, Q — cof/coc and E = F/OJ^. This scaling shows that the dynamics depends only on the three parameters, /C, Q, and e. Poincare surfaces of section for {2 = 1/2 indicate stable harmonic motion in the well [117]. As shown previously, for { 2 ^ 1 / 2 it is not possible to define a potential energy surface—simply ignoring the paramagnetic term is incorrect and results in a gauge dependent potential. However, by constructing a zero-velocity surface, [96,99] V = H

= 2

1 r

(xr + y) - Fx 2

(14) one can show that for Q < 1 it is still possible to produce an outer well at large distances from the nucleus. In this case the motion may be chaotic [117], but, provided tunneling is unimportant the electron will be confined in the well by the curves of zero velocity for all values of K below the saddle point. 4.3. Ionization in EP fields: 3 1/2 dof Unlike the LP and CP cases, the EP case supports no constants of motion, and has 3 1/2 dof- too many for the immensely useful diagnostics of the dynamics, the Poincare Surfaces of Section, to be of any use (unless close to the LP or CP limits). We saw before that the EP problem can also be mapped on the EXB problem, albeit with © Physica Scripta 2001

Rydberg Electrons in Crossed Fields: A Paradigm for Nonlinear Dynamics Beyond Two Degrees of Freedom time-dependent fields. That might contribute to the observation that, in the words of an authoritative review "the matter of polarization...is more complicated than previously realized" [118]. The ionization yield curves interpolate between the LP and CP cases in an uneven way, and their rich features depend very sensitively on the polarization [35]. A satisfactory classical-mechanical treatment of this problem is still outstanding because direct descendants of the treatments which were so successful in the LP and CP limits [38] experience difficulties in accounting for key features of the experiment in strongly perturbed regimes away from these limits. Of the classical-mechanical treatments which followed the publication of the EP results of Fu et al. [27], that of Griffiths and Farrelly [119] was the most promising because they used the generators of the SO(4) group to simplify the classical problem into a time-dependent coupled pair of coupled spins. Their classical phase space simulations yielded good qualitative agreement with the experiments of the time, though the multidimensional nature of the problem made analytical treatments impossible. The classicalmechanical approach was also adopted by Richards [38], who, in time-honored fashion, mapped the EP problem on a one-dimensional pendulum forced by terms of frequency 2
181

of view also predicts that after an adequate time averaging, the x component of the microwave field will appear as a combination of two crossed static fields, magnetic and electric. The effect of a magnetic field on ionization is well-known: It shuts off ionization in the plane perpendicular to it (in this case the plane of polarization). Of course, when an electric field is present in that plane, ionization becomes possible again. The relative magnitudes of these two fields hold the key to the explanation of the observed experimental data and our analytical predictions explain the recent experimental [35] and numerical results [35,38] which show how ionization yields in the EP problem interpolate between the high LP and low CP yields. While our quantal work [120] accounts accurately for the experimental observations, we hope that an ionization mechanism will emerge from further work with the clarity characteristic of classical mechanics. 5. Chaotic scattering in the crossed-fields problem Evidence for chaotic scattering in the crossed-fields problem came from an unexpected quarter when Main and Wunner detected Ericsson fluctuations in the computed spectrum of hydrogen in crossed fields [74]. This quantal discovery was soon confirmed by classical calculations [75] showing that the ionization times for the electron have fractal structure. Further evidence came from the examination of ionization of hydrogen in CP fields [128] - a problem intimately connected to the crossed fields. By concentrating on the threshold ionization dynamics [76], we were able to identify the classical mechanism that explains this and related observations for energies below, at and above threshold. In particular we found that the transition to chaotic scattering is caused by of a critical point in the Hamiltonian flow (the so-called Stark saddle point), which in turn arises from a velocity-dependent, Coriolis-like force in Hamilton's equations. The transition through this critical point shows clearly the changes in the dynamics as the energy is increased through the threshold: Just below the critical point the system shows soft chaos. At threshold, many bifurcations of prominent periodic orbits take place and the chaotic region expands. Just above the critical point, the large chaotic sea is drained by ionization, but some periodic orbits survive because they are protected by KAM tori. These show up in spectra as long-lived, prominent features above threshold and also contribute to delayed ionization.

We reformulated and solved the dynamics of this longstanding problem [77,78] by combining a concept from chemical physics, the Transition State (TS), with modern Qr = (ri + n")Q0I where ri + n" = 0, ± 1 , ± 2 . . . ± (n - 1) methods of chaotic transport theory to partition phase space (15) using dynamics [129]. Ours is a planar, 2dof model which is, however, representative of the full dynamics. While TST where ri is the quantized projection of \(L + A) on Br + Er, has been mainly used in chemical physics, it also offers conand n" is the projection of \(L — A) on Br — £ r [126] (for an siderable advantages in other problems that involve some illuminating derivation using the Old Quantum Theory, kind of transformation. In the EXB case one encounters see [127]). We took this derivation one step further: By some new challenges not usually found in chemical time-averaging [124], we reduced the problem to a Rydberg problems. Because of the paradigmatic nature of the atom subject to two effective static fields [120] in the EXB problem, our solution also provides the way to picture EXB configuration: a magnetic field, perpendicular to the the scattering dynamics in a large class of experimentally polarization plane, and an electric field in the polarization important problems, which have been enumerated in the plane. This expression amounts to viewing the one EXB section. component of the EP field (x-component, say) from a frame Transition state theory (TST) [130,131] postulates the moving the with the other (the z-component). This point existence of a minimal set of states that all reactive © Physica Scripta 2001

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trajectories must pass through and which are never encountered by any nonreactive trajectories-that is, the TS is a hypersurface of no return. Our understanding of the transition state as a geometrical structure in configuration space and in phase space two degree-of-freedom systems has been greatly aided by Pechukas who showed [130] that there is a distinguished periodic orbit whose projection into coordinate space connects two relevant branches of the equipotential surfaces. This (unstable) periodic orbit has the property that it bounds a surface separating reactants and products. Hence, it was given the name "periodic orbit dividing surface" or PODS. There is a close relationship between PODS and the turnstile based approach to chemical reaction dynamics developed by Davis [132] and Davis and Gray [133]. PODS are very important for the study of reaction dynamics in two degree-of-freedom systems for several reasons. First, they bound a global Poincare section. This is important because it implies that every trajectory (with finite initial condition) in a definite region of phase space will intersect the Poincare section transversely. In this way the PODS define the barrier between products and reactants. One could view these global Poincare sections as the two degree-of-freedom analogues of the potential wells that arise in one degree-of-freedom systems. Moreover, being global Poincare sections they provide not only a reduction in the dimensionality of the problem, but a complete and unambiguous description of the dynamics. We showed by construction that there is a TS in the planar crossed fields problem, which, when viewed as a Periodic Orbit Dividing Surface (PODS) [130,131], reveals the mechanism of chaotic scattering that characterizes these systems. Remarkably, in this case the TS is a boundary in phase rather than configuration space because of the lack of time-reversal symmetry in EXB. We used its stable and unstable manifolds to understand the progress of ionization [77]. The intersection of the stable manifold with a suitably chosen Poincare surface of section gives a fractal tiling [134-136] which in turn determines the rate of ionization, in that the scaling laws of the areas of these tiles are directly responsible for the exponential decay of ionization probabilities. The insights from this work will be essential for the challenging problem of partitioning the phase space of higher-dimensional systems through Normally Hyperbolic Invariant Manifolds [137] - a major unsolved problem of reaction rate theory. Can this 2dof picture be generalized to higher dimensions? In 3dof systems periodic orbits, and their stable and unstable manifolds, do not have the right dimensions to divide the phase space, i.e., the stable and unstable manifolds are not codimension-one in the energy surface and a one dimensional periodic orbit cannot act as the boundary of a Poincare section. Nevertheless, the nature of many multidimensional problems is such that one can find a higher dimensional structure that may have many of the features of the PODS that have played such an important role for two degree-of-freedom systems. Future work will have to develop the analytical and computational framework for the study of phase space transport in systems with three or more degrees-of-freedom by building on previous work [129,137] which began the development of a theory of higher dimensional dividing surfaces that act to confine trajectories, Physica Scripta T90

and higher dimensional separatrices in general. The answer to this question will provide the means of understanding possibly the main mechanisms underlying the nonlinear dynamics and geometry of transition state theory for general n degree-of-freedom systems. Acknowledgements I am grateful to my fellow researchers without whose contributions this survey would have been much more slender: First and foremost. D. Farrelly for the work on Rydberg atoms in CP fields and Trojan atoms, J. von Milczewski for work on Rydberg atoms in crossed fields. E. Oks for work on microwave ionization, and C. Jaffe for the transition state. This work was supported by the US National Science Foundation.

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Classical Analysis of Correlated Multiple Ionization in Strong Fields Bruno Eckhardt1 and Krzysztof Sacha1'2 1 Fachbereich Physik. Philipps Universitat Marburg, D-35032 Marburg, Germany " Instytut Fizyki im. Mariana Smoluchowskiego, Uniwersytet Jagiellonski, ul. Reymonta 4, PL-30-059 Krakow, Poland

Received August 14, 2000

Abstract We discuss the final stages of the simultaneous ionization of two or more electrons due to a strong laser pulse. An analysis of the classical dynamics suggests that the dominant pathway for non-sequential escape has the electrons escaping in a symmetric arrangement. Classical trajectory models within and near to this symmetry subspace support the theoretical considerations and give final momentum distributions in close agreement with experiments.

1. Introduction Classical models for atomic processes can provide useful insights in situations where quantum effects are not prominent, as for instance in the dynamics of highly excited states or in multiphoton absorption processes. Microwave ionization of hydrogen and other atoms with a single valence electron [1-3], localized wave packet dynamics or electron scattering off atoms come to mind [4]. They provide a natural starting point for semiclassical investigations that include, at least approximately, quantum interference effects. In particular the ionization of Rydberg atoms in linear and elliptically polarized microwave fields has received considerable attention and similarities and differences between classical and quantal behaviour have been sorted out in great detail [1-3]. In most cases only a single electron is considered and sufficient to interpret the observations. Interactions between electrons seem to play an important role in multiphoton multiple ionization in strong laser fields. Experiments show that the yield of multiply charged ions is much higher than can be expected on the basis of an independent electron model [5,6]. More recently, it has been noted that the electrons can escape non-sequentially and that they are correlated in their final state [7-10]. This correlation in the final state came as a big surprise and it is our main objective here to discuss a classical model for it.

2. The model The full process of multiphoton multiple ionization is quite complicated and involves many steps. A plausible model relevant for the field intensities of the experiments is the rescattering mechanism [11-18]. Before the pulse arrives, the atom is in its ground state. Then one electron escapes from the atom, most likely by tunnelling through the Stark barrier. This electron is then accelerated by the field and can be reflected back towards the atom. During this impact energy is transfered to other electrons, perhaps lifting them above the ionization threshold or bringing them close enough so that tunnelling is again possible. If not enough energy is provided at this stage, perhaps another rescattering © Physica Scripta 2001

process can follow until eventually multiple ionization takes place or the pulse disappears. However, before the escape to multiple ionization all excited electrons pass close to the nucleus where they interact strongly with the each other and with the Coulomb attraction. During this phase their (classical) motion is very fast compared to the field oscillations and an adiabatic analysis, keeping the field fixed, can be applied. Moreover, because of the strong interaction all memory of the previous motion is lost, so that the initial state for the multiple ionization event is a statistical distribution of electrons close to the nucleus. Our discussion starts once this intermediate cloud of excited electrons has formed. We do not consider the process by which it has been generated; for instance, one might imagine exposing ions to both an electric field and an electron beam. The arguments that follow focus on two electron escape but can easily be extended to discuss multiple ionization, as indicated below. The Hamiltonian then has as usual the kinetic energy of the electrons, their mutual repulsion, the attraction to the core and the potential due to the oscillating electric field. In many experiments the recoil momentum of the ion is measured, and given the extremely small momenta of the photons it is possible to calculate it as the sum of the momenta of the electrons. Initially, there is no field and the atom is in its ground state. In the final state, after the pulse is turned off, both electrons are free and have positive total energy. Not all the energy difference has to be provided by the impacting electron since there can be additional acceleration by the field after the electrons escape from the core region. However, within the adiabatic assumption motivated above, the energy content of the intermediate electron cloud has to be high enough to let both electrons escape from the immediate vicinity of the nucleus. Without field this implies positive energy, but if the field is on and non-zero, a Stark saddle forms some distance away from the nucleus and the electrons can escape over it. The rapid acceleration downfield will then pull the electrons out and feed in the energy needed to remain asympotically free once the pulse is gone. The Stark saddle that forms in the field provides a focus and a bottleneck for the electrons which they have to cross in order to leave the atom. All electrons see the same saddle and would like to cross it, but if they try to do simultaneously, as suggested by the observed electron correlations, their mutual repulsion gets in their way. Suppose that one electron is slightly ahead of the other when running up the hill towards the Stark saddle: the one that is ahead has the advantage that the repulsion from the companion pushes it uphill, whereas the one behind not only has to fight the attraction to the nucleus but also the repulsion from the Physica Scripta T90

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one ahead. Their interaction is perfectly balanced if they cross the saddle side by side, with reflection symmetry with respect to the field axis. The previous considerations suggest that deviations from this symmetric configuration are amplified and cannot lead to simultaneous ionization. The arguments used here are similar to the ones advanced by Wannier in his analysis of double ionization upon electron impact [19,20]. Therefore, we propose that near the threshold for double ionization the dominant path leading to non-sequential double ionization has both electrons escape symmetrically with respect to the field axis. If more than two electrons are ionized simultaneously the natural extension is that they form a regular n-gon in a plane perpendicular to the field axis. 3. Symmetric double ionization

4

1 = + — + 2Fxf(t) cos(o>? + ) 2

2v iy

(2)

and the pulse shape f{t) = sm\nt/Td)

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(a) 0.2

0.1

/ (i)

with potential energy V(x,y,t)

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With the field pointing along the .v-axis and two electrons confined to the plane z = 0 their coordinates in the symmetric subspace are (x, y, 0) and (x, —y, 0) in position and (px,Py, 0) and (px, -p y ,0) in momenta. The classical Hamilton function for this geometry then is (in atomic units) H(px, py, x, y, t) = p- + p- + V(x, y, t)

20.0

0.0 -5.0

^-w

-2.5 0.0 2.5 2 px (a.U.)

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Fig. 3. Final momentum distributions for F — 0.137 a.u. and initial energy E — -0.58 a.u.: (a) ion momentum parallel to the field axis and (b) perpendicular momentum of one electron.

(3)

line x = rs cos 9 and y = rs sin 9 with 9 = n/6 or 5n/6 and at a 2 where the duration of the pulse is taken to be four field distance r = \fh/\Ff{t) cos(a>t + )|. For the above cycles, Td = Sn/u>. The rescattering of the electrons leads mentioned field the saddle has an energy of Vs = —1.69 a.u.. Sample trajectories within the symmetric configuration to a highly excited complex of total energy E which every are shown in Fig. 2. It is evident that they cross the saddle now and then is close to the symmetric configuration during a maximum of the field and that once on the other described by the Hamiltonian (1). Any configuration on this energy shell (for some fixed time t) as well as any phase side the energy increases rapidly. This acceleration is accom of the field is equally likely, and the experimental obser- panied by a rapid separation from the nucleus, so that during field reversals the electrons will not return to the nuclues and vations are averages over initial conditions and phases. Figure 1 shows equipotential lines for the potential (3) at a will remain essentially free. A quantity of direct experimental interest is the distrimaximum of the field for F = 0.137 a.u., corresponding to an 14 2 bution of momenta of the ion, estimated as plon ~ — intensity of 6.6 • 10 W/cm . The saddle is located along the iPi +Pi) [7-10]. The many realization of the experiment and the unknown details of the initial conditions can be modelled by averaging over all initial conditions of prescribed energy and all phases of the field. The results for peak field strength F = 0.137 a.u. are shown in Fig. 3. For initial energy E = —0.58 a.u. the final distribution of momenta clearly shows the double hump structure also seen in experiment. The perpendicular momentum of a single electron show a clear suppression near zero momentum and a long tail. The suppression for small momenta is a clear sign of electron repulsion. We are not aware of experimental data on this distribution.

4. The saddle and three-dimensional motions

Fig. I. Adiabatic potential V(x, y, t) for fixed time / in the symmetric subspace. The saddle moves along the dashed line. Physica Scripta T90

Within the symmetric subspace mentioned before the position of the saddle separating trapped motion from ionized motion is clear. And as in many models of chemical reactions © Physica Scripta 2001

Classical Analysis of Correlated Multiple Ionization in Strong Fields it has one unstable direction that defines the reaction coordinate and a stable motion perpendicular to the reaction coordinate. However, in the space of six degrees of freedom of the full 3-d two electron motion and in the adiabatic approximation for the field the stability analysis of the saddle reveals two unstable directions and four stable ones. The stable directions are of minor importance: if excited they persist in the neighborhood of the saddle as some uncoupled finite amplitude motions. The second unstable direction besides the reaction coordinate is responsible for the importance of the symmetric subspace. Motion leading away from this symmetric subspace will typically have one electron escaping and the other returning to the nucleus. This corresponds to single ionization. The electron returned to the nucleus may have enough energy to ionize in the next step or may gain additional energy from the field to ionize later. Either way, the electrons do not escape symmetrically and simultaneously, so that there are no correlations between the two outgoing electrons and the ionization is sequential. Without going into the technical details of this analysis, we can illustrate some of these features with trajectories started slightly outside the symmetry plane (Fig. 4).

187

Fig 4(a) shows initial conditions on the saddle and symmetrically escaping electrons. For some deviation from symmetry, one electron escapes, the other remains trapped to the nucleus (Fig. 4(b)). It is possible, however, that the second electron picks up enough energy to ionize itself (Fig. 4(c)). In this figure the loss of correlation between the electrons is evidenced by their escape in opposite directions.

5. Triple and higher ionization The model is easily extended to the case of simultaneous removal of more than two electrons [9]. The key assumption is that the process is dominated by a symmetric configuration of the electrons with respect to the field polarization axis. Specifically, we assume that all electrons move in a plane perpendicular to the field and that they obey a CNv symmetry, which generalizes the Cjv symmetry of the previous case. The reflection symmetry limits the momenta to be parallel to the symmetry planes and thus confines the motion to a dynamically allowed subspace in the high-dimensional iV-body phase space. With the electric field directed along the z-axis the positions of the Af-electrons are zt = z, />, = p and ,,
= N

P'p+P'z

V(p, z, t),

(4)

+ NzFf{t)cos(a)t + 4>).

(5)

-50 with potential energy -100 -50

0

50

100

150

V =

N2 ^p2

+

;+

N(N - 1)

z2 "•" 4p

sin(n/N)

The equipotential curves look very much like the ones shown for two particles and the dynamics is similar. One interesting aspect of this many electron configuration is that it is limited to at most 13 electrons: for larger numbers of electrons the repulsion between them overweights the attraction to the nucleus and no saddle configuration can be found.

6. Final remarks

-20 -40

-20

0

20

40

x (a.u.) Fig. 4. Trajectories of electrons outside the symmetric subspace for E = -0.58 a.u. Initial positions are close to the saddle and marked by heavy dots; the electrons are distinguished by dotted and continuous tracks. Frame (a) shows for reference the ionization in the symmetric subspace. Frame (b) shows a case where outside this symmetric subspace one electron escapes and the other falls back to the ion. Frame (c) shows an example of sequential ionization of both electrons in opposite directions. © Physica Scripta 2001

The present considerations suggest that correlated, nonsequential multiple ionization in strong laser pulses proceeds through a saddle configuration with symmetrically moving electrons. The configurations can be seen analogous to the symmetrically escaping electrons in double ionization without field as discussed many years ago by Wannier. As in that case it is possible to derive a threshold law, which, however, is not only different from his but also much more difficult to observe because of the presence of the laser pulse. Further consequences of the model are under investigation.

Acknowledgements Financial support by the Alexander von Humboldt Foundation and K.BN under project 2P302B00915 are gratefully acknowledged. Physica Scripta T90

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References 1. Koch, P. M. and van Leeuwcn. K. A. H., Phys. Rep. 255, 289 (1995). 2. Delande, D.. in: "Les Houchcs Session LII, Chaos and Quantum Physics 1989", (editors M. J. Giannoni, A. Voros and J. Zinn-Justin), (North-Holland, Amsterdam, 1991), p. 665. 3. Delande, D. and Zakrzewski, J., in: "Classical, semiclassical and quantum dynamics in atoms", (H. Friedrich and B. Eckhardt, eds). Lecture notes in physics no. 485 (Springer, Berlin. 1997). 4. Rost, J. M., Phys. Rev. Lett. 72. 1998 (1994); Phys. Rep. 297, 271 (1999). 5. Fittinghof, D. K , Bolton, P. R., Chang, B. and Kulander, K. C . Phys. Rev. Lett. 69, 2642 (1992). 6. Walker, B. et al.. Phys. Rev. Lett. 73. 1227 (1994). 7. Weber, Th. et al.. Phys. Rev. Lett. 84. 443 (2000). 8. Weber, Th. et al., J. Phys. B: At. Mol. Opt. Phys. 33, LI (2000).

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9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Moshammer, R. et al.. Phys. Rev. Lett. 84, 447 (2000). Weber, Th. et al.. Nature 405. 658 (2000). Corkum, P. B., Phys. Rev. Lett. 71, 1994 (1993). Kulander, K. C , Cooper. J. and Schafer, K. J., Phys. Rev. A 51. 561 (1995). Walker, B., Sheehy. B., Kulander, K. C. and DiMauro, L. F., Phys. Rev. Lett. 77, 5031 (1996). Becker, A. and Faisal, F. H. M.. J. Phys. B 29, L197 (1996). Becker, A. and Faisal, F. H. M„ J. Phys. B 32, L335 (1999). Sheehy, B. et al.. Phys. Rev. A 58, 3942 (1998). Becker, A. and Faisal, F. H. M.. Phys. Rev. A 59, R1742 (1999). Becker, A. and Faisal, F. H. M„ Phys. Rev. Lett. 84, 3546 (2000). Wannier, G. H„ Phys. Rev. 90. 817 (1953). Rau, A. R. P.. Phys. Rep. 110, 369, (1984); Wigner, E. P., Z. Phys. Chemie B 19, 203 (1932); Trans. Faraday Soc. 3429, (1938).

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Physica Scripta. T90, 189-195, 2001

Classically-Forbidden Processes in Photoabsorption Spectra J. B. Delos1, V. Kondratovich1, D. M. Wang1, D. Kleppner 2 and N. Spellmeyer2* ' Department of Physics, College of William and Mary, Williamsburg, VA 23187, USA Research Laboratory of Electronics, George R. Harrison Spectroscopy Laboratory and Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA

2

Received 23 October 2000

PACS Ref: 03.65.Sq, 32.70.Cs, 32.80.Rm, 32.60.+i

Abstract We review two classically-forbidden processes that give rise to prominent structures in atomic photoabsorption spectra. In one, the electron undergoes classically-forbidden reflection above a potential-energy barrier. In the other, absorption occurs into a classically-forbidden region of the electron's motion.

1. Introduction It has been a pleasure for us to participate in this Y2K Nobel Symposium on Quantum Chaos. In this paper (based on talks by two of the authors), we would like especially to give honor to Martin Gutzwiller for his development of periodic orbit theory, and to Michael Berry for the many insights he has given into classical and quantum chaotic dynamics. Here we summarize some developments that were not mentioned in our lectures. We examine features in absorption spectra that are associated with classically-forbidden dynamical processes. We are considering one-photon absorption from the ground state (or any compact tightlybound state) to high-Rydberg states of an atom in strong fields. The specific cases of interest are hydrogen or lithium in electric, or crossed electric and magnetic fields. Photoabsorption is of course an intrinsically quantummechanical process - if we tried to describe this by classical or semiclassical mechanics, we would find that single-photon absorption from a low state into a Rydberg state would be classically forbidden. A classical electron in the lowest Bohr orbit of a hydrogen atom has a high orbital frequency, while in the excited state it has a low orbital frequency. In a classical theory, we would expect the frequency of light absorbed by the atom to match the orbital frequencies of the electron. However we learned from Bohr that the frequency at which the atom absorbs light does not match the orbital frequency of the electron; instead the frequency of light is such that the energy per photon matches the energy gap between the allowed states of the electron. Accordingly, the process cannot be described by classical mechanics. On the other hand, many aspects of the dynamics of the electron in its excited states can be described by classical or semiclassical mechanics. Recurrences are classical orbits of the electron that begin and end within a few Bohr radii of the atomic nucleus [1,2]. These recurrences are visible in real time if the electron is excited by a short pulse of light. They are also visible through interferences: the density of states and the absorption spectrum as a function of energy have fluctuations that are correlated with periodic orbits or with closed orbits. Each closed orbit produces an *Present address: MIT Lincoln Laboratory, Lexington, MA 02420. © Physica Scripta 2001

oscillation having wavelength Xg on the energy axis equal to 2nh/T, where T is the return time of the orbit. "Recurrence Spectroscopy" is the interpretation of measurements of absorption spectra as measurements of classical orbits of the active electron [2], Recurrence spectroscopy makes it possible to extract from spectroscopic data periods, classical actions, and bifurcations of closed orbits. Recently it has been learned that by applying a weak oscillating field to the atom and observing the resulting change in the recurrence strengths, it is even possible to obtain the position as a function of time q(t) for the electron on its orbit [3], In this paper we examine two classically-forbidden electron motions that have been observed in the spectroscopy of high-Rydberg states. For one of these the experimental observations were found to be in quantitative agreement with the theoretical predictions. For the other, theoretical calculations are presently incomplete.

2. Classically-forbidden above-barrier-reflection If a hydrogen atom is placed in an electric field F, then the potential-energy of the electron V(r)=-l/r

+ Fz

(1) X/1

has a saddle point at energy Es = —2F . Above the saddle-point energy, a classical electron has enough energy to escape. Nevertheless, many quasiperiodic orbits with energy above Es are bound forever to the atom. This is because the equations are separable in semiparabolic coordinates u = +Jr + z, v = yjr — z, with effective Hamiltonians [4] hu =p2J2 -Eir

+ Fu4/2

hv=p2J2-Ev2-Fv4/2=\-p

=\+p (?)

where /? is the separation constant (—1 < /? < 1). This separation constant P — cos 6 is related to the polar angle 0 which represents the initial direction of motion of the electron. Also, 1 ± fi plays the role of the effective energy associated with the u or v motions respectively (Fig. 1). The motion along the u coordinate is always bound (Figs. 1 and 2) However in the v-coordinate, there is an "effective" or "dynamical" potential-energy barrier. If an electron with energy above Es leaves the atom in a "downhill" direction, then cost? ~ —1, hv = 1 — /? is large, and the electron escapes over the barrier. These directlyescaping electrons generate a smooth continuum in the photoabsorption spectrum. If the electron leaves the atom Physica Scripta T90

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classical orbits remain constant while the size of each orbit varies so that the classical action around the orbit is proportional to SjW where S) is the scaled action of the y'th closed orbit. Then each closed orbit produces in the absorption spectrum a sinusoidal oscillation sin(SjW + const.). To extract these oscillations, we take a Fourier transform of the absorption spectrum over the variable w. The resulting recurrence spectrum has peaks at the scaled actions S) of the closed orbits. In the measurements [4], lithium atoms are excited to the 3s state by two-step resonant excitation 2s -*• 2p -*• 3s, Fig. I. Effective potentials for motion along u and v coordinates. The quanand then to an m = 0 Rydberg state by a tunable laser. tities 1 + p and 1 - /i play the role of effective energies. The u-motion is always bound; the v-motion can show tunneling or above-barrier reflections. The absorption spectrum (Fig. 3), was taken for scaled energy s = E/\J~F = — 0.125. This is far above the Stark saddle-energy, es = —2. The absorption spectrum consists of a reasonably flat continuum, a sequence of four narrow resonances, and a sequence of four less intense and broader resonances. These are associated respectively with escaping classical orbits, bound classical orbits, and quasiclassical orbits that undergo quantum above-barrier reflection. This connection is established by using semiclassical quantization conditions extended above the dynamical potential barrier by means of a uniform-WKB approximation [5]. The resulting quantization conditions give complex turning points, and they lead to complex quantized values of the energy E and the separation constant fi. Using these quantization conditions, the narrow peaks in the absorption spectrum (Fig. 3) are identified as levels with parabolic quantum numbers nu = 100 - 103 and nv = 0. These are ordinary above-saddle-energy Stark states. They are correlated with eigentrajectories which are similar to the bound trajectory shown in Fig. 2. The electron has enough energy to go over the saddle point and escape, Fig. 2. Bound trajectory (solid line) and escaping trajectories (dashed lines) in but too much of the energy is associated with motion transverse to the escape direction. A classical electron would Coulomb field combined with electric field along r-axis. The scaled energy, £=—1.9, is slightly above the saddle point energy, £saddie = — 2. All be bound forever to the atom, while a quantum electron trajectories have enough energy to escape, but those having ejection angle could only escape by tunneling through the dynamical bar9 < Bc are bound by the dynamical barrier in the v-coordinate. The rier shown in Fig. 1. energetically forbidden region is shaded. The broad resonances in Fig. 3 are of particular interest. They correspond to quantum states that are classically in an uphill direction, then cos 0 ~ 1, Av = 1 — /? is small, and unbound. For these states, the electron not only has enough the electron is trapped by the dynamical barrier. It has energy to escape, the energy is distributed between u and enough energy to escape, but it never finds the escape route v motions in such a way that a classical electron does escape: (Fig. 2). Quantum states associated with this classically- the effective energy (1 — /?) associated with v-motion is above bound motion are narrow resonances; their lifetimes are the top of the effective potential-energy barrier shown in governed by the tunneling rate through the dynamical bar- Fig. 1, and the electron orbits are similar to the escaping orbits shown in Fig. 2. rier. The quantum wave function has corresponding behavior. There is a sharp boundary between bound and free motions at (ic = cos dc — 1 - E2/2F. If the electron leaves The propagation of a quantum wave from the atom toward the atom just slightly downhill from this critical angle then and over the barrier corresponds in semiclassical ltv is just barely higher than the dynamical barrier. A classi- approximaton to this family of orbits. However, near the cal particle would move slowly over the barrier and escape: top of the barrier in the effective potential energy, the a quantum wave, however, is partially reflected and partially semiclassical approximation breaks down, and the ray transmitted. This classically-forbidden above-barrier reflec- description is no longer valid. The quantum solution near tion produces broad resonances in the absorption spectrum. the top of the barrier yields a transmitted wave and a Scaled energy spectroscopy [1,2] makes it possible to reflected wave. The transmitted wave corresponds to ordiobserve these phenomena experimentally. In this method nary Newtonian trajectories which are launched in the the excitation energy E and electric field F are simul- escape sector and go to infinity (Figs. 2 and 4). Although taneously varied so that the scaled energy e = E/Fl/2 is con- the reflected wave can also be correlated with Newtonian stant, and the spectrum is recorded as the scaling variable trajectories, the reflection itself is a quantum process that w = F~ 1 / 4 is varied. Under these conditions, the shapes of cannot be described by Newtonian mechanics. Physica Scripta T90

© Physica Scripta 2001

Classically-Forbidden Processes in Photoabsorption Spectra

191

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W Fig. 3. Scaled absorption spectrum of Li vs. w = F~['4 at £ = -0.125. experiment (solid line) and theory (dotted line). The parabolic quantum numbers («„, n„) are indicated. All peaks correspond to resonances that are far above the potential-energy saddle. States with n, = 0 are below the dynamical barrier; escape is classically forbidden and they decay by tunneling. Those with n, = 1 are above the dynamical barrier and escape is classically allowed; they have a lifetime because of quantum reflection above the dynamical barrier. The shadowed strips show the position and widths of these states.

Theory without reflections

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Fig. 4. An example of an above-barrier-reflected orbit. Heavy line: a below-barrier closed orbit which makes one cycle of u-motion and 5 cycles of amotion (1:5 orbit). Light line: above-barrier-reflected orbit with the same raio of periods. Classically, an electron would follow the dashed line and escape. (The critical angle 9C lies between the heavy and light lines.) On the below-barrier orbit, the electron makes a full stop and then retraces itself. Near the stop-point the trajectory is rectilinear and parallel to the stopping force (the sum of the Coulomb force and the force from the external field). The above-barrier closed orbit also retraces itself. However, at the endpoint, the trajectory does not have a stopping point - the reflection occurs with nonzero velocity. Other shapes of reflected orbits can also occur.

For the Stark system, Eq. (1), the dynamical barrier lies on a parabolic curve v = (E/F)[/2, which is indicated as the dot-dashed line in Fig. 4. An example of a closed orbit associated with above-barrier reflection is shown. These new orbits which arise from above-barrier reflection must be included in closed orbit theory. The formulas are similar to those for ordinary closed orbits, and we expect that usually only a few above-barrier orbits will be important. The recurrence spectrum of the data in Fig. 3 is shown in Fig. 5. We see here a sequence of evenly-spaced peaks having large scale modulations in their heights. The modulations arise from interference of waves which undergo classically© Physica Scripta 2001

Fig. 5. Recurrence spectra vs. the scaled action s. The nlh peak represents the effect of the « lh -return of the parallel orbit combined with all other orbits which have n oscillations of u-motion. Dotted line is the theoretical envelope of the modulations, induced by the interference of quasiclassical closed orbits undergoing above-barrier reflections with the ordinary closed orbits.

forbidden above-barrier reflection with waves that reflect below the barrier. At the scaled-energy considered here (s = —0.125), the action of every orbit is close to a multiple of the action of the parallel orbit. Therefore individual orbits are not resolved in Fig. 5: each peak corresponds to all orbits having the same number of oscillations of u-motion. The height of each peak is governed by the coherent sum of the amplitudes Physica Scripta T90

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E(cnT 1 ) Fig. 6. The measured delayed absorption spectrum at F = 1000 V/cm and B = 6.002 T. The quantum states are organized by the principal quantum numbers n and the angular quantum number n$ corresponding to the principal action and the azimuthal action. The states with the same quantum number n belong to a sequence and the states in the sequence are distinguished by the quantum number n$.

Familiar formulas for oscillator-strengths of electronic transitions involve the dipole matrix element between initial and final states, . In studies of absorption to high-Rydberg states, the initial state \i > is localized to within a few Bohr radii of the nucleus. The excited states are very large: if the principal quantum number n is around 33, then the radius of the state is about a thousand an, and if n is 100, the radius is ten thousand a0- Normally the initial state lies within a classically allowed region of the final state. If the initial state were to lie within a classically forbidden region, then we would expect that the overlap and the dipole matrix element between initial 3. Absorption into a classically-forbidden region and final states would be "exponentially small". However We now turn to another classically-forbidden process. Here in experiments we observe a sequence of states for which the process is H(2p) —• high-Rydberg states in crossed elec- calculations show that the initial state is in the classically forbidden region. tric and magnetic fields [6].

of the returning waves associated with all of these orbits. If we neglect above-barrier reflections, we get a sequence of peaks with slowly-decaying height. However when abovebarrier reflections are taken into account, modulations occur in the recurrence spectrum. The closed-orbit sum reveals that the large modulations in the recurrence spectrum result from the interference of classical, below-barrier orbits with nonclassical, above-barrier closed orbits. The major closedorbit contributions come from Tu : Tv = 1 : 4 and 1: 5 belowbarrier orbits and from 1 :5 and 1:6 above-barrier orbits.

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Classically-Forbidden Processes in Photoabsorption Spectra Fig. 6 shows the absorption spectrum to long-lived states for a hydrogen atom in crossed electric and magnetic fields [6]. The magnetic field is 6.002 T, and the electric field is 1000 V /cm. We see a regular sequence of narrow absorption lines. The quantum states associated with these lines correspond to a regular family of quasiperiodic orbits, which lie close to the plane perpendicular to the magnetic field. We will see that the corresponding orbits do not touch the region of the initial state. For the crossed field system, there are two elementary periodic orbits (S+, S-) shown in Fig. 7 [8]. They lie in the plane perpendicular to the magnetic field, and S+ goes around in a right hand sense relative to B. On this orbit, the Coulomb force and the Lorentz force both point inward. This orbit is stable and quasielliptical in the energy range we study. Quasiperiodic orbits oscillate about S+, and these orbits occupy a sufficient volume of phase space to support quantum states.

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In Einstein-Brillouin-Keller (EBK) theory, each quantum state corresponds to a torus having classical actions restricted to certain integer or half-integer values (these quantized tori are also called eigentrajectories or eigentori) The classical actions are defined as integrations around independent loops on the torus. A method for calculating these action integrals was invented by Martens and Ezra [9] By integrating equations of motion, we can obtain numerical data for [q{t),p{t)]. For this three-dimensional system, regular motion is quasiperiodic with three independent fundamental frequencies a>= (cut, w2> 03), so the variables can be expressed as [9]:

9(0 = J 2 9 k e x p t i A : ' ("" + Ml k

(3)

(4)

k eX P(t) = aJ^P Pt'* ' to* + W' From Fourier-transform of [q(t),p(t)], we can extract the frequencies co and the coefficients [qk,Pk] in this expansion. This Fourier expansion for the coordinates and momenta can be converted to an action-angle representation. The actions I = {I\,I2,h) are constant and the angles 0 = (6>i, 02, 03) evolve linearly in time:

0(s) = (os + fi.

(5)

Then the Fourier series representing the time-development can be reinterpreted as the transformation equations from {0,1) to (q,p) with fixed values of action variables:

q(0,I) =

J^qk(iy *

•

»

p(0,I) = J2Pk(IWk0

(6)

(7)

This is a parametric representation of a torus in phase space. Now three independent fundamental loops can be denned by allowing one one angle variable to vary from 0 to In while the other two angle variables are held fixed. Then the actions along the loops are

=

(\/2n)j>p-dq

=

(\/2n)j>p.^L&6i

(8)

-1.0

Fig. 7. The primary periodic orbits S+ and S- in a hydrogen atom in crossed electric and magnetic fields. © Physica Scripta 2001

To apply this method to the atomic system , we need a few refinements - the equations of motion are singular at the origin, and regularizing this singularity involves expanding phase space from six dimensions to eight dimensions. Details are given in [10]. A typical trajectory is shown in Fig. 8. It oscillates in a range about S+, staying always close to the z — 0 plane. Three loops associated with this torus are shown. One is associated with the small-amplitude z-motion along the magnetic field, one is associated with the azimuthal motion around the atom, and the third is associated with a combination of azimuthal motion and radial oscillation about S+. Accordingly, we call the action variables /,, 1$ and Ip+
J. B. Delos et al.

Fig. 8. The torus explored by the trajectory launched at E = -110 cm ' o r e = -0.5779, 6 = 87° and <j> = 140° and the three loops C, with ('=(1,2,3) used to calculate the actions. In (a), (b), we show the loops and the trajectory in two dimensional spaces xy and xz, then show all of them in a three dimensional graph below (a) and (b). (Note the small range of z motion) The first loop C\ touches two caustics, and includes one loop of angular motion and one loop of radial motion; the second loop Ci is a loop of angular motion and it touches no caustics; the third loop is one loop of z motion, touching two caustics.

we compute and record the energy and the action variables, and then we fit the data set (Ip+^, 1$, I:, E) to a smooth funtion E = a0 + ax Ip+(j> +a2I^,+ a}I: + a4lj+lj) + aslj + a6I: ailp+^Ify + a%lp+^lz + agl^L. Once the coefficients in this function are known, we can input the quantized values of the actions to find the corresponding energy eigenvalues [11], According to EBK quantization theory, the quantization conditions for Ip+^, 1$, L are Ip+(j> =np+(p + 1/2, / . = #!.+ 1/2,

on a graph of the measured absorption spectrum (Fig. 6). The states fall into several sequences. Each sequence is distinguished by the principal quantum number n, and the members of a sequence are distinguished by the quantum number n$. The accurate correspondence between calculated and observed values indicates that these regular quasidiscrete states in the absorption spectrum correspond to regular tori which stay close to the xy plane and oscillate around the stable periodic orbit S+. Now finally we notice that most of the tori in this family do not touch the origin of coordinates, and therefore the classically-allowed region of the wave function does not overlap the initial state: photoabsorption occurs into an evanescent part of the wave function in a classicallyforbidden region of space. The observed absorption lines are surprisingly strong. Can we construct an extended semiclassical formula that would describe them? This has not yet been done, but we can speculate that such a formula would combine two theoretical elements. The first element is extension of the semiclassical formula for the wave functions associated with each eigentorus into the classically forbidden region near the nucleus. In one-dimensional systems, semiclassical approximations in forbidden regions are presented in many textbooks [5]. In multidimensional systems, however, the issues involved are much more tricky. Some studies express the solution by analytically continuing classical mechanics into complex phase space [12]. Despite a number of successful implementations of such theories, analytic continuation of functions of several complex variables is nontrivial from either a physical or a mathematical point of view. The second element of the theory would be calculation of the dipole matrix element using the semiclassical wave function in the forbidden region. This problem is closely related to the semiclassical theory of oscillator strengths developed by Kondratovich and Delos [13]. They got a formula for oscillator strengths from low states to high-Rydberg states by using a combination of quantum, semiclassical and classical concepts. The quantum concept is that absorption of a photon creates outgoing electron waves having a certain angular distribution. The semiclassical concept is that each quantum state corresponds to an eigentorus. The classical concept is that each eigentorus has a characteristic "eigenejection-angle" 6„ such that electrons leaving the atom in that direction find themselves on the « th eigentorus. We find from these considerations that the strength of each absorption line is directly proportional to the intensity of outgoing electron waves in the direction that "feeds" the correspondin eigentorus.

where the half-integers result from the fact that the loops The analysis given in Ref. [13] presumed that the Cp+$ and C~ each touch caustics twice. We also define a total eigentorus overlaps the localized ground state of the atom. action / as This discussion now allows us to pose more clearly the new problem: can the formulas for the strength of absorp/ = Ip+

by the National Science Foundation and the © Physica Scripta 2001

Classically-Forbidden Processes in Photoabsorption Spectra References 1. Holle, A. et at, Phys. Rev. Lett. 56. 2594 (1986); Main, J. et at, Phys. Rev. Lett. 57, 2789 (1986). 2. Du, M. L. and Delos, J. B., Phys. Rev. Lett. 58, 1731 (1987); Du, M. L. and Delos, J. B., Phys. Rev. A 38, 1896 (1988); 38, 1913 (1988); Mao, J.-M. and Delos, J. B , Phys. Rev. A 45, 1746(1992); Gao, J. and Delos, J. B., Phys. Rev. A 46, 1455 (1992); Phys. Rev. A 49, 869 (1994); Main, J. etat, Phys. Rev. A 49, 847 (1994); Mao, J.-M. etal., Phys. Rev. A48, 2117 (1993); Wintgen, D., Phys. Rev. Lett. 58, 1589 (1987); Wintgen, D. and Friedrich, H., Phys. Rev. A36, 131 (1987); Eichmann, U., Richter, K., Wintgen, D. and Sandner, W., Phys. Rev. Lett. 61, 2438 (1988); Bogomolny, E. B., Sov. Phys. JETP 69, 275 (1989); Courtney, M. et at.. Phys. Rev. Lett. 74, 1538 (1995); Shaw, J. A. et at, Phys. Rev. A 5 1 , 3695 (1995); Spellmeyer, N. etal.. Phys. Rev. Lett. 79, 1650, 1997. 3. Haggerty, M. R., Spellmeyer, N., Kleppner, D. and Delos, J. B., Phys. Rev. Lett. 81, 1592 (1998); Haggerty, M. R. and Delos, J. B., Phys. Rev. A 61, 53406 (2000). 4. Kondratovich, V., Delos, J. B., Spellmeyer, N. and Kleppner. D., Phys. Rev. A 62, 43409 (2000).

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5. Child. M. S., "Semiclassical mechanics with molecular applications", (Oxford Univ. Press, New York. 1991), Sees. 3.2 and 3.5; Kemble, E. C , "The Fundamental Principles of Quantum Mechanics with Elementary Applications", (Dover, New York, 1958), Ch. Ill, Sec. 21j. 6. Freund, S. et at, Phys. Rev. A (submitted). 7. Actually the quasidiscrete peaks with m = 0 are narrower and higher than indicated in Fig. 3. We artificially broadened them in a manner consistent with the experimental resolution. 8. Flothmann, E. and Welge. K. H., Phys. Rev. A 54, 1884 (1996). 9. Martens, C. C. and Ezra, G. S., J. Chem. Phys. 83, 2990 (1985); Martens, C. C. and Ezra, G. S., J. Chem. Phys. 86. 279 (1987). 10. Wang, D. M., PhD thesis. College of William and Mary, 2000. 11. Noid, D. W. and Marcus. R. A., J. Chem. Phys. 62, 2119 (1975). 12. (a) Miller. W. H. and George. T. F., J. Chem. Phys. 56. 5668 (1972); Doll, J. D. and Miller, W. H.. J. Chem. Phys. 57, 5019 (1972). (b) Bartsch, T„ Main, J. and Wunner, G., Ann. Phys. (NY) 277, 19 1999. (c) Takada. S.. J. Chem. Phys. 104, 3742 (1996). 13. Kondratovich, V. and Delos, J. B., Phys. Rev. A 56, R5 (1997); 57, 4654 (1998).

Physica Scripta T90

Physica Scripta. T90, 196-201, 2001

Quantum Hall Effect Breakdown Steps due to an Instability of Laminar Flow against Electron-Hole Pair Formation L. Eaves* School of Physics & Astronomy, University of Nottingham, Nottingham NG7 2RD, UK Received July 28, 2000

PACS Ref: 73.40.Hm, 72.20.Ht, 72.20.My, 47.32.-y

Abstract The breakdown of the dissipationless state of the quantum Hall effect at high currents sometimes occurs as a series of regular steps in the dissipative voltage drop measured along the Hall bar. The steps were first seen clearly in two of the Hall bars used to maintain the US Resistance Standard, but have also been reported in other devices. This paper describes a model to account for the origin of the steps. It is proposed that the dissipationless flow of the quantum Hall fluid is unstable at high flow rates due to inter-Landau level tunnelling processes in local microscopic regions of the Hall bar. Electron-hole pairs are generated in the quantum Hall fluid in these regions and the electronic motion can be envisaged as a quantum analogue of the von Karman vortex street which forms when a classical fluid flows past an obstacle.

1. Overview Chaos and turbulence are two areas of modern physics that have captured widespread interest, in part because both phenomena impinge so directly on our everyday consciousness. Their relation to quantum mechanics is more subtle [1] and forms one of the themes of this Symposium. Several papers in the symposium discuss the chaotic motion of electrons- in semiconductor heterostructures with quantum dimensions: this has developed into an active research field during the last decade [2]. By measuring the electrical conductivity of these devices, it is possible to reveal aspects of quantum chaology, e.g. the effects of wavefunction scarring [3-5], that are difficult to observe directly in experiments on other systems. The quantum Hall effect (QHE) [6] seems at first sight to be unrelated to chaos and turbulence. The motion of the conduction electrons is of such well-regulated form that the ratio, axy, of the current to the Hall voltage is quantised precisely in units of e2//z. However, when the dissipationless state of the QHE breaks down at high currents, several effects are observed which seem analogous to those found in the breakdown of laminar flow in classical fluids at high values of the Reynolds number [7]. These similarities include the change in the flow characteristics and the presence of hysteresis and intermittent noise. The principal aim of the paper is to show that the occurrence of dissipative voltage steps in certain experiments on QHE breakdown can be understood in terms of inter-Landau level tunnelling processes caused by the presence of charged impurities in small localised regions of the sample. Here electron-hole pairs can be generated in the quantum Hall fluid and it will be shown that their motion is analogous to that of a von Karman vortex street which is formed when a classical fluid * e-mail: [email protected] Physica Scripta T90

flows around an obstruction with the Reynolds number above a critical value. Re « 60 [7], A connection between QHE breakdown and classical hydrodynamics is of further interest because of the intrinsically two-dimensional nature of the electron motion in the quantum Hall effect. In two dimensions, the problem of fluid flow is simplified: vortex stretching and tangling does not occur and vorticity is a conserved scalar quantity, even in the presence of viscosity, and obeys a diffusion equation. 2. Quantum hall effect breakdown In the integer quantum Hall effect (QHE), a twodimensional electron gas (2DEG) carries an almost dissipationless current along the voltage equipotentials when the applied magnetic field B is close to values which correspond to integral values of the Landau level filling factor v = nh/eB, where n is the electron sheet density [6]. However, when the current / flowing through the Hall bar exceeds a certain critical value / c , a sudden onset of dissipation occurs. This is observed as an increase in the longitudinal voltage drop, Vx, measured along the direction of current flow. (We define a set of cartesian coordinates so that x-y represents the plane of the 2DEG, z is along the direction of B and x is parallel to the long axis of the Hall bar sample. The _y-axis then corresponds to the direction of the Hall electric field). The increase in Vx corresponds to a dissipative current component, /, flowing perpendicular to the equipotentials. The increase in dissipation is usually accompanied by a breakdown in the quantisation of the Hall conductivity, axy, given by the ratio I/VH, where Vn is the Hall voltage. This phenomenon has been the subject of much experimental and theoretical work [8-33], and has been reviewed in detail in a recent article [8]. Let us focus on a particular aspect of QHE breakdown observed by Cage and coworkers in two Hall bar samples of great importance, those used to maintain the US Resistance Standard [9,10]. When a large current is passed through these samples and the magnetic field is swept over a narrow range close to filling factor values v = 2, it is found that Vx increases in a series of steps of regular height, A Vx « 5 mV as shown in Fig. 1. The breakdown region shows hysteresis with respect to the direction of field sweep, accompanied by intermittent noise. The latter effect is shown in more detail in Fig. 2 - it can be seen that, in the time domain, Vx jumps almost instantaneously between the discrete step values. Similar steps have also been reported recently for QHE breakdown at both v = 1 and 2 in two-dimensional hole © Physica Scripta 2001

Quantum Hall Effect Breakdown Steps due to an Instability of Laminar Flow against Electron-Hole Pair Formation

197

400

300 > •§- 200

100 -

0.5

1.0

0 -

1.5

Time (s) 10.5

11.0

11.5

12.0

300

B (T) w c o O

200

1

m

12.5

-

1

(b) _

i

" ft

w

V x (4,6)

-

12.26 T

-

:

100 n

[8]

[9]

•A,A.

34

38

42

Vx

[101 "

A"

46

50

(mV)

Fig. 2. (a) Plot of the dissipative voltage Vx as a function of time during a given sample period of just over 2 seconds at a magnetic field of 12.26 T for one of the NISTsamples. It shows intermittent jumps between different voltage steps, (b) Histogram of (a), showing the number of counts at the different voltage steps. The square brackets, [7] etc., correspond to the quantum number N in the hydrodynamic model presented here. The figures are taken from [9] by kind permission of Dr. M. E. Cage and NIST. Fig. 1. (a) A plot of longitudinal voltage Vx versus magnetic field strength B at T = 1.3 K and / = 2 1 0 / J A in the region of the v = 2 filling factor, for one of the US Resistance Standard samples at NIST. The arrows indicate the hysteresis. Cage et al have observed steps on upsweeps and downsweeps of B. (b) A detail of the breakdown curve, showing the large number of steps in Vx. The figures are taken from [9] by kind permission of Dr. M. E. Cage and NIST.

20 >

E

5>v9-2uA 4s! •

10

• >x

>^

o.o

v "-19 ">.

"

•

am

4.20

>

>v2

gas (2DHG) samples [11]. A typical series of breakdown curves for the 2DHG in the magnetic field range close to these two filling factors are shown in Fig. 3. The step height is A Vx =» 1 mV. Breakdown steps were also observed in experiments on a specially-designed Hall bar in which a current was passed through a short (10^m) and narrow (1 ^m) constriction [12]. In that case, however, it was found that the steps corresponded to regular increments (ARX« 200 Q) of the longitudinal magnetoresistance Rx = Vx/I, rather than of Vx. It is difficult to account for the results of these three experiments in terms of the known properties of the magnetoconductivity components, axx and axy, of the quantum Hall fluid (QHF) [13]. The model developed in the next section explains the breakdown steps in terms of inter-Landau level tunneling processes [14-20] in a small localised region or regions of the Hall bar which are characterised by a large value of the Hall electric field. The model involves strong local variations of the electric field and velocity field in the Hall bar and thus corresponds to a hydrodynamic picture of the current flow. This may account for the analogies between QHE breakdown and the breakdown of laminar © Physica Scripta 2001

3

1

4\

0

T,

(a)

,

4.15

4.20

B(T)

2.15 Fig. 3. (a) Plot of Vx versus B showing breakdown of the dissipationless state in the region of the v = 1 filling factor for a Nottingham 2D hole gas on a (311) surface [11]. The measurement was taken at T «s 300 mK. The inset shows in more detail voltage steps of height A Vx ~ 1 mV close to the breakdown threshold, (b) Similar data on the same sample at v = 2. Physica Scripta T90

198

L. Eaves

flow in classical fluids noted earlier [21,22] - see also refs. [23,24] for analogies between the QHF and classical fluids and Ref. [28] for analogies with superconductors. 3. A model for the steps: QHE breakdown due to inter-Landau level tunnelling

[18,34,35]. Here Ev can be significantly larger than the average value, Ey. A similar bunching of the equipotentials, i.e. a strong local £-field, could also arise for a more complicated distribution of nearby charge, e.g. a dipole due to oppositely-charged impurities or a small cluster of charged impurities. Figure 4(a) also shows the wavefunction of a state ao in the lowest filled Landau level (« = 0) and of a state fi{ in the n = 1 level. The two states have the same energy. The dispersion of the Landau levels over a range of energies is due to the spatial variation of <j>. For simplicity, we neglect spin. The orbit centres of ao and jS( are separated by a distance Ay = s. If the states have the same energy, we can write

Inter-Landau level tunnelling or, more precisely, interLandau level scattering, was proposed several years ago as a possible intrinsic mechanism for QHE breakdown [14-20]. In this process, an electron in the lower filled Landau level scatters elastically (or quasi-elastically with the emission of a low energy acoustic phonon) into the empty state of the upper unfilled Landau level. Since the wavefunctions of the Landau states are strongly localised mo)QS~/2 = Ho, the spatial overlap between the wavefunctions field is required for significant spatial overlap between is negligible. However, when con is sufficiently large, overlap the initial and final states of the same energy, i.e. the Hall occurs (see figure) and an electron in state ao can undergo electric field EH must be ~ Ao>c/e£B, where a>c = eB/m is the cyclotron frequency, m is the electron effective mass (= 0.07 me for GaAs) and £B = {h/eB)1 is the magnetic length. Alternatively, this condition can be expressed in the form mv\ ~ fiajc, where v<j = EH/B is the drift velocity of the cyclotron orbit centre. For the experimental conditions of Fig. 1, the average value of the component of the electric field E along y (the average Hall field) is relatively small, Ey = Vn/w ~ 7 x 103 Vm _1 , almost two orders of magnitude too small to produce significant wavefunction overlap. In other words, the average drift speed vd satisfies the relation mv\ , over a small localised region, as shown schematically in Fig. 4(a). Outside this region, the Hall field is assumed to be much lower. Over the length s along y, we write ecp = ma>ly2/2, where m is the effective mass of electrons in GaAs (w = 0.07m e ). This model potential is a function of only one unknown parameter coo, and corresponds to a large velocity gradient Q = dvx/dy = (d2(t>/dy2)/B = col/a)c for electron flow in this localised region of the Hall bar. A potential with a local quadratic variation along y would arise locally from the set of equipotentials shown in Fig. 4(b). The equipotentials are in the plane of the 2DEG. Due to the character of electron motion in crossed electric and magnetic fields, they also correspond to the directions of dissipationless current flow in this region. Equipotentials of this form arise from the electric field due to an unscreened charged donor impurity located in or close to the 2DEG layer, combined with the large Hall electric field Ev due to the charge distribution near the edge of the sample Physica Scripta T90

Fig. 4. (a) The schematic spatial variation of the energy of the two lowest Landau levels, n = 0 and n = 1 in the breakdown region. The eigenfunctions a0 and /?j are in the n = 0 (filled) and n = 1 (empty) levels respectively. They have the same energy and spatially overlap each other. The inset shows schematically the small breakdown region (cross-hatched) located somewhere between the two voltage probes (2 and 4) across which Vx is measured, (b) Diagram showing the voltage equipotentials in the local breakdown region. These correspond to the electric fields arising from a single ionised donor impurity located slightly above the 2D plane, combined with the large, locally uniform electric field that would occur due to the presence of edge charges, close to the physical edge of the sample. © Physica Scripta 2001

Quantum Hall Effect Breakdown Steps due to an Instability of Laminar Flow against Electron-Hole Pair Formation a quantum transition to the unfilled state fix. An interLandau level transition of this kind leads to the dissipative breakdown current, /, which flows perpendicular to the equipotentials. It requires a jump of orbit centre position. This can be induced by elastic scattering due to the presence of the charged impurity, which breaks the translational symmetry. Inter-Landau level scattering can also occur by emission of low energy acoustic phonons (quasi-elastic inter-Landau level scattering - QUILLS) [14-20] or possibly by excitation of magneto-plasmon modes [38]. In order to compare our model with the data, in particular the measured value of the voltage steps, AVX, shown in Figs. 1, 2 and 3, we need to determine the size of dissipative current / generated by inter-Landau level tunnelling. In principle, the tunnelling rate could be calculated using Fermi's Golden Rule. However, such a calculation is not straightforward even if we assume a relatively simple form of the electrostatic potential in the local region of breakdown, e.g. of the type shown in Fig. 4(a) due to a charged impurity and nearby edge charges. The problem is that inter-Landau level tunnelling creates an electron in the upper, previously empty. Landau level and an empty state, or hole, in the previously filled lower Landau level. The two particles experience a coulombic attractive interaction and also tend to screen the local potential which generated them, thereby perturbing the drift motion of other electrons in adjacent filled states of the lower Landau level. Since the interaction between the electron-hole pair (a magnetoexciton) and the QHF is quite complicated, even in the absence of a charged impurity or edge charges [39], we cannot set up a simple calculation based on an independent particle picture. To avoid this difficulty, let us consider the following semiclassical description, which seems to account for the essential physical features of the breakdown mechanism. For a sufficiently large value of current flow I and at a critical magnetic field, we assume that the electric field in a local breakdown region is large enough to induce inter-Landau level tunnelling. This process creates an electron-hole pair, the electron occupying state pY and the hole corresponding to the electron missing from state otoThe presence of the electron-hole pair acts to modify the electric field distribution in the local region due to the effect of screening. The locally strong electric field is partly screened by the presence of the electron and hole, thus inhibiting the inter-Landau level tunnelling process which generated them. However, due to the presence of the local electric field, the electron-hole pair drift away from the region in which they are formed. For the potential distribution shown in Fig. 1, assuming an independent particle description, the hole left in state ai0 has a high drift velocity, whereas the electron in p[ has a lower drift velocity, each given by the local value of the electric field. We can include the effect of the Coulombic attraction between the electron and hole on their drift motion by estimating the mean external electric field experienced by the pair. This is given by mw\slle. Therefore the electron and hole move along the direction of the equipotentials at a speed determined by the mean E x B drift of their orbit centres, given by (vd) « Jls/2. They move away a distance ~ s from the formation region in a time x = 2/Qoc, where a is a numerical factor, ~ 1. At this separation, they © Physica Scripta 2001

199

no longer screen effectively and the strong local electric field is restored, leading to the creation of another electron-hole pair. Assuming that the time x is the controlling factor in the formation of electron-hole pairs, we can take the rate of generation to be s» afl/2. On a longer timescale, given by the intra-Landau level energy relaxation time T£ ~ 10~10 [31], the electron-hole pair breaks up, the two particles diffusing away from each other with velocity components perpendicular to the equipotentials. This process can occur for each particle by emission of low energy acoustic phonons through intra-Landau level scattering processes. The dissipative backscattering current /' « eafi/2 causes an incremental increase in Vx given by A Vx = hi lie1 = haD./4e.

(2)

The series of steps shown in Fig. 1 would then correspond to successive contributions to the dissipative current of separate breakdown regions, each adding «s eo£l/2 to the dissipative current as B is swept away from the v = 2 value. In order to test the model, we need to compare the observed value of A Vx with the value of a>o required to generate inter-Landau level tunnelling. Using Eq. (1) and recalling that Q = col/a>c, we can write Eq. (2) in the form eAVx — nv.hr /ms~.

(3)

This equation allows us to relate the observed value of AVX = SmV to the orbit centre separation, s, rather than to tt>o. Setting a. « 1, we then obtain s2/l\ ~ nhcoc/eAVx, which gives s « 3.5£B, where £B = (h/eB)1. This value of s is in reasonable agreement with the condition for alignment of the classical turning points [16,17] of the wavefunctions ao and fi{, namely when JTP = (V3 + 1)^B- The wavefunction overlap condition corresponds to the threshold of the strong increase in the matrix element for inter-Landau level transitions [19]. The agreement provides support for our model. The measured step height determines Q. according to Eq.(2), and corresponds to a value of <x>o « 1013 s~l s» 0.4 cuc at 12T, so the quadratically varying potential (~ O>Q) makes only a small additional contribution to the magnetic confinement (~ col) 0I" t n e Landau level states. A value of COQ of this size is physically reasonable since it corresponds to the quadratic part of the potential caused by an unscreened ionised donor near the interface layer of a GaAs/(AlGa)As heterostructure. We recall that such an impurity combined with a strong local Hall field close to the sample edge acts to generate the equipotential distribution shown in Fig. 4(b). A rough estimate of the observed step width AB can also be made by relating AB to AVx using Eq. (1) of [27]. Assuming that in the region of breakdown the Landau level width is comparable to the cyclotron energy, ha>c, and since VH ( « 3 V) remains accurately quantised over the range of B where the steps are observed, we obtain AB/B « A Vx/ VH. This gives AB = BA Vx/ VH ~ 10" 2 T, in fair agreement with the typical observed values - see Fig. 1. The model can also be tested against the recently-reported voltage steps for QH breakdown of 2D hole gases grown on (311)A surfaces [11]. These data are shown in Fig. 3. Here the typical step height is A Vx « 1 mV for breakdown at both v = 1 and v = 2. For holes in a (311)A GaAs quantum well, Physica Scripta T90

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the hole cyclotron effective mass, m^ ~ 0.15me [40,41]. We can therefore use this value of mh and Eq. (3) to estimate s for hole gas breakdown. We also take into account the lower carrier density for the sample used in these experiments, which leads to a larger value of 1% at a given value of filling factor. This gives s = 2.3£B for breakdown at v = 2 (and 3.2£B at v = 1), which is quite close to the value of sTP and to the value of s deduced above from the step height observed in the two n-doped samples used by Cage and coworkers. The 2DHG data therefore provide further support for the model. To conclude this section, we note that the model may also be relevant to the results of experiments by Bliek etal. [12,17] in which the QHE breakdown was induced by passing current through a narrow constriction, 1 [im wide. The resulting Hall electric field in the constriction is sufficiently strong to induce inter-Landau level scattering [17]. These measurements are also of interest because steps appear as regular values of resistance Rx = Vx/I rather than of voltage Vx. Briefly, the model can account for this difference using the following argument (fuller details will be published elsewhere). In a very narrow constriction, we can assume that the form of the Hall voltage profile V(y) across the constriction is mainly determined by the edge charges. The form of V(y) is a logarithmic function, with a prefactor proportional to the total current / flowing through that channel [18,34]. The local velocity gradient, Q(y) = (d~(j)/dy2)/B, is then also proportional to /. From Eq. (2), it can be seen that this would then lead to increments AVX which are proportional to /, i.e. to constant increments of ARX. Further analysis based on this model, and taking account of the observed step height, ARX — 200 Q, indicates that the breakdown region in the channel is confined to a thin (~ 0.1) ^m layer close to one edge of the constriction, in agreement with model calculations of the high field region close to the edge [34-37].

4. Analogies with classical hydrodynamics and quantum fluids This concluding section discusses further the analogies between QHE breakdown and the breakdown of laminar flow in classical hydrodynamics. The expression that we have obtained for the rate of electron-hole pair generation in the local breakdown region is strikingly similar to the shedding rate of vortices in a von Karman vortex street [7]. For classical fluid flow of velocity U around a cylinder of diameter D at Reynolds numbers. Re, greater than ~ 60, vortices are shed at a rate given by fSU/D, or ~ fSQ. where Q, is the velocity gradient in the vicinity of the cylinder and /? is a numerical parameter of the order of unity. The resulting von Karman vortex street comprises a sequential line of vortex-antivortex pairs which drift along with the fluid flow. Let us develop the analogy by envisaging the electron-hole pair as a quantum analogue of a vortex-antivortex pair, with the additional idea that inelastic intra-Landau level scattering processes due to acoustic phonon emission cause the electron and hole to move apart in opposite directions in the Hall field. In Madelung's hydrodynamic description of the Schrodinger wave equation [42], the wavefunction can be written as ij/ = RelS, where R is a real amplitude Physica Scripta T90

and S is the action. By substituting this into the Schrodinger equation for an electron moving in crossed electric and magnetic fields, the electron velocity, v, associated with a given quantum state can be represented by the semiclassical equation v = (hS/S — eA)/m, where the first term represents the drift motion, v\, of the cyclotron orbit centre and the second term describes an effective internal cyclotron motion [22]. It is easily shown that the velocity field v satisfies an Euler equation - that is, the hydrodynamic equation of motion of inviscid fluid. (In this description, the velocity represents the motion of a small volume of fluid - a fluid particle or material body of fluid in the language of classical hydrodynamics). For the equipotentials shown in Fig. 3(b), S is not single-valued due to the presence of the charged impurity, so that v\ has a finite curl. The curl of the second term is simply ajc = eB/m, which represents the internal cyclotron motion around the orbit centre. In terms of our analogy, we can envisage the elastic inter-Landau level tunnelling process which generates the electron-hole pair as analogous to the formation of a vortex-antivortex pair with vorticity ±coc. The antivortex is effectively the missing electron, i.e. hole, in the lower Landau level state ao, and the vortex is the electron transferred to the upper Landau level state /?,. As the tunnelling process involves no dissipation, vorticity is conserved, as for an inviscid classical fluid obeying the Euler equation. The subsequent diffusion of the electron-hole pair is controlled by the inelastic scattering time t e , which corresponds to an an electron diffusion coefficient D ~ t\/i£ ~ H/moicxE- and is somewhat analogous to the kinematic molecular viscosity in a classical fluid vK. In classical hydrodynamics, the formation of vortices gives rise to an additional dissipative viscous drag force which can greatly exceed that due to v^. This can be represented in terms of an eddy viscosity vE ~ s2Q, where s is the so-called mixing length, which gives the typical eddy size [7]. Note that the concept of eddy viscosity applied to the QHF also provides us with an alternative way of deriving an expression for the dissipative current / [43]. In the QHE breakdown, the formation of electron-hole pairs in a local region can be thought of as giving rise to eddy viscosity increments AvE ~ s2D. ~ s2u)l/coc ~ 2h/m, according to Eq. (1). For an electron in GaAs, h/m~\Qr2 m 2 s_1, ~ 105 larger than the quantum of circulation {hiMHe) for superfluid He. In this sense the QH fluid is relatively stable against vortex pair formation at high flow speeds, compared to the analogous case of an ideal superfluid He film [44,45]. We can introduce a critical value dimensionless parameter to represent the local breakdown condition. This is given by s2Q/D « a»cTs ~ 103. This value is not very different from the critical Reynolds number Rec ~ 102 for fluid flow around a cylinder. In summary, this paper has proposed that the voltage steps observed in the breakdown of the integer quantum Hall effect can be understood in terms of inter-Landau level scattering processes in localised microscopic regions of the Hall bar. This leads to a hydrodynamic picture: the scattering generates electron-hole pairs in the quantum Hall fluid, analogous to the formation of a classical von Karman vortex street. This description may also help to account for qualitative similarities (intermittency and hysteresis) © Physica Scripta 2001

of Laminar Flow against Electron-Hole Pair Formation Quantum Hall Effect Breakdown Steps due to an Instability Oj between QHE breakdown experiments and the breakdown of laminar flow in classical fluids. Acknowledgements I am grateful to Dr. M. E. Cage and NIST for permission to use Figs. 1 and 2, and to my colleague Professor F. W. Sheard and Dr. M. E. Cage of NIST for several helpful discussions. This work was partly supported by the Engineering and Physical Sciences Research Council (UK).

References 1. Stockmann, H.-J., "Quantum Chaos - An Introduction", (Cambridge University Press, 1999). 2. Chaos, Solitons, Fractals 8: Special Issue, "Chaos and Quantum Transport in Mesoscopic Cosmos", (edited by K. Nakamura and M. S. El Naschie), (1997). 3. Wilkinson. P. B. et at. Nature 380, 608 (1996). 4. Eaves. L. et at. Nobel Symposium 99 on Heterostructures in Semiconductors, Arild. Sweden, June 1996. Physica Scripta T68, 51 (1996). 5. Fromhold, T. M. et at. Chaos, Solitons. Fractals 8. 1381 (1997). 6. von Klitzing, K., Dorda, M. and Pepper, M., Phys. Rev. Lett. 45, 494 (1980). 7. For a recent text on hydrodynamics, see for example Faber, T. E., "Fluid Dynamics for Physicists", (Cambridge University Press, 1995). 8. Nachtwei, G., Physica E 4, 79 (1999). 9. Cage, M. E., J. Res. Natl. Inst. Stand. Technol. 98, 361 (1993). 10. Lavine, C. F., Cage, M. E. and Elmquist, R. E.. J. Res. Natl. Inst. Stand. Technol. 99. 757 (1994). U. Eaves. L. et ai., Proc. EP2DS-13, Ottawa, Canada, August 1999, Physica E 6, 136 (2000). 12. Bliek, L. et ai. Surf. Sci. 196, 156 (1988). 13. Komiyama, S., Takamasu, T., Hiyamrzu, S. and Sasa, S., Solid State Commun. 54, 479 (1985). 14. Stormer, H. L. et ai, Proc. of the 17th Int. Conf. on the Physics of Semiconductors (17th ICPS 1984), (edited by J. D. Chadhi, W. A. Harrison), (Springer, Berlin, 1985), p. 267. 15. Heinonen, O. et at., Phys. Rev. 30, 3016 (1984). 16. Eaves, L. et ai, J. Phys. C: Solid State Phys. 17, 6177 (1984). 17. Eaves. L. and Sheard, F. W., Semicond. Sci. Technol. 1, 346 (1986). 18. Balaban, N. Q., Meirav, U., Shtrikman, H. and Levinson, Y., Phys. Rev. Lett. 71, 1443 (1993). 19. Chaubet, C , Raymond, A. and Dur, D., Phys. Rev. B 52, 11178 (1995). 20. Liu, Z. H.. Nachtwei, G„ von Klitzing, K. and Eberl, K., Semicond. Sci. Technol. 14, 357 (1999). 21. Eaves, L., Physica B 256-258, 47 (1998).

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22. Eaves, L., Physica B 272, 130 (1999). 23. Avron, J. E„ Seiler, R. and Zograf, P. G., Phys. Rev. Lett. 75. 697 (1995). 24. Aoki. H., 4th Tsukuba Int. Workshop on Chaos/Turbulence, November 1996. 25. Ebert, G., von Klitzing, K., Ploog, K. and Weimann, G., J. Phys. C 16, 5441 (1983). 26. Kuchar, F., Bauer, G., Weimann, G. and Burkhard, H., Surf. Sci. 142, 196 (1984). 27. Ando, T„ Matsumoto, Y. and Uemura, Y„ J. Phys. Soc. Jpn. 39, 279 (1975). 28. Rigal, L. B. et ai, Phys. Rev. Lett. 82, 1249 (1999). 29. Kawaji, S. et at, J. Phys. Soc. Japan 63, 2303 (1994). 30. Komiyama, S., Kawaguchi, Y., Osada, T. and Shiraki, Y., Phys. Rev. Lett. 77, 558 (1996). 31. Komiyama, S. and Kawaguchi, Y.,Phys. Rev. B 61, 2014 (2000). 32. Riess, J., Proc. 12th Int. Conf. on High Magnetic Fields in the Physics of Semiconductors II, Wiirzburg, 1996, vol. 1, (eds. G. Landwehr and W. Ossau), (World Scientific 1997), p. 165. 33. Tsemekhman, V., Tsemekhman, K., Wexler, C , Han, J. H. and Thouless, D. J., Phys. Rev. B 55, R10201 (1997). 34. MacDonald, A. H., Rice, T. M. and Brinkman, W. F., Phys. Rev. 28, 3648 (1983). 35. Thouless, D. J., J. Phys. C 18, 6211 (1985). 36. Chklovskii, D. B., Shklovskii, B. I. andGlazman, L. I., Phys. Rev. B46. 4026 (1992). 37. Beenakker, C. W. J. and van Houten, H. in "Solid State Physics: Semiconductor Heterostructures and Nanostructures", (eds. H. Ehrenreich and D. Turnbull), (Academic Press. 1991). pp. 1-228. 38. Wexler, C. and Dorsey. A. T., Phys. Rev. B 60, 10971 (1999). 39. Kallin, C. and Halperin, B. I., Phys. Rev. B 30, 5655-5668 (1984). 40. Hawksworth, S. J. et at, Semicond. Sci. Technol. 8, 1465 (1993). 41. Cole, B. E. et ai, Phys. Rev. B 55, 2503 (1997). 42. Madelung, E., Z. Phys. 40, 322 (1926). 43. Eaves, L., "An Eddy Viscosity Model of the Dissipative Voltage Steps in Quantum Hall Effect Breakdown", Proc. 11th Int. Winterschool New Developments in Solid State Phys. "Low-Dimensional Systems: Fundamentals and Applications", February 2000, Mauterndorf, Austria. To be published in Physica E. 44. Muirhead, C. M., Vinen, W. F. and Donnelly, R. J. Phil. Trans. R. Soc. Lond. A 311, 433 (1984). This paper also describes the analogy between the dynamics of an electron moving in two dimensions in crossed electric and magnetic fields and that of a vortex in a superfluid He film. 45. Feynmann, R. P., "Application of quantum mechanics to liquid helium," in Progress in Low Temperature Physics I (ed. C. J. Gorter), (North-Holland, 1955), Chapter 2.

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Dynamical and Wave Chaos in the Bose-Einstein Condensate William P. Reinhardt* and Sarah B. McKinney** Departments of Chemistry and Physics, University of Washington, Seattle. WA 98195-1700, U.S.A. Received August 25, 2000

PACS Ref: 05.45.Mt, 05.30.Jb, 05.45.Pq, 05.30.Yv, 05.30.-a

Abstract Within the past five years Albert Einstein's concept of a dilute atomic Bose Condensate has been realized in many experimental laboratories. Temperatures in the nano-Kelvin regime have been achieved using magnetic and optical trapping of laser and evaporatively cooled atoms. At such temperatures the relative de Broglie wavelengths of the gaseous trapped atoms can become long compared to their mean spacing, and through a process of bosonic amplification a "quantum phase transition" takes place involving 103 to 10" atoms, most of which end up in identical single particle quantum states, whose length scales are determined by the external trap. Rb, Na, Li and atomic H have been trapped at variable densities of the order of 1013/cm\ and in traps of varying geometries. Of these all but Li have an effective repulsive atomic pair interaction, but utilization of molecular Feshbach resonances allows the interactions of other species to be tuned over wide ranges of strengths, including control of the sign of the effective atomic pair interactions. Fully quantum and macroscopic systems at such low densities are a theorist's dream: simple Hartree type mean field theory provides a startlingly accurate description of density profiles, low energy excitation frequencies, and such a description, commonly called the Non-linear Schrodinger Equation (NLSE) or the Gross-Pitaevskii (GP) Equation, will be explored here. The NLSE appropriate for attractive atomic interactions is known to lead to chaotically unstable dynamics, and eventual implosion of the condensate should the local number density exceed a critical value. In this work we illustrate this type of chaotic collapse for attractive condensates, and then explore the types of chaotic dynamics of solitons and vortices, which are the signatures of dynamical non-linearity in the repulsive case. Finally the implications of the symmetry-breaking associated with phase rigidity are explored in model simulations of repulsive condensates: condensates with repulsive atomic interactions break into phase domains when subject to weak shocks and, perhaps surprisingly, break into chaotic "laser-speckle" type patterns as the shock level increases. The fully quantum mechanical NLSE thus displays a full range of chaotic types of motion: from particle-like chaotic collisions of solitons and vortices to fully developed time dependent wave chaos.

1. Introduction to Bose-Einstein condensation In 1924 and 1925 the concept of particles with fully symmetric wavefunctions was introduced by Bose [1] and elaborated and extended by Einstein [2,3] who introduced the concepts of stimulated emission of photons (spin 1 bosons) resulting in the well known properties of the laser. Einstein also suggested [3] that if composite bosonic atomic systems were cooled, or otherwise increased in density, to the point where de Broglie wavelengths became long compared to mean atomic spacings it might be possible to create a matter wave condensation, provided there were some prohibition preventing such many body systems from becoming liquid or solids. In 1959 Hecht [4] suggested that gases might be Bose-Einstein condensed in magnetic fields, in what we nowadays refer to as a BEC. In 1976 Stawally and Nosarow [5], unaware of Hecht's earlier suggestion, noted that spin polarized H atoms cannot bind to form molecular hydrogen. There is, of course, a long range attractive van der Waals interaction between H atoms in the molecular triplet state, *e-mail: rein(ofchem.washington.edu **e-mail: [email protected] Physica Scripta T90

but the resulting well in the otherwise repulsive BornOppenheimer potential is very shallow, and far too shallow to bind the atoms. Thus while singlet, or spin paired HT binds very strongly, Stawally and Nosarow suggested that in a strong magnetic field, and in the absence of spin flipping interactions with the container, or via three body collisions, that it might be possible to cool H atoms to essentially 0°K with no chance for the formation of a liquid or solid. Success has been achieved for the H atom condensation only in 1998 [6], as atomic H is a recalcitrant system [7,8], In parallel efforts laser manipulation and cooling of alkali atoms proceeded very differently [9], and more rapidly. The high oscillator strength transitions in Li, Na, K, Cs, Rb are in the visible region of the spectrum where tunable dye lasers allowed development of optical cooling techniques [10]. Groups at JILA, MIT and NIST were working hard in the mid 1990s to Bose condense triplet alkali atoms, in spite of the fact that these species do have triplet molecular bound states. Estimates of three body recombination rates suggested [11] that in suitable magneto-optical traps (MOTs) cooling might be sufficient to allow condensation in any case. This was proved to be correct in 1995 by the JILA groups of Wieman and Cornell [12] which were able to Bose condense Rb, using optical cooling and MOTs combined with forced evaporative cooling. The Ketterle [13] group at MIT succeeded soon after, and introduced nondestructive methods for probing the condensate cloud and its dynamics. There are now BECs in more than 20 scientific laboratories in many countries [14], and many reviews have appeared [15-20]. Einstein's predicted atomic condensate is thus at hand: what does one expect to be the experimental consequences and novelty of these systems, and what role will theory have in describing experiment and leading experimenters into new areas? In particular: will these systems show aspects of quantum chaos? The experimental consequence of the existence of a gas phase quantum fluid are great indeed. In condensed matter physics [21], superfluid bosonic ^He has been known since the early 1900s; paired fermions allow quantum condensation of electrons (Cooper pairs) in superconductors, and more recently superfluidity has been found in ,He. Unlike these, nowadays, familiar condensed matter systems, the newly achieved gaseous Bose systems allow a range of control undreamed of in the condensed phase. Solids and liquids have densities which are not easily variable, the interactions between electrons in Cooper pairs, or between two iHe's cannot be readily tuned. In strong contrast, the gaseous BECs have now been created over a range of densities of about three orders of magnitude, and in traps of different geometries and topologies. Changing the density allows tuning of the quantum fluid "healing length", dis© Physica Scripta 2001

Dynamical and Wave Chaos in the Bose-Einstein cussed below, which sets the scale size of solitons and vortices. In the condensed phase these are often too small for direct observation [21]: in the gaseous BEC solitons [22,23] and vortices [24-29] have now been directly imaged optically, at appropriately low densities. Optical pumping allows manipulation of the nuclear spins, while leaving the outer electrons spin polarized in the trapping fields: this leads to mixtures of atoms with distinguishable components which may either mix, or phase separate [30,31], The number of superfluid components in the condensate is thus an experimental variable. In 87 Rb and 85 Rb a nearby Feshbach resonance [32-34], originally pointed out by Heinzen [32], allows control of the effective pair interaction over many orders of magnitude, including sign reversal. Rb atoms with a zero field effective "repulsive" interaction, may have that interaction suddenly or adiabatically tuned. Tuning to an attractive value can result in complete collapse of the condensate, as predicted theoretically early on [35,36], and recently observed in the laboratory for a 85 Rb condensate [37]. 7Li condensates have a naturally attractive effective interaction, and implode ([38,39] and references therein) on reaching a critical density [40]. Such control of density, numbers of components, and the sign and strength of the interactions themselves in the gaseous BEC are without parallel in the condensed phase. Finally, it must be mentioned that simulation of the condensate has turned out be be a playground for computational physicists: a very large fraction of what has been seen experimentally has been found to follow the predictions (and postdictions) of a very simple form of mean field theory. This is, again, a great difference between the gaseous BEC and condensed matter systems where a simple and easily implemented computational picture does not exist. Theory has, of course, also played an enormously important qualitative role in the discovery of the condensate: Bose and Einstein were theorists, themselves. The simple mean field theory referred to above is due to Gross [41] and Pitaevskii [42], who in the early 1960s implemented the ideas of Hartree combined with use of the Fermi contact pseudopotential [43,44] to describe the effective pair interaction. In Section 2 we turn to a description of mean field theory, which is far more appropriate as a qualitative description of the dilute gaseous condensate than for the condensed phase superfluids which motivated its original development. The organization of the rest of the paper is then as follows: the chaotic nature of the collapse of a "noisy" attractive condensate is explored numerically in Section 3 as atoms are slowly added to a trap. In the remainder of the paper the focus is on condensates with effective repulsive atomic pair interactions and which dominate the experimental literature. Phase imprinting, or phase engineering, which relies on the appropriateness of a single particle description of the condensate, is introduced in Section 4 and shown to allow production of solitary waves in box and harmonic traps. The key role of the condensate phase in determining the nature of the solitons is introduced and illustrated [45,46]. Such solitons are stable in "quasi-one-dimension" which is the limit where transverse confinement is the order of the healing length [47]. However, as seen in Section 5, outside of this quite interesting limit the solitons become unstable [47,48], and in the presence of noise decay into vortex dipoles, which © Physica Scripta 2001

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may subsequently collide. Such collisions take place in a manner highly suggestive of the collisions of classical particles with hard core interactions, namely chaotically. We refer to this type of chaos as dynamical as it results from the dynamics of the collisions of the emergent particle like excitations of the BEC. In Sections 6, 7, and 8 a different and less familiar type of chaos is seen to follow from the phase rigidity of the condensate. This covers a range of behaviors from the fracturing of the condensate into domains of near constant phase (Sections 6 and 7) when weakly "shocked" to the emergence of a type of fully developed wave chaos reminiscent of laser speckle as such shocks become stronger, as illustrated in Section 7. We refer to this as Bose-Einstein speckle (BES) and compare (Section 8) its statistical properties with those of simulated laser speckle. Section 9 contains a brief summary and conclusions. 2. The non-linear Schrodingcr equation Perhaps surprisingly the macroscopic structure of the BEC is determined by simple one and two body aspects which actually dominate the current physical and theoretical descriptions. The highly dilute BECs now available for study may be described by a single Hartree or mean field "one body" wave function. This immediately gives factorization of the first order reduced density matrix y(r, r1) = <^*(r)^(/), giving rise to a text book example [21,49] of the off-diagonal-long-range-order, or ODLRO, often thought of as a sine qua non for description of a substance as a superfluid. The mean field picture has been found to describe densities and energetics of ground states and low lying collective excitations [17,50-52] of gaseous BECs, as well as solitons [45,46] and vortices [28,29] now experimentally observed, and confirming the validity of the mean-field description for even such high energy excitations [22,23], A brief and very elementary description of this theory follows [41,42]. In a dilute T = 0 Bose system the N-body wave function can be written as a symmetric product of one body functions,
(

h2

N

C

\

= t"P(ri).

(1)

Further, for the low relative momenta involved in atomic pair interactions, invoking the Fermi "contact" pseudopotential approximation [43,53] V(rt, r,-) ~ {Awxtr/m) 5{rt — rj), gives the Non-Linear Schrodinger (NLSE hereafter), or Gross-Piteavskii, Equation, where we assume that N~N-l:

h2

- ^ V,2 + V^'in)

A

+ {Anoih2/m)N\(p(rl)\- \
win) (2)

for the single particle ground state wave function (p(r = r,), fi Physica Scripta T90

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William P. Reinhardt and Sarah B. McKinney

are known to be unstable in two and three dimensions [35-37,40]. An initial Gaussian condensate will collapse to a ^-function under the dynamics of the NLSE which, at high enough density, will no longer give a complete description of the BEC. We consider a somewhat different physical situation: adiabatic growth in particle number for a condensate in the presence of noise, this latter modeling interactions of the growing condensate with non-condensed tr \ atoms. The initial state was taken as the usual particle in a box ground state for a single particle, and then the magnitude of the nonlinear term adiabatically increased = indt(p(r, i). from 0 as a continuous function of time: — (t2/\0)\(r) as the particles being added to the condensate in a stochastic Landau complex order parameter for the condensate [21] manner. Actual integration of Eq. (4) between the major and the resulting equations are then said to be pheno- time intervals was carried out by a variable step 4th order menological; or, as a coherent state expectation value of Runge-Kutta integrator, using a pseudospectral co-ordinate a Heisenberg boson field operator [54], q>{r, t) = [\p(r, t))\ basis appropriate to the box boundary conditions. The or a Dyson amplitude [55], (N - 1, 0\\j/(r, t)\N, 0); or even pseudospectral basis combined with use of fast fourier transa natural orbital [56]. While one can certainly join the argu- forms gives a very efficient algorithm for time evolution of ment as to the proper physical interpretation of ) as by time "1.6" a highly and healing length [21] obtained by dividing Eqs. (2) or (3) non-uniform collapse of the condensate has begun, 10 the local density is seen to vary over a range of e , with through by the coefficient of the non-linearity. The remaining + and — distinguish repulsive and attractive con- localized peaks approaching singularity. At times larger than 2.0 the system has collapsed to essentially a random densates. This gives, pattern of ^-functions. The precise nature of the dispersed and collapsing condensate is highly sensitive to initially small inhomogeneities, and displays the same type of (or = /J.(p(rj)) in the time independent case, where it is under- "chaotic" and random coagulation of matter as seen in stood that the trap potential energy, chemical potential and models of the early universe [58]. being the chemical potential. Here, a, the scattering length, which parameterizes the effective pair interaction, is chosen to reproduce the exact s-wave pair scattering amplitude as relative wave number approaches zero [43]. Collective single particle excitations may be described by the natural time dependent extension of Eq. (2), dropping the subscript,

time-scales are suitably re-scaled. In one dimension, and in the absence of a potential, the stationary states of Eq. (3) have been elucidated for "box" and "periodic" boundary conditions, in terms of Jacobi Elliptic functions [57], and their stability properties studied [47,48]. In the presence of an actual trapping potential, and/or in higher dimensionality analytic solutions are not usually available, and one resorts to study of the dynamics of condensates via numerical solutions of Eq. (4) which, along with its multicomponent generalizations [31], has been found to predict and to account for a surprisingly large range of static and dynamical behavior of the gaseous condensate experiments near T = 0.

3. Chaotic collapse of a growing attractive condensate To illustrate the use of Eq. (4), and give a first example of chaotic behavior in the BEC we consider growth of a condensate with a negative scattering length ( - 1 in our units) starting with a single particle trapped in a two dimensional square box of side 20£ x 20^, where £ corresponds to the healing length for the final particle number corresponding to a coupling constant of unity. Such attractive condensates Physica Scripta T90

We now switch attention to the chaotic dynamics of condensates with repulsive atomic pair interactions, namely those with a positive values of the scattering length, a. 4. Phase imprinting as a control of condensate dynamics Superconductors are often used to measure very small magnetic fields using SQUIDS [21] which make use of the Josephson effect [21,54,59]. In the Josephson effect the phase difference between two bulk superconductors controls the direction and magnitude of a supercurrent confined to a tunneling junction [21] connecting the bulk quantum fluids. Reinhardt and Clark [45] pointed out that solitons, originally predicted for both attractive and repulsive solutions of the NLSE by Zakharov and Shabat [60,61], moved as density notches in repulsive condensates with a notch velocity dependent on the phase offset between the adjoining and weakly connected parts of the condensate [45,62], in direct analogy to the Josephson effect. Thus the condensate phase profile controls the soliton velocity. This immediately suggested [63] that solitons might be created if an initially phase uniform condensate were phase imprinted, or "phase engineered", if such quantum phase control could be applied © Physica Scripta 2001

Dynamical and Wave Chaos in the Bose-Einstein Phase Evolution

Density Evolution

t = 0.0

t = 0J

t = 1.6

Max Fig. I. Adiabatic addition of atoms with naturally attractive interactions leads to chaotic collapse of the growing condensate cloud. Here a single particle in a two-dimensional box seeds a growing condensate described by the time dependent NLSE with stochastic white noise added each 0.1 natural time unit. By / = 1.6 time units, the phase and log of the density show a chaotic collapse of the cloud. By / = 2 using the range of densities shown, all that would be seen would be a scattering of "red" pixels, indicating complete collapse into multiple singular points in the density. The full physics is then no longer descried by the NLSE [37].

on a time-scale short compared to that for the condensate density to respond. Phase imprinting is very easily accomplished in simulations, and with rather more difficulty in the experimental lab! Figure 2 shows a sequence of time slices of the time evolution of both phase and density of

Phase Evolution

Condensate

205

a two dimensional condensate time evolved via Eq. (4). The initial state was taken to be the ground state of a two-dimensional condensate in a box of lengths 6c x 30<J, but with an artificially imposed tanh-like phase profile resulting in a phase jump of n/2 over a length of « 2£, with the jump occurring at the mid point of the long axis of the box. The subsequent dynamics show that a deep density notch propagates to the left, and that the initial phase jump profile is an approximate constant of the motion, as would be expected for solitons in the BEC [45]. The scale size of the soliton is «* 2q. Clearly seen are also supernumerary solitons and transient density oscillations of lesser depth resulting from the initial shock imposed by the suddenly imprinted phase profile in a region where the initial density was non-zero. Cleaner soliton creation would result from imprinting a phase offset across a region of zero or low density [45]. Phase imprinting has been realized in the lab: in a tour de force the Phillips group [22] at NIST was able to coherently separate an initially coherent condensate into two approximately equivalent parts, imprint a phase gradient on one of the pieces, and following coherent recombination, verify that the expected phase engineered profile had been achieved. The phase offset was imposed by simply "illuminating" half of the condensate in near resonant laser light. The resulting a.c. Stark shift gives an accumulating phase offset between the dark and illuminated parts of the condensate via their usual Schrodinger phase: exp(—\E(x,y, z)t/h), E(x,y,z) being spatially non-uniform following illumination of only " h a l f of the condensate. The NIST group [22] and in a similar experiment, the Hannover group [23], submitted papers on 19 and 20 October 1999, respectively, showing that such phase imprinting could indeed create solitons of the type discussed by theoretically by Zakharov and Shabat [60,61] and by Reinhardt and Clark in the context of solitons being driven at velocities determined by phase offsets in the BEC [45,46,63]. In addition to creation of primary solitons, the Phillips experiments also clearly show transients and secondary solitons, such as those seen in Fig. 2.

5. Dynamical chaos in the BEC: soliton instabilities The NLSE is manifestly "non-linear" and one might expect that its particle like excitations, which obey classical like

Density Evolution

Fig. 2. Dynamics at a series of times following sudden imposition of a non-uniform tanh-like phase profile on the ground state of a repulsive condensate in a two dimensional box. The phase jumps smoothly from 0 to n/2 over a length of about 2i in a box of length 30c;. The imposed phase jump quickly results in the formation of a dark, or density notch, soliton moving to the left. As the phase gradient was imposed in a region of non-zero density, other, less deep solitons as well as transients arc produced. © Physica Scripla 2001

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William P. Reinhurdt and Sarah B. McKinney

W

Phase Evolution

Density Evolution

Phase Color Code

Density Color Code 0

2JI

(b)

Phase Evolution

Max

Density Evolution

i

2JC

0

Fig. 3. Evolution of a condensate with two planar nodes in a box with longitudinal period boundary conditions. A phase offset of n/6 between the nodes and 0.1% white noise have been added at t = 0. The former states the nodes moving towards one another at a fixed velocity (45], the later causes them to begin to decay into vortices. However, before vortex formation is complete the two solitons collide, in a manner that shows an analog of chaos as it might arise in the collision of hard spheres: small changes in initial conditions (in this case simply different realizations of the noise, in (a) and (b)) yield quite different phase and density profiles at t = 48.

equations of motion in the semi-classical limit [45,64], might show chaotic behavior under appropriate circumstances. In two or three dimensions planar solitons become unstable through the well known snake instability [62] if their transverse length exceeds the healing length by more that about a factor of In [47]. Figure 3 illustrates this type of instability, wherein initially planar solitons, while beginning to decay into vortices, subsequently collide. In this simulation a condensate in a box of length 40c; and width 12c. with longitudinal periodic boundary conditions and transverse box boundary conditions, was initially chosen to be in a two node excited state of this two-dimensional "ring" system, but with an additional phase offset of 7t/6 from the value of n which would give rise to a stationary state of the system [47,57]. This phase offset causes the planar solitons to approach one another [45]. An instability was induced by addition of 0.1% white noise in the fourier transform of the initial wave function. The planar solitons then begin to decay into vortices [47] which continue to translate at the original phase induced velocity of the solitons. A complex and chaotic collision then ensues. The difference between the temporal evolution in Figs. 3(a) and (b) is simply a different realization of the same level of the white noise. An analog of one of the usual signatures of classical chaos is present: extreme sensitivity to initial conditions leads to quite distinct final outcomes from initially visually indistinguishable initial conditions. We refer to this as dynamical chaos as it results from the sensitivity of actual dynamical objects (solitons [45] or vortices [28,29]) behaving Physica Scripla T90

like particles which are an emergent property of solutions of the NLSE. The non-trivial, and to some unexpected, fact that the time dependent NLSE solutions of this type can actually correspond to experimental reality is confirmed in great detail in the NIST phase imprinting experiments [22]. Chaos of this same type, but at a far higher level of complexity, has been studied, also using the NLSE, by Nore, Abid, and Brachet in studies of turbulence in superflows [65]. 6. Symmetry breaking and phase rigidity In the traditional theory of superfluidity and superconductivity [21,54] phase gradients generate supercurrents, and thus a superfluid ground state must have constant phase. Leggett [66], Anderson [54], and others [21] have pointed out that this constant phase is not predetermined by any tendency for it to assume a particular value, and in fact uniformly shifting the overall phase of a superfluid condensate would have no physical implication. What then sets the phase at time "0"? The answer is undoubtedly the fine details of the preparation of the condensate, but none-theless the condensate ends up with a definite phase at a specific time. As all phases 0, 0 < < 2%, are equivalent the fact that a single value of the phase is actually present represents a "gauge symmetry breaking" of the system [66]. Anderson [54] has argued that one of the physical consequences of this symmetry breaking, and the constancy of the phase in the ground state, implies a "phase rigidity", in analogy with the rigidity of a crystalline solid, where the broken symmetry © Physica Scripla 2001

Dynamical and Wave Chaos in the Bose-Einstein Condensate Phase Evolution

207

Density Evolution t=o t=2 1=4 t=6

|t=8 I t= 10 11= 12

Density Color Code Max

o

Fig. 4. Sudden stopping of a repulsive condensate in the pseudo-one- dimensional two-dimensional box trap. At f = 0 the density corresponds to the ground state in the trap, while the phase is that of a Galilean boost, namely that corresponding to a uniform translation of the box. which is then suddenly stopped at I = 0. The subsequent dynamics illustrate the idea of the phase rigidity of the condensate, which by ( = 6 has broken into droplets of nearly constant phase. As these expand and translate, new phase gradients appear.

is "translational". The breaking of translational symmetry, as might occur in a crystal, implies a physical rigidity, consistent with the physical observation that a crystal will shatter if shocked. By analogy, Anderson has argued [49] that a shocked quantum fluid (or a broken symmetry liquid crystal) should shatter into domains of constant phase. Following such a fracturing of a repulsive BEC the domain walls, being density notches, will begin to propagate with velocities determined by the phase offsets of the adjoining pieces of uniform condensate [45]. It is remarkably simple to demonstrate, at the level of the NLSE description of the condensate, that these ideas are quite correct: Figure 4 shows a condensate (trapped in the pseudo-one-dimensional limit, where the transverse confinement is the order of c,) initially in uniform translational motion (note the initially constant phase gradient) suddenly, at time t = 0, finds itself trapped in a stationary box of length 40c;. The moving condensate shatters into droplets of near constant phase, nicely illustrating the reality of phase rigidity.

x

Phase

Intensity

Phase Color Code

Intensity Color Code 2;t

0

Max

Fig. 5. Phase and intensity of simulated laser speckle. This is a slice of a superposition in three dimensions of 10.000 plane waves with equal wavelengths, random directions and phases, and gaussian random amplitudes. The length of the box is 20 wavelengths. The red has been stretched on this and all subsequent intensity/density color scales to bring out the details of the structure.

7. Wave chaos: production of Bose-Einstein Speckle (BES) Laser speckle is a chaotic intensity pattern produced by coherent light reflecting from a rough surface or propagating through a medium with random refractive index fluctuations [67]. One can reproduce a speckle pattern numerically by modeling it as a superposition in three dimensions of a large number of coherent monochromatic plane waves with random directions and phases. Figure 5 shows the phase and intensity patterns produced by taking a slice through such a three-dimensional superposition. Speckle should be distinguished from the quasi-linear ridge structures known as "scars", which result from the superposition of monochromatic plane waves in two dimensions, as discovered by O'Connor, Gehlen, and Heller [68], and reproduced in Fig. 6. The reason for the difference is that what we see as speckle is actually a projection of waves in three dimensions onto a two-dimensional screen (or retina), so that although the light is monochromatic, there is a spread in the magnitudes of the projected wave vectors. A comparison of Figs. 8(a) and (b), which show the k-space intensity for scars and speckle, respectively, reveals just such a spread for the speckle pattern, while the ridge pattern consists of only a single magnitude for the wave vector. A superposition © Physica Scripta 2001

x

Phase

Intensity

Phase Color Code

Intensity Color Code 2K

0

Max

Fig. 6. Phase and intensity of simulated scars. This is a superposition in two dimensions of 10.000 plane waves with equal wavelengths, random directions and phases, and gaussian random amplitudes. The length of the box is 20 wavelengths. Physica Scripta T90

208

William P. Reinhardt and Sarah B. McKinney

of non-monochromatic waves in two dimensions produces a pattern identical to laser speckle. To investigate phase rigidity beyond the pseudo-onedimensional results of Section 6, we produced the ground state of a two-dimensional condensate in a square box via complex time evolution. We then shocked the condensate by imprinting a phase ramp to give it an initial velocity, oriented such that the condensate would collide with the rigid walls at an oblique angle. We varied the strength of the shock by varying the initial velocity from approximately 1 to 20 times the Bogoliubov speed of sound [69], cs = J Antra pi m2. We found that the condensate breaks up into domains of nearly constant phase, and the density pattern exhibits scattered small regions of high density, as shown in Fig. 7. Comparison of Fig. 7 with Fig. 5 reveals that the appearance of the condensate after it has settled is strongly reminiscent of laser speckle. We thus refer to it as Bose-Einstein speckle, or BES. The characteristic scale

Phase Evolution

Density Evolution

of the pattern, which in laser speckle is determined by the wavelength of the light, is here determined by the strength of the shock. Even though our BEC is two-dimensional, we do not see a ridge pattern. The reason for this becomes clear from looking at the density of the fourier transform of the wave function (Fig. 8(d)). Although the condensate starts out with a single uniform momentum, the non-linear interactions quickly cause dispersion in the momentum, producing an effect similar to that of projecting a three-dimensional superposition of waves onto a two-dimensional screen. The k-space density of our shocked BEC does not exhibit the same azimuthal symmetry as speckle, but remains in four clumps centered around the initial momentum and its reflections. The asymmetry is more pronounced for stronger shocks, and is most likely because the square enclosure leads to separable wave dynamics. Presumably with an irregularly shaped box the wave vectors would be distributed more evenly, in analogy with particle dynamics in a twodimensional chaotic billiard, and the k-space intensity of the shocked BEC would look even more like that of laser speckle. The size of the box, on the other hand, does not affect the results. Although making the box too small will produce edge effects, we saw no qualitative changes in the results when we varied our box length from 30c to 10c.

(a) Scars

x

Phase Color Code 0

Density Color Code 2jt

0

Max

Fig. 7. Phase and density evolution of a two-dimensional repulsive condensate shocked at an oblique angle in a square box with rigid walls. The length of the box is 10c, and the initial speed is 16«J/r, where t is the natural time unit, or approximately 14c,,, where cs is the Bogoliubov speed of sound. As soon as the condensate hits the wall, it breaks up into small regions of high density and domains of nearly constant phase, which slosh back and forth before settling into a pattern which is strongly reminiscent of laser speckle. The color scale has been normalized to the maximum density in each frame, which is a factor of ten larger at times later than at / = 0. Physica Scripla T90

(b) Speckle

Fig. 8. A:-space intensity/density of scars, speckle, and shocked condensates at / = 60. The color scale is the same as that used in Figs. (5-7).Thc units of k are inverse wavelengths for Figs, a and b, and inverse healing lengths for Figs, c and d. Scarred patterns (a) contain only one value of \k\, while speckle (b) and shocked condensates (c and d) contain a distribution of values, due to projection onto a two-dimensional screen and non-linearity, respectively. The clumping of intensity around four peaks seen in (d) is an artifact of the square enclosure. BECs exhibit k-space relaxation for weak shocks (c), but not for strong shocks (d). © Physica Scripla 2001

Dynamical and Wave Chaos in the Bose-Einstein Condensate

energy, since the total NLSE energy is conserved. As the strength of the shock increases, so does the time required for relaxation, until, for an initial speed of 10cs, there is no relaxation for time scales on the order of 100 natural time units. This is illustrated in Fig. 8(c), which shows the k-space density for a weak shock (v, = 4cs) after it has relaxed, and in Fig. 9, which shows the evolution of the &-space density versus \k\ for weak and strong shocks.

The size of the phase domains and regions of high density is inversely proportional to the healing length, and for phase domains smaller than | , it is independent of q, indicating that the kinetic energy is dominating the dynamics, and interactions are important only in causing the k-space dispersion discussed above. One might expect that the repulsive interactions would prevent structure from forming at a scale smaller than the healing length, but this does not appear to be the case. We were able to obtain regions of high density with a diameter on the order of cj/4, with only the number of grid points in the simulations imposing a limit. In a real BEC, three-body recombination would presumably occur much earlier and the condensate would undergo forced collapse and rebound, in analogy with the attractive case [37]. The results thus point to a situation in which the NLSE should completely break down as a description of the BEC. For initial speeds smaller than about 10cs, or phase domains larger than Q/2, there is a kind of relaxation in k-space, that is, the peak of the momentum distribution gradually shifts towards zero, and the phase domains and regions of high density dissolve. This appears to be a result of kinetic energy being converted into repulsive potential

t=o

209

8. Statistical properties of BES: first investigations Both scar and speckle patterns are produced by Gaussian random fields [68,67], which makes their statistical analysis particularly simple. All gaussian random fields obey the same first order statistics, and all higher order statistics can be expressed in terms of products of the first and second order moments [70]. The intensity distribution given of these patterns is given by [67] (5)

/>(/)= ^ e x p ( - ^

t = 50

t = 100

?

'. \

(a)

-2

\ %

-6

10 k

20

2

\ (b)

\\

-2

\

6

10 k

(c)

__ 2

(d)

^ >. "o5

/

aen o

-J

2 - 2 «*••

\

/\ /

X

\

••v

\

^

S-

^s.

-6

10 k

"\ 20

-6

^ 10 k

20

Fig. 9. Logio of &-space density vs. \k\. The units of k are inverse healing lengths, (a) For v, = 6cs the relaxation is complete by t = 5Q (b) For v, = 9cs, the relaxation is complete by / = 10Q (c) However, for a slightly stronger shock, v, = 12cs, the relaxation is just beginning at / = 100. (d) For a very strong shock, v, = 20cs, there is clearly dispersion, but no relaxation, with the peak remaining at the same value of \k\. © Physica Scripta 2001

Physica Scripta T90

210

William P. Reinhardt and Sarah B. McKinney \(a) Scars

\-

\ (a) Speckle

\

5.

5-3

'•"-. .".-. 0 Intensity

(c) BEC with v, = 4c

2

4 Intensity

\ ( d ) BEC with Vj = 11c

5-3

Density

Density

Fig. 10. Logio of intensity/density distribution for scars, speckle, and shocked condensates at t = 60. Scars (a), speckle (b), and strongly shocked condensates (d) are produced by gaussian random fields and thus have exponential intensity distribution functions. As the strength of the shock is decreased, the condensate distribution diverges from exponential (c).

Fig. 11. Autocorrelation function versus separation distance <5 for scars, speckle, and shocked condensates at t = 60. The units of delta are wavelengths for Figs, a and b. and healing lengths for Figs, c and d. Scars have structure on a longer length scale than speckle. The autocorrelation functions of shocked BECs exhibit less structure, but that of the strongly shocked BEC is similar to that of speckle.

dissipating through collision with the shallow end of the harmonic trap, and converting to vortex trains and phonons, 2i For strongly shocked condensates, the density distribution is has thus displayed a remarkable ability to describe coherent indistinguishable from an exponential, with the standard modes of the BEC rather far from the ground state. The deviation given by Eq. (6). Although the density patterns essence of this success, in regimes where the ground state condensate is surely "depleted" rests on the observation of the weakly shocked condensates do resemble speckle that, although it is not the traditional ground state which patterns at a glance, their distributions differ slightly from has macroscopic occupancy, there is nonetheless a single exponentials, but approach exponentials as the strength particle state with such full occupancy. Said another way, of the shock increases. The density (intensity) distributions ODLRO is preserved but generalized to give rise to a time are shown in Fig. 10. dependent factorization of the 1-matrix in terms of the The differences between scars, speckle, and our shocked solitonic (or other time dependent) solutions of the NLSE. condensates are more apparent in the second order Thought about in this manner there is still a macroscopically statistics, namely the autocorrelation function, C(d) = occupied single quantum state, and it shares many of the / i//(r)il/(r + <5)dr, which is equal to the inverse fourier trans- stability properties of the ground state: namely single parform of the fc-space intensity [70]. Figure 11 shows that ticle excitations into low lying states always raise the local scarred patterns, whose autocorrelation function is energy. Further, as seen in Section 5, such non-linear quanproportional to Jo(kS) [68], exhibit correlation on much tum excitations may well behave chaotically. longer length scales than speckle patterns. The autoAs now seen in Sections 6, 7 and 8 of the present paper the correlation function of the shocked condensate is similar time dependent NLSE also confirms P. W. Anderson's to that of speckle, but exhibits less structure. It will be expectation of condensate phase rigidity. Namely the dilute interesting to see how this depends on the shape of the gaseous BEC is a rigid object, via the gauge symmetry enclosure. breaking of the condensate phase, and shatters if shocked at velocities greater than the sound speed. Here, it must be immediately acknowledged, the arguments of the 9. Summary and conclusions preceding paragraph may well fail. It will, thus, be most The time-dependent non-linear Schrodinger equation pre- interesting to determine the range of validity of the NLSE, and whether the speckle-like wave chaos it predicts will dicts a wide range of properties of the BEC from collapse resemble the actual response of a BEC to shock, this latter of attractive condensates, low frequency phonon spectra, yet to be observed in the laboratory. to the existence and persistence of solitons and vortices. All of these are now confirmed experimentally, and found to be remarkably well described by the time-dependent NLSE. This simple NLSE quantitatively describes the Acknowledgements solitons observed by the Phillips group [22], at least over The authors gratefully acknowledge highly stimulating interactions with their their experimental lifetime, this being limited by the soliton theoretical collaborators Charles Clark, Lincoln Carr, Mary Ann Leung,

where the standard deviation is related to the intensity as

Physica Scripta T90

© Physica Scripta 2001

Dynamical and Wave Chaos in the Bose-Einstein and especially Joachim Brand, the support of the U.S. National Science Foundation. We have benefited greatly from discussions with Jerry Seidler about the properties of laser speckle. We also wish to acknowledge the hospitality of the Clark group at NIST, where a final version of the manuscript was prepared, and especially the comments of David Feder and Keith Burnett, who called Ref. [65] to our attention. The organizers of the Nobel Symposium on Quantum Chaos arc to be congratulated on setting an excellent program and for their gracious hospitality.

References 1. Bose, S. N., Z. Phys. 26, 178 (1924). 2. Einstein, A., Sitzungsberichte der Preussischen Akademie der Wissenschaften 1924, 261 (1924). 3. Einstein, A., Sitzungsberichte der Preussischen Akademie der Wissenschaften 1925, 3 (1925). 4. Hecht, C. E., Physica 25, 1159 (1959). 5. Stwalley. W. C. and Nosarow, L. H., Phys. Rev. Lett. 36, 910 (1976). 6. Fried, D. G. et al. Phys. Rev. Lett. 81. 3811 (1998). 7. Bagnato, V., Prichard, D. E. and Kleppner, D., Phys. Rev. A 35, 4354 (1987). 8. Greytak, T. J. and Kleppner, D.. in "New Trends in Atomic Physics, Proc. Les Houches Summer School. Session XXXVIII, Les Houches, France, 1993", (Edited by G. Greenberg and R. Stora), (North Holland. Amsterdam, 1994). 9. "Bose-Einstein Condensation", (Edited by A. Griffin, D. W. Snoke and S. Stringari), (Cambridge University Press, New York, 1995). 10. "Laser Manipulation of Atoms and Ions", (Edited by E. Arimondo, W. D. Phillips and F. Strumia), (North-Holland, Amsterdam, 1992). 11. Roberts, J. L., Claussen, N. R., Cornish, S. L. and Wieman, C. E.. Phys. Rev. Lett. 85, 728 (2000), and references therein. 12. Anderson, M. H. et al.. Science 269, 198 (1995). 13. Andrews. M. R. et al.. Science 273, 84 (1996). 14. An up to date BEC website is maintained at Georgia Southern University: http://amo.phy.gasou.edu/bec.html. 15. Clark, C. W. and Edwards, M., Eds. "Special Issue of: Journal of Research of the National Inst. For Standards and Technology 101", 4, 1996. 16. "Bose-Einstein Condensation in Atomic Gases", (Edited by M. Inguscia, W. Stringari and C. Wieman), (IOS Press, Amsterdam 1999). 17. Dalfovo, F., Giorgini, S.. Pitaevskii, L. P. and Stringari. S., Rev. Mod. Phys. 71. 463 (1999). 18. Ketterle, W., Physics Today 52, 30 (1999). 19. Burnett, K., Clark. C. W. and Edwards, M„ Physics Today 52, 37 (1999). 20. Special Issue on "Coherent Matter Waves", J. Phys. B: At. Mol. Opt. Phys., September (2000). 21. Tilley, D. R. and Tilley. J.. "Superfluidity and Superconductivity", 3rd ed. (IOP, Bristol 1990), an overview of "traditional" quantum fluids. 22. Denschlag, J. et al.. Science 287, 97 (2000). 23. Burger, S. et al., Phys. Rev. Lett. 83, 5198 (1999). 24. Matthews, M. R. et al, Phys. Rev. Lett. 83, 2498 (1999). 25. Williams, J. E. and Holland, M. J., Nature 401, 568 (1999). 26. Raman. C. et al, Phys. Rev. Lett. 83. 2502 (1999). 27. Madison. K. W., Chevy, F„ WohJleben. W. and Dalibard, J., Phys. Rev. Lett. 84, 806 (2000). 28. Feder, D. L., Clark, C. W. and Schneider, B. I., Phys. Rev. Lett. 82. 4956 (1999). 29. Svidzinsky. A. A. and Fetter, A. L., Phys. Rev. A 58, 3168 (1998). 30. Myatt, C. J. et at, Phys. Rev. Lett. 78. 586 (1997). 31. Pu, H. and Bigelow, N. P., Phys. Rev. Lett. 80, 1130 (1998).

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32. Courteille, Ph. et al., Phys. Rev. Lett. 81, 69 (1998). 33. Stenger, J. et at, Phys. Rev. Lett. 82, 2422 (1999). 34. Roberts, J. L., Claussen, N. R., Cornish. S. L. and Wieman. C. E., Phys. Rev. Lett. 85, 728 (2000). 35. Kosmatov, N. E., Shvets, V. F. and Zakharov. V. E., Physica D 52, 16 (1991). 36. Tsurumi, T., Morise, H. and Wadati, M., Int. J. Mod. Phys B 14, 655 (2000). 37. Cornish. S. L. et al., Phys. Rev. Lett, (submitted), cond-mat/ 0004290. 38. Bradley, C. C , Sackett, C. A. and Hulet. R. G., Phys. Rev. A 55, 3951 (1997). 39. Bradley, C. C , Sackett, C. A. and Hulet, R. G., Phys. Rev. Lett. 78, 985 (1997). 40. Bohn, J. L., Esry, B. D. and Greene, C. H„ Phys. Rev. A 58, 584 (1998). 41. Gross, E. P., Nuovo Cimento 20. 454 (1961). 42. Pitaevskii. L. P., Sov. Phys. JETP 13, 451 (1961). 43. See, for example, the discussions of Burnett et al, in Physics Today. [19]. 44. Huang, K. and Yang. C. N„ Phys. Rev. 105. 767 (1957). and K. Huang, in Griffin [9]. 45. Reinhardt. W. P. and Clark, C. W., J. Phys. B: At. Mol. Opt. Phys. 30. L785 (1997). 46. Reinhardt, W. P., in "Tunneling in Complex Systems". (Edited by S. Tomsovich), (World Scientific, Singapore, 1998), p. 277. 47. Carr, L„ Leung, M. A. and Reinhardt, W. P., J. Phys. B: At. Mol. Opt. Phys., special issue on coherent matter waves September (2000). 48. Carr, L., Kutz, N. and Reinhardt, W. P., Phys. Rev. E (submitted July 2000). 49. Anderson, P. W., "Basic Concepts of Condensed Matter Physics", (Addison-Wesley, Reading 1984). 50. Edwards, M. et at, Phys. Rev. A 53, R1950 (1996). 51. Edwards, M. et at, Phys. Rev. Lett. 77, 1671 (1996). 52. Stringari, S., Phys. Rev. Lett. 77. 2360 (1996). 53. Lee, T. D. and Yang. C. N„ Phys. Rev. 105, 1119 (1957). 54. Anderson, P. W., Rev. Mod. Phys. 38, 298 (1966), reprinted in [49]. 55. Lifshitz, E. M. and Piteavskii, L. P., "Statistical Physics, part 2", Vol. 5 of "Landau/Lifshitz Course in Theoretical Physics", 3rded. (Pergamon Press, New York, 1980). 56. Lowdin, P.-O., Phys. Rev. 97, 1474 (1955). 57. Carr, L., Clark, C. W. and Reinhardt, W. P., Phys. Rev. A 62, xxx and yyy (2000), cond-mat/9911178 & 9911177. 58. Hu, W„ Barkana, R. and Gruzinov, A., Phys. Rev. Lett. 85, 1158 (2000). 59. Josephson. B. D., Phys. Lett. 1, 251 (1962). 60. Zakharov, V. E. and Shabat, A. B.. Sov. Phys. JETP 34, 62 (1972). 61. Zakharov, V. E. and Shabat, A. B., Sov. Phys. JETP 37, 823 (1973). 62. Kivshar, Y. and Luther-Davies, B., Phys. Reports 298, 81 (1998). 63. Reinhardt. W. P. and Clark, C. W„ Bull. APS. Abstract QB16.10. March 1999 (Centennial) Meeting of the APS. Atlanta, GA. 64. Kosevitch, A. M., Physica D 41, 253 (1990). 65. Nore, C , Abid, M. and Brachet, M. E., Phys. Fluids 9, 2644 (1997). 66. Leggett, A. J., in [9]. 67. Goodman, J. W., in "Statistical Properties of Laser Speckle Patterns", Vol. 9 of "Topics in Applied Physics", 2nd ed., (Edited by J. C. Dainty), (Springer-Verlag, Berlin, 1984), Chap. 2, pp. 9-75. 68. O'Connor, P.. Gehlen, J. and Heller, E. J., Phys. Rev. Lett. 58, 1296 (1987). 69. Bogoliubov, N. N„ J. Phys. USSR 11, 23 (1947). 70. Bendat, J. S. and Piersol, A. G., "Random Data: Analysis and Measurement Procedures", 2nd ed. (Wiley, New York, 1986).

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Wave Dynamical Chaos: An Experimental Approach in Billiards* A. Richter Institut fur Kernphysik, Technische Universitat Darmstadt. D-64289 Darmstadt, Germany Received August 8, 2000

PACS Ref: 05.45.+b

Abstract

analogous to the stationary Schrodinger equation

Two experiments with super- and normal conducting microwave cavities which are used as an analog to two-dimensional, infinitely deep quantum potentials are presented. The eigenvalues of such a potential can be measured directly through the resonance frequencies of the cavity, while the eigenvectors can be determined by measuring the field distributions inside the cavity. In the case of open systems another information - the imaginary part of the eigenvalues - is observable by measuring the widths of the resonances. As examples for this experimental approach to quantum chaos the semiclassical reconstruction of length spectra of billiards with varying chaoticity based of various trace formulas proposed by Gutzwiller, by Ullmo, Grinberg and Tomsovic and by Berry and Tabor for systems with chaotic, mixed and regular dynamics, respectively, and the first observation of square root singularities - so called exceptional points recently proposed by Heiss - in the energy spectrum of coupled cavities are presented.

V 2 f = -k2V

1. Introduction In this talk I will present results from two recent analog experiments performed at Darmstadt with microwave billiards [1,2]. The key idea in these experiments is to exploit the well known analogy between the stationary Schrodinger equation for a two-dimensional (2D) and infinitely deep potential and the stationary Helmholtz equation describing the electric field inside an appropriately shaped microwave cavity [3-6] - an experimental ansatz, which has just recently been applied with great success in the more general fields of wave dynamical problems (see e.g. [7,8]), nuclear physics [9] and solid state physics [10]. In general the stationary Helmholtz equation describing the electric field inside a microwave cavity V2E = -k2E

(1)

with

*=S,

(4)

with k =^

l2m£

(5)

for a particle of mass m and energy S trapped inside a 2D potential with infinitely high walls. For sufficiently flat microwave resonators the solutions of Eq.(l), or equivalent^ Eq. (4), are discrete eigenvalues k2 with corresponding eigenfunctions E and f, respectively. The classical billiard, i.e. a plane 2D area in which a point-like particle moves according to the laws of Newtonian motion frictionless on straight line paths and interacts only with the boundary of the area accordingly to the law of reflection (upper part of Fig. 1), has then as an analog the quantum billiard (lower left part of Fig. 1) which can be modelled through an electromagnetic cavity (lower right part of Fig. 1). It was this connection between a classical and a quantum billiard that has triggered the initial interest in quantum billiards: In the case of a classical billiard the dynamics of the particle can be either regular, chaotic or mixed depending only on the shape of the boundary, which leads to a classification of classical billiards into regular, chaotic and mixed billiards, and the question, whether this classification can also be applied to the corresponding quantum systems has been in the focus of intensive research for more than 20 years [11-13]. For time-reversal invariant systems, the relevant ensemble from Random Matrix Theory (RMT) to describe the spectral fluctuations properties is the Gaussian Orthogonal Ensemble (GOE), for a recent review of RMT in quantum systems see [14]. The other major

(2)

c

/ denoting the frequency, c the velocity of light, and E the electric field which vanishes at the boundary, is a vectorial wave equation. Nevertheless, for a cavity height d < c/2/max w ' t n /max being the highest accessible frequency, E will be always perpendicular to the bottom and the top of the microwave cavity [7]. In this case the vectorial part of the electric field can be separated by using the ansatz £(f) = <j>(r)e:,

(3)

and Eq. (1) becomes a scalar wave equation which is then * Work supported by the Deutsche Forschungsgemeinschaft under contract No. RI 242/16-1 Physica Scripta T90

Semiclassicai / Theory' /

Experiment

Fig. I. The two model systems - classical and quantum billiard - and the experimental system - the electromagnetic cavity - considered in this talk. The quarter of a Bunimovich stadium billiard is taken as an example (from [7])© Physica Scripta 2001

Wave Dynamical Chaos: An Experimental Approach in Billiards approach, called Periodic Orbit Theory (POT), towards the understanding of the fluctuation properties rests on the pioneering work of Gutzwiller [15] who derived a semiclassical relationship between the density of states of a chaotic quantum system and the periodic orbits of the corresponding classical system. The talk is organized as follows: In Section 2 a brief description of the experimental methods used to measure the resonance frequencies (eigenvalues) of the superconducting microwave billiards considered as a first example for an application of RMT and POT (Section 3) and of the electric field distributions (wave functions) to study certain wave dynamical problems as a second example in a parametric, normal conducting billiard (Section 4) is given. Finally a short outlook is presented in Section 5. 2. Experimental methods

213

the boundary of the billiard and a top plate on which the antennas are mounted. To overcome conductance fluctuations between the different plates it is in general not sufficient to just supply some external force - virtually pressing the different plates together - but indium wire should be used to establish a good electrical contact between the different parts of the resonator. In the second example to be presented in Section 4 of this talk I will discuss experiments with such a nc cavity (a sketch of the cavity is given in Fig. 4). While nc cavities are usually convenient to set up and to handle, their resolution is insufficient for experiments involving small frequency splittings (see e.g. [16]) or a large number of levels (see e.g. [7] and [1] for further examples) as can be seen from the transmission spectra in the upper part of Fig. 2. The resolution is tied to the full width at half maximum f of a resonance, in the sense that a large r leads to overlapping resonances that can not be resolved properly. One often uses the quality factor Q =f/T as a figure of merit for the achievable resolution, with Q « 103 for a typical nc cavity. The reason for this relatively low g-value is of course the finite conductance of the cavities copper walls leading to energy losses in those walls and subsequently to broader resonances [17].

While quantum billiards were treated initially only theoretically or in numerical calculations (see e.g. refs. given in [7]) the analogy described above and summarized in Fig. 1 made the quantum billiard accessible to the experimentalist. A typical microwave billiard (i.e. a 2D microwave cavity as A way to overcome these problems is to use sc cavities. described above) has a height of d — 7mm leading to a frequency / m a x % 21GHz, before turning three-dimensional. They are commonly manufactured of electron beam welded Microwave power, produced by an rf-source, is coupled into niobium (Nb) sheets and become superconducting at temthe cavity using dipole antennas that penetrate the cavity peratures below Tc — 9.2 K. The increased conductance of only slightly (in general less than 0.5 mm). The microwave the cavities walls yields quality factors of g » 103 to 107 power reflected at this antenna or transmitted to another but requires a liquid helium bath cryostat to reach temperaantenna is then analyzed by a network analyzer leading to tures below Tc. While the first experiments discussed in the corresponding reflection or transmission spectra (Fig. 2). [7] have been performed in the cryostat of the supFrom the viewpoint of an experimentalist one has to dis- erconducting Darmstadt electron linear accelerator tinguish between two types of microwave cavities: Normal- S-DALINAC [18], the experiments are now carried out in (nc) and superconducting (sc) cavities. A nc cavity is usually a cryostat dedicated to microwave cavity experiments which composed of several copper plates, namely a bottom plate, is sketched together with the microwave equipment in Fig. 3. an inset that is appropriately shaped and that constitutes I shall discuss such an experiment in detail in Section 3. For now I would like to point the reader to the lower part of Fig. 2 where a typical transmission spectrum for a sc cavity is shown and the increased resolution is evident. Just recently a new type of sc cavity has been investigated at our laboratory [19] which combines the advantages of the modular setup of nc cavities and the high g-values of sc cavities. The cavity is composed of a thin lead coating _ V \ * PQ that becomes superconducting below 7.2 K and which is » V 1 \ \ T3 - 8 0 , 1 \ I * \ \ applied onto copper plates. The cavities are build in the same 300 K • I (I I >l \ I—I—I » | | I—I I l| I I I—1—)—I—I—I—I—; manner as nc-cavities, with the difference that soldering wire OH - 2 0 r with a high lead content is used to established the electrical contact between the different plates. Although the Q-values reached with these cavities is slightly lower (£> ~ 104 to 106) than the one reached with Nb, they are still superior to nc cavities while they have the advantage of being 4.2 K modular. 17.75 17.80 17.85 17.90 17.95 18.00 I will now talk about the different sets of data that a microwave cavity experiment yields before I will present f (GHz) two typical experiments in Sections 3 and 4. The foremost information that can be gathered from e.g. a transmission Fig. 2. Typical eigenmode spectra of a Limacon billiard (see Section 4) from 17.75 to 18 GHz. In the upper part the cavity is at T = 300 K (i.e. normal spectrum like the one in Fig. 2 are the resonance frequencies of the cavity - either by searching for local maxima in the conducting) and only the high resolution superconducting measurement (T = 4.2 K, lower part) reveals the proper sequence of eigenfrequencies. Note transmission spectra (a technique usually applied in sc the frequency shift (indicated by the dashed lines for six groups of experiments) or by describing parts of the transmission eigenmodes) is mainly due to the contraction of the billiard cavity during spectrum with formulae given e.g. in [17] and extracting the cooling down from 300 K. to 4.2 K (the lower part of the figure is from the resonance frequencies. The sequence of measured [I])© Physica Scripta 2001

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A. Richter other possibilities exist [20], in the experiments discussed in [7] and in Section 4 a method initially suggested by Slater [21] is applied to measure the field distributions. A small (i.e. V <SC i \ where V is the volume of the perturbating body and A the wavelength of the microwave coupled into the cavity) metallic body perturbating the electromagnetic field inside the resonator at the position r will lead to a frequency shift Af=fo(aE2(r)-bB2(r)),

Fig. 3. Sketch of the experimental equipment to measure the eigenmode spectra of superconducting cavities. A billiard of the Limacon family discussed in Section 4 is housed in the liquid helium bath cryostat shown on the right side while the microwave and data acquisition equipment is indicated on the left side (from [7]).

eigenfrequencies \f\,f2,fi,—,fi,fi+i,—} which correspond directly via Eqs. (2) and (5) to the eigenvalues of the analogous quantum system provide the backbone of any statistical analysis. Examples for various statistical tests that can be applied to the sequence of eigenfrequencies for sc billiards investigated at Darmstadt are described in great detail in [7] so I will omit further discussions here and would like to point the interested reader instead to this article and the references therein. Besides the eigenfrequencies several additional resonance parameters can be gained from describing a cavity resonance with the appropriate formulas [17], the most important one being the width r of the resonance. As I pointed out above F is tied to the microwave energy dissipated in the cavity. In a sc cavity the power is dissipated only through the antenna channels [17], while in nc cavities the main cause for power dissipation are the surface currents in the finite conductive copper walls of the cavity. In the second example that I will discuss later in this talk in Section 4 a cavity has been set up to allow the investigation of a system with complex eigenvalues. Looking at reflection spectra [17] one can show that in this case the real part of an eigenvalue is given by the resonance frequency and the imaginary part by the resonance width [2] - in other words by determing both resonance parameters, systems with complex eigenvalues can be investigated in an experiment. To conclude this section on experimental methods I will discuss how to measure the squared electrical field distribution |£(i*)|2 which is, due to the analogy between Eq. (4) and the scalar Helmholtz equation, closely related to the eigenvectors of the Schrodinger equation. Although Physica Scripta T90

(6)

with fo being the resonance frequency of the unperturbed field, a and b constants depending on the geometrical and material properties of the perturbating body [16,2] and E(r) and B(r) denoting the electric and magnetic field at the position i\ respectively. The geometry of the perturbating body can be chosen in such a way that b ~ 0 [21,16] so that virtually only the squared electrical field E2(r) is detected. In Fig. 4 I show a sketch of the experimental setup. The perturbating body can be moved inside the cavity by a steering magnet placed outside the cavity and positioned by a PC (left hand side of Fig. 4). Combining the position of the steering magnet and the frequency shift Af yields the desired field distribution (right hand side of Fig. 4). In what follows I shall discuss in detail for the limited time and space for this talk and its writeup only two of the experiments performed in Darmstadt. It should nevertheless be clear to the reader by now that the information one can get from a microwave cavity experiment is going well beyond the sequence of eigenvalues and that those experiments provide a unique opportunity to test a wide variety of theoretical expectations in quantum and wave dynamical chaos in general with an excellent precision.

3. First example: test of trace formulas for spectra in billiards of varying chaoticity As a first example from our recent experiments with microwave cavities I will present the experimental test of semiclassical trace formulas for spectra in billiards which are regular, mixed (neither completely regular nor chaotic) and fully chaotic. I hereby follow closely the presentation given in [1]. The semiclassical relationship between the density of states of a chaotic quantum system and the properties of the periodic orbits (pos) of the corresponding classical system is known for nearly 30 years [11,22], The so-called Gutzwiller trace formula expresses the density of states by a weighted sum over all individual classical pos. Integr a t e , i.e. regular, systems can be described by EinsteinBrillouin-Keller (EBK) quantization [23]. A trace formula for such systems was first derived by Gutzwiller [15] and later in a different way by Berry and Tabor [24]. Beside these two limiting cases of chaotic and regular dynamics, respectively, systems with intermediate, mixed behavior have attracted more and more attention in recent years. Testing the various trace formulas in experiments with billiards requires a family of them that covers the whole range of possible dynamics, i.e. from regular via mixed to chaotic. Billiards of the Limacon type can be constructed for that by varying a single control parameter L The family includes the integrable and regular circular billiard [25] and the fully chaotic cardioid [26] as limiting cases which © Physica Scripta 2001

Wave Dynamical Chaos: An Experimental Approach in Billiards

215

Magnetic Body

Magnetic Body

£

7 A

i

n

'i

/

ibles

DO Steering Magnet

A

A * Shift At Network Analyzer

Fig. 4. Experimental setup to measure squared field distributions (|£| 2 and/or |B|2) with a perturbating body. On the left side a sketch of the nc cavity with the perturbating body and the steering magnet is shown. On the right side the whole setup including the rf- and the data acquisition equipment is sketched together with a typical field distribution (the r.h.s. of the figure is from [16]).

tem are not known. Thus, a generalization of formulas (7) and (8) and a method to evaluate the parameters entering these formulae are needed. Ullmo et al. [27] started their evaluation of a trace formula for mixed systems with the Berry-Tabor expression. However, they did not use the propagator formalism of [24], but the energy dependent Green's function and the perturbative ansatz. Nevertheless, they went beyond Ozorio de Almeidas work: Instead of truncating the Fourier expansion of the corrected actions, which results in the damping Bessel term in Eq. (8), they mapped the problem onto the pendulum. In addition they introduced an action which is a composition of the mean action S = (5 U + Ss)/2 and the difference action AS = {Su - 5 s )/2 of the two remaining periodic orbits (stable and unstable). Entering these relations into the integral which describes (7) PM(S) the dephasing of the po contribution of the family M under nh^'Ml'1 \g"E 11/2 cos *)• a perturbation (see [27]), one is able to modify the expression with T being the period of the po, g'E the curvature of the line of the density of states for the integrable case (Berry-Tabor of constant energy H(l\, />) — E,S the action of the po and r\ description, Eq. (7)). This yields the following contribution its Maslov index. to the density of states for each pair of stable and unstable To move from the regular towards the near-integrable po. case, one can follow a perturbative ansatz from Ozorio de Almeida [28]: Adding a small perturbation to an inteiS inn i7t\ 1 Re e x p i r y grable system changes the density of states p^ in such a PUM(S) J) m'M^'i way, that only the first order correction to the action has to be added, still, the resonant tori, on which the periodic (9) x [T[J0(S) - iaJiis)] orbits of the regular system existed are destroyed. According to the Poincare-Birkhoff theorem only two periodic orbits +iAT\jl(s) + \a. T[Jo{s)-J2{s)} per torus will survive: one stable (s) and one unstable (u). Thus for near-integrable systems Ozorio de Almeida [28] with 51 = AS/H being the normalized correction to the action, found a modified Berry-Tabor expression for the density T the averaged period (half of the sum of two periods) and of states AT their difference. The quantities Jo, Ji and / i are the standard Bessel functions. The quantity a is the ratio of the s T p°{S) = p M Jo(AS/h) (8) determinants of the monodromy matrices of the stable where AS is the difference of the action of the stable and and the unstable po. For a -*• 0 one obtains just the result unstable orbit, respectively. In a typical case however the of Ozorio de Almeida (Eq. (8)). The Maslov index is denoted unperturbed Hamiltonian and the perturbation of the sys- by r\ and g"E is evaluated in accordance with [29]. The can be well described by the respective trace formulas. For a proper treatment of billiards with mixed dynamics a trace formula has recently been proposed by Ullmo, Grinberg and Tomsovic [27] and I shall present in the following the experimental test of this formula using superconducting billiards of the Limacon family [1]. Before presenting the experimental data I will briefly recall the salient features of trace formulas in general and then treat the formula of Ullmo et al. especially. In the case of an integrable system the contribution of classical periodic orbits with topology M = (M\, Mi), specifying the individual winding number of the po on the tori, to the density of states is given by the Berry-Tabor formula [24], which is based on the EBK. quantization.

1J

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interested reader will find a more detailed description of these calculation in e.g. [27]. To apply Eq. (9) to actual spectra small replacements are necessary: The action is given by S — Hkl, with k the wave number and / the length of the po. The period of the po can be expressed by its length and the term M-,g"E can be evaluated by using expressions given in [27]. Ullmo et al. have tested their trace formula numerically by applying it to a quartic oscillator for which they have calculated the first 12000 eigenvalues. They found good agreement between the simulated quantum spectrum and its reconstruction with Eq. (9). As an additional test, the two limiting cases, the Berry-Tabor result for integrable systems (Eq. (7)) and the Gutzwiller result for chaotic systems, are easily reproduced from Eq. (9). One obtains the first one for AS —»• 0, while the other results from the asymptotic expression for the Bessel functions, 0

G

( i ) 4 E fTTCOS^ - « £ ) , /nr^|det(M„-l)|1'2 \h '"2/

distribution P(s) and the number variance I2(L), respectively. The chaoticity of the systems can be quantified e.g. in terms of the so-called Berry-Robnik distribution [31]. This model interpolates between the two limiting cases of pure Poissonian and pure GOE behavior for a classical regular resp. chaotic system [7]. Therefore a mixing parameter q is introduced, which is directly related to the relative chaotic part of the invariant Liouville measure of the underlying classical phase space in which the motion takes place. For q = 0 one has a regular and for
(10) P' 1 U C (0

with M being the monodromy matrix and r\ the Maslov index of the periodic orbits [22,11]. However, they did not test their formula on actual experimental data. From what I stated in Section 1 it should be clear that sc cavities provide an excellent tool to verify and check the results of Ullmo et al. Consequently a one-parameter family of superconducting two-dimensional microwave resonators with the shapes shown in Fig. 5 has been manufactured. They all belong to the family of Limacon billiards, which have been numerically studied in [30]. Their boundary is defined as the quadratic conformal mapping of the unit disc onto the complex w-plane: w — z + AZ2, where / e [0, 1/2] controls the chaoticity of the system. Four billiards of different chaoticity with parameters 1 = 0, k = 0.125, / = 0.15 and A = 0.3 have been built and investigated. All billiards, except the first one, are desymmetrized. For A = 0, one has a circle, which is known to be integrable, i.e. regular. Investigations of the classical Poincare surface of section of the other configurations have shown, that the fraction of the chaotic phase space is 55% (A = 0.125), 66% (A = 0.15) and nearly 100% (A = 0.3). The two limiting cases (A = 0 and A = 0.3) provide systems that can be used to gauge our experiment against the well established traced formulas by Berry and Tabor and Gutzwiller, respectively.

= /

Akek'[P{k) - pWe5"(£)],

(in

where p(k) is the measured level density of the system, pWcyl(k) its smooth part and [kmm, i m a x ] is the wave number interval in which the data were taken. Note, Eqs. (7), (9) and (10) are already describing the fluctuating part of the level density. In Fig. 7 the length spectrum of the A = 0.125 billiard and the identified periodic orbits up to a length of about 1.4 m are shown. The length of the periodic orbit corresponds to the position of the peak in the Fourier

The billiard cavities were excited with frequencies up to 20 GHz, so that one gets a total number of more than 1000 resonances for each billiard (about 660 resonances for the circular billiard). The following test of the trace formula for mixed systems is solely based on these measured sets of eigenfrequencies. In Fig. 6 the short- and long-range correlations of the measured eigenvalue sequences are investigated in terms of RMT by using the nearest neighbor

350 mm

481 mm

458 mm

431 mm

Fig. 5. Shapes of the investigated billiards of the Limacon family. All billiards are desymmetrized, except the first one (from [1]). Physica Scripta T90

Fig. 6. Nearest neighbor distribution P(s) (left side) and number variance £2{L) (right side) of the four investigated systems. Beside the data the two limiting distributions (Poisson and GOE) are displayed as well as the fit to the data (dashed line) based on the Berry-Robnik model. ;C! Physica Scripta 2001

Wave Dynamical Chaos: An Experimental Approach in Billiards (c)

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(a)

m w ^

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Fig. 7. Length spectrum of the A = 0.125 billiard. On the top the first periodic orbits of the system are displayed and identified in the spectrum below.

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spectrum and the stability of the periodic orbit to the amplitude of the peak. By applying the trace formula in Eq. (9) to the spectra of the investigated biltiards, the calculations were restricted to the first periodic orbits up to a length of 1.4 m. The reconstruction of the spectrum of the circular billiard was done with the help of the Berry-Tabor formula, one limiting case of Eq. (9), using a symbolic code through which all periodic orbits are straightforwardly determined [32]. For the three other billiards the properties of each po (length, number of reflections, curvature of the boundary at the reflection point, Maslov index) were calculated numerically. The so found characteristic values for each po form - together with Eq. (9) - the basis for the reconstruction of the experimental spectra on the theoretical side. A view at the squared Fourier transform of the fluctuating part of the density of states - the so called length spectrum is given in Fig. 8. Here a comparison between the experimental data and the numerical reconstruction for the four investigated systems is presented. For the circle the reconstruction is in very good agreement with the measurement. The reconstruction for the two billiards belonging to the mixed regime (A = 0.125 and 1 = 0.15 billiard) is for the shortest periodic orbits also in good agreement with the measured data, whereas for the following periodic orbits with a length / > 1.3 m small deviations become visible. These deviations do not occur in the positions of the periodic orbits but in the height of the reconstructed peak. The same situation is found for the chaotic X — 0.3 billiard, where the predictions of Gutzwiller's trace formula have been compared to the data. In Fig. 9 the real and imaginary part of the experimental and theoretical Fourier transformed fluctuating part of the level density for the A = 0.125 billiard around the periodic orbit pair at 1.21 m / 1.22 m are compared with the theoretical reconstruction calculated by using Eq. (9). The figure clearly displays that beside the predicted positions also the phases of the periodic orbits are in good agreement with the experimental data. © Physica Scripta 2001

0.5

0.0 0.4

, „A .A, 0.6

0.8

1.0

1.2

1.4

1 (m) Fig. 8. Comparison between the measurement (solid line) and the reconstruction (dashed line) of length spectra with the help of the trace formula given by Eq. (9). The insets for the X = 0.125 and X = 0.15 billiard show a magnification of the first periodic orbit at / «* 0.47 m (from [1]).

Fig. 9. Real and imaginary part of the length spectrum of the X = 0.125 billiard. The measured spectrum is drawn as solid and the reconstruction as dashed line. The peak belongs to the periodic orbit pair at 1.21 m /1.22 m (from [1]).

To ensure that not experimental uncertainties are the reason for these deviations the measured eigenfrequency sequences for the two limiting cases (regular and chaotic) have been cross checked with the respective trace formula and numerical simulations [33], which confirmed the excellent quality of our data (see Fig. 8). The small deviations found in the completely chaotic case (lowest part of Fig. 8) are indeed connected to special mechanical problems Physica Scripta T90

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of the respective microwave cavity, particularly to the cusp at the lower left corner. The small deviations found for the billiard systems with mixed dynamics (which do not have such a cusp), / = 0.125 and 0.15, could be explained by two reasons: Equation (9) has been derived for the case of small perturbations of a regular system, i.e. the near-integrable case, while the two investigated billiards already constitute highly mixed systems. Furthermore the number of eigenvalues considered in our comparison (around 1000) is much smaller than the set that Ullmo et al. have used for their test (around 12000), so that the experiment might be still too far away from the true semiclassical regime. However, the results for the mixed systems obtained with the trace formula of Ullmo et al. is much more satisfying than using Gutzwiller's trace formula straightforwardly without taking the Poincare-Birkhoff theorem into account.

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4. Second example: wave functions in a parametric billiard and the topological structure of exceptional points As a second example for the usefulness of microwave cavity experiments for the investigation of wave dynamical problems I will focus on the recent observation [2] of branch root singularities in the energy spectrum of a parametric cavity a completely different experiment than the one discussed in Section 3, in the sense that it actually requires the operation of cavities at room temperature and that in addition to the resonance frequencies, the widths and the field distributions need to be observed too. These experiments are based on theoretical predictions of Heiss [34]. As an introduction I will discuss the essential aspects of degeneracies and singularities by looking at a two-level model. In the case of a hermitian Hamiltionian two real parameters are needed to force a degeneracy [35]. To simplify the discussion further I chose the two parameters to be a coupling s and an additional external parameter 5 which leads to a 2 x 2 Hamiltonian of the form

0

i

*•

s

Fig. 10. The eigenvalue trajectories of HD? plotted above the two-parameter space spanned by s and S leads to a double cone structure. As an example two cuts for i- = 0 and s ^ 0 are shown.

the path, i.e. (13) Following the respective eigenvectors ip{ and i//2 along the closed loop one finds that after completing the loop one of the eigenvectors picks up a sign change [36], which can be summarized as

;M=i;l-

(14)

In numerical simulations [36] and experiments [37] this sign change of the pair of eigenvectors is usually the feature (12) H ^(S, s) which is used to show that a closed loop contains a DP. Ei For E[ ^ E2 the eigenvalues EI and ?,2 of HDP are only The sign change is commonly described by an additional degenerate under a variation of S if s = 0. For all other phase of n - the geometrical phase or Berry's Phase - which space. couplings s ^ 0 an avoided crossing of the eigenvalues will depend on the trajectory in the parameter DP In contrast to the basic model of H real physical syscan be observed [35,36]. In the parameter space spanned tems are in general open systems, i.e. in the case of by S and s one finds a single point of degeneracy at microwave cavities the power dissipation at the antenna (3 = E2/E[,s = 0) and if the eigenvalue trajectories are plotted above the parameter space a double cone structure channels and - especially in normal conducting cavities at the cavity walls. This can be incorporated in Eq. (15) [36] emerges (Fig. 10). Due to the resemblance of the double cone with a diabolo, by adding widths Ti and f2 to the eigenvalues, i.e. the point of degeneracy is also known as Diabolic Point (DP) SEi+ifi EP (15) [36]. While the observation of the diabolo structure is poss- H {5,s) = E2 + \r2 ible by looking at the eigenvalue dynamics, the direct investigation of a DP in an experiment or in a numerical The addition of complex widths leads to a non-hermitian simulation is difficult, because an exact degeneracy will only Hamiltonian (as opposed to its hermitian counterpart occur at a single point (SDP, sDP) in the parameter space. A HDP) and to a different topology in the eigenvalue space, common method to prove the existence of such a single-point which is now spanned by the real and imaginary parts of degeneracy is to surround the degeneracy in the parameter the eigenvalues. Looking only at the real part of the space (a possible circular loop is sketched in Fig. 10) and eigenvalues (i.e. in the case of microwave cavities the frelook at the development of the eigenvalues and eigenvectors quencies f\ and f2), one now finds a crossing for a broad along the trajectory. In the case of a DP this will lead to range of couplings.? - there is no single point of degeneracy two consecutive avoided crossings of the eigenvalues along between f\ and f2 as it used to be the case of HDP and U

6E{ s

Physica Scripta T90

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© Physica Scripta 2001

Wave Dynamical Chaos: An Experimental Approach in Billiards

1

1

'

1

'

1

(ii) the EP is a fourth order branch point for the eigenvectors and (//'/) a different orientation of the loop in the parameterplane yields a different phase behavior [2].

'

/ / /

*

EP

U) •

/

/

/

\

-

* \ \

•

X

*>*

r*' 1

1

1

1

1

Rejej Fig. 11. Real and imaginary parts of the complex eigenvalues of HE? (;.'], c2 under a variation of i) for two different couplings s\ and .v?). The point, where the complex eigenvalues coalesce is called Exceptional Point (EP) (from [2]).

the diabolo structure is destroyed. However, when looking at the trajectories of the complex eigenvalues si and s2 (Fig. 11) for different couplings s one sees that there still is a single point in the eigenvalue spectrum, where the two complex levels coalesce. In Fig. 11 I show the eigenvalue trajectories of / / E p under a variation of S for two different settings s\ < Rc{sEP} and s2 > Re{JEP}, where

r.

^EP

r-. /£% - SEi — ±I

(16)

is the complex coupling for which the two complex levels coalesce. Detailed calculations [34,38] show that the EP is a second order pole of the eigenvalue spectra and not just a degeneracy as the DP, which also implies a different behavior if the singularity is encircled. Following the eigenvalue trajectories sketched in Fig. 11 one finds that two closed loops in the parameter space (spanned by S and s) are required for one closed loop in the energy space, so that after the first loop in the parameter space the eigenvalues are interchanged, i.e. for the real and imaginary parts one has

[«

and

kWr-

(17)

From what I stated above it should be clear that a microwave cavity should be an ideal system for the experimental observation of EPs: the real and imaginary parts of the eigenvalues can be measured directly as resonance frequencies and widths and the eigenfunctions are accessible by measuring the squared field distributions. The knowledge of the squared field distribution is enough to observe the crucial geometric phase if one follows a path laid out by Berry and Wilkinson [36] which I will outline below. In the presence of a geometric phase Eq. (3) has to be modified to

E(r, t) = 4>(r)e~

(19)

where y(8, s) is the geometric phase. Accessible to the experiment is \(j>(r)~\, but changes in y{S,s) can be observed if y(S, s) is defined at one point S0l so in the parameter space and S and .s are then varied in steps small enough so that the geometric phase for the resulting wave function can be deduced from the initially defined y(So, so) [36]. To set up a system where EPs can be encircled, two semi-circular copper cavities of slightly different size which can be coupled by adjusting the opening of a slit by an amount s between them have been used (Fig. 12). The cavity is composed of three copper plates - namely a bottom plate, a circular inset with the variable slit and the top plate where the dipole antennas are mounted. A teflon (i;r ~ 2.1) semi-circle is placed in one side of the cavity which yields the second parameter 6, i.e. the distance between the centers of the cavity and the semi-circle. To assure a uniform electrical contact even at higher frequencies between the different parts of the cavity, indium wires with a diameter of 1 mm have been placed close to the inner edge of the cavity as sketched in Fig. 12. If the EP is to be encircled, S will have to be adjusted in a way that the imaginary part of the EP changes its sign, since the actual observation of the system is restricted to a real coupling s= Rejs}. Encircling the EP then requires four steps [2]:

The trajectories also imply that a crossing (avoided crossing) of the real parts of the eigenvalues - the projection of E,(<5, S) on the x-axis - always implies an avoided crossing (crossing) of the imaginary parts - the projection of e,(<5, s) on the v-axis. The two eigenvectors i/*, and \j/2 are not just interchanged like their corresponding energies but only one of them undergoes a sign change [2,34], i.e.

x\>2 M -
219

Antenna fSaT T

Xc

(18)

0.1m

y X Indium Wire

As an immediate consequence one can conclude that: (/)

four closed loops in the parameter space are needed to obtain a complete loop in the eigenvector space.

© Physica Scripta 2001

Fig. 12. Sketch of the microwave cavity as seen from the top (left) and from the side (right). In the side view the indium wire which is used to establish the electrical contact between the different copper plates is enlarged (from [2]). Physica Scripta T90

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2

3

4

A. Richter Assuming in the first step that s < (/"[ — r^)/! and looking at the EP with Im{S'BF} > 0 one first varies 6 so that the imaginary part of sEP will be less than zero. During this process a frequency crossing and a widths anti-crossing should be observed [34]. In the second step the coupling s is varied to s > (T[ — r i ) / 2 . This changes the position of the system in the s plane but leaves the position of the EP fixed. Now S is set back to its original value, i.e. Im{JEP} > 0. Because of the enhanced coupling we will now observe a frequency anticrossing and a widths crossing [34]. The last step is again moving the position of the system in the s plane to the initial reduced coupling, thus closing the path in the parameter plane.

The behavior of the complex energy levels in the presence of an EP and the associated repulsion of the complex levels is the first feature that should be used to establish the presence of an EP. In Fig. 13 the spectra of reflected power from the cavity in the range of 2.72 to 2.84 GHz for s = 58 mm under a variation of S are shown. A frequency crossing of the initially eights and ninths mode, for which numerical simulations with the program MAFIA [39] suggested an EP that can be encircled, is clearly visible, meaning that this setup represents the s < Re{Jbp} case. The exact resonance parameters were determined by fitting the resonances and the background with the expressions given in [17]. In Fig. 14 the resonance frequencies and widths for s = 58 mm are shown. Consistent with [34] and Fig. 11 we found a crossing of/i and / i yet the complex eigenvalues do not cross which can be seen when looking at Ti and TT that show an avoided crossing. For an enhanced coupling (s = 66 mm) the situation is reversed (Fig. 15): The crossing of /Y2 now implies an avoided crossing of f\ ,2 which again ensures that the complex eigenvalues do not cross. The two cases considered constitute steps one and three of the closed path in the parameter plane. The eigenvalues depend only weakly on s so I omit presenting the data for steps two and four. The behavior presented in Figs. 14 and 15 shows that there is a repulsion between two complex energy levels - a phenomenon that has been observed in a wide variety of experiments (see e.g. [40,41]). To establish the existence of a singularity between the two levels I will now focus on the behavior of the field distributions. In accordance to Berry's method, the signs of the measured squared field distributions have been chosen at do = 30 mm and so = 58 mm and a variation of AS = 5 mm is small enough to keep track of the initially defined sign along the trajectory. Figure 16 shows that along the closed path in parameter space \j/\ is changed to ij/2 in accordance to the behavior of the eigenvalues. The variation of \jjy is small when s is set to 58 mm, since the two levels simply cross without interaction [35,36] and I omit presenting the field distribution for 3 = 40 mm since the two resonance frequencies are almost exactly degenerate for this setup and \ji{ could not be resolved anymore. For s = 66 mm the variations are more dramatic since the two modes are now coupled more strongly. Figure 17 shows the corresponding development of i/>-> - again the squared field patterns are simply interchanged, but when looking at the signs of the final state the additional geometric phase of n is evident, clearly conPhysica Scripta T90

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Wave Dynamical Chaos: An Experimental Approach in Billiards

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firming the physical reality of EPs and distinguishing them from the DPs. The observed phase behavior is uniquely connected to the EP since, despite the fact that the cavity parameters where changed in the same way for both modes, only one of them exhibits a sign change along the path. This leads directly to the chiral properties of an EP [2], which can be deduced from the development of the eigenvalues during four closed loops in the parameter space, i.e.

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slit is opened to s = 66 mm, then S is changed to 50 mm, s is set back to 58 mm and S is changed back to its initial value of 30 mm. The effects on the sign of the field distributions are summarized in Fig. 18 where only the initial and the resulting distributions are shown. It is evident, that the orientation of the closed loop defines which field distribution exhibits a sign change. It has nevertheless to be noted that it is not possible to determine the orientation of the loop by observing the development of a single field pattern - only if the signs of i//l and ij/2 are chosen initially the two orientations can be distinguished. While the repulsion of complex levels has been established by looking at the resonance parameters for different settings of S and s, the appearance of a geometrical phase and the chirality that has been found while encircling the EP clearly confirms for the first time the physical reality of EPs.

5. Conclusion To conclude this talk I would like to summarize some essential aspects of microwave cavity experiments in the field of wave dynamical chaos. From looking at the two experiments presented in this talk and additional examples (see e.g. [7,42,43]) it should be clear that microwave cavities are an excellent tool to investigate wave dynamical chaos and to verify theoretical calculations. Even the use of nc cavities yields already very precisive results and with the advent of sc cavities the investigation of tiniest frequency splittings or extremely large sets of data is possible. The information gained from such experiments is almost complete: One can measure the resonance frePhysica Scripta T90

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A. Richter

quencies and widths - which yield the real and imaginary part of the eigenvalue - and the squared field distributions, which are connected to the eigenvectors. I exemplified the necessary techniques through two experiments: Firstly, the semi-classical reconstruction of a length spectrum based on trace formulas from POT has been discussed. The respective experiments have been performed with a family of superconducting resonators with varying chaoticity and yielded very large sets of eigenfrequencies that form the basis for the further analysis. It could be shown that a trace formula for mixed systems recently suggested by Ullmo et al. [27] describes the length spectrum well but not perfectly. In the second example a normal conducting cavity has been considered. To observe singularities - so called Exceptional Points (EPs) investigated in detail recently by Heiss [34] - in the eigenfrequency space the resonator allows the variation of two parameters and the resonance frequencies, widths and field distributions need to be measured. The trajectories of two complex eigenvalues showed repulsion - a behavior that can be observed by looking at frequencies and widths and that is of a fundamental interest [34,40,41]. To prove the existence of an EP between the two levels the presumed singularity has been surrounded in the two-parameter space and the development of the field distributions has been recorded. Along the trajectory a predicted [34] geometric phase appears that acts only on one member of the pair of eigenvectors connected at the EP. This new and peculiar behavior of a geometric phase leads to a chirality attached to the EP that has been observed for the first time by changing the orientation of the loop in the parameter space. Acknowledgements The results presented in this talk are based on a tremendous amount of work and effort of the collaborators in my group at Darmstadt, which consist at present of the following members: C Dembowski, H.-D. Graf, A. Heine, H. Rehfeld and C. Richter. It is a great pleasure to thank all of them for their numerous contributions. I am particularly grateful for the time and effort spent by C. Dembowski and H. Rehfeld in helping me to produce this manuscript. On the particular aspects of the trace formulas discussed I acknowledge the many discussions I had in the past with O. Bohigas, M. Gutzwiller, S. Tomsovic and D. Ullmo, and on the matter of the Exceptional Points with D. Heiss. Finally I thank S. Abcrg, Karl-Fredrik Berggren and Par Omling for having organized a symposium with unusually high scientific standards in a most pleasant surrounding.

References 1. Dembowski, C. et al.. Phys. Rev. Lett., submitted. 2. Dembowski, C. et al., Phys. Rev. Lett., submitted. 3. Stockmann, H.-J. and Stein, J.. Phys. Rev. Lett. 64. 2215 (1990).

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4. Sridhar. S., Phys. Rev. Lett. 67. 785 (1991). 5. So. P., Anlage. S. M.. Ott, E. and Oerter, R. N„ Phys. Rev. Lett. 74. 2662 (1995). 6. Graf. H.-D. et al.. Phys. Rev. Lett. 69, 1296 (1992). 7. Richter. A., in ''Emerging Applications of Number Theory", The IMA Volumes in Mathematics and its Applications, (edited by D. A. Hcjhal. J. Friedman. M. C. Gutzwiller, and A. M. Odlyzko), (Springer, New York. 1999), Vol. 109, p. 479. 8. Alt, H. et al.. Phys. Rev. Lett. 79. 1026 (1997). 9. Alt, H. et al.. Phys. Rev. Lett. 81. 4847 (1998). 10. Dembowski, C. et al.. Phys. Rev. E 60, 3942 (1999). 11. Gutzwiller, M. C , "Chaos in Classical and Quantum Mechanics", (Springer, New York. 1990). 12. Berry, M. V.. Proc. R. Soc. Lond. A 413 183 (1987). 13. Bohigas. O.. in "Chaos and Quantum Physics", (M.-J. Giannoni, A. Voros. and J. Zinn-Justin, eds), (Elsevier, Amsterdam, 1991), p. 87. 14. Guhr, T., Miiller-Groeling, A. and Weidcnmuller. H. A., Phys. Rep. 229, 189 (1998). 15. Gutzwiller, M. C , J. Math. Phys. 11. 1792 (1970). 16. Dembowski, C. et al.. Phys. Rev. Lett. 84, 867 (2000). 17. Alt, H. et al. Phys. Lett. B 366, 7 (1996). 18. Alrutz-Ziemssen. K. et al.. Part. Ace. 29, 53 (1990). 19. Dembowski, C. et al., Phys. Rev. E 62, R4516 (2000). 20. Stein, J. and Stockmann. H.-J., Phys. Rev. Lett. 68. 2867 (1992). 21. Maicr, L. C . Jr. and Slater, J. C , J. Appl. Phys. 23, 1352 (1968). 22. Gutzwiller. M. C , J. Math. Phys. 12. 343 (1971). 23. Einstein, A., Verhandlungendcr Deutschcn Physikalischen Gesellschaft 19. 82 (1917); Brillouin. L.. Le Journal de Physique et le Radium 7, 353 (1926); Keller. J. B., Ann. Phys. (N.Y.) 4, 180 (1958). 24. Berry, M. V. and Tabor. M., Proc. R. Soc. Lond. A 349, 101 (1976); Berry. M. V. and Tabor, M., J. Phys. A 10, 371 (1977). 25. Berry, M. V.. Eur. J. Phys. 2, (1981). 26. Backer, A.. Steiner. F. and Stifter, P., Phys. Rev. E 52. 2463 (1995). 27. Ullmo, D., Grinberg, M. and Tomsovic. S., Phys. Rev. E 54, 136 (1996); Tomsovic, S., Grinberg, M. and Ullmo, D., Phys. Rev. Lett. 75, 4346 (1995). 28. Ozorio de Almeida. A. M., "Hamiltonian Systems; Chaos and Quantization", (Cambridge University Press. Cambridge. 1988). 29. Bohigas, O., Tomsovic. S. and Ullmo, D.. Phys. Rep. 223, 43 (1993). 30. Robnik. M., J. Phys. A 16. 3971 (1983); J. Phys. A 17, 1049 (1984). 31. Berry. M. V. and Robnik, M., J. Phys. A 17, 2413 (1984). 32. Balian, R. and Bloch. B., Ann. Phys. 64, 76 (1971). 33. Hesse. T.. Dissertation, University of Ulm. Germany (1997). 34. Heiss, W. D., Eur. Phys. J. D 7. 1 (1999); Phys. Rev. E 61. 929 (2000). 35. von Neumann, J. and Wigner, E.. Z. Phys. 30. 467 (1929). 36. Berry, M. V. and Wilkinson, M., Proc. R. Soc. Lond. A 392, 15 (1984). 37. Lauber, H.-M., Weidenhammer, P. and Dubbers. D.. Phys. Rev. Lett. 72, 1004 (1994). 38. Kato. T., "Perturbation theory of linear operators", (Springer. Berlin. 1966). 39. Weiland, T„ Numerical Modelling 9, 295 (1996). 40. The interchange of two complex eigenvalues z^ an Zk+\ has recently been discussed by P.v. Brentano and M. Philipp in Phys. Lett. B 454, 171 (1999) and experimentally observed by Philipp, M. et al.. Phys. Rev. E. 62. 1922 (2000). 41. Latinne, O. et al., Phys. Rev. Lett. 74, 46 (1995). 42. Stockmann, H.-J., "Quantum Chaos: An Introduction", (Cambridge University Press, Cambridge. 1999). 43. Sridhar, S., contribution to this symposium.

Ci Physica Scripta 2001

Physica Scripta. T90, 223-230, 2001

Acoustic Chaos C. Ellegaard, K. Schaadt and P. Bertelsen Center for Chaos and Turbulence Studies, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen 0, Denmark Received 19 October 2000

PACS Ref: 62.30.+d, 03.65.Sq, 05.45.-a, 43.20.Ks, 43.20.Tb, 43.20.Wd, 43.40.At

Abstract We use high resolution acoustic spectroscopy as an analog system to quantum systems. We study transitions between regularity and chaos and specific symmetry breaking. We compare with the predictions of Random Matrix Theory that was developed for the study of the complicated compound nuclear states. The acoustic system is interesting in its own right. It is governed by a vectorial wave-equation in contrast to the scalar Schrodinger equation, and we study the complex interplay of different resonant modes arising from this fact.

1. Introduction In several contributions to this conference we have heard about spectroscopy of the resonances of microwave cavities [1-3]. In the two-dimensional limit the Helmholz equation governing these systems becomes exactly equivalent to the Schrodinger equation. The present contribution deals with acoustic resonant systems. These are also governed by a wave equation, but it is not the Schrodinger equation. The Schrodinger equation is a scalar equation, the equation for the elastic wave is a vectorial equation. It is the equation for the displacement of material, and the forces involved are derived from the elasticity tensor. Only in special cases will it reduce to a scalar equation (namely for flexural modes in a thin plate). Nevertheless acoustic systems are useful as analogues for quantum systems, and we shall see that many features follow the predictions of RMT (Random Matrix Theory). See Ref. [5] for a recent revue. In that sense we can say we extend the testing ground of RMT to this system with a much more complex wave motion. In addition we concentrate on identifying features that are specific to the acoustics. The main reason for using the acoustic systems is the quality of the data produced, although it is impressive what is shown during this conference from the micro-dot experiments and the Rydberg atoms. Our g-value is typically 10,000 if we use aluminium and around 100,000 when we use quartz and even over 1 million in special cases, which allows for the resolution of many thousand resonances. This paper gives a short introduction to acoustics. Then gives some typical examples of the measurements we do on spectral statistics and on wave functions. Finally it goes into more details with a discussion of the separation of modes by means of the widths of states.

to the surroundings as small as possible. Furthermore most experiments are done in vacuum to reduce damping. In principle we hit the resonator and listen to it. In fact our first experiments [4] were carried out in this fashion. The resonator was given a sharp tap by bouncing a ball-bearing ball on it. This sharp tap is a delta function in time and can excite all the resonances. One of the points is placed on a small piezo-electric plate. The transient signal from this is recorded in an analyzer that performs Fourier analysis (FFT) to yield the resonance spectrum. In the most recent experiments the resonator is placed on three piezo-electric transducers, one of which is a transmitter for exciting the resonator. The two others are receivers for recording the response. The kind of objects can be 3-dimensional (3D) or 2D. They can be quartz crystals, fused quartz or aluminium. A typical spectrum is shown in Fig. 1. It is a short section of a spectrum around 600 kHz from a fused quartz plate. The widths of the peaks are typically 5 Hz giving a Q-value in excess of 100 000.

2.1. Acoustic waves in 3 dimensions The equation of motion for the displacement u in an isotropic solid is: + ii)V{Vu) + pV1u,

p-^={X

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224

C. Ellegaard, K. Schaadt and P. Bertelsen

Fig. 2. Nearest Neighbour Distributions and A3 statistics for Poisson, GOE and 2 GOE respectively.

The in-plane modes have the displacement u = (u, v, 0) and are described in the Poisson model:

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Thus in a finite object a standing wave becomes a complicated mixture of modes. 2.2. 2-dimensional models For plates the wave motion further differentiates: The transverse modes are very distinct depending on whether the displacement is perpendicular or parallel to the plane of the plate. The wave equations are obtained by suitable approximations for each case. The flexural modes are completely decoupled from the other modes and the displacement is characterised by: u = (0, 0, w). The flexural modes are described by the Kirchoff-Love model: ph—T=-DA~w,

(4)

with D = .®' 2> and h = thickness. The stationary equation is scalar and biharmonic: (A + k2)(A - k2)w(x, y) = 0,

(5)

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Physica Scripta T90

1

(7) V( VH) - - V x V x u 1— v 2 As in 3D it separates into two wave equations with the dispersion relations k\ = (W/KC{) and k, = (a»/ct) for longitudinal and transverse waves respectively. These will, in general, mode convert at boundaries. For further details see refs. [6,7]. df-

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3. Symmetries Figure 2 shows the spectral statistics we will be discussing: The nearest neighbour distribution (NND), and the A3 statistics. The figure shows the curves for three types of statistics, (1) Poisson statistics which is expected for integrable systems or systems with many symmetries, (2) the GOE which is expected for chaotic systems with no symmetries [10], and (3) for 2 noninteracting GOE which is expected for systems with exactly 1 symmetry. This shows up very specifically in several of the following examples. 3.1. Symmetry breaking in quartz The first example is from a 3D single-crystal quartz resonator. It is an example of a specific symmetry breaking. The resonator is a rectangular block and because of the anisotropics of the quartz crystals it possesses only one symmetry: 180° flip around the .x-axis. (This experiment has been called isospin breaking by T. Guhr, see [11], because of this single symmetry). We very gradually break this symmetry by machining away an octant of a sphere at one corner, thus creating an octant of a 3D Sinai billiard. The first section of Fig. 3 shows the NND of the intact block. The data follows exactly the curve for 2 GOE as expected from the symmetry. The spectra from which Fig. 3 is derived each contain 1,500 resonances. Section b shows the NND for an octant of just 0.5 mm radius removed. One immediately sees the level repulsion by the dip towards zero at zero levelspacing. This is immediately compensated for in a peak at the nearest non-zero levelspacing, whereas the rest of the distribution remains unchanged. In the following Sections d-f the dip deepens and the peak broadens until the distribution already for a © Physica Scripta 2001

Acoustic Chaos

225

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Fig. 4. The same sequence as for Fig. 3, but showing the A3 statistics. Fig. 3. A sequence of nearest neighbour distributions (NN D) recorded during the breaking of the single symmetry described in the text. Section a shows the distribution for a rectangular quartz block of dimensions 40 x 25 x 14 mm with the crystal .v-axis along the longest side and the r-axis along the shortest side. In the figure the dotted curve represents the Wigner (or 1 GOE) distribution, and the dashed curve the 2 GOE distribution. Sections b through f show the NND after cutting away an octant of a sphere in one corner with a radius of 0.5, 0.8, 1.1. 1.4 and 1.7 mm, respectively. In section x an octant of radius 10 mm has been machined away.

radius of 1.7 mm closely resembles the 1 GOE distribution. This sequence shows the great sensitivity of this method to defects in a symmetric system. The mass removed in Fig. 3(b) is one part in 10\ Figure 4 shows the A3 distribution for the same sequence. This distribution measures long-range correlations among levels and is not as sensitive to the small perturbations. Only when a large piece (radius 10 mm, section x) is removed, does it go all the way to 1 GOE. Details of this work is found in [12], and the work has been discussed in [13]. 3.2. Symmetry breaking in plates We start by looking at the most symmetric object, the circle. Figure 5 shows the statistics for a circular aluminium plate. © Physica Scripta 2001

The circular plate is semiclassically integrable [7], but the statistics do not correspond to the Poisson distribution. This is due to the fact that many eigenmodes are degenerate, and since our plate is not a perfect circle, some of these degenerate modes split op into several close-lying eigenfrequencies. This is observed as a "Shnirelman peak" in the left figure. Instead, the statistics follow a scaled Poisson distribution, which has been theoretically predicted by [8,9]. The scaled curves in Fig. 5 corresponds to a spectrum where 33% of the levels are degenerate levels, which have been split up. Figure 6 shows a sequence of symmetry breaking in 2D aluminium plates. Experimental results for 5 different plates are included, and the shape of each plate is shown inside the plot. The sequence starts at the top (a) with a very regular shape, a square, and the statistics show the Poisson distribution. Down through the sequence, more and more symmetries are broken until in section (d), the Sinai stadium, no apparent symmetries are left. However, the statistical distributions for this plate correspond exactly to 2GOE, and this is due to the fact, that two non-interacting classes of resonances exist for a plate of any given shape. These two classes, flexural and in-plane, are illustrated in Fig. 7. Physica Scripta T90

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The flexural modes are antisymmetric with respect to the midplane of the plate, while the in-plane modes are symmetric. The mirror symmetry of the plate in the thickness direction is responsible for the 2GOE behaviour we observe. This symmetry may be broken by cutting a slit in one side of the plate. The result of this is shown in section (e) at the bottom of Fig. 6. After cutting the slit, the statistics become exactly 1GOE because the last symmetry is broken. The statistics for the intermediate shapes, (b) and (c), in Fig. 6 are very difficult to interpret. One has to keep track of many classes of modes with different properties and different level densities. Therefore, no clear conclusion can be made without some sort of separation of the modes. This will be further discussed in Section 5.

4. Wavefunctions In contrast to quantum physics we can in the acoustic system measure wavefunctions in 2D plates.This is done by adding an additional transducer above the plate that can be moved across the plate to scan the entire area, measuring the amplitude as it goes. Two simple examples of wavefunctions in a rectangular quartz plate are shown in Fig. 8. The distribution of amplitudes for these is shown in Fig. 9. For the simple single-sinus wave the normalised distribution drops off at 2 as expected. The distribution for the double sinus drops off at 4. Figure 10 shows examples of wavefunctions for the chaotic shape - the so-called Sinai stadium. Notice that the two very closelying resonances 510.6 and 513.1 kHz have very different wavelengths because they obey different dispersion relations. The node pattern is chaotic and the amplitude distributions show the Porter-Thomas distribution as shown in Fig. 11. This distribution is found irrespective of the dispersion relation of the wavefunction.

5. Width distributions We want to study the specific acoustic proporties in more detail, and to do this we have developed a method for Physica Scripta T90

separating, experimentally, the different modes. When we determine the frequencies we actually fit the peaks in detail, so we get frequencies, amplitudes and widths. If we look at the amplitudes. Fig. 12, we see that the amplitude distribution fits nicely with the Porter-Thomas distribution. Not surprising considering the amplitude distribution found for the wavefunctions. Our transducers are just sampling the wavefunction of each state at a random position. In RMT one expects to see the same for the widths. What we see is Fig. 13 where we instead find a relatively sharp peak. It looks like a Gaussian, but with a tail. This is of course because the resonances decay into many other channels than assumed in RMT. We loose energy to the transducers, we loose energy to other internal degrees of freedom in the crystal and finally to heat. The curves fitted to the data are #2-like distributions with a large number of degrees of freedom. These data are from large sets of data on aluminium plates. In the acoustic system we can have control over the damping and this can be different for different modes. Normally the experiments are carried out in vacuum in order to keep the damping as low as possible. But to gain control over the damping we introduce varying amounts of air pressure. In Fig. 14 we see a small section of a spectrum as a function of pressure. It is seen how some peaks retain their narrow width whereas others broaden significally as a function of pressure. The flexural modes couple strongly to the surrounding air whereas the in-plane modes do not. The flexural modes are damped and thereby broadened, and may be selected by measuring the width as a function of pressure. Figure 15 gives the distribution of widths as a function of pressure and it is clearly seen how one group separates completely from the rest. These are the flexural modes. We can then select the modes one by one and show the spectral statistics separately for each mode class. This is done in Fig. 16. Where we in Section 3 mixed the two classes to make a transition from 2 GOE to 1 GOE, we see here how we instead can separate them and study them independently. © Physica Scripta 2001

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Fig. 7. Illustration of the two symmetry classes.

419.6kHz

437.7kHz _

30

30

2

20

1 °

>, 10

10 0

0 0

20

40 x (mm)

60

20

40 x (mm)

60

Fig. 8. Examples of simple wavefunctions of flexural type in a rectangular plate of fused quartz. © Physica Scripta 2001

Physica Scripta T90

228

C. Ellegaard. K. Schaadt and P. Bertelsen

i

Z = AV

- A7'

Fig. 9. The histogram shows the normalised amplitude distributions for the two wavefunctions of Fig. 8. The smooth curve is the Porter-Thomas distribution.

510.5kHz

513.1kHz

Fig. 10. Examples of chaotic wavefunctions in Sinai-stadium shaped aluminium plates. The dispersion relation for the one on the right corresponds to a tlcxural mode. The one on the left to in-plane longitudinal.

5 1 3 1kHz

5 :C.5KHZ 0.0000

^ \

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-

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0 01 0 0

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;

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=

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Fig. II. The histogram shows the normalised amplitude distributions for the two wavefunctions of Fig. 10. The smooth curve is the Porter-Thomas distribution.

,0^-

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amplitudes

1.5

£

1

-

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Fig. 12. Amplitude distribution for the total It)4 resonance peaks in the spectra of the aluminium stadium plate. The solid line is the Porter-Thomas distribution. Physica Scripta T90

Pi

0.5

I

i

^^CTir^-^n—i_^—=

1 1.5 2 normalized width, x

2.5

Fig. 13. Experimental width distribution compared to r ' V . Both data and function are normalised such that the mean width is unity. © Physica Scripta 2001

Acoustic Chaos

229

0.08

F

0.04 0.02

1

Vacuum

0.06

-

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404

406

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404

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Fig. 14. Acoustic resonance spectra of a 2 mm aluminium plate as function of pressure.

Fig. 15. The distribution of widths of the resonances as function of pressure,

Fig. 16. Top: the NND and A3 statistics for the flexural modes alone. Bottom: The NND and A3 for the in-plane modes alone. © Physica Scripta 2001

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C. Ellegaard, K. Schaadt and P. Bertelsen

The result for the flexural modes shows pure 1 GOE as expected. The result for the in-plane modes is similarly consistent with 1 GOE. See also Ref. [14]. 6. Conclusions We have shown how the acoustic system lends itself to high quality investigations of the predictions of the Random Matrix Theory. The system is very flexible and allows the study of specific symmetry breaking in a detailed and controlled manner. The elastic wave equation leads to a more complex set of modes than the Schrodinger equation, but we find that the results are still well described by RMT. We show how we, experimentally, can begin to disentangle the various acoustic modes and how we can exploit this to study the specific properties of each symmetry class of the acoustic system. This work is continuing. Acknowledgement This work is supported by the Danish National Research Council.

Physica Scripta T90

References 1. Richter. A., Physica Scripta T90, 212 (2001). 2. Lu, W. T., Pance. K.. Pradham, P. and Sridhar. S.. Physica Scripta T90, 238 (2001). 3. Stockmann, H-J., contribution to this conference. 4. Ellegaard, C , Guhr. T., Lindemann. K., Nygard, J. and Oxborrow, M., Phys. Rev. Lett. 75. 1546 (1995). 5. Guhr, T., Miiller-Groeling, A. and Weidenmuller, H. A., Phys. Rep. 52, 189 (1998). 6. Bertelsen, P., Ellegaard. C. and Hugues, E., Eur. Phys. J. B 15, 87 (2000). 7. Bogomolny, E. and Hugues, E„ Phys. Rev E 57 (1998). 8. Chirikov, B. V. and Shepelyansky. D. L.. Phys. Rev. Lett. 74, 518 (1995). 9. Biswas, D., Azam. M. and Lawande. S. V., Phys. Rev. A 43 5694 (1990). 10. Bohigas. O., Giannoni, M. J. and Schmit, C . Phys. Rev. Lett. 52, 1 (1984). 11. Guhr. T. and Weidenmuller, H. A., Ann. Phys. 199, 412 (1990). 12. Ellegaard. C , Guhr, T.. Lindemann, K.., Nygard. J. and Oxborrow. M., Phys. Rev. Lett. 77, 4918 (1996). 13. Leitner, D. M., Phys. Rev. E 56 4890 (1997). 14. Schaadt. K.., Simon, G. and Ellegaard, C , contribution to this conference.

© Physica Scripta 2001

Physica Scripta. T90, 231-237, 2001

Ultrasound Resonances in a Rectangular Plate Described by Random Matrices K. Schaadt1, G. Simon1,2 and C. Ellegaard1 'Center for Chaos and Turbulence Studies, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen 0, Denmark. Permanent address: Department of Atomic Physics, Eotvos Lorand University, H-1117 Budapest, Hungary

2

Received October 19, 2000

PACS Ref: 05.45.+d

Abstract We present experimental results for the ultrasound transmission spectra and standing wave patterns of a rectangular plate offused quartz. We demonstrate that mode conversion for the in-plane modes leads to complicated behavior and the fluctuation statistics are described by the Gaussian orthogonal ensemble (GOE) of random matrix theory (RMT). Studying the distribution of normalized frequency shifts upon a temperature increase of 5'C, and observing the corresponding measured standing wave patterns, it is revealed that a small number of special resonances are protected from mode conversion and therefore do not take part in the mixing. Such states serve as reference points in our data analysis and allow us to compare our results to a random matrix model for symmetry breaking.

1. Introduction Not too many physicists would believe that studying the free resonant vibrations of a rectangular plate of isotropic and homogeneous material is a justifiable occupation in science nowadays. The truth is that this classic problem, as many other problems of elastodynamics, remains unsolved. Renewed interest and insight in these problems have come from the application of methods used in the field of quantum chaos. It was recently seen [1] that the fluctuation properties of spectra from quantum billiards and from the flexing of thin plates appear to be identical. This result was confirmed experimentally in Ref. [2], which also presented the theoretical Weyl formula and found agreement with the experimental result. Also, very recently, cavity scattering in elastodynamics was investigated [3]. The conjecture of Ref. [9] states that spectral fluctuations of quantum chaotic systems obey RMT, and Ref. [10] states that also the motion of the energy levels of quantum chaotic systems, under a perturbation of an external parameter, obeys RMT. Experimental results with acoustic systems [4-8] strongly suggest the applicability of these two conjectures to a wider range of systems than quantum chaotic systems [11]. In that capacity, these experiments have served not just as analogue systems of quantum billiards , but more generally, to promote problems of elastodynamics as interesting problems in their own right. In this paper, we focus on the set of plate vibration modes known as the in-plane resonances, for which the displacement is in the plane of the plate and the wave equation is vectorial rather than scalar. For these resonances, mode conversion is important: On the boundary of the plate, a purely transverse or purely longitudinal incoming wave is converted into two outgoing waves, one of each type. In Ref. [12] it was found in a numerical simulation of ray splitting in classical billiards that chaoticity is enhanced. A natural © Physica Scripta 2001

question to ask is whether mode conversion gives rise to chaos for a simple system like the rectangle, which is otherwise integrable. We try to answer this question experimentally by presenting the fluctuation statistics of the in-plane resonances, the distribution of normalized frequency shifts, which corresponds to the distribution of level velocities, due to a perturbation, and finally a gallery of measured in-plane standing wave patterns. One of the interesting aspects of our results is that we find both mixed states and states that are "bouncing-ball"-like [13]. The latter serve as reference points in our data analysis and allow us to compare our results to a random matrix model for symmetry breaking [14].

2. Experimental setup We measure ultrasound transmission spectra of a rectangular plate using piezoelectric transducers, see Fig. 1. There are three such transducers, of which one is a transmitter and two are receivers. The temperature of the system is kept constant to within 0.005°C using a temperature controller, such that the eigenfrequencies are not affected by fluctuations in room temperature. The pressure of the air surrounding the plate can be controlled and kept at a low value where air damping of the plate vibration is insignificant. For a more detailed description of our setup, see Ref. [15]. The rectangular plate is made offused quartz. Such plates have several features that make them appealing for our experiments. First, they are isotropic and homogeneous. Second, they are flat to very high precision, which guarantees the reflection symmetry through the middle plane of the plate. Third, the surfaces are optically polished, which reduces noise in our scanning experiments, where we map out the vibration amplitude of the standing waves. Lastly, the acoustic g-value is between 105 and 106 at a typical frequency of 500 kHz. The rectangular plate has side lengths 36.24 mm and 70.90 mm. The thickness of the plate is 2.000 mm. Previous experiments [6,7] on single-crystal quartz blocks reported Q-values of around 106 and utilized this high quality for measuring many eigenfrequencies. Fused quartz has the same g-value as single-crystal quartz, although plate modes show a Q-value around 210 5 . During measurements, the plate is resting only on three tiny spikes, making the vibrations as close to free as possible. We are thus using Physica Scripta T90

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K. Schaadt, G. Simon and C. Ellegaard

525.4 0 -20

-

-60

525.8 526.0 526.2 Frequency [kHz]

'

•

-40 7

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526.4

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526.4

526.6

525.4

525.6

525.8 526.0 526.2 Frequency [kHz]

526.4

526.6

,

,

,

,

,

Fig. I. Measured spectra of the rectangular plate, showing the effect of both pressure change and temperature change. The pressure change allows a labelling of the resonance peaks, as either flexural (F) or in-plane (I), since the flexural modes are strongly affected by the presence of air, which is not the case for the in-plane modes.

an isotropic plate and free boundary conditions apply, so the symmetry through the middle plane of the plate is intact.

3. Dispersion relations and Weyl law For a finite isotropic plate, if nothing breaks the mirror-symmetry through the middle plane of the plate, there are two uncoupled classes of vibration modes: flexural, for which the displacement is mainly perpendicular to the plate, and in-plane, for which the displacement is mainly in the plane of the plate, see Refs. [16,17] for details. For the data analysis, it is necessary to know the wavelength and the number of modes as a function of frequency. In the low-frequency limit where the wavelength is much larger than the thickness of the plate, the flexural modes obey a biharmonic equation for the scalar displacement perpendicular to the plate. In the present study, however, we consider frequencies where the low-frequency approximation breaks down, and it is necessary to use an expansion, recently worked out in Ref. [2], of the exact dispersion relation: 12 1/4 r~ ? KF ss — _ Vf2(l + avQ + a2Q2), Physica Scripta T90

(1)

given in terms of the dimensionless wavenumber K = kh and frequency Q = lufh/cu where h is the thickness of the plate, / is frequency, ct is the velocity of transverse waves, and k is the wavenumber. Here, K = ^/2/(l — v), v being Poisson's ratio. The expansion coefficients a\ and a2 are functions of v only and are also given in Ref. [2]. For fused quartz, v = 0.16 [18] and we get a{ = 0.177 and a2 = -9.40 • 10" 3 . The flexural staircase function, i.e. the number of flexural modes as a function of the wavenumber, is also given in Ref. [2] and reads NF(KF) =

4nV

+fi

K

*mK*>

*

(2)

where 5 is the area of the plate, L is the circumference and ft is a constant depending only on v. The expression forft=is given in Ref. [2]. Inserting v = 0.16 we get /?F = 1.6. We note that Eqs. (1) and (2) are valid for Q < n, i.e. at frequencies below the critical frequency where there is not room for one half shear wavelength in the thickness direction. For a fused quartz plate of thickness 2 mm, the critical frequency is around 900 kHz. In this paper we consider frequencies no higher than 750 kHz, and Eqs. (1) and (2) are valid. In-plane modes are, in general, mixtures of longitudinal and in-plane transverse waves, that propagate independently inside the plate but couple upon reflection at a boundary, where mode conversion takes place, see e.g. Ref. [16] for details. For the in-plane modes, there is obviously no single dispersion relation, because they are mixtures of longitudinal and transverse wave motion. The dispersion relation for in-plane transverse traveling waves is Kt = Q.

(3)

The dispersion relation for in-plane longitudinal traveling waves, to first order in Q, reads Q

K\

Kt

(4)

For the frequency range considered in this paper, it is sufficient to use this first-order approximation, which is just the first term of the expansion given in Ref. [2]. The number of in-plane modes as a function of dimensionless wavenumber is [2]: N

M

=

^ (

l

+

^

+

P

^ -

(5)

Again, ft is a constant depending only on v and inserting v = 0.16 we get ft = 2.2.

4. Ultrasound transmission spectrum A HP 3589A spectrum /network analyzer is used to measure the ultrasound transmission spectrum in the frequency range between 450 kHz and 750 kHz. First, we measure the spectrum at 35°C and atmospheric pressure. Then, the measurement is repeated at 35°C, but this time in vacuum. The temperature is then increased to 40°C, and the spectrum is measured again in vacuum and at atmospheric pressure. A section of the transmission spectrum is plotted in Fig. 1 for all four combinations of temperature and pressure. Notice that the temperature increase makes the resonance © Physica Scripta 2001

Ultrasound Resonances in a Rectangular Plate Described by Random Matrices peaks shift upwards in frequency. This comes about because the thermal expansion is negligible for fused quartz compared to the dv/dT and dc t /d7\ that are both positive, see Eqs. (6) and (7). Also, notice how the change of air pressure strongly influences the flexural modes and allows a separation of the modes into flexural and in-plane. All resonance peaks are fitted with a so-called skew-Lorentzian [19] using interactive software that we have developed in the programming language IDL. The fit-parameters include the eigenfrequency and the width. We find 628 resonances. This must be compared to the Weyl law Ni + Np, given in Eqs. (2) and (5), which gives 612. We comment on the deviation in the following section. 5. Mode separation The separation of the modes is made on the basis of the distribution of the widths of the resonance peaks. In Fig. 2 we show this distribution for vacuum and atmospheric pressure. In vacuum, the width distribution is just one slim peak with a mean value of about 3 Hz. There is no additional structure and it is not possible to distinguish flexural modes from in-plane modes. Notice that the mean of 3 Hz gives a value for the average quality Q = 6-105 Hz/3 Hz — 2-105. The distribution at atmospheric pressure is drastically different. It is clear that the widths fall into two groups, one situated around T = 6 Hz, the other one at about T = 34 Hz. Importantly, these two groups are completely separated, and we use this to separate the modes into flexural (T > 20 Hz) and in-plane (r < 20 Hz). So, going from vacuum to atmospheric pressure, the widths of the in-plane resonance peaks grow by a factor of 2, whereas the widths of the flexural resonance peaks grow by more than an order of

0.40

0.30

0.20

0.10

0.00

0.40

P = 1 atm 0.30

0.20 -

In-plane

Flexural

0.10

0.00

20 r [Hz] Fig. 2. Distribution of resonance peak widths, measured in vacuum and at atmospheric pressure © Physica Scripta 2001

233

magnitude. A similar scheme of mode separation is presented in Ref. [17]. The separation immediately leads to a mode count for each of the two types of modes. We find that 315 resonances are flexural and 313 are in-plane. The theoretical predictions of Eqs. (2) and (5), lead to 298 flexural and 314 in-plane resonances. The deviations have two sources. First, the constants of elasticity are not precisely known. We have used the values c\ = 3750 m/s and v = 0.16 from Ref. [18] since these values give nice agreement for the number of in-plane modes. Second, since we have used only the analytic part of the expansion of the flexural dispersion relation, given in Ref. [2], and this expansion converges slowly, we must be prepared to accept a significantly larger deviation than what was found in Ref. [2],

6. Spectral fluctuation statistics We can now study the properties of the two mode classes separately and start by looking at spectral fluctuation statistics. In Fig. 3 we present the spacing distribution and the spectral rigidity for flexural modes and in-plane modes separately. Our results show that the flexural modes obey Poisson-like statistics on short scale, whereas we note a deviation towards increased rigidity for long scale fluctuations. It is well known that the flexural modes obey a scalar wave equation at low frequency, i.e. when the thickness of the plate is much smaller than the wavelength, and since the problem is separable for the rectangle, Poissonian behavior is expected. At higher frequency, as in our case, it is not evident that this would be the case. In fact, it is known that a full three-dimensional analysis is necessary to get the correct dispersion relation at high frequency, i.e. when the wavelength and the plate thickness are comparable. At 750 kHz, which is the highest frequency in our experiment, the flexural wavelength is 3.8 mm, which is comparable to the plate thickness of 2 mm. For the in-plane modes the results are more clear, since both the spacing distribution and the spectral rigidity show that the resonances behave as four independent classes, each with fluctuation properties described by the GOE. This is in agreement with the numerical work in Ref. [12], where it is found that mode conversion enhances chaoticity. On the other hand, the rectangle is not classically chaotic, even in the presence of ray-splitting. In Ref. [20] it is found for the square billiard that only 3 momentum directions come into play if the ratio of the two wave velocities is greater than \ / 2 ~ 1 . 4 1 . For fused quartz, this ratio is KCt/ct = K ~ 1.54, and we find numerically for the rectangle that 3 momentum directions are present. This means that phase space is left unexplored and there is no ergodicity. Interestingly, it is found in Ref. [20] that the exponential proliferation of orbits, which is expected for K < V5, persists at K ~ y/2, signifying the presence of topological chaos. We find that the in-plane modes are strongly coupled within each of the four independent classes produced by the symmetries of the rectangle. As we shall see in the following, however, the rectangle does allow the existence of "bouncing ball"-like modes, that are not influenced by mode conversion. It seems that such resonances, that we shall also refer to as "special states" or "special resonances", Physica Scripta T90

234

K. Schaadt, G. Simon and C. Ellegaard 1.4 1.2

* Flexural d a t a - Poisson •• 4 GOE

1.0 0.8

x Flexural data 1 GOE 2 GOE 4 GOE -•-" Poisson

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x

x-

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y x xX!-x--:

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Fz'g. 3. The spacing distribution and spectral rigidity for the flexural (upper plot) and in-plane (lower plot) modes of the rectangular plate. The curves labelled "2 GOE" and "4 GOE" correspond to a superposition of two and four independent GOE spectra, respectively.

are too few in number to make a difference for the fluctuation statistics.

7. Mode mixing and special states We may think of mode conversion as a symmetry-breaking mechanism that acts to mix transverse and longitudinal wave motion. These obey different dispersion relations, see Eqs. (3) and (4), and their wavelengths change by different amounts in response to a change of temperature. In our experiments, this translates into shifts in eigenfrequency, which we can measure. That effect can be used to investigate to which degree a particular mode behaves as a transverse or a longitudinal mode. Inserting K = kh and Q = 2nfh/ct into Eq. (3) and differentiating with respect to the temperature, we see that: d/t /t

=

-o-

J_d£t dT ' ctdT

7tdf

dT,

(6)

where T is temperature and a is the thermal expansion coefficient. For fused quartz, a can be ignored. Similarly, for longitudinal waves:

dfi

1 ,dv 4K dT

+

ldcA 7tdf)dT>

(7)

and we note the additional term, proportional to dv/d7\ compared to Eq. (6). It was described in section 4 that we measured the transmission spectrum at 35°C and at 40°C. In Fig. 4 we show the resulting shift in eigenfrequency for the in-plane modes. The data points display the expected /-proportionality, and seem to define a lower limit and Physica Scripta T90

an upper limit with a densely populated region closer to the lower limit. Normalizing the shift to the frequency, as suggested in Eqs. (6) and (7), we now study the distribution of df/f, see Fig. 5. Since in this paper, df/f serves merely as a label, for convenience we leave out a factor of 10~3 whenever we give numbers for df/f. To interpret the result, let us first consider what we would find if mode conversion was not present. In that case, we would expect two d-like peaks, one at df/f coming from the transverse modes, and one at df/f coming from the longitudinal modes. What we observe is that these <5-like peaks almost vanish as the symmetry is destroyed by mode conversion, producing instead a large peak of mixed modes. However, two small peaks do remain where the <5-like peaks would have been. These are special resonances in the sense that they are protected from mode conversion and therefore do not participate in the mixing. With our interpretation of Fig. 5, we estimate df/f = 0.37 and df/f = 0.48. We have not been able to find values for dc t /dT and dv/dT for fused quartz in the literature, and therefore cannot directly check these numbers. Except for the special states, this scenario closely resembles the results obtained in Refs. [14,17] in a random matrix model for symmetry breaking. There, a so-called asymmetry number is denned, which measures to what extent an eigenvector belongs to one or the other of two, originally uncoupled, subspaces. Here, the normalized frequency shift plays the role of the asymmetry number. Moreover, it is a property of the above-mentioned model that the mean asymmetry number is conserved as the symmetry is broken, and equals the weighted average of the © Physica Scripta 2001

Ultrasound Resonances in a Rectangular Plate Described by Random Matrices 8. Standing wave patterns

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500

550

650

600

700

750

f [kHz] Fig. 4. In-plane eigenfrequency shift as a function of frequency for a temperature increase of 5 C from 3 5 C to 40UC. Each "+" represents an eigenfrequency. For the ten data points marked with additional symbols (A. • . O) we also measured the corresponding standing wave pattern, see Fig. 6.

30

Mixed

n

25

80

— 60 |

20 r

40

°- 20 0

1

0.44 df/f

=3 15

0.55

Transverse 10

Longitudinal

5 0 0.30

235

, r 0.35

0.40

0.45

0.50

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df/f Fig. 5. Distribution of normalized frequency shift for the in-plane modes, corresponding to a temperature increase of 5°C. Inset: Same distribution, but plotted for the flexural modes.

asymmetry number for the individual subspaces when the symmetry is good. Denoting by p t and p^ the density of states for transverse and longitudinal modes, respectively, we find ptdft/f + p\df\/f\ = 0.403, since pl/p] = K2 = 2.38. In comparison, the mean value of the distribution plotted in Fig. 5 is 0.407. The nice agreement indicates that our interpretation of Fig. 5 is correct. The inset to Fig. 5 shows the P{df/f) for the flexural modes, which is a single peak centered at df/f = 0.437. Note that if the P(df/f) was plotted for all the modes together, it would be very difficult to give a clear interpretation of the P(df/f). It is thus necessary to carry out the separation of the resonances such that one can study the in-plane resonances alone. From Fig. 5 we also see that the peak of mixed resonances is fairly slim, and we expect this feature to be much more pronounced for chaotic geometries, where the 'wings' of special states would not exist. This means that mode conversion is a strong effect that thoroughly mixes transverse and longitudinal waves to produce in-plane modes with a fairly homogeneous character. © Physica Scripta 2001

We now turn to study the standing wave patterns of the eigenmodes of the rectangle. The experimental technique used is a bit different from that presented in [15], which was designed for measurements on aluminium plates, where the Q-va\ue is around 104. Since a plate of fused quartz has a Q-value between 105 and 106, an eigenstate also has a considerable life time of several seconds. The technique presented in [15] was based on measuring the resonance peak in question for every point in a pre-defined grid on the plate surface. Roughly, one must then wait for two life times in order to obtain the vibration amplitude in one point, and the total measurement time can be days. Instead, we use a phase-locked loop technique which ensures that the plate is always excited at the resonance frequency. As we scan the plate, a HP 89410A FFT analyzer records the signal for about one second in every point and measurement time is greatly reduced. Between 550 and 650 kHz a number of in-plane resonance peaks were selected for such a measurement of the standing wave pattern. We want to emphasize that the displacement field is a vector field in all three spatial dimensions. For the flexural modes, the main component of the displacement is perpendicular to the plate, but this is not the case for the in-plane modes. In our experiments the vibration is measured using a piezo-electric component with which we essentially measure an amplitude and a phase. For flexural modes we are confident that this amplitude is proportional to the actual vibration amplitude of the plate, see Ref. [15]. For in-plane modes, the measured amplitude is some unknown function of the full displacement field. Nevertheless, it is possible to extract valuable information from the measured standing wave patterns. In Fig. 6 we show grey-scale plots of ten measured in-plane standing waves. The first thing to notice about these standing wave patterns is how some modes seem very ordered whereas others look highly irregular. Ordering the modes with respect to increasing df/f reveals a picture which is in perfect agreement with what we have found in the previous section of this paper: The modes of smallest df/f, labelled "A", look regular because they belong to the special transverse resonances that are protected from mode conversion and therefore do not take part in the mixing. The modes of intermediate df/f, labelled " • " . look complicated because they are mixtures of transverse and longitudinal wave motion. Finally, modes of high df/f, labelled V are again ordered since they belong to the special longitudinal resonances. Notice also the lack of symmetry for these measured standing wave patterns. This is particularly pronounced for the mixed modes, but is also evident for the regular modes if one looks at the details. We believe that this is an artifact due to the measurement technique, as described earlier in this section. One of the measured standing wave patterns stands out as a clean "bouncing-ball" mode, namely the one labeled 'o3'. In this case, we can compare the wavenumber to the dispersion relation of Eq. (4) to check if this is really a longitudinal mode. For the transverse modes, there is no equally pure picture among our measured standing wave patterns, and the "A2"-mode is our best candidate. In Fig. 7 we compare Physica Scripta T90

236

K. Schaadt, G. Simon and C. Ellegaard

«*v „.,*.,.. .Vo.-n-

WfflHi/t •..*••

1 2 5 k [1/mm]

4

1

2 3 k [1/mm]

fig Z Left: Average FFT of horizontal sections of the standing wave pattern labelled "A2". Right: Average FVV of vertical sections of the standing wave pattern labelled "<>3". A long dashed line is placed at twice the wavenumber of transverse and longitudinal wave motion, respectively. Shorter dashed Sines indicate multiples of that value.

Fig. 5. Grey-scaSe plots of ten in-plane standing waves. The grey-scale is chosen logarithmic because this enhances contrast. The plots are ordered according to increasing value of the normalized frequency shift, df/f, and carry the same labels ( A . Q O) as in Fig. 4. Note that going from top left plot to lower right plot, one moves from transverse (A), through mixed ( • ) , into longitudinal CO), modes.

of our data analysis: First, the spectral fluctuation statistics which show that the in-plane modes follow the statistics of four superposed GOE spectra. Second, the distribution of normalized frequency shifts that arise from heating the plate by 5 °C, which shows that almost all resonances are mixed by the mode conversion, except a few special modes that remain either transverse or longitudinal Third, the standing wave patterns that look complicated for the mixed modes and display a high degree of order for the special modes, even revealing the wavenumber for a transverse and a longitudinal mode, which agrees perfectly with the corresponding dispersion relation. In addition, we have seen that mode conversion is a strong effect that thoroughly mixes transverse and longitudinal waves into in-plane resonances with a rather homogeneous character. Our work shows that the rectangle is an interesting system when mode conversion is present.

Acknowledgements the wavenumbers coming from the dispersion relations to The authors are grateful to Andrew D. Jackson, Anders Andersen, Preben the experiment, using FFT. Bertelsen, Gregor Tanner, Thomas Guhr, and Mark Oxborrow. This work To calculate the wavenumbers, we insert the resonance was supported by the Danish National Research Council. frequencies, given in Fig. 6, into the relevant dispersion relation. For longitudinal wave motion at 561.7 kHz, we i n d k = 2nf/Kct = 0.61 1 /mm. Since the standing wave patterns represent the absolute value of the amplitude, the basic References length-scale is half the wavelength, corresponding to twice 1. Bogomolny, E. and Hugues, E., Phys. Rev. E, 57, 5404 (199S) the wavenumber, i.e. 1.22 1/mm. Similarly, for transverse 2. Bertelsen, P., Ellegaard, C. and Hugues, E., Eur. Phys. J. B. 15, 87-96 (2000). wave motion at 587.5 kHz, we expect 1.97 1/mm. In both 3. Cvitaeovic, P., Ssndergaard, N. and Wirzba, A., to 'be published. cases, Fig. 7 shows that there is a large peak at the expected 4. Weaver, R. L.. J. Acowst. Soc. Am. 85, 1005 (1989). wavenumber. This gives further support to the interpret5. Ellegaard, C. et al, Phys. Rev. Lett. 75, 1546 (1995). ations we gave in Section 7. 6. Ellegaard, C , Guhr, T., Lindemann, K.., Nygard, J. and Oxborrow, M., 9. Conclusion We have measured acoustic transmission spectra and standing wave patterns of a rectangular fused quartz plate. We have demonstrated how the resonances can be separated into iexural and in-plane types using the pressure of the surrounding air. It is made clear that this separation is absolutely necessary in order to study the interesting behavior of the in-plane modes. We find that a very clear picture emerges when we compare the three main parts Physica Scripta T90

Phys. Rev. Lett. 77, 4918 (1996). 7. Bertelsen, P., Ellegaard, C , Guhr, T., Oxborrow, M. and Schaadt, K., Phys. Rev. Lett. S3, 2171 (1999). 8. Schaadt, K. and Kudrolli, A., PRE 60. R3479 (1999). 9.. Eohigas, O., Giannoni, M. J. and Schmit, C , Phys. Rev. Lett. 52, 1 (1984). 10. Simons, B. D. and Altshuler, B. L., Phys. Rev. Lett. 7©, 4063 (1993). 11. Guhr, T., Miller-Grading, A. and Weidenmiiller, H. A., Phys. Rep. 299, 189 (1998). 12. Couchman, L., Ott, E. and Antonsen, T. M., Jr., Phys. Rev. A 46,6193 (1992). 13. McDonald, S. W. and Kaufman, A. N., Phys. Rev. Lett. 42, 1189 (1979). © Physica Scripta 2001

Ultrasound Resonances in a Rectangular Plate Described by Random Matrices 14. Andersen, A., "Topics in Random Matrix Theory", M.Sc. Thesis, Niels Bohr Institute. Copenhagen, 1999. http://www.nbi.dk/~aanders/ thesis.ps 15. Schaadt, K., "The Quantum Chaology of Acoustic Resonators". M.Sc. Thesis, Niels Bohr Institute, Copenhagen, 1997. http://www.nbi.dk/ ""schaadt /speciale / speciale. ps.gz 16. Graff, K. F., "Wave Motion in Elastic Solids", (Dover Publications,

© Physica Scripta 2001

Inc., New York 1975). 17. Andersen. A.. Ellegaard, C . preparation. 18. Weast, R. C , (editor), "CRC 59th edition, (1979). 19. Alt, H. et at.. Nucl. Phys. A, 20. Biswas. D.. Phys. Rev. E 54.

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Jackson, A. D. and Schaadt. K... in Handbook of Chemistry and Physics", 560, 293 (1993). 1232 (1996).

Physica Scripta T90

Physica Scripta. T90, 238-245, 2001

Quantum Correlations and Classical Resonances in an Open Chaotic System Wentao T. Lu, Kristi Pance, Prabhakar Pradhan and S. Sridhar* Physics Department, Northeastern University, Boston, Massachusetts 02115, USA. Received October 22, 2000

PACS Ref: 05.45.Mt, 05.45.Ac, 03.65.Sq, 84.40.-x

Abstract We show that the autocorrelation of quantum spectra of an open chaotic system is well described by the classical Ruelle-Pollicott resonances of the associated chaotic strange repeller. This correspondence is demonstrated utilizing microwave experiments on 2-D n-disk billiard geometries, by determination of the wave-vector autocorrelation C(K) from the experimental quantum spectra S2i(k). The correspondence is also established via "numerical experiments" that simulate S2\(k) and C(K) using periodic orbit calculations of the quantum and classical resonances. Semiclassical arguments that relate quantum and classical correlation functions in terms of fluctuations of the density of states and correlations of particle density are also examined and support the experimental results. The results establish a correspondence between quantum spectral correlations and classical decay modes in an open system.

1. Introduction The correspondence between quantum and classical mechanics has turned out to be particularly rich for systems that are classically chaotic [1], This correspondence is usually considered in terms of the asymptotic behavior of quantum properties. Statistical analysis of the quantum spectra of closed chaotic systems display short-range correlations that are universal and can be described by random matrix theory (RMT) [2,3]. The long range behavior of spectral fluctuations require semiclassical treatments that incorporate periodic orbits [4]. In this paper we establish a new connection between the statistics of quantum spectra and classical dynamical properties of an open chaotic system, utilizing physical and numerical experiments on the well-known n-disk billiard geometry. The n-disk system [5] is a paradigm of open hyperbolic systems, with an associated strange repeller [6] whose strange properties arise from the Cantor set of the unstable periodic orbits (PO). For hyperbolic repellers, Ruelle [7] and Pollicott [8] showed that the dynamical evolution of repellers toward equilibrium can be expressed in terms of complex poles, leading to the so-called Ruelle-Pollicott (RP) resonances. The n-disk billiards have been extensively addressed theoretically, both classically and semiclassically [5,9,10,11] and only recently have been studied experimentally [12,13]. Our microwave experiments on the n-disk billiards measure the quantum transmission spectrum SSi {k), which is essentially the two-point Green's function G(ri,r2,k), and from which the spectral autocorrelation C(K) is determined. The small wave vector K (long time) behavior of the spectral autocorrelation provides a measure of the quantum escape rate, and is in good agreement with the cor* e-mail: srinivas(a>neu.edu Physica Scripta T90

responding classical escape rate ycl [12,13], as predicted by RMT analysis of this universal behavior. For large K > yd (short time), non-universal oscillations of the autocorrelation C(K) are observed, that can be understood completely in terms of classical RP resonances. This behavior is similar to that of the variance Z2(L) of closed chaotic systems, which also displays leading universal behavior for small L, followed by non-universal behavior for large L that can be described in terms of system specific periodic orbits [4], The present work demonstrates that C(K) of open systems displays a similar trend from short-range universality to long-range non-universality, however the non-universal contributions to quantum correlations are described in terms of other classical phase space structures of the associated repeller, viz. the RP resonances [14]. A concise account of the main results was presented in Ref. [15]. This paper further explores the details of the experimental aspects, as well as confirming the results via numerical experiments using simulated quantum spectra. Semiclassical arguments that demonstrate this correspondence are discussed in terms of fluctuations of the density of states and correlations of the particle density. The wider implications of the present results are also discussed. 2. Experimental results The quantum dynamics of the 2-D n-disk system can be realized in a microwave experiment exploiting the mapping between the Helmholtz equation and the Schrodinger equation in their stationary forms and in the 2-D limit. This is because under the conditions of the experiment, the Maxwell-Helmholtz equations reduce to (V2 + k2)¥ = 0 with f = E: the microwave electric field. The experimental geometry consists of thin copper disks sandwiched between two large copper plates of size 55 x 55 cm2 in area. In order to simulate an infinite system, microwave absorber material ECCOSORB AN-77 was placed between the plates at the edges. Microwaves were coupled in and out through antennas inserted in the vicinity of the scatterers. All measurements were carried out using an HP8510B vector network analyzer which measured the complex transmission amplitude (S2i) and reflection amplitude (S\i) S-parameters of the coax + scatterer system. An example of the measured transmission T(k) = \S2\(k)\2 of a 4-disk system is shown in Fig. 1. For more details of the microwave experiments, see Ref. [13]. An open system can be represented by an effective Hamiltonian consisting of two parts, H = Hc + i W, where Hc © Physica Scripta 2001

Quantum Correlations and Classical Resonances in an Open Chaotic System We next compare the experimental resonances semiclassical calculations.

3

2.5 -

,

,N£ •••

•<:!

2

;1.5 -

m. 1

1-

0.5 -

n-

i

^AJV

i

k

Ur V V I A —

8

12

16

20

with

2.1. Calculation of Quantum Resonances Although any chaotic system can be quantized numerically by directly diagonalizing the Hamiltonian truncated in certain expansion space, for a large number of systems, one uses the techniques based upon semiclassical periodic orbit theory [1], such as the cycle expansion [17], Fredholm determinant [18] or harmonic inversion [19]. Except for the lowest eigenvalues, the semiclassical quantization gives very accurate results. Using Gutzwiller's trace formula [1], the semiclassical Ruelle zeta-function can be derived as an Euler product over all the prime PO which are the PO without repetition [10,11]

WGHz> Fig. 1. Experimental transmission T(k) of the 4-disk system in the fundamental domain with R/a = 4^/2 and a = 5cm. Here / = (c/2n)k. Inset: Geometry of the 3- and 4-disk billard. The solid lines represent the fundamental domain in which the experiments were carried out.

239

no

'/>.sc

(2)

with tp.se the semiclassical weight tp,K =

{-l)''exp(\kLp)/\Ap\l'2Alp.

(3)

Here lp is the number of collisions of the PO with the disks, is the Hamiltonian for the closed system and W = W^ rep- Lp is the length of the PO and Ap the eigenvalue of the instaresents the decay to open channels. Since the total Ham- bility matrix Jp. j comes in from the decomposition of the iltonian is not Hermitian, the eigenfunctions i//n of H trace of the Green's function into Ruelle zeta-functions with with eigenvalues E„ do not form an orthogonal set. Instead, j = 0, 1, • • •. The quantum resonances are the poles of the let <j>n be the eigenfunction of the adjoint operator H^ with Ruelle zeta-function which is directly related to the trace eigenvalues £*, and the i/f's and <^'s form a bi-orthonormal of scattering matrix or the trace of the Green's function. set with /\j/*m{r)(j)n(r)dr = Sm„. The Green's function for If a symbolic dynamics exists for the system, only a few an open system is [16] G{r\, r>, k) — J2n ^^j1*^. For the prime POs will be needed in the zeta-function to give quite n-disk system, with the quantum resonances in wave vector accurate eigenvalues because the curvature in the cycle space kn — sn + is'n, we have expansion will decay exponentially with increasing PO length [17]. As an example, consider the non-chaotic 2-disk system l^„(n)^n(r») with disk separation R and disk radius a. There is only \G(ri,r2,k)\2=^-J2 1 2 2 h V [(* + *n) + tf ][(* - *n) + **] one unstable prime PO between the disks. The symmetry 2m y-y, i^;(/-i)(/) n (r2)i/> m (i-i)^(r2) group of PO is Ci, with two one-dimensional irreducible + h- n,tn {k2_kl){k2_k2mT representations, symmetric A{ and anti-symmetric A2. The semiclassical resonances in wave vector space are with A = a-l + The second sum with the prime includes the off-diagonal ka =[wt + i(l/2)ln/l]/( J R-2a) terms with n ^ m. Their contribution can be neglected if y 3, there is no anaof the system, so that S2\(k) = A(k)G(ru r2, k) with A(k) a lytical expression of the semiclassical quantum resonances. slowly-varying modulating function [13]. Setting To do the numerical calculations of the quantum resonances, cn = (2m/h2)\A{k)^B(r{)(f>B{r2)\2/[{k + sn)2 + s'2l and ignor- use can be made of the symmetry of the system and the cycle ing the off-diagonal terms, we get the transmission expansion in the fundamental domain [10,11]. We have calcoefficient T(k) expressed as a sum of Lorentzian peaks culated numerically the quantum resonances of the 3-disk and 4-disk systems with different disk separation ratio a by searching the poles of the Ruelle zeta function (2). An (i) example of the results of the calculation for the •s ,^ for n ^(k-sny+s* a 3-disk system is shown in Fig. 2. The experiments only The coupling cn in Eq. (1) depend on the location of the access a small range of the quantum resonances with probes. The k dependence of ca is locally weak near a 0 < ReOO < 20GHz and -0.3GHz < Im
nv-'E

(© Physica Scripta 2001

Physica Scripta T90

240

Wentao T. Lu, Kristi Pance, Prabhakar Pradhan and S. Sridhar Table I. Experimental and theoretical RP resonances of the 4-disk system in the fundamental domain with R/a = A\fl and a — 5 cm. The RP resonances are in units of cm~~[.

-0.1

-0.2

' cxp

5=-

0.000 0.187 0.378 0.515 0.632 0.842 1.016 1.060 1.254

-0.3

If

, 40

50

60

Fig. 2. Quantum resonances in the ,-fi representation of the 3-disk system with R/a = \yft with a = 5cm in the fundamental domain. The quantum resonances we observed experimentally are in the range 0 < / < 20 GHz. See [13] for details.

2.2. Experimental Spectral Autocorrelation C(K) The spectral autocorrelation function is determined from the experimental spectra as CT(K) = [T(k — (K/2))T(k + (K/2)))k with average carried out over a band of wave vector centered at certain value k0 and of width Ak [13]. The spectral autocorrelation can be fitted well by a superposition of Lorentzians :

*w =±,n=0 E 'n^+ (K ± YnY

0.049 0.053 0.051 0.165 0.079 0.055 0.045 0.096 0.060

0.0599 0.0980 0.0910 0.1682 0.0647 0.0841 0.0785 0.1067 0.0718

I 100

Re(f)(GHz)

c

0.000 0.242 0.386 0.468 0.629 0.868 1.027 1.125 1.255

(4)

with y'n ± r/n' the classical RP resonances in wave vector space. A semiclassical derivation will be given in Sec. 3. The experimental RP resonances are obtained by fitting the experimental autocorrelation as shown in Fig. 3 with Eq. (4). Similar comparisons between experiment and theory for the /i-disk system with n = 2, 3, 4 are presented in Ref. [15]. The coupling bn in the above equation which should depend on the location of the probes determine the decay probability of the classical RP resonances. Since we do not have knowledge of them experimentally, they are chosen to optimize the fitting. The experimental RP resonances (y'„,y'a) obtained from the Lorentzian decomposition of the experimental C(K) as shown in Fig. 3 are displayed in Table I. They can be compared with theoretically calculated RP resonances, which are the poles of the classical Ruelle zeta-function, and whose calculation will be explained in the next subsection. Although the position of the peaks of the oscillations in the experimental autocorrelation in Fig. 3 is quite accurately

0.8

K^cnr1) Fig. 3. Autocorrelation function C(K) VS Klcmr1) of the 4-disk system in the fundamental domain with R/a = 4V2 and a = 5cm. The grey lines is the experimental C(/c) calculated from the experimental trace shown in Fig. 1. The data show the small n universal behavior followed by the non-universal oscillations for large K. The entire autocorrelation C(K) can be described in terms of the RP resonances using Eq. (4) (thick solid line). The thin lines represent the Lorentzian decomposition into individual RP resonances. Physica Scripta T90

© Physica Scripta 2001

Quantum Correlations and Classical Resonances in an Open Chaotic System given by the imaginary part yj,' of the RP resonances, the experimental half-width of the oscillation is almost always smaller than the calculated real part yn of the RP resonances. This is possibly because the absorber used is not ideal, and leads to quantum resonances that are slightly sharper than calculated. This results in systematically sharper widths of the experimental RP resonances. 2.3. Calculation of Classical resonances The RP resonances are essential ingredients of a Liouvillian description of the classical dynamics, and can be calculated as the poles of the classical Ruelle zeta-function. The classical dynamics of Hamiltonian flow in symplectic phase space can be described by the action of a linear evolution operator which is also the Perron-Frobenius operator C{y, x) — 6{y —f'(x)). Here f'(x) is the trajectory of the initial point x in phase space. The trace of the Perron-Frobenius operator is tr£ r = fS(x-f'(x))dx= J2P TP Y^L\ K* ~ rTP)\ det(Z — / p | . The first summation is over all the prime classical PO/> with period Tp, the second one is over the repetition r of the prime PO. Jp again is the instability matrix. The evolution operator has a Lie group structure C' = e~Al with A the generator of the Hamiltonian flow. This generator A has the classical resonances as its eigenvalues, yn = y'n ± iyJJ, which we call the RP resonances [7,8]. We have tr£' = J2Z=o SnS~y"', with gn the multiplicity of the resonances. These resonances determine the time evolution of any classical quantity. The trace formula for classical flows is obtained from the Laplace transform of the above expression [21] /KX)

tr£(s) = /

dtes'trC' = tr{A - si)'

= X>,E

QTSTP

(5)

~|det(/-/p

In order to calculate the RP resonances, the classical Ruelle zeta-function is introduced as <=/j.cl

n^-w1-

(6)

Here the product is over all the prime PO with tpci the classical weights of the periodic orbits Wi =

^V>{sTp)/\Ap\Al

(7)

Here the integer ft > 0 comes in from the expansion of the One can see that the above determinant det(7 - / p classical Ruelle zeta-function is very similar to the semiclassical one (Eq.(2)). The classical Ruelle zeta-function is exact and can be directly derived from the above trace formula (5) of the Perron-Frobenius operator. The topological pressure P(P) can be defined as the simple pole of the classical Ruelle zeta-function [22] with fi extrapolated to the entire real space. All the characteristic quantities of the classical dynamics, such as the Kolmogorov-Sinai entropy, escape rate, fractal dimensions, can be derived from P(P) [9]. For the hard-disk system, the classical velocity u of particles is constant. So Tp = Lp/v. For simplicity, one can set v = 1. The RP resonances are then calculated in wave-vector space. © Physica Scripta 2001

241

For example, the classical RP resonances of the 2-disk system are yn = [In A ± inn]/(R — 2a) with A given before. Here n is even for A[ representation and n odd for A2 representation. Similar to the situation of the quantum resonances 5n + iyn, there is no analytical expression of the RP resonances y'n ± iyj,' for the chaotic «-disk system. They can be calculated numerically as the poles of the classical Ruelle zeta-function Eq. (6) [9], 14 prime PO up to period 3 were used in our calculation of the RP resonances of the 4-disk system, the results of which are shown in Table I. 2.4. Numerical simulation To gain further insight into the established experimental correspondence, we have utilized the results of the calculations of the quantum and classical resonances to carry out a numerical simulation of the physical experiment. We calculate numerically a close approximation to the experimental quantum spectrum using the quantum resonances. We then determine the spectral autocorrelation of the simulated spectrum, and compare its decomposition with the calculated classical resonances. Writing Eq. (1) explicitly, we have the following form of the transmission T(k) ~ (2m/H2)\A(k)\2 £ n l'ftn(' l)<M'-2)| f-—T—-nr; '—7n- Here we assume the presence of the ,

2

[(k+Sa)~ +S*][(k -.!•„)- +tf] F probes poses a small perturbation and will not change the q u a n t u m spectrum. For a closed chaotic system with time-reversal symmetry, the wave density is expected to follow the Porter-Thomas (PT) distribution in R M T [23]. F o r a very open system and when the probes are far away from the scatterers, the wave density will not follow the PT distribution. But in the vicinity of the scatterers, one m a y assume the density to follow the PT distribution. Setting pln = I ^ C n ) ! 2 and p2n = |n(i-2)|2, one has, P(pn) = (2npn~p) "exp(—pa/2~p) with p being the ensemble averaged density. The densities at the location of different antennas were found not to be correlated in closed cavities [24]. We assume the same is true for an open chaotic system in the vicinity of the scatterer disks. For simplicity, we consider the ensemble average of the transmission coefficient T{kj ~ (2m/fi2)\A{k)\2plp2 E n \/[{k + sa)2 + J2] [(k - sn)2+ 2 s' ]. F r o m this expression, one can see that the contribution from broad resonances is suppressed. The main contribution is from sharp resonances. This has already been observed in our experiments [12,13]. For these sharp resonances, if they are far away from the origin, (k + sn)' + s"2 ~4k2. Thus, for large k»/n, the above transmission can be approximated as T(k) ~ (2m/n')\A(k)\2(pl'p2/4k2) ^n

[(k — sn)~ + s'2] . The function A(k) is found to be proportional to k [25,26], For convenience, we set A(k) = 2k and also (2m/^2)p[„p2« = 1- We thus get

T(k)^J2 ^[(k-sn)2 + sZY

(8)

The k dependence of A(k) can be understand as follows. In the experimental setup, one of the antenna can be regarded as a dipole radiating electromagnetic waves. The radiated electrical field of a dipole is proportional to k if an alternating current with constant amplitude / j n was maintained on it [27]. The voltage picked up on another antenna is just the electrical field at the antenna location times the length Physica Scripta T90

242

Wentao T. Lu, Kristi Pance, Prabhakar Pradhan and S. Sridhar

d of the antenna inside the cavity. The transmission amplitude S21 measured by the analyzier is the ratio of the output voltage on one antenna to the input current of another antenna, S21 = d^om/Tin* thus A(k) = d£~.in//in oc k. The expression (11) is thus used to get the averaged transmission and then to calculate the autocorrelation CT(K) numerically. We have done that for the w-disk system with n = 2, 3, 4 using a = 5 cm. Here we just present the results for the 3-disk system as shown in Fig. 4 and 5. Note that / = (c/2n)k, 1GHz is equivalent to 0.2096cm -1 . The symmetry group for the chaotic 3-disk system is Civ [10], which has two one-dimensional irreducible representations A\, An, and one two-dimensional irreducible representation E. In the fundamental domain, only the resonances of the Ai representation will contribute to the transmission [12]. The semiclassical resonances of A2 representation in

Fig. 4. Numerical spectral transmission T(k) in the fundamental domain of the 3-disk system for R/a = 4A/3 and a = 5 cm.

the range 0 < Re ka < 100 and -0.5 < Im ka < 0 for the disk separation R/a = 4-\/3 are obtained from C/,iC(iA:) withy = 0. About 200 resonances are obtained as shown in Fig. 2. The corresponding classical resonances shown in Fig. 5 are from the Ai representation of the classical Ruelle zeta-function (6) with /J = 1. 8 prime PO up to period 4 were used in our calculations of the quantum and RP resonances. The numerical transmission and the autocorrelation are shown in Fig. 4 and 5, respectively. One can see that the oscillations in the autocorrelation are fully determined by the classical RP resonances as predicted by Eq. (4).

3. Quantum and classical correlations In the above section, we have demonstrated experimentally and numerically that the quantum auto-correlation and hence the statistics of quantum resonances are determined by the RP resonances of the underlying classical dynamics. Here we explore the theoretical justification for this correspondence. For chaotic open scattering systems, the distribution and correlation of the quantum resonances have been discussed in the framework of RMT [28]. However, the RMT results are the ensemble averages of different systems with certain symmetry in common. Consequently, RMT does not make contact with system specific features of the underlying classical dynamics. Thus it is unable to make any concrete connection between the quantum and classical dynamics, for which it is necessary to go beyond RMT. The appearance of nonuniversality requires the use of structures such as periodic orbits within semiclassical calculations. To proceed, we first consider the correlation of the density of states. In the semiclassical periodic orbit theory, the fluctuation part of the density of states is [1] V{k) =

--\mtxG(ry,r2,k) ^irkL„

(9)

—\**Y.T,± r = i | d e t ( 7 - / ; ) |11/2'

The trace of the Green's function is the integration over the configuration space with r\ = *•?• The autocorrelation is

c iK

- ^jKCdkv(k+iHk-^

K K

» -

Substituting the semiclassical expression (9) of V(k) into the above equation, one gets [29] c.rsT„

V

P

7?i det(/-/p Vn

7? + (K + /n)" 0.8

1.2

K, / ( c m - 1 ) Fig. 5. Numerical spectral autocorrelation C(K) (top) and RP resonances / ± \y" (bottom) for the 3-disk system with R/a = 4 ^ 3 and a = 5cm in the fundamental domain. C(K) was calculated from the numerical spectrum shown in Fig. 4 for interval Afc = 6cirT' with the central values k0 in cm - 1 indicated in the figure. Physica Scripta T90

•+ •

-Retr

1 A-si (10)

/„

y?+(K-y'n)~

with s = WK. Since the spectrum of the generator A is v(y'n ± iy'n'), one can see that in the semiclassical theory, the autocorrelation of the density of states is determined solely by the classical RP resonances. The transmission spectrum T(k) is a projection of the density of states, one expects that the autocorrelation of the transmission is also determined by the RP resonances. The approximation in © Physica Scripta 2001

Quantum Correlations and Classical Resonances in an Open Chaotic System Eq. (10) is valid only for hyperbolic systems with finite symbolic dynamics. For these systems as explained in Sec. 2.1, the contribution to the autocorrelation C-P(K) is mainly from a few fundamental prime POs. For those PO, one has T2 ~TTP with T the average period. For weakly open or closed system, the approximation will break down. The experimental quantity whose correlations we examined in Sec. 2 is the Green's function, while theoretical arguments usually examine the density of states or the time delay [30] which are however not directly measurable experimentally. To develop the correlation theory for the transmission we measured, consider the stationary Green's function G(r, ro, k) which is the solution of the following equation (V2 + k2)G(r, r0, k) = S(r - r 0 ), with appropriate boundary conditions. The quantum mechanical propagator is the Fourier transform of the Green's function. One can construct a time evolution of a wave packet K(r, i*o, t) with AT(r, ro,*) = (2nh\Tx f^ G(r, r0, y/lmE/fr^'^dE, with e = h~k2/2m. Here the integration is performed around e0 = h~k\llm, with the range Ae = nvAk and v = hko/m the group velocity of the wave packet. The integration in the E space can be changed into that in the k space. We get K(r, r0, t\= e-,e'>'^{v/2m) fAk G(r, r0,k0+ k)e~wktdk. The propagator K(r, ro, t) is just the wave function at point r due to a (5-function excitation of the system at point ro and time to = 0. The particle density p(t) is thus

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cancelled. Thus the above expression is valid for both closed and open systems. The above arguments closely parallel Agam's [32] recent application of the diagrammatic techniques for disordered systems to open chaotic systems. We remark that the above arguments differ somewhat from those of Agam, whose results are valid when the two probes are far away from each other which implies (7"2) = 2{T}2. This is the case in the diffusive limit with a Poisson distribution of the transmission coefficient T. The above correspondence between the autocorrelation derived from semiclassical theory and the classical resonances is obviously not exact, but requires corrections due to quantum interference. These corrections are small for open systems, but are expected to dominate for closed systems. 4. Discussion and concluding remarks

The results presented here establish an alternative path from quantum to classical achieved by considering the correspondence of quantum correlations and classical decay modes. As Fig. 3 shows, the small K behavior of C(K) is universal, while the large K behavior shows non-universal oscillations that are completely described by the classical RP resonances. This demonstrates that in an open system, quantum correlation functions are intimately related to decay laws of classical survival probabilities or evolution of particle p(t) = \k(r, r0, r)|"= - ^ / G(r, r0, h + k)e~wk'dk ensembles. 1 ' 47T- J M Most theoretical calculations in quantum chaos using x f G*(r,r0,k0+k')e'vk''dk', ensemble averaging are restricted only to the leading order JAk consistent with RMT, which probes only the average properwith the average given by (p(/))(= lim^oo T~{ ties of the system, such as the average escape rate y. The f0 p(t)dt = (v2/2nL)fdk\G(k)\-. A probability measure is leading RMT contribution to the correlation C(K) is assumed for the above average to converge [7]. Here L is Lorentzian [33,34,35,36,12] C(jc) = C(0)/(1 + (/ c o s y > , (12) cal particle ensembles. The notion of ensemble averaging «=o that is common in quantum mechanics is also useful in classiwhere the coefficients b'n are the coupling, and y'n ± iy„ the RP cal treatments of chaotic systems. This is because while the resonances evaluated in wave-vector space. The Fourier prediction of individual particles is sensitive to initial conditions, the dynamics of appropriate averages of a perturbed transformation of (11) and (12) gives system relaxing towards equilibrium is well defined. The RP resonances determine the decay modes of the dynamical systems evolving from a nonequilibrium state to equilibrium or steady state. For an open system, Gaspard and Nicolis [39] with bn = 2nL2b'n/v4Ak. The autocorrelation is usually have derived an escape rate formula to relate the coefficient normalized as Cr(0) = 1 so that the system size L will be of diffusion and the escape rate of large open systems. While © Physica Scripta 2001

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Wentao T. Lu, Kristi Pance, Prabhakar Pradhan and S. Sridhar

for an open system, the escape rate y0 is nonvanishing, for closed system, y0 = 0, and the corresponding eigenstate is just the equilibrium state. The RP resonances are essential ingredients of a description of irreversibility and thermodynamics based upon microscopic chaos [40,9,41]. However these resonances have remained mathematical objects. The present experiments have provided physical reality to them. It is remarkable that we are able to observe classical resonances in a quantum experiment. Because microwaves do not interact (collide) with each other, ensemble averaging is easier to achieve in microwaves than in fluids. Thus theories that employ ensemble averaging are ideally applicable to the microwave experiments. This is true for both open systems as shown here, and for disordered billiards [42]. The present work shows that the information concerning the classical resonances is coded into the quantum spectrum. This is somewhat anticipated in periodic orbit theory, where one can observe the strong similarity between the classical (Eq.(6) and (7)) and semiclassical (Eq.(2) and (3)) Ruelle zeta-functions. A one-to-one correspondence exists between kn and yn for the 2-disk system as discussed in Sec. 1. Another example where an exact relation between the classical and quantum resonances exists is the system of a free particle sliding on a compact surface of constant negative curvature, the RP resonances were found by Biswas and Sinha [37] to be y0 = 0, yn=\± i/?n for n > 1 with real pn the solution of a certain equation while the quantal excitation eigenvalues are E„ = p\ + \. One would also expect a relation between the distribution of the classical and semiclassical resonances. The RP resonances have recently been observed in the distribution of the zeroes of the Riemann zeta function [38].

[46] calculated the level correlation and showed that it can be fully determined by the classical spectrum of the Perron-Frobenius operator. Bogomolny and Keating [47] arrived at a similar conclusion using the periodic orbit approach. The applicability of these theories for closed systems is currently a matter of debate [48], but the applicability of the supersymmetry theory to disordered systems is well established, and is strongly supported by microwave experiments on disordered billiards [23,42]. As pointed out by Berry [4], the two limits T —> oo and h —>• 0 do not commute with each other, thus the long-time quantum evolution is fundamentally different from long time classical evolution. The probability Pq that the quantum chaotic system decays at time t after its formation follows a power law [49,24] Pq(t) ~ (1 + 2Tty2-M2, for / larger than the equilibrium time of the system. Here T is the average decay width, M the number of open channels. As the system is opened up with more channels, the decay will be almost exponential since the above equation can be approximated by Pq(t) ~ exp[—(4 + M)Tt\, for large M and the weight of the algebraic tail becomes negligible. More precisely, as pointed out in Ref. [50], there exists a new quantum relaxation time scale tq < /H, with ?H = #/A the Heisenberg time. Beyond that time, the quantum correlation decays algebraically. For shorter times, the quantum correlation decays exponentially in the same way as the classical correlation does. For an open system, since the spectrum is continuous, the Heisenberg time is actually infinite, and so is the quantum relaxation time tq. This is one reason that the classical resonances are clearly visible in the present experiments, and the Lorentzian decomposition of the spectral autocorrelation works so well.

The transmission spectrum measured in the microwave experiment directly corresponds to the conductivity Our work demonstrates that suitable quantum correlations diffuse just like classical observables in an open measured in electronic quantum dots. In quantum dot expersystem. An interesting question is what will happen when iments though, the magnetoconductivity
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Quantum Correlations and Classical Resonances in an Open Chaotic System References 1. Gutzwiller, M. C , "Chaos in Classical and Quantum Mechanics", (Springer, New York, 1990). 2. Bohigas, O., "Chaos and Quantum Physics". (Edited by Giannoni, M.-J., Voros, A., and Zinn-Justin, J.) (Elsevier Science Publishers, 1991), p.87 and references therein. 3. Guhr, T., Miiller-Groeling , A. and Weidenmuller, H. A.. Phys. Rep. 299, 189 (1998) and references therein. 4. Berry, M., "Chaos and Quantum Physics". (Edited by Giannoni, M. -J., Voros, A., and Zinn-Justin, J.) (Elsevier Science Publishers, 1991), p251. 5. Gaspard, P. and Rice. S. A., J. Chem. Phys. 90, 2225 (1989); 90, 2242 (1989); 90, 2255 (1989); 91, E3279 (1989). 6. Kadanoff, L. P. and Tang, C , Proc. Natl. Acad. Sci. USA 81, 1276 (1984). 7. Ruelle, D., Phys. Rev. Lett. 56, 405 (1986); J. Stat. Phys. 44, 281 (1986). 8. Pollicott, M., Inv. Math. 81, 413 (1985); ibid. 85, 147 (1986); Ann. Math. 131, 331 (1990). 9. Gaspard, P. and Alonso Ramirez. D„ Phys. Rev. A 45, 8383 (1992); Gaspard, P., "Chaos, Scattering and Statistical Mechanics", (Cambridge University press, Cambridge, 1998). 10. Cvitanovic. P. and Eckhardt. B., Nonlinearity 6, 277 (1993); Cvitanovic, P., Artuso, R., Mainieri, R., and Vattay. G.. "Classical and Quantum Chaos: a Cyclist Treatise", http://www.nbi.dk/ Chaos Book/. 11. Gaspard, P., Alonso, D., Okuda, T. and Nakamura, K., Phys. Rev. E 50, 2591 (1994). 12. Lu, W., Rose, M., Pance, K. and Sridhar, S„ Phys. Rev. Lett. 82, 5233 (1999). 13. Lu, W., Viola, L„ Pance, K., Rose, M. and Sridhar. S., Phys. Rev. E 61, 3652 (2000): ibid. 62, 4478 (2000) and references therein. 14. As discussed in subsequent sections, the POs underly the description of both the quantum and classical resonances. 15. Pance, K... Lu, W. and Sridhar, S„ Phy. Rev. Lett. 85, 2737 (2000). 16. Datta, S., "Electronic Transport in Mesoscopic Systems", (Cambridge University Press, Cambridge, 1995). 17. Cvitanovic, P. and Eckhardt. B., Phys. Rev. Lett. 63, 823 (1989). 18. Cvitanovic, P., Rosenqvist, P. E. and Vattay, G., Chaos 3, 619 (1993). 19. Main. J., Phys. Rep. 316, 233 (1999). 20. Wirzba, A., Chaos 2. 77 (1992). 21. Cvitanovic. P. and Eckhardt, B., J. Phys. A 24, L237 (1991). 22. Ruelle, D., "Thermodynamics Formalism", Encyclopedia of Mathematics, Vol. 5 (Addison-Wesley, Reading, MA, 1978). 23. Kudrolli, A„ Kidambi, V., and Sridhar, S., Phys. Rev. Lett. 75, 822 (1995). 24. Alt, H. et al. Phys. Rev. Lett. 74, 62 (1995).

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25. Stein, J., Stockmann, H.-J. and Stoffregen, U., Phys. Rev. Lett. 75, 53 (1995). 26. Stockmann, H.-J., "Quantum chaos: an introduction", (Cambridge University Press, Cambridge, 1999). 27. Jackson, J. D., "Classical Electrodymanics", second edition, (John Wiley & Sons, Inc., 1975). 28. Fyodorov, Y. V., Titov, M. and Sommers, H. -J„ Phys. Rev. E 58, R1195 (1998); Fyodorov, Y. Y. and Khoruzhenko, B. A., Phys. Rev. Lett. 83, 65 (1999), Fyodorov, Y. V. and Sommers, H. -J., J. Math. Phys. 38, 1918 (1997). 29. Cvitanovic, P., private communication. 30. Eckhardt, B., Chaos 3, 613 (1993). 31. Christiansen, F.. Paladin, G. and Rugh, H. H., Phys. Rev. Lett. 65,2087 (1990). 32. Agam, O., Phys. Rev. E 61, 1285 (2000). 33. Bliimel, R. and Smilansky, U., Phys. Rev. Lett. 60. 477 (1988). 34. Jalabert, R. A., Baranger, H. U. and Stone, A. D., Phys. Rev. Lett. 65, 2442 (1990). 35. Lewenkopf, C. H. and Weidenmuller, H. A.. Ann. Phys. 212, 53 (1991). 36. Lai. Y.-C. Bliimel, R.. Ott, E. and Grebogi, C , Phys. Rev. Lett. 68, 3491 (1992). 37. Biswas, D. and Sinha, S., Phys. Rev. Lett. 71, 3790 (1993). 38. Bohigs, O., Leboeuf, P. and Sanchez, M. J., arXiv: nlin.CD/0012049. 39. Gaspard, P. and Nicolis, G., Phys. Rev. Lett. 65, 1693 (1990). 40. Hasegawa, H. H. and Saphir, W. C . Phys. Rev. A 46, 7401 (1992). 41. Dorfman, J. R., "An introduction to chaos in nonequilibrium statistical mechanics", (Cambridge University Press, Cambridge, 1999). 42. Pradhan, P. and Sridhar, S., Phys. Rev. Lett. 85, 2360 (2000). 43. Efetov, K. B., Adv. Phys. 32, 53(1983); Efetov, K. B„ "Supersymmetry in Disorder and Chaos", (Cambridge University Press, Cambridge, 1997). 44. Andreev, A. V. and Altshuler, B. L., Phys. Rev. Lett. 75, 902 (1995). 45. Mirlin, A. D., Phys. Rep. 326, 259 (2000). 46. Agam, O., Atshuler, B. L. and Andreev, A. V . Phys. Rev. Lett. 75.4389 (1995). 47. Bogomolny. E. B. and Keating, J. P., Phys. Rev. Lett. 77, 1472 (1996). 48. Prange, R. E., Phys. Rev. Lett. 78, 2280 (1997). 49. Harney, H. L., Dittes, F.-M., and Miiller. A.. Ann. Phys. (N.Y.) 220. 159 (1992). 50. Casati, G., Maspero, G., and Shepelyansky, D. L., Phys. Rev. E 56, R6233 (1997); Savin, D. V. and Sokolov, V. V., ibid. 56, R4911 (1997); Fraham, K.. M., ibid. 56. R6237 (1997). 51. Agam, O., Phys. Rev. A 60, R2633 (1999). 52. Weber, J., Haake, F. and Seba, P., Phys. Rev. Lett. 85. 3620 (2000). 53. Ketzmerick. R., Phys. Rev. B 54, 10841 (1996).

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Physica Scripta. T90, 246-247, 2001

Why do an Experiment, if Theory is Exact, and any Experiment can at best Approximate Theory? H.-J. Stockmann* Fachbereich Physik der Universitat Marburg, Renthof 5, 35032 Marburg, Germany Received July 12, 2000

PACS Ref: 03.65.Ge, 03z.65.Sq, 05.45.Mt, 73.20Fz, 73.23.-b

Abstract A personal view is given on the interplay between classical wave experiments and theory.

When I was invited to my first conference on the quantum mechanics of classically chaotic systems, organized by Fritz Haake and Robert Graham 1990 in Nordkirchen, the number of experimental contributions was below 10 percent. Besides me, presenting our first results on chaotic microwave cavities [1], the only other experimentalists were Peter Koch who gave a talk on his Rydberg atoms in strong microwave fields [2], and Karl-Heinz Welge reporting on the hydrogen atoms in strong magnetic fields [3]. Meanwhile the situation has changed completely. Now there are two further groups working very actively on wave chaos in microwave resonators, headed by Srinivas Sridhar [4] and Achim Richter [5] (not to speak of other groups contributing with a number of isolated but impressing results [6-9]). The experiments with classical waves have been completed by Clive Ellegaard and his coworkers on vibrating blocks and plates [10]. A completely new type of quantum chaos experiments came up, when semiconductor microstructures of submicron size became available. Here in particular by Charlie Marcus with his experiments on quantum dot billiards has to be mentioned [11], followed by a number of other groups, several of them presented at this symposium (for a more complete account on the billiard experiments see Chapter 2 of Ref. [12]). This enormous increase of experimental activities has left its traces in the present meeting as well. Now about one third of the talks have been given by experimentalists, and one further third were on topics directly influenced by experimental work. But are the studies of classical waves experiments at all? A number of colleagues object that in particular the microwave billiards are just some specialized analog computers. They argue that stationary Schrodinger equation and Helmholtz equation are completely equivalent, and that therefore nothing can be learnt from experiment, which could not have been obtained equally well or even better from theory. From an extreme standpoint one might even ask the question I have chosen for the title of this contribution, which has been taken from a referee report on one of our recent publications (ironically, the theory was not even exact in this case, but based on second order perturbation theory only!). Interestingly enough, this question has never been asked in the context of hydrogen atoms in strong magnetic fields. Here it has become accustomed to plot experimental and

theoretical spectra as mirror images of each other to demonstrate the agreement between both spectra [13] (on this workshop we have seen a number of examples of this type). Is hence the hydrogen atom in a magnetic field nothing but an analog computer? Maybe the people working in the field are partly responsible themselves for this classification. Until recently I had no problems to speak of analog experiments in the context of the microwave billiards. This changed when Fritz Haake asked me to write together with him a popular article for the Physikalische Blatter on the implications of spectral level dynamics on the random matrix behaviour of chaotic spectra [14] (a topic touched by Martin Zirnbauer in his talk, see also Ref. [15]). In the draft of my part Fritz commented any occurrence of the term "analog experiment" with expressions like "abominable", "shame", "disgusting", etc.. I promised to improve and to use the term "classical wave experiment" from now on. Hopefully all of us will agree on the point that the label we attach to these types of experiments is not of relevance. The only thing that counts is whether they are able to produce interesting new results or not. In this respect the classical wave experiments must not be afraid of any comparison. Take as an example the experiments presented in the session, of which I had the honor to be the chairman. Srinivas Sridhar showed results of the localization-delocalization transition of wavefunctions in disordered systems and compared them with theoretical predictions. The theoretical efforts to describe wavefunctions and all types of mutual correlations increased enormously in recent years [16,17] which undoubtedly is also due to the fact that these quantities are now accessible to the experiment, which was not the case prior to the introduction of the microwave techniques. In addition there is an increasing number of theoretical works proposing to check theory in a microwave experiment [18,19]. In the second talk of this session Clive Ellegaard presented recent results on the spectra of vibrating blocks and plates. These experiments are quite far from quantum mechanics, since both transversal and longitudinal vibration modes are present, and boundary conditions are very complicated due to mode conversion during reflection. In spite of this fact the spectra are in complete accordance with random matrix predictions (the same is true, by the way, for three-dimensional microwave resonators, where there is no direct correspondence to quantum mechanics as well [20,21]). This shows that the range of applicability of the Bohigas-Giannoni-Schmit [22] conjecture is much wider as it was originally foreseen by the authors. This fact should be taken into account in all attempts to prove this conjecture

* e-mail: stoeckmann(oJphysik.uni-marbug.de Physica Scripta T90

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Paper Title (see the talk by Martin Zirnbauer). Though it was pointed out by Michael Berry on this meeting that the wave equation for vibrating blocks can be mapped again to the Schrodinger equation, it does not seem obvious to me whether the same is true for the boundary conditions as well. The best what can happen to an experimentalist is an outcome of the experiment which is not in accordance with the theoretical prediction. We met this situation twice, first when we studied different types of spectral level dynamics in chaotic and disordered systems [23], second when looking for resonance widths in a billiard with one attached open channel in dependence of the opening. According to theory in the limit of large openings the widths of all but one resonance should approach zero. Some years ago I became convinced from discussions with colleagues from theory that it should be impossible to approach this regime in a microwave billiard. Therefore I was somewhat hesitating when Emil Persson, a coworker of Ingrid Rotter, Dresden, who joined our group for three months, proposed to do just this experiment. My reservations were unfounded, however, since in a very short time Emil succeeded in demonstrating the effect [24], When presenting the results to Hans Weidenmiiller in the poster session of this meeting, he commented them with the words: "Never believe a theoretician!" The organizers asked me to give a personal account of my impressions of the symposium. I decided to focus on the topic concerning me most, namely the mutual interplay between theory and experiment. I would like to end with a thank to the organizers, Karl-Fredrik Berggren, Par Omling, and Sven Aberg for inviting me to this beautiful

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and exciting symposium, and the Nobel Foundation for the generous support. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

Stockmann, H.-J. and J. Stein. Phys. Rev. Lett. 64, 2215 (1990). Galvez, E. et at. Phys. Rev. Lett. 61, 2011 (1988). Holle. A, et at. Phys. Rev. Lett. 56, 2594 (1986). Sridhar, S., Phys. Rev. Lett. 67, 785 (1991). Graf, H.-J. et at, Phys. Rev. Lett. 69, 1296 (1992). Doron. E„ Smilansky, U. and Frenkel, A., Phys. Rev. Lett. 65. 3072 (1990). Lauber, H.-J., Weidenhammer, P. and Dubbers, D., Phys. Rev. Lett. 72. 1004 (1994). Sirko, L„ Koch, P. and Bliimel, R.. Phys. Rev. Lett. 78, 2940 (1997). Hersch, J., Haggerty, M. and Heller. E., Phys. Rev. Lett. 83. 5342 (1999). Ellegaard, C. et at, Phys. Rev. Lett. 75. 1546 (1995). Marcus, C. et at. Phys. Rev. Lett. 69, 506 (1992). Stockmann, H.-J., "Quantum Chaos - An Introduction", (University Press, Cambridge. 1999). lu. C. et at, Phys. Rev. Lett. 66, 145 (1991). Haake. F. and Stockmann, H.-J.. Phys. Bl. 56, 6:27 (2000). Braun, P. et at, arXiv:nlin.CD/0006022. Janssen, M.. Phys. Rep. 295, 1 (1998). Mirlin, A., Phys. Rep. 326, 259 (2000). Tschersich, A., Efetov, K. B., arXiv:cond-mat/9911284. Narevich, R., Prange. R. and Zaitsev, O., Phys. Rev. E 62, xxx (2000). Deus. S., Koch, P. and Sirko. L., Phys. Rev. E 52, 1146 (1995). Alt, H. et at, Phys. Rev. Lett. 79, 1026 (1997). Bohigas. O., Giannoni, M. and Schmit. C , Phys. Rev. Lett. 52, 1 (1984). Barth, M., Kuhl, U. and Stockmann, H.-J., Phys. Rev. Lett. 82, 2026 (1999). Persson, E., Rotter, I., Stockmann, H.-J. and Barth, M., Phys. Rev. Lett. 85, 2478 (2000).

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Physica Scripta.Vol. T90, 248-262, 2001

Wave-Chaotic Optical Resonators and Lasers A. Douglas Stone Applied Physics, Yale University, P.O Box 208284, New Haven CT 06520. USA Received November 22, 2000

PACS Ref: 05.45-a, 05.45.Mt, 42.55.Sa, 42.60.Da

Abstract Deformed cylindrical and spherical dielectric optical resonators and lasers are analyzed from the perspective of non-linear dynamics and quantum chaos theory. In the short-wavelength limit such resonators behave like billiard systems with non-zero escape probability due to refraction. A ray model is introduced to predict the resonance lifetimes and emission patterns from such a cavity. A universal wavelength-independent broadening is predicted and found for large deformations of the cavity. However there are significant wave-chaotic corrections to the model which arise from chaos-assisted tunneling and dynamical localization effects. Highly directional emission from lasers based on these resonators is predicted from chaotic "whispering gallery" modes for index of refraction less than two. The detailed nature of the emission pattern can be understood from the nature of the phase-space flow in the billiard, and a dramatic variation of this pattern with index of refraction is found due to an effect we term "dynamical eclipsing". Semiconductor lasers of this type also show highly directional emission and high output power but from different modes associated with periodic orbits, both stable and unstable. A semiclassical approach to these modes is briefly reviewed. These asymmetric resonant cavities (ARCs) show promise as components in future integrated optical devices, providing perhaps the first application of quantum chaos theory.

1. Introduction The body of modern research in quantum mechanics which has come to be known as "quantum chaos" has two main strands. One strand is the application of statistical methods to complex and/or disordered quantum systems at T = 0. Historically this approach began in the 1950's with the work of Wigner, Dyson and others [1,2] who developed random matrix theories (RMT) to describe statistically the complex spectra of compound nuclei. This approach has had a recent renaissance and further development in the study of mesoscopic condensed matter systems [3-6]. The other strand is the attempt to extend semiclassical methods to treat quantum hamiltonians with fewer constants of motion than degrees of freedom, for which at least some regions of phase space have chaotic dynamics. This approach is not as old as the statistical approach, having been pioneered by Martin Gutzwiller in the late 60's [7-11], and extensively developed in the last two decades. Two major links between the approaches were established in the 80's with the Bohigas-Giannoni-Schmidt conjecture [12] and the semiclassical derivation by Berry of the A3 energy-level statistics for the chaotic and integrable limits [13]. The BGS conjecture hypothesized that fully chaotic classical dynamics led to RMT statistics for the corresponding quantum system; the Berry work derived this result from semiclassical considerations and unitarity for a specific statistical quantity. As discussed by Zirnbauer [14], the precise statement and range of validity of the BGS conjecture is still unclear, but the qualitative correctness of the conjecture is certainly well-established by numerical studies. Physica Scripta T90

The statistical methods have the attraction of allowing in many cases explicit analytic calculations of statistical quantities which are universal in the sense that they describe all fully chaotic hamiltonians, at least in some parameter regimes. The semiclassical approach has a different emphasis. First, there is the basic question of what are the fundamental limits of accuracy of the semiclassical description of chaotic or mixed hamiltonians. Equally important however is the goal of explaining quantum properties in terms of classical dynamics in order to gain a better intuitive understanding of quantum systems with sufficiently low symmetry that no exact analytic wavefunctions and energy eigenvalues can be found. Research in this direction has produced a number of approximate methods for calculating quantum properties of interest, valid in the short wavelength (Planck's constant goes to zero) limit, the most notable of these being periodic orbit theory based on Gutzwiller's Trace formula [7] and its recent reformulations and extensions [11]. These methods have been applied to explain experimental results in atomic and condensed matter physics over the past two decades. It has been recognized for some time that the same concepts and approximation methods should also apply to classical linear wave equations such as Maxwell's equations or the wave equation of acoustics; hence the term "wave chaos" has been introduced to include this wider class of applications. Extensive work has been done on both the semiclassical and statistical analysis of resonances in 2D microwave cavities [15,16], however with a strong emphasis on confirming previous theory, and not motivated by the intrinsic interest of these systems. In the current paper we will review the work done by the Yale group applying classical and semiclassical methods to understand and predict experimental properties of a class of dielectric optical resonators and lasers which are of great interest to optical physicists but for which very little previous theory existed. Optical resonator theory is particularly well-suited to the application of semiclassical methods. Photons are non-interacting (or weakly-interacting in a non-linear medium), one has rather well-controlled experimental systems to study, and solutions of the wave equation in highly non-symmetric geometries may be of great practical interest for technology. Indeed our work reported below has led to what we believe are the first patented inventions arising from research in quantum/wave chaos [17]. Finally, "theory" in classical electromagnetism has been in most cases reduced to pure numerical solution of the wave equation. We hope through our work to demonstrate that there is still a value to approximate analytic theories and intuitive physical © Physica Scripta 2001

Wave-Chaotic Optical Resonators and Lasers reasoning in these areas, but that some new concepts need to be introduced. The formal analogy we are pursuing is thus between the quantum mechanics of particles and classical electromagnetism. The classical electric and magnetic fields will be the analog of the wavefunctions of the Schrodinger equation; we will not treat here any phenomena of what is conventionally called quantum optics. Maxwell's wave equation and its reduction to the Helmholtz equation of course already include such "quantum" effects as tunneling and interference. Finally, in the extreme X -* 0 limit of the wave equation one can apply ray optics and this limit is the analog of classical mechanics for the Schrodinger equation. It should be noted however that the formal analogy is not perfect. The interaction of light with matter is described by the dielectric function in the wave equation which multiplies the eigenvalue, k2, of the Helmholtz equation, unlike the potential in the Schrodinger equation which does not multiply the energy eigenvalue. This means that if one regards the dielectric function as the analog of the potential term in the Schrodinger equation, then in the k -»• 0(k -*• oo) limit a dielectric discontinuity does not become negligible. This adds a fundamentally stochastic aspect to the dynamics exemplified by the phenomenon of Fresnel reflection and refraction at a dielectric interface. This disanalogy is not important for experiments within a region of uniform dielectric constant, as in most of the microwave experiments, but is important for the resonator physics we will be discussing. Another important difference between the wave equation and the Schrodinger equation is the vector nature of the wave equation; here however we shall primarily describe geometries for which this difference is not crucial and the polarization degrees of freedom are separable.

2. Optical resonators and ARCs Optical resonators for conventional lasers are typically made using metallic mirrors, usually in the parallel linear configuration known as the Fabry-Perot resonator [18].

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The directional properties and Q-values of such resonators are controlled by the transmissivity of the mirrors and their size and spacing. However in attempting to make high-g resonators on the few-micron size scale many groups (see Fig. 1) have employed dielectric resonators which use total internal reflection to trap light incident at oblique angles [19]. It is such dielectric resonators that we will attempt to describe with the models presented below. If such a resonator is cylindrical or spherical simple geometric considerations imply that the angle of incidence, x- of a circulating ray is conserved and if Snell's law were rigorously obeyed a ray with sin x>\/n (where n is the index of refraction) would remain trapped indefinitely (see Fig. 1). Wave solutions corresponding to such a trajectory are known as whispering gallery (WG) modes, in analogy to similar modes, of acoustic cavities analyzed by Lord Rayleigh. A simple analysis of the wave equation [20] for the spherical or cylindrical dielectric cavity shows that light is not trapped forever by total internal reflection (which only holds rigorously for an infinite flat surface); due to the curvature of the surface the light will eventually escape by evanescent leakage, a process which is equivalent to the tunneling of photons through the angular momentum, barrier. It is this process which determines the g-value of the resonances of ideal dielectric cylinders or spheres (although in practice impurities, absorption or other imperfections may dominate). The smallest micro-lasers ever made were fabricated in the early 90's based on cylindrical (disk-shaped) dielectric cavities and were shown to provide low threshold and relatively high Q [21,22]. However the emission from WG modes is isotropic unless additional guiding elements are added to the resonator, and these microdisk lasers did not provide controlled directional emission or adequate output power for applications. The conservation of the angle of incidence of a ray mentioned above is equivalent to the conservation of angular momentum implied by rotational symmetry; therefore the dynamical barrier to ray escape is due to the existence of constants of motion which make the ray dynamics integrable. Thus it is natural to ask what will be the effect

Fig. I. (Upper left) Quantum cascade semiconductor micro-cylinder laser from ref. [46]; (upper right) Ethanol micro-droplet containing rhodaminc dye lasing under optical pumping (see ref. [19.43.44]). Both lasers emit from whispering gallery modes of the type depicted in the schematic (center). For a cylindrical or spherical boundary the angle of incidence•/_is conserved. In the current work we will analyze the effect of smooth deformations of cylinders (bottom left) and spheres (bottom right) on the resonance spectrum: in this case the angle of incidence will fluctuate in a manner determined by the mixed ray dynamics. A generic deformation we will study extensively is the quadrupolar cylinder ARC with a boundary given by R{<j>) = R(\ + ECOS2) © Physica Scripta 2001

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of breaking this rotational symmetry. If the symmetry is broken by a rough deformation of the boundary then the assumption of specular reflection will fail (one will have internal Rayleigh scattering instead) and the ray-dynamical description will not be very helpful. However if the boundary is deformed in a smooth manner one will obtain a smooth KAM-like transition to chaotic ray dynamics. It is an interesting and challenging problem in wave chaos theory to understand the corresponding change in the resonance properties of this system. We refer to generic smooth, convex deformations of the boundary from circular symmetry as ARCs (asymmetric resonant cavities - see Fig. 1). We develop a resonator theory for ARCs below. By assumption, a resonator has some coupling to the external world, so instead of orthogonal normal mode solutions at discrete real frequencies, the wave equation for a resonator has "resonant" modes (sometimes called quasi-normal modes [23]) centered around a frequency to with a width y = r~', the Q-value of the mode is defined by Q = an. For light to = ck, where c is the speed of light in vacuum and k = 2%jX is the wavevector in vacuum, and we will henceforth give results in terms of the dimensionless frequency, kR, (R being the mean radius of the resonator) and dimensionless width, T = yR/c. Resonances appear in linear optics as rapid variations with frequency of the elastic lightscattering cross-section from an object illuminated monochromatically from outside. In rotationally symmetric geometries each such resonance can be labelled by its angular momentum (or ^-component thereof)- In a non-linear medium it is possible to have states which emit with no incoming wave, for example the lasing modes of an active medium under external pumping. It is known that the width of the scattering resonances in linear optics can also be obtained by assuming no incoming wave and solving the boundary conditions at discrete complex k = q — iT [18,23]. This is the method we use in all of our numerical results and in the semiclassical method sketched at the end. In an active medium one may think of these resonances as being pulled " u p " to real wavevectors by the gain. Although for describing the laser modes discussed below we should in principle find the radiating solutions of the non-linear wave equation with an active medium [24], we will focus on the easier problem of the linear resonances in the current work, an approximation which is often used in laser theory [18]. A key point that differentiates this work from most previous applications of semiclassical theory to chaotic systems is that our focus is on the width r and the anisotropic emission pattern 1(6) of the resonant levels induced by the dynamics and not on their real energies (wavevectors).

3. ARCs as open quantum billiards The wave equation within a smooth convex boundary is generically non-separable (the ellipse being the only exception [25]) and no solution in terms of special functions arising from one-dimensional ODE's is possible. Moreover, typical cavities of interest are in the 5-50 micron size range, and their mode-spacing at optical wavelengths is very small compared to the perturbation induced by the deformations of interest (in the range of 5-20% of the unperturbed radius). Physica Scripta

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Hence one may not treat these deformations by applying perturbation theory to the undeformed case. The small parameter in the problem is not the strength of the deformation but rather the ratio of the light wavelength to the perimeter of the cavity, (kR)~\ motivating a semiclassical treatment. In fact a great deal of insight can be obtained by beginning with the extreme short wavelength limit for which a ray optical description is valid [26,27]. In this limit the system is similar to the well-studied classical billiard problem except that a ray has a non-zero probability of escape at each reflection from the boundary. As noted above, this escape probability will be exponentially small for angles of incidence above the critical (Snell) angle since it is due to tunneling, but will become order unity below the Snell angle. A model treatment of this refraction process from a curved surface due to Nockel [27] indicates that for typical values of kR the sub-critical reflection probability is very close to the Fresnel formula for a flat interface. We employ this model in the ray simulations below. There is an enormous literature on billiards as model dynamical systems. Much work has focused on specific shapes which can be proven to generate fully chaotic dynamics, such as the Sinai billiard or Bunimovich stadium billiard. In this work we will be interested in families of billiards which are characterized by a deformation parameter, E, and which reduce to a circle when e, —>• 0 (the billiard represents the cross-section of a deformed cylindrical resonator, see Fig. 1). Since the hamiltonian of a billiard is not analytic at the boundary these dynamical systems do not satisfy all of the conditions of conventional KAM theory for the smooth transition to chaos; nonetheless, due to work of Lazutkin and others [28,29], it is well-established that such billiard families do make a smooth transition to chaos as the deformation is increased from zero according to the KAM scenario. This scenario is illustrated by the surface of section plots in Fig. 2. Here we take the SOS coordinates to be tfi, sin/, where 4> is the azimuthal angle at which the ray collides with the boundary and sin/ is the angle of incidence at the boundary with respect to the normal. Here and elsewhere (unless otherwise specified) the boundary will be defined by the quadrupolar deformation, R(4>) = ^o(l + £ cos2^>); this is the smoothest pure deformation possible within a multipole expansion (the dipole is a shift of the center of mass to order e). Initially we discuss the ideal billiard problem, without the refractive escape process which occurs in the corresponding problem in ray optics. At zero deformation (e = 0) one sees the straight lines corresponding to conserved angular momentum (L oc sin / ) . At intermediate deformation (e = 0.05) one sees chaotic layers appearing along with stable periodic orbits, but also surviving KAM tori and limited motion in sin/. Orbits with large initial values of sin/ remain near their initial values; only orbits in the chaotic layer associated with the unstable period-two orbit along the major axis can reach normal incidence (sin/ = 0) and even reverse their sense of rotation. In contrast, at deformations larger than 10% much of phase space is chaotic, essentially all the KAM tori have been broken, and most orbits will explore almost all possible angles of incidence (although there do remain some stable islands which can play a role as well). (Q Physica Scripta

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Wave-Chaotic Optical Resonators and Lasers Surfccs of Sacilon for the Quqdruoole at c— 0

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/•<£. 2. Four surface of section plots for the quadrupole billiard with £ = 0,0.05, 0.1, 0.2 using the coordinates sin/, (j>, where/ is the angle of incidence and <^>is the angular position of the bounce point of a ray on the boundary. No escape processes are included at this point. Straight lines at zero deformation correspond to conserved angular momentum and large values of sin/ correspond to whispering gallery modes (negative values of sin/ corresponding to opposite sense of circulation are not shown - the SOS has reflection symmetry). With increasing deformation one sees the growth of chaos according to the K.AM scenario and the destabilization of orbits in the whispering gallery region. Note the large chaotic separatrix region around the stable 2-bounce orbit at ij> = ±ir/2.

4. Refractive escape The qualitative consequences of these ray dynamical properties for the resonances of ARCs may be guessed using "classical-quantum" (ray/wave) correspondence. There are three regimes illustrated by the three trajectories in Fig. 3. At zero deformation we have conventional whispering gallery modes, and a direct connection between orbits and modes of the wave equation can be established following Einstein-Brillouin-Keller semiclassical quantization (if we neglect escape) [30]. The Q-values of these resonances for sin# > \jn are determined by the tunneling process mentioned above. For intermediate deformation rays at high angle of incidence remain confined to the region of phase space sinj; -> 1* (the blue trajectory in Fig. 3) and we expect there still to be high-Q whispering gallery-type modes, even though the deformation mixes many angular momenta. Their g-values will still be determined by tunneling, but (we shall see) a more complex type of tunneling process now appears to be relevant. For large deformations the ray dynamics is ergodic on the large chaotic component and does not allow trapping at large angles of incidence (the black trajectory in Fig. 3), hence we expect much lower Q resonances. In this regime there is a fundamentally new decay mechanism for the resonance which we call refractive .£) Physica Scripta 2001

escape. Instead of tunneling out of the resonator, light is emitted by chaotic diffusion from high to low sin^ which (if this process is slow enough) can lead to chaotic whispering gallery modes which are localized in phase space at high sin# but have lifetimes controlled by the time it takes to diffuse to the critical value of sin# = l/«(seeFig. 3). These modes were first identified from our earlier work [26,27], and it is the lifetime and emission pattern of such modes that we are able to describe within the ray model defined in the next section. At larger deformations and high index of refraction new modes appear which are not of the whispering gallery type and which we will describe in later sections of the paper.

5. Summary of the ray model The goal of the ray model is to calculate the widths and emission patterns of ARC resonances based on ray dynamics in the regime of large deformation where decay occurs by chaotic diffusion and refractive escape. We put forward the essential idea of the model in 1994 [26]; the more precise model discussed here was proposed and tested experimentally in 1996 [31]and both its successes and limitations were demonstrated in comparison to numerical results by Nockel and Stone in 1997 [32]. Here is a summary of the model. Physica Scripta T90

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A. Douglas Stone Whispering gallery orbits in the quadrupole

e-

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Fig. 3. Three trajectories corresponding to the same initial conditions but different degrees of quadrupolar deformation. At zero deformation (red) sin/ is conserved and we have a conventional quasiperiodic whispering gallery orbit. At 5% deformation the orbit (blue) is deformed but follows an unbroken KAM curve with bounded variations in sin/. At 10% deformation the initial region of phase space is in the chaotic sea and the trajectory undergoes chaotic diffusion to lower values of sin/. When the value of sin/ fall below l/« (the green region) refractive escape will occur. One sees that despite the chaotic motion, due to a dominant flow pattern in phase space, the escape is going to occur predominantly near <j> = 0, n. (See the discussion of directional emission below). The corresponding real space trajectories (without escape) are shown below.

1.

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A correspondence is made between a set of initial 6. Resonance widths for chaotic WG modes conditions for rays in phase space and a set of WG resonances via an adiabatic approximation. In If the ray model is valid, then there exist a class of chaotic convex billiards there exists an approximate WG modes which are peaked at high angular momentum (adiabatic) conservation law [28] which becomes exact as are conventional WG modes, but which emit not by as sinx —* 1, leading to motion along the curves tunneling but by diffusion to lower angular momentum sin#() = [1 — (XK((j))2/i]l/2((x is the adiabatic constant (sin^ < !/")• Moreover these modes are concentrated in in this motion, K(<1>) is the curvature of the boundary phase space along the curves sin/((/>) determined by the adiaat angle <j>). In Figs. 4 and 6 we show examples of these batic approximation. In Fig. 4 we show the numerically adiabatic curves for the quadrupole. determined Husimi distributions for two resonances correWidths r for these resonances and emission patterns sponding to chaotic whispering gallery modes. The width 1(0) are calculated by propagating an ensemble of rays of such resonances according to the ray model is just the with the prescribed initial conditions and allowing inverse of the diffusive pathlength L for escape and is indeescape at each collision according to an escape rule pendent of the wavevector k of the resonance. This is a strong [27,32] which includes (approximately) both refractive prediction of the model and is in marked contrast to conand evanescent emission. Specifically, the resonance ventional WG resonances for which tunneling leads to an width (in A:-space) is given by T =< R/L >, where L exponential sensitivity to wavevector. This "universal is the length within the resonator of escaping broadening" of chaotic WG modes will only set in above trajectories and R is the radius of the undeformed the critical deformation, ec, at which KAM barriers to phase cavity. The angular intensity pattern of emitted space diffusion have been broken. This prediction is verified radiation 1(0) is given by the angular distribution of by the numerical data in Fig. 5a, where we see that the the escaping rays of this ensemble taking into account resonance lifetime in the diffusive regime is universal and semiquantitativery given by the ray calculation. Note also refraction upon escape.

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* Fig. 4. Husimi (phase space) distributions for two numerically determined resonances compared to the adiabatic invariant curves of the quadrupole which arc plotted on the SOS. Horizontal lines indicate the escape angle for n = 2 (left) and n= 1.36 (right). Clearly these states arc localized at high angular momentum and follow approximately the invariant curves, but with significant inhomogeneity along the curve. The n = 2 resonance has high emission amplitude at 4> = 0,n. representing the generic behavior; whereas the ra= 1.36 resonance has high emission amplitude displaced from (j> = 0.n. by the presence of the stable four-bounce islands. This leads to the dynamical eclipsing effect discussed below (sec also Figs. 6.7.8 below).

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Deformation e Fig. 5. (a) Logarithm ofthcdimcnsionlcss resonance width'//? vs. deformation calculated numerically for two resonances with kR = 12.1 (solid) and kR = 44.6 (dotted) and calculated using the ray model for kR — 12.1 (dashed). The width is seen to be approximately independent of A: above the deformation (gray region) for which refractive escape can occur, and in good agreement with the ray model. There is a large discrepancy between ray and wave calculations around E = 0.05 which we attribute to chaos-assisted tunneling (see text), (b) Logarithm of the width for a resonance with width kR = 33.2 and < s i n / > = 0.9 (solid) as compared to the ray model for this resonance (dashed). The reduction in the actual width compared to the ray model is attributed to dynamical localization effects (sec text). (© Physica Scripia 2001

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that even for deformations approaching 20%, the g-values of these chaotic WG resonances are > 1000 which is considered a useful value for semiconductor optics. We shall see below that these resonances also give highly directional emission as required for most applications.

to explain a well-known series of experiments on the microwave ionization of hydrogen [38]. It had not been demonstrated in chaotic billiards prior to our work, but subsequently a number of papers appeared showing how to generalize the theory to such systems [39,40]. The localization length scales with the effective diffusion coefficient of the classical map; as shown in a contribution 7. Quantum/wave corrections to the ray model to this volume by Nockel [41], this diffusion constant rapidly tends to zero as sin# —>• 1 for billiard maps of this type. That Although the results of Fig. 5a generally support the explains why localization effects are more important for usefulness of the ray model, they also indicate a major states localized at large sin^; they will have shorter shortcoming. At deformations below e = 0.07 refractive localization lengths. Obviously if the localization length ray escape is not possible due to KAM barriers and escape must be by photon tunneling. The ray model includes in angular momentum is longer than the distance to the crititunneling in its escape rule but only the direct tunneling cal value for refractive escape, localization effects will be through the angular momentum barrier. This direct negligible. Nockel also presents numerical evidence [41] tunneling rate depends strongly on wavevector and in the of localization effects for such modes of open resonators case of Fig. 5a the rate is chosen to agree with the which scale correctly with kR. A similar model for which kR =12.1 resonance. To be precise, each resonance of dynamical localization effects can be more quantitatively the circular cavity has a calculable g-value and can be evaluated is the "rough billiard" introduced by related to an angle of incidence by the semiclassical relation: Shepelyansky and Frahm [40]. We have recently generalized sin;*; = m/nkR [29,30] (where m is the angular momentum this model to treat dielectric optical resonators [42]. The index of the mode). If we interpret the Q-value value as existence of such modes is quite striking from the point arising from repeated reflections then each resonant value of view of resonator theory. Such a mode strongly localized of kR generates a reflectivity R (and transmissivity at high angular momentum (sin^ ~ 1) will have T = 1 — R) vs. angle of incidence sin# [27]. This function exponentially large Q values even though geometric optics R(smx, kR) is used in the ray simulations for the deformed indicates that it is impossible to trap a classical ray in case as well. By construction it correctly describes escape the resonator. However the pseudo-random nature of these from the circular resonator, for which sin# is conserved; states make them unsuitable for applications. for deformed shapes with kR;» 1 it should describe the probability of direct tunneling at each collision with the boundary. One sees however that at deformations around 8. Directional emission from ARCs e = 0.05 the actual tunneling rate is much higher than that For quadrupolar ARCs with index of refraction n < 2 the predicted by the ray model with this escape rule. There is ray model is an excellent predictor of the directional emia natural explanation for this discrepancy. The ray model ssion properties; the quantum/wave corrections described neglects "internal tunneling". Each time the wave reflects above do not seem to play a significant role. For any from the curved boundary it emerges with some spread ARC with a large quadrupolar component the phase space, of its momentum around the value corresponding to specualthough partially chaotic, can still be usefully analyzed with lar reflection; thus part of its amplitude can leak through reference to the structures found in the integrable ellipse KAM barriers into regions of phase space where no barriers billiard. In the ellipse billiard phase space (Fig. 6a) is divided exist, and from which escape by refraction (or by a shorter by a separatrix between trajectories which circulate outside tunneling process) is possible. This process is very similar the foci, giving rise to WG orbits with elliptical caustics, to the phenomenon of "chaos-assisted tunneling" already and trajectories which stay between the foci and oscillate known in the quantum chaos literature [33-35]. This explaback and forth forming a hyperbolic caustic (sometimes nation for the discrepancy found in Fig. 5a is very natural, called "librational" orbits) [25]. For the quadrupole billiard but remains speculative at present; quantitative methods for calculating these effects do not yet exist for general the separatrix curve is replaced by a broadened chaotic layer (see Fig. 2) and higher order stable periodic orbits appear shapes. which are absent in the ellipse, but one can still think roughly Another discrepancy between the ray model and exact of dividing phase space into rotational and librational numerical solutions is illustrated by the results in Fig. 5a,b. regions. Notice in 5a that at large deformations the ray model slightly As deformation increases chaotic regions expand (see Fig. overestimates the escape rate; whereas as noted above the 2), but the pattern of phase space flow describing this model underestimates the escape rate in the tunneling rotational motion remains the same and trajectories tend regime. This effect is more marked for states with high aver- to follow the adiabatic curves (see Fig. 6b) for intermediate age angular momentum as shown by the data in Fig. 5b times (with one important exception to be discussed below). for which the mean angular momentum corresponds to The chaotic diffusion which allows the trajectory to hop ransinj; > 0.9. It was proposed in [32] that such modes are domly between adiabatic curves is slow. If n < 2 escape beginning to feel the effects of dynamical localization, which occurs before the librational region is reached and this picis the suppression of phase space diffusion due to destructive ture of slow diffusion transverse to the rapid rotational interference, in close analogy to Anderson localization in motion along adiabatic curves is a good description. Thererandom systems. This phenomenon has been known for a fore the generic trajectory starting in the WG regime will long time in the context of the quantized standard map diffuse "downward" in the SOS (i.e. from higher to lower [36,37] (kicked rotor) and was invoked in a crucial way values of sin^) until it is moving along the adiabatic Physica Scripta T90

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Wave-Chaotic Optical Resonators and Lasers

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invariant curve which is tangent to the critical line sin/ = 1 /n and then the trajectory will rapidly reach the region near the minima at which most of the escape will occur (these minima are denoted as escape points in Fig. 6b for the case n = 2). By the definition of the adiabatic invariant curves given above, these minima occur at the points of maximum curvature on the boundary. A crucial point is that since the diffusion transverse to the adiabatic curves is slow, trajectories will not "jump" below the critical line before escaping and so one will have predominantly critical emission near the points of highest curvature, leading to well-collimated emission in the tangent direction and a highly directional output in the far-field [31,32]. This generic behavior is shown in Fig. 7 below. The generic behavior is quite intuitive since one expects high emission from the regions of highest curvature, and tangent emission is characteristic of whispering gallery modes. However the model has more content than just this intuitive prediction; it also predicts a robust class of exceptions to the generic behavior[31,32]. If the islands associated with a stable periodic orbits are intersected by the critical line, then trajectories of the type we have been discussing, which start outside the islands at higher sin/, cannot reach the points of highest curvature and depart strongly from the adiabatic behavior. An important example of this is shown in Fig. 4 where for n = 1.5 a stable period-four orbit exists in the vicinity of the points of maximum curvature at the critical angle. Trajectories inside the islands cross the critical line rapidly and would not correspond to high-Q modes; trajectories outside the islands circulate around them and cross the critical line displaced from the points of maximum © Physica Scripta 2001

curvature. As a results one has primarily tangent emission from four points on the boundary (not two as in the generic case) and four displaced peaks in the far-field emission pattern as opposed to the two at 0 = ±n/2 which one has in the generic case. We refer to this effect as dynamical eclipsing [31] since it is a ray-dynamical property of the given shape which prevents emission from the points of highest curvature. Dynamical eclipsing illustrates the power of the phase-space method to predict and explain emission patterns which would be counter-intuitive and puzzling if found experimentally or by numerical solution of the wave equation. A final strong prediction of the ray model for emission directionality is that the emission pattern from these chaotic WG modes should depend only on the index of refraction and not on either the wavevector or the g-value of the resonance. This is because the directionality is controlled primarily by the phase-space flow in the region of the critical line. This prediction is verified by the numerical data of Fig. 7 below. 9. Experimental evidence for chaotic whispering gallery modes The rather surprising conclusion drawn from the ray model and numerical studies is that there exist modes concentrated in the chaotic region of the mixed phase space of the cavity which nonetheless have highly directional emission. We call these modes chaotic whispering gallery modes. The fundamental reason that one may have directional emission in the presence of chaotic ray dynamics is that the slow Physica Scripta T90

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A. Douglas Stone

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diffusion means that all of the chaotic component of phase space is not explored with a few iterations of the map. The dynamical eclipsing effect is a particularly clear signature of chaotic WG modes; a wavefunction must have a significant component in the chaotic component in order to be distorted by the presence of stable islands (see e.g the right Husimi distribution in Fig. 4). Conversely one would expect no such effect to occur in an elliptical cavity which has no stable islands at the points of highest curvature on the boundary where the adiabatic curves have their minima (see Fig. 6a). A recent experiment by Chang et al. [43] measured the emission pattern from lasing microdroplets and found strong evidence for the dynamical eclipsing effect. In the experiment ethanol droplets in the 30-50 micron range are generated by forcing the liquid through an orifice with a segmenting device (a so-called Berglund-Lui droplet generator). The droplets also contain a small amount of Rhodamine B dye which provides an active medium for lasing when the droplets are pumped optically. As the droplets fall freely they are axially symmetric but initially Physica Scripta T90

quite aspherical and oscillate between a prolate and oblate deformation. This oscillation is damped and the slowest decaying component of the shape distortion is the quadrupole moment. Hence by studying the lasing emission from these droplets at a specific distance below the orifice it is possible to observe the lasing emission from a purely quadrupolar deformation of a sphere. Because the problem is in 3D with axial symmetry it is equivalent to a series of 2D problems corresponding to different values of the conserved z-component of angular momentum. We have looked at this system some time ago [44] and showed that the effective 2D dynamics is that of a "centrifugal billiard"; i.e. when the motion is projected into 2D the dynamics consists of specular reflection from the boundary and parabolic motion between collisions due to the angular momentum barrier. The analysis of the complete set of possible lasing modes is rather complex, however for the modes near L: — 0 the angular momentum barrier disappears and the behavior reduces to that of the 2D quadrupole billiard and we can apply the previous concepts to that subset of the resonant modes. These modes are the only ones that can emit from © Physica Scripla 2001

Wave-Chaotic Optical Resonators and Lasers

Equatorial image

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30° sub-equatorial image

Fig. 8. CCD images of the laser emission from prolate lasing droplets taken from the experiment of rcf. [43]; left image is taken in the equatorial plane, right image is taken 30" below the equatorial plane. The equatorial image shows no lasing emission from the poles whereas the sub-equatorial image sees significant emission. We attribute this pattern to dynamical eclipsing from the four-bounce stable orbit (blue dashed curve) which deflects the whispering gallery emission from the equatorial plane as shown in the schematic. The bright emission points in the equatorial image arc expected from non-zero L- modes according to the adiabatic model for this case (yellow lines in the schematic).

the "polar" regions of the droplet and therefore we focus on that aspect of the experimental data. The points of highest curvature on the surface of the prolate spheroid are at the poles and thus by our above model the generic emission is from the poles in the tangent direction (the equatorial plane). However (well before our theory was developed) it had been observed that there was almost no lasing emission visible from the poles in the equatorial plane. It was then understood that this was due to the dynamical eclipsing effect described above, which is significant at the index of refraction n = 1.36 of ethanol. A brief graphical summary of this aspect of the experiment is given in Fig. 8; the full details are given in reference [43].

10. High Index ARCs and semiconductor microlasers One of the exciting implications of our work which was recognized early on [17,26] was that chaos theory might be used to guide the design of micro-lasers of practical value. The workhorse of optical telecommunications is the semiconductor diode laser (recognized e.g. by the Nobel Prize in Physics in 2000), and there is strong interest in micron scale lasers for integrated optical devices and interconnects. In fact the development of efficient semiconductor micro-lasers is one of the primary technological motivations for research in dielectric micro-cavities. Typical semiconductor lasers are made of semiconducting heterostructures involving doped and undoped III-V semiconductors such as GaAs, InAs,GaN with the insulating AlAs also used © Physica Scripla 2001

as needed. The earliest micro-disk lasers were conventional semiconductor diode lasers [21,22], but recently micro-disk lasers were fabricated using a multiple-quantum-well structure for the active region in which the lasing transition is between electronic sub-band levels [45,46]. These lasers, known as quantum cascade lasers, were pioneered by Capasso, Faist, Gmachl and coworkers at Bell Labs; they emit from inter-sub-band transitions in the infrared (3/rni < X < 30/rni, tunable with appropriate fabrication) and have many advantages over conventional lasers in that wavelength range [47]. In 1997 a collaboration was formed between Yale and Bell Labs to study ARC microcylinder quantum cascade lasers. To test the effect of deformation on the lasing properties, a set of cylinder lasers of increasing deformation were fabricated and measured on the same chip [46]. Initial expectations were that the lasing would be from chaotic WG modes with qualitative properties following the ray model discussed above, perhaps with some decrease in directionality due to the higher index of refraction (« = 3.3 is the average index of the layered structure). However the measured emission properties were quite different from those expected from chaotic WG modes. Instead of a gradual onset of directional emission, very little anisotropy was found for E < 0.14, whereas above this value a large (30:1) anisotropy rapidly developed (See Fig. 9 below). The peak in the emission pattern was not at 90° as expected from generic whispering gallery emission, but rather was at 45° Moreover with index 3.3 and this large deformation Physica Scripla T90

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0 30 60 90 Far-field angle (degree)

Fig. 9. (Top left) Angular dependence of the emission intensity from deformed cylindrical quantum cascade semiconductor micro-lasers for £ = 0 (triangles). £ = 0.14 (open circles) and t — 0.16 (filled circles). Left inset show lasing intensity vs. wavelength at high pump power, regularly spaced peaks agree well with spacing of bow-tic modes. Right inset shows schematic of bow-tie orbit and defines the angular coordinate system (reprinted from ref. [46]). (Top right) Comparison of the experimental data for E = 0.16 to numerically determined far-field patterns for the ground (dotted), first excited (dashed) and second excited transverse mode (solid) of the bow-tie resonance. The peak transmission is clearly better described by the second excited transverse mode. Bottom panels: intensity patterns corresponding to the ground (left), first excited (center) and second excited (right) bow-tie resonance.

there was no stable orbit at the points of highest curvature to generate dynamical eclipsing near the critical line for escape (see Fig. 2). The main difference between semiconductor ARCs and the lower index (glass and liquid) ARCs analyzed above is that due to the higher index refractive escape can only occur at much lower values of sin^. This gives rise to two differences in the dynamics. First the adiabatic model becomes questionable. If the index of refraction is significantly larger than two, the trajectory will have to pass from the rotational region of phase space into the chaotic separatrix region mentioned above (Figs. 2,6) in order to escape. This is the region of strongest chaos in quadrupolar billiards and here we found that the adiabatic approximation to the ray motion breaks down completely for deformations above 10% [46]. In agreement with this result of the ray dynamics, we were unable to find any highly directional whispering gallery modes numerically [46]. This ruled out chaotic WG modes as the source of the onset of directional emission at large deformations in the Bell Labs experiments. The second effect of high index is that orbits in the librational region may be well trapped and can play a role. It was this possibility which we looked at to explain the experimental data The experimental lasers emitted at X = 5.2/rni, for which direct imaging of the mode was difficult, so the relevant mode had to be inferred from the far-field pattern and spectral data. Physica Scripta T90

The rather sudden onset of the directional emission suggested a mode which (unlike the WG modes) was not continuously connected to zero deformation. There was one clear candidate. As noted, at zero deformation there are no librational modes (other than the degenerate case of the Fabry-Perot mode based on the period-two orbit which has very low Q) and there are no stable periodic orbits of the librational type until e *=» 0.11. At this point the stable period-two orbit undergoes a period-doubling bifurcation generating two stable period-four orbits with the geometry of a bow-tie (the four islands associated with the bow-tie are visible in the SOS for e = 0.2 in Fig. 2) and two unstable period-four orbits with the shape of the letter vee. Until e « 0.14 the angle of incidence of the bow-tie orbit remains below the critical angle and presumably the corresponding modes are too short-lived to lase; as it crosses the critical angle one expects a large increase in its Q-value allowing it to lase. Since the bow-tie modes are strongly localized in space they give highly directional emission. Finally, pumped well above threshold several modes are observed to lase (inset Fig.9) and their observed spacing agrees with that expected from bow-tie modes to better than 3% accuracy.

11. Mode selection and the power puzzle Modes such as the bow-tie, which are associated with stable islands, are less novel from the point of view of quantum © Physica Scripla 2001

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chaos theory than the chaotic whispering gallery modes discussed above. The analogous states of a closed billiard have been well known for many years, and they also may be regarded as generalized gaussian modes of the type known from Fabry-Perot theory. Such modes for the closed cavity may be semiclassically quantized and characterized by two quantum numbers (mode indices) associated with longitudinal and transverse motion. Specifically one finds:

bow-tie lasers which allows it to optimize its output power. The experiment provides some modest support for this conjecture as the circular lasers are consistently found to lase on several modes simultaneously, whereas the ARC lasers are single mode unless they are pumped very hard [46]. Work is underway to analyze mode competition in chaotic lasers and verify these speculations.

kn,m = kl; + (m + \/2)kT

12. Scarred lasing modes

(1)

where k\ = 2n(n + n)/L (L is the length of the orbit, fi is a Maslov index) and kJ — 2nw/L (w is the winding number associated with the transverse oscillations around the central fixed point). The mode spacing observed in the lasing spectrum (Fig. 9,inset) was the longitudinal mode spacing Ak = 2n/L, indicating that it was a single transverse mode which was lasing for different values of n. However the peak position in the far-field was not that predicted by ray optics (i.e. by refracting a ray out of the bow-tie orbit). In fact the peak location corresponds to the second transverse excitation of the bow-tie (m = 2 in Eq. (1) above), see Fig. 9. Although numerical work and physical arguments allowed us to identify this mode after the experiment, we have no predictive theory for the mode selection in this case. The second excited bow-tie mode is not the highest Q mode of the cold cavity, nor is it particularly selected by the peak of the gain curve, which is broad enough to allow other modes to lase. The issue of mode selection in ARCs is made more salient because of the second major finding of the Bell Labs experiment. Not only did the bow-tie laser provide highly directional emission, improving the brightness (power into a given solid angle) of the laser by a factor of order 30; but the deformed lasers produced more than a thousand times the output power of the identically-fabricated circular lasers [46]. The demonstrated high output power and directional emission solve the major problems with earlier semiconductor microdisk lasers and makes their use technologically promising. The difference in peak output power between the ARC and circular cylinder lasers is interesting theoretically and not yet explained. The peak output power certainly depends on the non-linear properties of the lasing and is not a property of a given mode of the linear wave equation. The standard and well-verified model of the power output of a single-mode Fabry-Perot laser [18] finds that the power output is optimized for a given pump power when the external cavity loss (which is the width we are calculating) equals the internal cavity loss (neglected in our model). One may conjecture then that the bow-tie optimizes the power output, even though it is not the highest Q mode. However this observation is not sufficient to explain the experiment; the circular lasers have a range of (^-values corresponding to different radial quantum numbers for a given angular momentum. In particular, there should exist modes with Q very near that of the bow-tie. The main difference between circular and ARC lasers is that in the circular case this mode will strongly overlap in space with other higher Q modes; whereas in the ARC chaos has wiped out the competing higher Q modes in the vicinity of the bow-tie. Therefore we conjecture that it is the lack of mode competition in © Physica Scripta 2001

In dielectric optical resonators the key requirement for a high-g mode is a resonance with average angular momentum above (or near) the critical value for total internal reflection. We have discussed two types of modes which meet this requirement; chaotic whispering gallery modes which are trapped by slow diffusion and modes based on stable periodic orbits. One of the interesting and well-studied results of quantum chaos theory is the existence of localized modes based on unstable periodic orbits, known as scarred states [48]. If such an orbit is trapped by total internal reflection and supports scars one may imagine it can lead to an unstable lasing mode. Exactly such a possibility seems to be realized in semiconductor diode micro-lasers fabricated and studied by Rex, Chang et al. [49,50]. The lasers in question are quadrupolar ARCs based on a GaN heterostructure with an effective index of n « 2.6, which means that the bow-tie modes are not as well trapped as for the quantum cascade lasers. Again directional lasing emission is found to set in above 10% deformation, but not as suddenly or dramatically as in the quantum cascade lasers of reference [46], Moreover these lasers emit in the optical and by imaging the boundary the experimenters where able to localize the points of bright emission, which were at « ±15°, 180 ± 15° (where (j) is measured from the major axis). This ruled out the bow-tie modes which would emit at larger values of <j>. As there are no stable periodic orbits with bounce points near (j> = 15°, it was natural to look for short unstable periodic orbits which might support scars. We have found [50] such modes based on the two symmetry-related triangle orbits (See Fig. 10, due to their geometry Berry has suggested the term "Scar of David" for these orbits). Scarred states of these orbits exist and account rather well for the observed far-field pattern (Fig. 10). Detailed analysis of these experiments in order to explain the full range of observations is still in progress. We reiterate that there is at present no theory which explains the selection of this scarred mode as the lasing mode. This and the previous experiments indicate the importance of understanding the non-linear wave equations (or rate equations) which apply to chaotic cavities with an active medium.

13. Semiclassical theory for ARCs The importance of modes based on periodic orbits in ARC lasers showed the necessity of a theoretical approach beyond the ray model. A full analytic semiclassical theory of the linear wave equation would be quite difficult, since the semiclassical quantization of the mixed phase space for a closed system is still not a solved problem in general. Moreover resonance widths can be exponentially smaller than Physica Scripta T90

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-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Fig. 10. (Upper left) Electric field intensity pattern for a resonance which scars the triangular orbits ("Scar of David", sec inset). (Upper right) Husimi distribution showing the intensity maxima of this resonance near the bounce points of the triangle orbits (denoted by white triangles). (Lower left) Far field intensity pattern of the measured lasing emission from a GaN semiconductor microcylinder laser with a quadrupolar deformation of z = 0.12 (open circles) compared to far field (solid line) calculated from the state at upper left. (Lower right) Measured near field pattern (open circles) and calculated pattern (solid) for scarred resonance. Experimental data from ref. [50].

the local level spacing, requiring a degree of energy resolution which is beyond current semiclassical methods for chaotic systems based on periodic orbit sums. However we have recently made progress on this problem [51] by treating the emission from the resonator semiclassically, assuming that one has some information about the bound states of a corresponding closed resonator. Space limitations do not permit a detailed explanation of the theory, so we merely sketch the main ideas here. The first step is to note that while the effects of deformation cannot be treated perturbatively (in the interesting regime), for sufficiently narrow resonances we can treat the leakage perturbatively. We do this in a manner which is similar (but not identical) to the R-matrix approach familiar in nuclear physics. However the really interesting step is to then evaluate the perturbative expressions semiclassically, which is an excellent approximation in general for these systems, and which leads to relatively simple analytic formulas with a transparent physical interpretation. Physica Scripta T90

In the first specific implementation of the approach, we treat an infinite dielectric cylinder of index of refraction, n, with quadrupole-deformed cross-section. The TM modes for such a geometry have the electric field parallel to the cylinder axis and the boundary conditions that the electric field and its derivative are continuous at the surface of the cylinder. As noted above, the quasi-bound states of this problem will satisfy these boundary conditions with no incoming wave from infinity at discrete, complex values of the wavevector k. The boundary conditions can be reformulated as a set of linear parametric equations for the coefficients of an angular momentum expansion of the internal solution (we denote these coefficients by the vector |a >, which is finite-dimensional as we can truncate the space for sufficiently high angular momentum). These equations take the form: M(k) \a > = 0

(2)

where the non-hermitian matrix M is constructed from © Physica Scripta 2001

Wave-Chaotic Optical Resonators and Lasers appropriate Bessel and Hankel functions integrated around the boundary R(0). Equation (2) will not have physical solutions for real k, but assuming that there exist narrow resonances it will have many solutions for small \m{k}=-Y. Therefore it is natural to seek a decomposition of M into an unperturbed part, H(), which has purely real roots q and a perturbation, V, and expand k = a - iT around its real part. q. Equation (2) then determines Tin terms of the eigenstates of Ho with eigenvalue zero denoted by |ao >: r = < a 0 |V|a 0 > < ao\dH0/dk \<XQ ->l

3 C8.

(3)

Similar arguments give an expression for the far-field emission intensity 1(0) of the quasi-bound state in terms of the eigenstates of H 0 which we may regard as describing the "closed" resonator (a more precise interpretation of Ho is the object of current investigation). These expressions are then evaluated semiclassically, i.e. by stationary phase approximations to the angular momentum sums and spatial integrals. One then finds quite simple expressions; for example «»,2 US) = l ^ e x p t i m ^ ) E„,(0)a\711

261

o.o Fig. II. Comparison of far-field emission pattern /() for a bow-tie mode as calculated numerically (dark curve) and analytically using the semidassical method of ref. [41] (light curve). The semidassical method captures the main features, but not the fine structure. Note that this bow-tie mode is a lowest order transverse mode, but for n = 1.5. so its far field pattern differs from that of Fig. 9 (reprinted from ref. [51]).

(4)

where <^± are the points on the boundary where a ray with internal angular momentum m can escape from the resonator and refract in direction 6 while satisfying Snell's law (this statement is a geometric consequence of the two non-trivial stationary phase conditions). E„,(0) is the semidassical amplitude for emission which we have calculated analytically [51 ] and depends only on geometric properties of the boundary, on (/>*, and on the parameters q, n, R (the mean radius), aj"' is the amplitude for having a particular angular momentum m in the closed resonator. This expression has nice physical interpretation: The intensity at 0 in the far-field is given by the square of a sum of amplitudes, where each amplitude is the product of the amplitude to have angular momentum m multiplied by the emission amplitude in direction 0 for that value of angular momentum. Initial numerical tests of this semidassical formula and the semidassical approximation for the width Tin comparison with both the exact and perturbative formulas are promising [51]. In particular, for the case of bow-tie modes, the semidassical wavefunctions localized on the stable islands can be calculated analytically (equivalent to obtaining the a)"' in Eq. (3) above) and a full analytic formula for the intensity distribution can be obtained. The comparison of this analytic formula with exact numerics is shown below (Fig. 11); reasonable agreement is found.

ssion pattern of resonant modes of the cavity. For low-index of refraction (glass, liquids) whispering gallery-type modes in the chaotic region of phase space determine the lasing properties of such resonators. For the higher index of refraction characteristic of semiconductors, modes associated with periodic orbits, both stable and unstable, are most relevant. It is striking that many, if not all, of the novel effects associated with quantum/wave chaotic systems appear relevant to the understanding of these resonators in various regimes. For example we have found effects associated with the breaking of KAM tori and diffusive phase space transport, with chaos-assisted tunneling, with dynamical localization, and with both stable island states and unstable scarred states. There remain many challenges for theorists to obtain a complete semidassical theory of such resonators, and particularly to generalize or modify the current concepts to describe both the linear and non-linear regime of the wave equation in an active medium. Major improvements in the performance of semiconductor micro-cylinder lasers have been demonstrated using these Asymmmetric Resonant Cavities (ARCs), and they are being seriously pursued for applications, thus demonstrating once again the value of curiousity-driven fundamental research in generating novel approaches to problems of technological interest.

Acknowledgments 14. Summary and conclusions Methods from the theory of classical and quantum chaos in conservative systems are well-suited to describe non-symmetric dielectric optical resonators and lasers. These methods are particularly useful for resonators based on smooth deformations of cylinders or spheres (ARCs). Such resonators may be regarded as billiards with escape by refraction or tunneling. An analysis of the phase-space structure and phase-space flow of the billiard then allows physical insight and simple models for the lifetime and emi© Physica Scripla 2001

I wish to thank my collaborators on the ARC theory and experiments. The ray model was developed in collaboration with Jens Nockel. with important input from Richard Chang and Attila Mekis. Work on the theory of bow-tie modes and the semidassical method was done with Evgenii Narimanov. Gregor Hackenbroich and Philippe Jacquod. Recent theoretical work on scarred modes was done with HakanTureci and Harald Schwefel. Experimental work on ARCs in collaboration with my group was done by the group of Richard Chang, specifically Paul Chen. Seongsik Chang. Andrew Poon and Nathan Rex; we have also very much enjoyed the collaboration on quantum-cascade ARC lasers with Fcdcrico Capasso. Claire Gmachl and their coworkers at Bell Labs. Helpful conversations are acknowledged with Henrik Bruus, Martin Gutzwillcr and Dima Shcpclyansky. Work at Yale was supported by NSF grants DMR-0084501. PHY-9612200. and DMR-9215065 and US. Army Research Office grant DAAH04-93-G-0009. Physica Scripla T90

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References 1. Porter, C. E., "Statistical Theory of Spectra: Fluctuations" (Academic Press, NY, 1965). 2. Mehta, M, L., "Random Matrices and the Statistical Theory of Energy Levels", (Academic Press, NY, 1967). 3. Efetov. K. B„ Adv. Phys. 32. 53 (1983). 4. Altshuler, B. L. and Shklovskii, B. I.. Sov. Phys. -JETP, 64, 127 (1986). 5. "Mesoscopic Quantum Physics", Courses 1,8,9, Les Houches 1994, Session LX1, (Eds. E. Akkermans. G. MontambauxJ.-L. Pichard and J. Zinn-Justin), (North-Holland, Amsterdam, 1995). 6. Beenakker, C. W. J„ Rev. Mod. Phys. 69, 731, 1997. 7. Gutzwiller, M. C , J. Math Phys. 12. 343 (1971). 8. "Chaos and Quantum Physics", Les Houches 1989, Session LII, (Eds. M.-J. Giannoni. A. Voros and J. Zinn-Justin) (North-Holland, Amsterdam, 1991). 9. Gutzwiller, M. C , "Chaos in Classical and Quantum Mechanics", (Springer-Verlag, New York, 1990). 10. Reichl. L., "The Transition to Chaos in Conservative Classical Systems: Quantum Manifestations". (Springer-Verlag. New York, 1992). 11. Brack, M. and Bhaduri, R. K., "Semiclassical Physics", Frontiers of Physics, Vol. 96, (Addison-Wesley, NY, 1997). 12. Bohigas, O., Giannoni, M. J. and Schmidt, C , Phys. Rev. Lett. 52. 1 (1984). 13. Berry. M. V.. Proc. Roy. Soc. A 400. 229 (1985). 14. Zirnbauer, M. R., Invited talk at the symposium. 15. Lu, W. T., Pance, K... Pradham, P. and Sridhar, S.. Physica ScriptaT90, 238 (2001). 16. Stockmann, H.-J.. Physica Scripta T90, 246 (2001). 17. "Asymmetric resonant optical cavity apparatus". Stone. A. D„ Chang. R. K. and Nockel. J. U.. US patent # 5.742.633; issued April 21. 1998; "Solid-state Laser for Operation in Librational Modes", F. Capasso et al.. U.S. patent # 6 , 134, 257, issued October 17, 2000. 18. See for example, "Lasers", Siegman. A. E., (University Science Books, 1986), pp. 558-891. 19. See for example, "Optical Processes in Microcavities", Chang, R. K. and Campillo, A. J., eds.), (World Scientific, 1996). 20. Johnson, B. R„ J. Opt. Soc. Am. 10. 343 (1993). 21. McCall. S. L. et al., Appl. Phys. Lett. 60. 289 (1992). 22. Levi, A. J. etai, Appl. Phys. Lett. 62, 561 (1993); Appl. Phys. Lett. 62, 2021 (1993). 23. Leung, P. T., Liu. S. Y. and Young, K., Phys. Rev. A 49, 3057 (1994).

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24. Harayama, T., Davis, P., and Ikeda, K. S., Phys. Rev. Lett. 82. 3803 (1999). 25. Berry, M. V., Eur. J. Phys. 2, 91 (1981). 26. Nockel, J. U„ Stone, A. D. and Chang, R. K., Optics Lett 19, 1693 (1994). 27. Nockel, J. U.. PhD Thesis, Yale University (1997). 28. Lazutkin, V. F., "K.AM Theory and Semiclassical Approximations to Eigenfunctions", (Springer-Verlag, Berlin, 1993). 29. Robnik, M., J. Phys. A 16, 3971 (1983). 30. Keller. J. B. and Rubinow, S. I., Ann. Phys. 9, 24 (1960). 31. Nockel, J. U., Stone, A. D., Chen, G., Grossman, H. L. and Chang, R. K., Optics Lett 21, 1609 (1996). 32. Nockel, J. U. and Stone, A. D., Nature 385, 45 (1997). 33. Bohigas, O., Boose, D., Carvalho, R. E. and Marvulle. V., Nucl. Phys. A 560. 197 (1993). 34. Tomsovic. S. and Ullmo. D„ Phys. Rev. E 50, 145 (1994). 35. Doron, E. and Frischat, S. D„ Phys. Rev. Lett. 75, 3661 (1995). 36. Grempel, D. R„ Fishman. S., and Prange. R. E., Phys. Rev. Lett. 49. 833 (1982). 37. Shepelyansky, D. L., Phys. Rev. Lett. 56. 677 (1986). 38. Casati, G., Chirikov, B. V., Guarneri. I. and Shepelyansky. D. L., Phys. Rev. Lett. 56, 2437 (1986). 39. Borgonovi, F., Casati, G. and Li, B. W., Phys. Rev. Lett. 77. 4744 (1996). 40. Frahm, K. M. and Shepelyansky. D. L., Phys. Rev. Lett. 78. 1440 (1997). 41. Nockel. J. U., Physica Scripta T90, 263 (2001). 42. Starykh, O. A., Jacquod, P., Narimanov, E. E. and Stone, A. D., Phys. Rev. E. 62. 2078 (2000). 43. Chang, S. C , Nockel, J. U„ Chang, R. K. and Stone, A. D.. J. Opt. Soc. Am. B 17, 1828 (2000). 44. Mekis. A., Nockel, J. U., Chen, G.. Stone. A. D. and Chang, R. K.. Phys. Rev. Lett. 75, 2682 (1995). 45. Gmachl, C. et al., IEEE J. Quantum. Elec. 33, 1567 (1997). 46. Gmachl, C. et al.. Science 280, 1556 (1998). 47. Faist. J. et al.. Science 264. 553 (1994). 48. Heller, E. J., Phys. Rev. Lett. 53, 1515 (1984). 49. Rex, N. B„ Chang, R. K.. Guido. L.. J.. Bour, D. and Kneissl, M., abstract, CLEO/QELS 2000. 50. Rex, N. B., Tureci, H„ Schwefel, H., Chang, R. K. and Stone, A. D„ unpublished. 51. Narimanov, E. E., Hackenbroich, G., Jacquod, P. and Stone. A. D., Phys. Rev. Lett. 83, 4991 (1999).

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Angular Momentum Localization in Oval Billiards Jens U. Nockel Nanovation Technologies, 1801 Maple Avenue, Evanston, IL 60201, USA Received August 7, 2000

PACS Ref: 05.45.Mt, 42.15.-i, 42.25.-p

Abstract Angular momentum ceases to be the preferred basis for identifying dynamical localization in an oval billiard at large excentricity. We give reasons for this, and comment on the classical phase-space structure that is encoded in the wave functions of "leaky" dielectric resonators with oval cross section.

Geometric optics is an important engineering tool because of its explanatory and predictive power, even when wave effects are present, as is the case in resonant cavities. Nevertheless, quantum chaos has not been widely recognized as an issue in optical resonators until recently [1], because the engineer often has the freedom to choose geometries for which either the ray picture is simple or the wave equation is separable (up to small perturbations). This is a luxury that we do not usually have in naturally occuring, "self-assembled" optical resonators such as, e.g., aerosol droplets [2,3] or microcrystallites [4]; these examples typically have mixed phase spaces. What we learn from such systems in turn allows us to accept chaotic ray dynamics as a way to introduce added freedom into the design of optical devices in a wide range of material systems, such as semiconductor microdisks [5,6], polymers and glasses [7-9]. One of the essential phenomena that makes chaotic resonators useful in this respect is dynamical localization, because it allows cavity resonances with decay rates that exceed the values expected from classical ray considerations. Mixed dynamics does not necessarily make it impossible to identify localization [10], provided the classical phase space stucture is properly taken into account. In this paper, we discuss how localization can be characterized in oval dielectric cavities. From the quantum-chaos perspective, dielectric microcavities allow us to study the ray-wave duality in a class of open billiard systems bounded by "penetrable" walls which introduce an escape condition in phase space [11]. This openness arises because the internal and external region are coupled across the dielectric-air interface. In many cases this interface can be considred as abrupt on the scale of the wavelength, in which case one arrives at a set of polarization-dependent dielectric boundary conditions which in the ray limit correspond to Fresnel's laws of reflection. The latter have two basic consequences:

internal-reflection condition may still continue along an internal trajectory with attenuated amplitude [12]. In fact, for large refractive index «, the limit of a closed cavity with reflectivity R = 1 is approached. In the quantum-classical transition under such circumstances, the competition between the internal time scales (as set most prominently by the density of levels) and the state-dependent decay rates must be taken into account [13-15]. This becomes especially interesting in cavities with mixed phase spaces because of their intricate temporal evolution [16,17]. The main objects of study in microlaser design are single, isolated resonances. The reason is that the properties of a laser are typically determined by the spatial and emission characteristics of only one or a few quasibound states. In a single-mode laser, it is the state whose k lies closest to the real axis [18]. In contrast to the random-wave assumption that is justified in the presence of hard chaos [19], highly anisotropic intensity patterns of wave functions are typical for mixed systems. These are in fact desirable in a laser because anisotropy can translate to focused emission [20]. Individual quasibound states can be studied in great detail in microlaser experiments, because one can make spatially and spectrally resolved images of the emitter under various observation angles [3]. The numerical aspects of the electromagnetic scattering problem are challenging and have several decades of history, particularly in atmospheric sciences. If the dielectric constant can be assumed piecewise constant in the spatial domains defining the scatterer, one computational method is that of wavefunction matching: in each dielectric region, a "Treftz basis" is introduced [21], consisting of free-space stationary solutions at a fixed wavenumber k. The unknown expansion coefficients of a true wave solution in this basis are determined by imposing the dielectric boundary conditions. In the present study, we are interested in quasibound states of a cylindrical dielectric surrounded by air. For simplicity, we specialize to the case where the electric field is polarized parallel to the cylinder axis, so that Maxwell's equations reduce to a scalar wave equation [22]. The internal and external wave functions


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Here, r and (p are polar coordinates, e the outward normal; k hand, one is often forced by the intended applications to is the free-space wavenumber, and m labels angular go beyond this analogy [33]. As was first argued in Ref. [20], the Poincare map of the momentum. Implicit in the form of \j/eM is the condition that only outgoing cylindrical waves be present in the exterior, billiard, Eq. (4), contains the essential information deteri.e. the solutions will represent emission without any mining the emission characteristics of the corresponding incoming wave [26] - appropriate for fluorescence or lasing. resonator. Therefore, it is necessaery to understand the The resulting homogenous system of Eq. (3) for the Am, properties of this map in detail. It can be written in the form Bm has a nontrivial solution only at a set of discrete, complex sin / = sin / + F(.s, sin 7), (5) values of k. One advantage of Eq. (1) is that it permits a reformulation in terms of the internal scattering matrix of the 2D s = s + G(s, sin / ) , (6) cross-sectional billiard, in the sense of the "scattering approach to quantization" [23]. However, it is by no means where s is the arc length along the boundary and the bar clear that these expansions in angular momentum eigen- denotes the new coordinates after one iteration of the map. functions with coefficients Am, Bm converge. The assumption The functions F and G contain the nonlinearity of the that this is in fact the case is known as the Rayleigh hypoth- dynamics, as shown in Fig. 1 where we plot F versus final esis [24,25]. For definiteness, the convex boundaries which position s for three fixed values of sin %. Note that the amplishall serve as our model systems are parametrized in polar tude of the nonlinearity goes to zero as sin x —* 1. To quancoordinates by two-dimensional multipoles of constant area. tify this. Fig. 2 plots the root-mean-square of F versus starting sin);. A fit with (1 - s i n 2 / ) 3 / 2 shows good agreement. The reason for this functional form with its /-() = / ? ( l + e c o s ( v 0 ) ) / V l + e : / 2 , (4) non-analyticity at sin % — 1 is understandable from very genwhere v = 1, 2 . . . and E measures the fractional deform- eral considerations: If one considers F as a function of cos x instead of sin#, ation. The simplest cases are the Limacon shape (v = 1) and the quadrupole (v = 2). Convergence problems arise then a Taylor expansion around cosx = 0 yields a mapping if the cross section is too strongly deformed, so that the radii equation of the form of convergence of the inner or outer expansion, Eqs. (1) and 2 (7) (2) intersect the boundary; this can make it impossible to cos x &Ms) +fi(s) cosx +fi{s) cos x + formulate the matching conditions. As long as the shape is convex, however, numerical experience shows [26] that the problem can be regularized over a wide range of deformations by performing the wavefunction matching at a discrete number N of points in real space and F(sinx,<)>) making the number M of angular momenta m in Eq. (1) for fixed smaller than N. The resulting rectangular matrix problem sinx can then be solved by singular-value decomposition [27]. Additional improvment can be achieved by finding two island or more choices of origin for the polar coordinates such that chains the respective domains of convergence for all resulting versions of Eq. (1), taken together, cover the boundary completely. The additional unknowns in these expansions Fig. I. A narrow strip of the Poincare section for a quadrupole with s = 0.1, are then connected by analytical continuation. A simple magnifying the whispering-gallery region. Superimposed on the high-order example for this analytical-continuation approach is the island chains and invariant KAM curves are plots of the kick strength annular billiard [28,29], in which a circle with an eccentric functions. circular inclusion permits expansions of the type Eq. (1), centered either at the inner or outer circle; the connection between the two expansions is given analytically via the 0.6 » J I • 1 • 1 ' 1 ' addition theorems for Bessel functions. There are no truly bound states in this finite-sized 2D photonic system; the same is true for the generalization 0.4 ^^oc(l-sm2x)3/2 to a three-dimensionally confined cavity of finite extent. This is the reason for having to permit complex k above, assuming < n is real. The fact that all states are metastable distinguishes 0.2 these systems from the otherwise similar subject of attractive wells in quantum mechanics. This is a reminder of the . 1 , i . i f i iS inequivalence between optical and quantum-mechanical 0 0 0.2 0.4 0.6 0.8 1 wave equations. Microwave experiments [30-32] can under certain restrictions emulate of Schrodinger's equation, if sin x the genuinely electrodynamic aspects of the resonator (resulting from the vectorial nature and hyperbolic charac- Fig. 2. RMS deviation of sin / from its initial value for one iteration of the teristics of Maxwell's time-dependent wave equations) are billiard map for the quadrupole at £ = 0.08, averaged over final position not important. In dielectric optical cavities, on the other s. Circles are from the exact ray dynamics, solid line is a fit with (1 - sin2 x) /2Physica Scripta T90

© Physica Scripta 2001

Angular Momentum Localization in Oval Billiards We expect fo = 0 since a trajectory starting with cos / = 0 must end up with c o s / = 0 (corresponding to "rolling" or "grazing" motion along the convex surface. Using this expansion for sin / ->• 1 in Eq. (5), we obtain sin / « sin / -f{(s)f2(s){\

- sin2 x)3/2

(8)

The resulting sin / - dependence in Fig. 2 suggests that the map of the convex billiard should exhibit invariant curves in the "whispering-gallery" (WG) region (sin x —»• 1) at arbitrarily large deformations, where other KAM curves (such as the one belonging to the inverse golden mean winding number) have been broken up. The last statement is in fact implied by Lazutkin's theorem [34], which however requires 553 continuous derivatives of r{(j)) to prove that invariant WG tori exist with nonzero Lebesgue measure. Here, we can go beyond the mere existence statement and ask what consequences the existence of a stable WG region has for the neighboring phase space. We shall find that only a much smaller number of 3 continuous derivatives enters the physical considerations describing the phase space of the convex billiard. We base our argument on an adiabatic approximation used in Ref. [35], which will be discussed further below. There, the unknown multiplier f\{s)fi(s) of Eq. (8) is determined from geometric considerations to yield

265

J. This leads to za, P)=PS+T—{\-

3/2.

Py>-

(14)

+ dp)

3K(S)

where c(p) is the integration constant which may still depend on p. Applying the second of Eqs. (13) to this result, we arrive at the position mapping equation, 2p , = 5+ cV)-~(l-^)'

/ 2

K(S)

(15)

.

This can in principle be inverted to get s as a function of/? and s. We dispose of the arbitrary c'(p) in such a way that Eq. (15) reduces to the exact expression in the circular billiard where K = 1. The result is, reinstating sin / for p, s = s — 2 arccos(sinz) + 2 ( 1 V

— ) s i n / ( l - sin 2 /) 1 / 2 . *(•*)/ (16)

In contrast to the analogous result in Ref. [35], this position map remains well-defined over the whole range of | sin/| = 0 . . . 1. The billiard shape enters in Eqs. (9) and (16) only through the curvature as a function of J. The "effective map" as defined through Eqs. (9) and (16) reproduces the global structure as well as local detail of the true Poincare sections for the Limacon and quadrupole 2K'(S) 3/2 billiards [26]. Although some additional rescaling of the (1 sinx) (9) srn/ = s i n / - 3K2(S) deformation parameter e is required for best agreement, where K(S) is the curvature and K' its derivative. In Ref. [35], one can use the effictive map to understand classical properthis together with a similar approximation for the position ties of the billiard, such as the existnce of Lazutkin's mapping function G(s, sin x) is used to convert the invariant tori. The utility of this approach consists in amplitudes of the two mapping equations, Eq. (5) and breaking the billiard problem up into two distinct subproblems: the geometric analysis leading to the Poincare Eq. (6), into a differential equation: mapping on the one hand, and the nonlinear dynamics of that map on the other hand. We now wish to apply Eqs. i/sin/ s i n / - sinx F(s, sinx) F(j, sin/) (10) (9) and (16) to our understanding of Lazutkin's theorem. G(s, sin x) G(s, sin x)' ds s—s The adiabatic approximation leading to Eq. (9) relies on which can be solved by separation of variables to obtain an the fact that for trajectories in the whispering-gallery region, adiabatic invariant curve there is a separation of time scales between slow changes in the average sin / and a fast circulation in arc length s around p(s) » J\-(\-
az

s = •

(13)

This definition guarantees that the map is area preserving. The first equation above is just what we already obtained in Eq. (9), so we can infer Z(s, p) by integrating Eq. (9) over © Physica Scripta 2001

"^-Ih^-^mh

(17)

where L is the circumference of the boundary. The assumption of a diffusive growth in the variance of sin / leads us Physica Scripta T90

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Jens U. Nockel

to make the random phase approximation [36] for s, as contained in the integral over s above. In fact, this approximation is hard to justify in generic billiards with a mixed phase space, but improvements can in principle be made by including an average over a finite number of mapping steps in the definition of D. The main conclusion from Eq. (17) is that the diffusion constant follows the sin % dependence of Fig. 2 (squared) and hence vanishes for s i n / —>• 1. Now define the (discrete) diffusion time in p, xp to be the number of iterations it takes to diffuse across the whole allowed p-interval; define further the phase randomization time xs as the number of reflections necessary before s has wrapped once around the boundary. Then from Eq. (17),

tized values of a are then translated back to angular momentum by using the approximate semiclassical relation known from the circle, m = n kR sin %.

(20)

We use this together with our classical considerations to estimate the spread in m that is evident in the wavefunctions. The interval of m on the horizontal axis corresponds to the range 0 < sin x < 1; the dashed vertical lines indicate the "ionization border" sin/ = l/«. At e = 0.08 in the quadrupole, unbroken KAM curves exist only above s i n x « 0.9. The sharp falloff in \Am\ at large m is due to the classical inaccessibility of the high-sin x region by diffusion - it directly shows the imbalance in the nonlinearity between large and small sin/. Exponential localization is identifiable in Fig. 3 only to the left of a plateau surrounding the semiclassical maxima, of and from Eq. (16), width Am « 4 in (a) and Am « 10 in (b). The explanation for this is that the adiabatic curve, Eq. (11), for a = 0.8 oscillates between sin %mdK == 0.89 and sin xmm — 0.72. The latter translates to an angular momentum spread of Am w 4, 9, and 14, respectively, for the states quantized The proportionality constants are functions of the at kR w 11.97, 27.26, and 44.16. The agreement with the figdeformation alone. As the whispering-gallery limit is ure confirms that the semiclassical maximum, ionization approached, xp diverges much faster than TS, SO that the border and width of the plateau in m all scale proportional existence of invariant curves can be inferred as sin x —*• 1. to kR. This leaves us with an m interval 6m to the right Since no more than the first derivative of K appears in of the escape threshold, of width 6m « nkR (sin xmm — the "kick strength" of the analytic mapping of Eqs. (9) sinx c ) = nkR(0.72 — 0.5), corresponding to a classically and (16), this suggests that only three continuous derivatives diffusive region in which a probability decay should be of the boundary suffice to explain Lazutkin's tori. observed. In the figure, an exponential decrease away from As an example of how the above remarks on phase space the semiclassical plateau is discernible for kR = 11.97 (a) transport properties in a convex billiard help us understand and 27.26 (b), whereas the state at kR = 44.16 (c) exhibits the quasibound states of the corresponding resonator. Fig. 3 shows how dynamical localization can be discerned in the numerical solutions of the wave problem. The same states displayed here have been investigated in Ref. [37] as a function of the deformation e. There, an adiabatic quantization based on the invariant curves Eq. (11) was introduced, according to which all three states were found to correspond to approximately the same value of
© Physica Scripta 2001

Angular Momentum Localization in Oval Billiards large angular momentum components over the whole classically confined region. The localization lengths Q estimated from the observed slopes for the two lower-kR states are approximately in a ratio of 2:1, in reasonable agreement with the factor of two between the respective wavelengths. This is the expected behavior [38], because the diffusion constant entering c, is the same in all resonances. Angular momentum as a prefered basis for measuring dynamical localization is a useful initial choice in the oval billiard, but as the existence of the plataeus above indicates, it is not strictly the correct one. Recall that a, and not sin/ or angular momentum, is the adiabatic invariant in the semiclassical quantization for the states in Fig. 3. This can be made very clear by comparing to the special case of the ellipse where the adiabatic curve Eq. (11) becomes exact. Then er, which in the circle is the angular momentum, acts as a constant of the motion while sin/ still oscillates. Clearly, there is no diffusion although the angular momentum decomposition shows a spread Am whose width is determined by the eccentricity. For the oval billiard, this means that a should be considered as the diffusing variable. Classically, the transformation from sin/ to a is given by Eq. (11). Wave-mechanically, the goal will be to project the true wave function t// onto a Treftz basis defined by these adiabatic invariant curves. Judging by the results presented here, this is a worthwile program for future work because a better-adapted basis significantly expands the interval over which one is allowed to assume the separation of time scales which leads to classical diffusion in the first place. In billiards that remain close to a circle, such as a short stadium [38] or rough billiard [39,40] this problem does not arise. However, in oval billiards, these classical considerations apply. Moreover, the breakdown of the Rayleigh hypothesis makes it not only desirable but necessary to abandon angular momentum as the basis in which to detect dynamical localization.

3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

23.

24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

Acknowledgement

34.

I would like to thank Steve Tomsovic for valuable discussions.

35. 36. 37. 38. 39. 40.

References 1. Stone. A. D.. contribution in this issue. 2. Mekis. A. et al. Phys. Rev. Lett. 75, 2682 (1995).

(Q) Physica Scripta 2001

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Chang, S. et al, submitted to J. Opt. Soc. Am. B (2000). Vietze, V. et al. Phys. Rev. Lett. 8t, 4628 (1998). Yamamoto, Y. and Slusher, R. E„ Physics Today 46(6), 66 (1993). Zhang, J. P. et al, Phys. Rev. Lett. 75. 2678 (1995). Dodabalpur, A. et al. Science 277, 1787 (1997). Collot, L., Lefevre-Seguin. V., Raimond, M. and Haroche, S., Europhys. Lett. 23. 327 (1993). Serpengiizel. A., Arnold, S. and Griffel, G., Opt. Lett. 20. 654 (1995) Robinson, J. C. et al. Phys. Rev. Lett. 75. 3963 (1995). Borgonovi. F., Guarneri, I. and Shepelyansky, D. L., Phys. Rev. A 43, 4517 (1991). Kohler, A. and Blumel, R., Ann. Phys. New York 267, 249 (1998). Prigoginc, I.. Phys. Rep. 219. 93 (1992). Kohler, S. et al, Phys. Rev. E 58. 7219 (1998). Braun. D., Braun, P. L. and Haake, F., Physica D 131, 265 (1999). Zaslavsky, G. M., Physics Today 52(8), 39 (1999). Huckestein, B., Ketzmerick, R. and Lewenkopf. C. H., Phys. Rev. Lett. 84, 5504 (2000). Misirpashaev. T. Sh., Beenakker, C. W. J., Phys. Rev. A 57. 2041 (1998). Eckhardt. B.. Dorr, U., Kuhl, U. and Stockmann, H.-J., Europhys. Lett. 46, 134 (1999). Nockel, J. U.. Stone, A. D. and Chang. R. K., Opt. Lett. 19, 1693 (1994). Pctrov, P. K. and Babenko, V. A.. J. Quant. Spectrosc. Rad. Trans. 63, 237 (1999). Nockel, J. U. and Stone, A. D., in: "Optical Processes in Microcavities", (edited by R. K.. Chang and A. J. Campillo), (World Scientific, Singapore, 1996). Smilanski, U., in: Proc. 1994 Les Houches Summer School LXI Mesoscopic Quantum Phys., (edited by E. Akkermans, G. Montambaux, J. L. Pichard and J. Zinn-Justin), (Elsevier, Amsterdam, 1995), p. 373. van den Berg, P. M. and Fokkema, J. T., IEEE Trans. Anten. Propag. AP-27, 577 (1979). Barton, J. P., Appl. Opt. 36. 1312 (1997). Nockel, J. U.. Dissertation Thesis, Yale University (1997). Penrose, R., Proc. Cambridge Phil. Soc, no. 52 (1956). Doron, E. and Frischat, S., Phys. Rev. Lett. 75, 3661 (1995). Hackenbroich. G. and Nockel. J. U., Europhys. Lett. 39, 371 (1997). Stein, J. and Stockmann, H.-J., Phys. Rev. Lett. 68, 2867 (1992). Alt, H. et el. Phys. Rev. E 54, 2303 (1996). Sridhar, S. and Heller, E., Phys. Rev. A 42. R1728 (1992). Hentschel, M. and Nockel, J. U., to be published in Proc. Royal Netherlands Academy of Arts and Sciences, 2000. Lazutkin, V. F., " K.AM Theory and Semiclassical Approximations to Eigenfunctions", (Springer. New York, 1993). Robnik, M. and Berry, M. V., J. Phys. A: Math. Gen. 18. 1361 (1985). Lichtcnberg, A. J. and Lieberman, M. A., Physica D 33, 211 (1988). Nockel, J. V. and Stone. A. D., Nature 385, 45 (1997). Borgonovi. F., Casati, G. and Li, B., Phys. Rev. Lett. 77, 4744 (1996). Frahm, K. and Shepelyansky, D. L., Phys. Rev. Lett. 79, 1833 (1997). Starykh, O. A. et al, cont-mat/0001017.

Physica Scripta T90

Physica Scripta. T90, 268-277, 2001

Chaos and Time-Reversed Acoustics Mathias Fink Laboratoire Ondes et Acoustique, Ecole Superieure de Physique et de Chimie Industrielle de la Ville de Paris, Universite Denis Diderot, UMR CNRS 7587, 10 Rue Vauquelin, 75005 Paris, France Received October 9, 2000

PACS Ref: 3.40 Kf, 3.65 Ge, 3.65 Sq, 5.45b, 43.20 Fn 43.20 Ks, 43.25 Rq

Abstract The objective of this paper is to show that time-reversal invariance can be exploited in acoustics to accurately control wave propagation through random propagating media as well as through chaotic reverberant cavities. To illustrate these concepts, several experiments are presented. They show that chaotic dynamics reduces the number of transducers needed to insure an accurate time-reversal experiment. Multi-pathing in random media and in chaotic cavities enhances resolution in time-reversal acoustics by making the effective size of time reversal mirrors much larger than its physical size. Comparisons with phase conjugated experiments will show that this effect is typical of broadband time-reversed acoustics and is not observed in monochromatic phase conjugation. Self averaging properties of time reversal experiments conducted in chaotic scattering environments will be emphasized.

1. Introduction Time-reversal invariance can be exploited in acoustics to accurately control wave propagation through disordered media, whether the disorder is due to multiple scattering from random inhomogeneities or due to chaotic ray trajectories in reverberant cavities. Surprisingly, this study proves the feasibility of time-reversal in wave systems with chaotic ray dynamics. Paradoxically, chaotic dynamics is not only harmless but also even useful, as it guarantees ergodicity and thus reduces the number of acoustic channels needed to insure an accurate time-reversal experiment. The spatial complexity of the acoustic field resulting from chaotic multi-path propagation increases the spatial resolution of the time-reversed wave. The acoustic wave equation in a non-dissipative heterogeneous medium is invariant under a time reversal operation. Indeed, it contains only a second-order timederivative operator. Therefore, for every burst of sound p(r, t) diverging from a source- and possibly reflected, refracted or scattered by any heterogeneous media- there exists in theory a set of waves p{r, —t) that precisely retraces all of these complex paths and converges in synchrony, at the original source, as if time were going backwards. This idea gives the basis of time reversal acoustics. Taking advantage of these two properties the concept of time reversal cavity (TRC) and time reversal mirror (TRM) has been developed and several devices have been built which illustrated the efficiency of this concept [1-3]. In such a device, an acoustic source, located inside a lossless medium, radiates a brief transient pulse that propagates and is distorted by the medium. If the acoustic field can be measured on every point of a closed surface surrounding the medium (acoustic retina), and retransmitted through the medium in a time-reversed chronology, then the wave will travel back to its source and recover its original shape. Note that it requires both time reversal invariance and Physica Scripta T90

spatial reciprocity [4] to reconstruct the exact time-reversed wave in the whole volume by means of a two-dimensional time-reversal operation. From an experimental point of view a TRC consists of a two-dimensional piezoelectric transducer array that samples the wavefield over a closed surface. An array pitch of the order of k/2 where / is the smallest wavelength of the pressure field is needed to insure the recording of all the information on the wavefield. Each transducer is connected to its own electronic circuitry that consists of a receiving amplifier, an A / D converter, a storage memory and a programmable transmitter able to synthesize a time-reversed version of the stored signal. In practice, TRCs are difficult to realize and the TR operation is usually performed on a limited angular area, thus limiting reversal and focusing quality. A TRM consists typically of some hundred elements, or time-reversal channels. Two types of time-reversal experiments conducted with TRMs, will be discussed. It will be shown that the wave reversibility is improved if the wave traverses a random multiply scattering medium before arriving on the transducer array. The multiple scattering processes allow to redirect one part of the initial wave towards the TRM, that normally miss the transducer array. After the time-reversal operation, the whole multiply scattering medium behaves as a coherent focusing source, with a large angular aperture for enhanced resolution. As a consequence, in multiply scattering media, one is able to reduce the size and the complexity of the TRM. The same kind of improvement may be obtained for waves propagating in highly reverberant media such as closed cavities or waveguides. Multiple reflections along the medium boundaries significantly increase the apparent aperture of the TRM and a set of experiments conducted in chaotic cavities will be presented. It will be shown that, for a reflecting cavity with chaotic boundaries, a one channel time reversal mirror is sufficient to ensure reversibility and optimal focusing.

2. Time reversal cavities and mirrors The basic theory employs a scalar wave formulation (j)(r, t) and, hence, is strictly applicable to acoustic or ultrasound propagations in fluid. However, the basic ingredients and conclusions apply equally well to elastic waves in solids and to electromagnetic fields. In any propagation experiment, the acoustic sources and the boundary conditions determine a unique solution (f>(r, t) in the fluid. The goal, in time-reversal experiments, is to modify the initial conditions in order to generate the dual solution <j)(r, T — t) where T is a delay due to causality requirements. D. Dowling, D. Jackson and D. Cassereau © Physica Scripta 2001

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[4,5] have studied theoretically the conditions necessary to Spatial reciprocity and time-reversal invariance of the wave insure the generation of <j>(r, T — t) in the entire volumeequation (1) yield the following expression: of interest. 0 tr (r,, /,) = G{rx, T-tx\ i-0, t0) - Giruh \r0,T- to). (4) 2.1. The time-reversal cavity Although reversible acoustic retinas usually consist of discrete elements; it is convenient to examine the behavior of idealized continuous retinas, defined by two-dimensional surfaces. In the case of a time-reversal cavity, we assume that the retina completely surrounds the source. The basic time-reversal experiment can be described in the following way: In a first step, a point-like source located at ro inside a volume V surrounded by the retina surface S, emits a pulse at t = to > 0. The wave equation in a medium of density p(r) and compressibility n(r) is given by (Lr + L,)cj>(r, t) = -AS{r - r0)S(t - t0), 1 L, = -K(r)d„ L V V

" {W,

This equation can be interpreted as the superposition of incoming and outgoing spherical waves, centered on the initial source position. The incoming wave collapses at the origin and is always followed by a diverging wave. Thus the time-reversed field, observed as a function of time, from any location in the cavity, shows two wavefronts, where the second one is the exact replica of the first one, multiplied by - 1 . If we assume that the retina does not perturb the propagation of the field (free-space assumption) and that the acoustic field propagates in an homogeneous fluid, the free-space Green's function G reduces to a diverging spherical impulse wave that propagates with a sound speed c. Introducing its expression in (4) yields the following formulation of the time-reversed field:

(1) 0 t r (n ,ti) = K(n - r0, ti-T

+ to),

(5)

where A is a dimensional constant that insures the compati- where the kernel distribution K(r, t) is given by bility of physical units between the two sides of the equation ; for simplicity reasons, this constant will be omitted in the (6) 6\ / - • 4n\r\ V c) 47t|r| following. The solution to Eq. (1) reduces to the Green's function G(r, t \ ro, to). Classically, G(r, t | #"o, to) is written The kernel distribution K(r, t) (the propagator) corresponds as a diverging spherical wave (homogeneous and free space to the difference between two impulse spherical waves that case) and additional terms that describe the interaction of respectively converge to and diverge from the origin of the field itself with the inhomogeneities (multiple scattering) the spatial coordinate system, thus the location of the initial and the boundaries. source. It follows from this superposition that the pressure We assume that we are able to measure the pressure field field remains finite for all time throughout the cavity, and its normal derivative at any point on the surface S although the converging and diverging spherical waves show during the interval [0, T]. As time-reversal experiments a singularity at the origin. are based on a two-step process, the measurement step must The time-reversed pressure field, observed as a function of be limited in time by a parameter T. In all the following, time, shows two wavefronts, where the second one is the we suppose that the contribution of multiple scattering exact replica of the first one, multiplied by — 1. If we consider decreases with time, and that T is chosen such that the in- a wide-band excitation function instead of a Dirac distriformation loss can be considered as negligible inside the vol- bution 6(t), the two wavefronts overlap near the focal point, ume V. therefore resulting in a temporal distortion of the acoustic During the second step of the time-reversal process, the signal. It can be shown that this distortion yields a temporal initial source at rg is removed and we create on the surface derivation of the initial excitation function at the focal point. of the cavity monopole and dipole sources that correspond If we now calculate the Fourier transform of (6) over the to the time-reversal of those same components measured time variable /, we obtain during the first step. The time-reversal operation is described by the transform t —*• T — t and the secondary sources are 1 sin(to|r|/c) 1 sin(A:|r|) K(r, co) = (7) 2/7T JX k\. s(r, t) = G(r, T-t\ro, t0), (2) where X and k are the wavelength and wavenumber, d„4>s{r, t) = dnG{r, T-t\r0,toX respectively. As a consequence, the time-reversal process In this equation, 9„ is the normal derivative operator with results in a pressure field that is effectively focused on the respect to the normal direction n to S, oriented outward. initial source position, but with a focal spot size limited Due to these secondary sources on S, a time-reversed press- to one half-wavelength. The size of the focal spot is a direct ure field lr(r\,t[) propagates inside the cavity. Using a consequence of the superposition of the two wavefronts modified version of the Helmoltz-Kirchhoff integral one and can be interpreted in terms of the diffraction limitations (loss of the evanescent components of the acoustic fields). finds A similar interpretation can be given in the case of an inhomogeneous fluid, but the Green's function G now takes f+OO jd into account the interaction of the pressure field with the <M'i.fi)= M [G^OIMR&M) J—OQ J (3) inhomogeneities of the medium. If we were able to create d"r a film of the propagation of the acoustic field during the first -<j>s(r,t)dnG(ri,tl)\r,t)]-r step of the process, the final result could be interpreted as a

pir)

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projection of this film in the reverse order, immediately followed by a re-projection in the initial order. 2.2. The acoustic sink: focusing below the diffraction limit The apparent failure of the time-reversed operation that leads to diffraction limitation can be interpreted in the following way: The second step described above is not strictly the time-reversal of the first step: During the second step of an ideal time-reversed experiment, the initial active source (that injects some energy into the system) must be replaced by a sink (the time-reversal of a source). An acoustic sink is a device that absorbs all arriving energy without reflecting it. Taking into account the source term in the wave equation, reversing time leads to the transformation of source into sinks. For an initial point source transmitting a waveform s(t), the wavefield obeys the wave equation with a source term (Lr + L,)(r, t) = -AS(r - r0)s(t)

(8a)

and is transformed, in the time reversal operation in (Lr + L,)<j>(r, -t) = -AS(r -

rQ)s(-t).

(8b)

To achieve a perfect time reversal experimentally, the field on the surface of the cavity has to be time reversed, and the source has to be transformed into a sink. Therefore one may achieve time-reversed focusing below the diffraction limit. The role of the new source term S(r — r0)s(—t) is to transmit a diverging wave that exactly cancels the usual outgoing spherical wave. Taking into account the evanescent waves concept, the necessity of replacing a source by a sink in the complete time-reversed operation can be interpret as follows. In the first step a quasi-point source of size quite smaller than the transmitted wavelengths radiates a field whose angular spectrum contains both propagating waves and evanescent waves. The evanescent wave components are lost after propagation over some wavelengths. In the time-reversed step, the time-reversed field retransmitted by the surface of the cavity does not contain evanescent components. The role of the sink is to be a source modulated by s(—t) that radiates exactly, with the good timing, the evanescent waves that have been lost during the first step. Therefore the resulting field contains the evanescent part that is needed to focus below diffraction limits. With J. de Rosny, we have recently built such a sink in our laboratory and we have observed focal spot size quite below diffraction limits (typically with dimension A/20) [6]. 2.3. The time-reversal mirror This theoretical model of the closed time-reversal cavity is interesting since it offers an understanding of the basic limitations of the time-reversed self-focusing process; but it has several limitations, particularly compared to classical experimental setup that usually works without acoustic sink: • It can be proven that it is not necessary to measure and time-reverse both the scalar field (acoustic pressure) and its normal derivative on the cavity surface: measuring the pressure field and re-emitting the time-reversed field in the backward direction yields the same results, on the condition that the evanescent parts of the acoustic fields have vanished (propagation along several wavelengths) [7]. Physica Scripta T90

This comes from the fact that each transducer element of the cavity records the incoming field from the forward direction, and retransmits it (after the time-reversal operation) in the backward direction (and not in the forward direction). The change between the forward and the backward direction replaces the measurement and the time reversal of the field normal derivative. From an experimental point of view, it is not possible to measure and re-emit the pressure field at any point of a 2D surface: experiments are carried out with transducer arrays that spatially sample the receiving and emitting surface. The spatial sampling of the TRC by a set of transducers may introduce grating lobes. These lobes can be avoided by using an array pitch smaller than Amin/2 where Am;n is the smallest wavelength of the transient pressure field. In this latest case, each transducer senses all the wave vectors of the incident field. The temporal sampling of the data recorded and transmitted by the TRC has to be at least of the order of Tmm/8 (Tmm minimum period) to avoid secondary lobes [8]. It is generally difficult to use acoustic arrays that surround completely the area of interest, and the closed cavity is usually replaced by a T R M of finite angular aperture. This yields an increase of the point spread function dimension that is usually related to the mirror angular aperture observed from the source

3. Times reversal experiments 3.1. Time-reversal through random media A. Derode et al. [9] carried out the first experimental demonstration of the reversibility of an acoustic wave propagating through a random collection of scatterers with strong multiple scattering contributions. In an experiment such as the one depicted on Fig. 1, a multiple scattering sample is placed between the source and an array made of 128 elements. The whole set-up is in a water tank. The scattering medium consists in a set of 2000 parallel steel rods

'-SI'

—§||s#»«-#>

•4r-

m4**+¥*~

(A)

A'iV

|#||NI*t4*»«*™^*(B)

(first step) !••:>••

tip

Fig. 1. Sketch of experiment. IQ Physica Scripta 2001

Chaos and Time-Reversed Acoustics

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Fig. 3. Patterns of the time reversed pressure peak in the focal plane, in water (thin line), through a multiple scattering sample (thick line). 20

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than in a purely homogeneous medium. High spatial frequencies that would have been lost otherwise are redirected towards the array, due to the presence of the Fig. 2. Experimental results obtained through the multiple scattering sample. scatterers in a large area. A - Signal transmitted through the multiple scattering sample and received on element of the array, B - Time reversed signal observed at the source. This experiment shows also that the acoustic time-reversal experiments are surprisingly stable. The recorded signals (diameter 0.8 mm) randomly distributed. The sample have been sampled with 8-bit analog-to-digital converters thickness is L = 40 mm, and the average distance between that introduce quantization errors and the focusing process rods is 2.3 mm. The source is 30 cm away from the TRM still works. This has to be compared to time-reversal experand transmits a short (1 fis) ultrasonic pulse (3 cycles of a iments involving particles moving like balls on a elastic billiard of the same geometry. Computation of the direct 3.5 MHz). Figure 2(a) shows one part of the waveform received on and reversed particle trajectory moving in a plane among the TRM by one of the element. It spread over more than a fixed array of some thousand concave obstacles (a Lorentz 200 /us, i.e. ~ 200 times the initial pulse duration. After gas) shows the that the complete trajectory is irreversible. the arrival of a first wavefront corresponding to the ballistic Indeed, such a Lorentz gas is a well known example of wave, a long incoherent wave is observed, which results from chaotic system that is highly sensitive to initial conditions. the multiply scattered contribution. In the second step of the The finite precision that occurs in the computer leads to experiment, the 128 signals are time-reversed and trans- an error in the trajectory of the time-reversed particle that mitted and an hydrophone measures the time reversed wave grows exponentially with the number of scattering around the source location. Two different aspects of this encounters. problem have been studied: the property of the signal recR. Snieder and J. Scales [10] have performed numerical reated at the source location (time compression) and the simulations to point out the fundamental difference between spatial property of the time-reversed wave around the source waves and particles in the presence of multiple scattering by location (spatial focusing). random scatterers. In fact, they used time reversal as a diagThe time-reversed wave traverses the rods back to the nostic of wave and particle chaos: in a time reversal source, and the signal received on the source is represented experiment, a complete focusing on the source will only take on Fig. 2(b): an impressive compression is observed, since place if the velocity and positions are known exactly. The the received signal lasts about 1 /is, against over 300 ^s degree 8 to which errors in theses quantities destroy the qualfor the scattered signals. The pressure field is also measured ity of focusing is a diagnostic of the stability of the wave or around the source, in order to get the directivity pattern particle propagation. Intuitively, the consequences of a of the beam emerging from the rods after time-reversal slight deviation 3 in the trajectory of a billiard ball will and the results are plotted on Fig. 3. Surprisingly, multiple become more and more obvious as time goes on, and as scattering has not degraded the resolution of the system: the ball undergoes more and more collisions. Waves are indeed, the resolution is found to be six times finer (solid much less sensitive than particles to initial conditions. line) than the classical diffraction limit (dotted line)! This Precisely, in a multiple scattering situation, the critical effect does not contradict the laws of diffraction, though. length scale S that causes a signification deviation at a time The intersection of the incoming wavefront with the sample t in the future decreases exponentially with time in the case has a typical size D. After time reversal, the waves travel of particles, whereas it only decreases as the square-root on the same scattering paths and focus back on the source of time for waves in the same situation. as if they were passing through a converging lens with size Waves and particles react in fundamentally different ways D. The angular aperture of this pseudo-lens is much wider to perturbations of the initial conditions. The physical than that of the array alone, resulting in an improvement reason for this is that each particle follows a well-defined in resolution. In other words, because of the scattering trajectory whereas waves travel along all possible sample, the array is able to detect higher spatial frequencies trajectories, visiting all the scatterers in all possible comTime (|is)

© Physica Scripta 2001

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Mathias Fink

bination. While a small error on the initial velocity or position makes the particle miss one obstacle and completely change its future trajectory, the wave amplitude is much more stable because it results from the interference of all the possible trajectories and small errors on the transducer operations will sum up in a linear way resulting in small perturbation.

of the array to an observation point r\ different from the source location ro, the signal recreated in r\ at time tx = 0 writes:

(M'i.0) = JA/0>»;(0df,

(10)

Notice that this expression can be used as a way to define the directivity pattern of the time-reversed waves around the 3.1.1. Time-reversal as a matched-filter or time correlator. source. Now, due to reciprocity, the source S and the As any linear and time-invariant process, wave propagation receiver can be exchanged, i.e. h'Ai) is also the signal that through a multiple scattering medium may be described would be received in r\ if the source was the y'-th element as avlinear system with different impulse responses. If a of the array. Therefore, we can imagine this array element source, located at ro sends a Dirac pulse S(t) they'-th trans- is the source, and the transmitted field is observed at two ducer of the TRM will record the impulse response hj(t) that points r\ and ro. The spatial correlation function of this corresponds, for a point transducer, to the Green function wavefield would be (hj(t)h'j(t)) where the impulse responses G(r, t | ro, 0). Moreover, due to reciprocity, hj(t) is also product is averaged on different realizations of the disorder. the impulse response describing the propagation of a pulse Therefore Eq. (10) can be viewed as an estimator of this from the y'-th transducer to the source. Thus, neglecting spatial correlation function. Note that in one time-reversal the causal time delay T, the time-reversed signal at the experiment we have only access to a single realization of source is equal to the convolution product hj{t) * hj(—t). the disorder. However, the ensemble average can be replaced This convolution product, in terms of signal analysis, is by a time average, a frequency average or by a spatial avertypical of a matchedfilter. Given a signal as input, a matched age on a set of transducers. filter is a linear filter whose output is optimal in some sense. In that sense, the spatial resolution of the time-reversal Whatever the impulse response hj(t), the convolution mirror (i.e. the —6 dB width of the directivity pattern) is hj(t) * hj(—t) is maximum at time t = 0. This maximum is simply an estimate of the correlation length of the scattered always positive and equals f hj(t)dt, i.e the energy of the sig- wavefield [11]. nal hj{t). This has an important consequence. Indeed, with an This has an important consequence. Indeed, if the resoliV-elements array, the time-reversed signal recreated on the ution of the system essentially depends on correlation source writes as a sum properties of the scattered wavefield, it should become independent from the array's aperture. This is confirmed by j=N the experimental results. Figure 4 presents the directivity ^fo, f) = £ > ( 0 * A,-(-f). (9) patterns obtained through a 40 mm thick multiple scattering sample, using either 1 array element or the whole array (122 Even if the hj(i) are completely random and apparently elements) as a time-reversal mirror. In both cases, the spatial uncorrelated signals, each term in this sum reaches its maxi- resolution at —6dB is the same: ~ 0.85 mm. In total conmum at time t = 0. So all contributions add constructively tradiction with what happens in a homogeneous medium, around / = 0, whereas at earlier or later times uncorrelated enlarging the aperture of the array does not change the contributions tend to destroy one another. Thus the —6 dB spatial resolution. However, even though the number re-creation of a sharp peak after time-reversal on an ,/V of active array elements does not influence the typical /V-elements array can be viewed as an interference process width of the focal spot, it has a strong impact on the background level of the directivity pattern (~ — 12dB for between the N outputs of N matched filters. 28dB for N = 122), as can be seen in Fig. 4. The robustness of the TRM can also be accounted for N =\, through the matched filter approach. If for some reason, Finally, the fundamental properties of time-reversal in a the TRM does not exactly retransmits hj{—t) but rather random medium rely on the fact that it is both a space hj{—t) + rij(t), where «,(?) is an additional noise on channel and time correlator, and the time-reversed waves can be j , then the re-created signal is written: viewed as an estimate of the space and time auto-correlation j=N

j=N

J2 h}(t) * hj(-t) + J2 hj(t) * n(t). 7=1

j=\

The time reversed signals hj(-t) are tailored to exactly match the medium impulse response, which results in a sharp peak. Whereas an additional small noise is not matched to the medium and, given the extremely long duration involved, it generates a low-level long-lasting background noise instead of a sharp peak. 3.1.2. Time reversal as a spatial correlator. Another way to consider the focusing properties of the time-reversed wave is to follow the impulse response approach and treat the time-reversal process as a spatial correlator. If we note h'j(t) the propagation impulse response from they'-th element Physica Scripta T90

-"10

-5 0 5 Distance from the source fmml

10

Fig. 4. Directivity patterns with N — 128 transducers (thick line) and N = 1 transducer (thin line). © Physica Scripta 2001

Chaos and Time-Reversed Acoustics functions of the waves scattered by a random medium. The estimate becomes better with a large number of transducers in the mirror. Moreover, the system is not sensitive to a small perturbation since adding a small noise to the scattered signals (e.g. by digitizing them on a reduced number of bits) may alter the noise level but does not drastically change the correlation time or the correlation length of the scattered waves. Even in the extreme case where'the scattered signals are digitized on a single bit, A. Derode has shown recently [12] that the time and space resolution of the TRM were practically unchanged, which is striking evidence for the robustness of wave time-reversal in a random medium. 3.2. Time-reversal in bounded media In the time-reversal cavity approach, the transducer array samples a closed surface surrounding the acoustic source. In the last paragraph, we have seen how the multiple scattering processes in a large sample widens the effective TRM aperture. The same kind of improvement may be obtained for waves propagating in a waveguide or in a cavity. Multiple reflections along the medium boundaries significantly increase the apparent aperture of the TRM. The basic idea is to replace one part of the TRC transducers by reflecting boundaries that redirect one part of the incident wave towards the TRM aperture. Thus spatial information is converted into the time domain and the reversal quality depends crucially on the duration of the time-reversal window, i.e. the length of the recording to be reversed. Experiments conducted by P. Roux in rectangular ultrasonic waveguides have shown the effectiveness of the TR processing to compensate for multipath effects [13]. Impressive time recompression has been observed, that compensated for reverberation and dispersion. Besides, as in the multiple scattering experiment, the TR beam is focused on a spot which is much thinner than the one observed in free water. This can be interpreted by the theory of images in a medium bounded by two mirrors. For an observer, located at the source point, the TRM seems to be escorted by an infinite set of virtual images related to multipath propagation and effective aperture 10 times larger than the real aperture have been observed. Acoustic waveguides are also currently found in underwater acoustic, especially in shallow water, and TRMs can compensate for the multipath propagation in oceans that limits the capacity of underwater communication systems. The problem arises because acoustic transmission in shallow water bounce off the ocean surface and floor, so that a transmitted pulse gives rise to multiple copies of it that arrive at the receiver. Recently, underwater acoustic experiments have been conducted by W. Kuperman and his group from San Diego University in a sea water channel of 120 m depth, with a 24 elements TRM working at 500 Hz and 3.5 kHz. They observed focusing and multipath compensation at a distance up to 30 kms [14]. 3.2.1. Time-reversal in chaotic cavities. In this paragraph, we are interested in another aspect of multiply reflected waves : waves confined in closed reflecting cavities such as elastic waves propagating in a silicon wafer. With such boundary conditions, no information can escape from the © Physica Scripta 2001

273

system and a reverberant acoustic field is created. If, moreover, the cavity shows ergodic and mixing properties and negligible absorption, one may hope to collect all information at only one point. C. Draeger et al. [15-17] have shown experimentally and theoretically that in this particular case a time-reversal can be obtained using only one TR channel operating in a closed cavity. The field is measured at one point over a long period of time and the time-reversed signal is re-emittted at the same position The experiment is 2D and has been carried out by using elastic surface waves propagating along monocrystalline silicon wafer whose shape is a D-shape stadium. This geometry is chosen to avoid quasi-periodic orbits. Silicon was selected for its weak absorption. The elastic waves which propagate in such a plate are Lamb waves. An aluminum cone coupled to a longitudinal transducer generates these waves at one point of the cavity. A second transducer is used as a receiver. The central frequency of the transducers is 1 MHz and its bandwidth is 100%. At this frequency, only three Lamb modes are possible (one flexural, two extensional). The source is isotropic and considered point-like because the cone tip is much smaller than the central wavelength. A heterodyne laser interferometer measures the displacement field as a function of time at different points on the cavity. Assuming that there are nearly no mode conversion between the flexural mode and other modes at the boundaries, we have only to deal with one field, the flexural-scalar field. The experiment is a two step-process as described above: In the first step, one of the transducers, located at point A, transmits a short omnidirectional signal of duration 0.5 /is into the wafer. Another transducer, located at B, observes a very long chaotic signal, that results from multiple reflections of the incident pulse along the edges of the cavity, and which continue for more than 50 milliseconds corresponding to some hundred reflections along the boundaries. Then, a portion of 2 milliseconds of the signal is selected, time-reversed and re-emitted by point B. As the time reversed wave is a flexural wave that induces vertical displacements of the silicon surface, it can be observed using an optical interferometer that scans the surface around point A (Fig. 5). One observes both an impressive time recompression at point A and a refocusing of the time-reversed wave around the origin (Figs. 6(a) and 6(b)), with a focal spot whose radial dimension is equal to half the wavelength of the flexural wave. Using reflections at the boundaries, the time-reversed wave field converges towards the origin from all directions and gives a circular spot, like the one that could be obtained with a closed time reversal cavity covered with transducers. The 2 millisecond time-reversed waveform is the time sequence needed to focus exactly on point A. The success of this time-reversal experiment is particularly interesting with respect to two aspects. Firstly, it proves again the feasibility of time-reversal in wave systems with chaotic ray dynamics. Paradoxically, in the case of onechannel time-reversal, chaotic dynamics is not only harmless but also even useful, as it guarantees ergodicity and mixing. Secondly, using a source of vanishing aperture, we obtain an almost perfect focusing quality. The procedure approaches the performance of a closed TRC, which has an aperture of 360°. Hence, a one-point time-reversal in Physica Scripta T90

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Mathias Fink

Fig. 5. Time reversal experiment conducted on a silicon wafer.

a chaotic cavity produces better results than a TRM in an open system. Using reflections at the edge, focusing quality is not aperture limited, and in addition, the time-reversed

(a)

(b)

wafer scanned region IS mm

collapsing wavefront approaches the focal spot from all directions. Although one obtains excellent focusing, a one-channel time-reversal is not perfect, as a weak noise level throughout the system can be observed. Residual temporal and spatial sidelobes persist even for time-reversal windows of infinite size. They are due to multiple reflections passing over the locations of the TR transducers and they have been expressed in closed form by C. Draeger [17]. Using an eigenmode analysis of the wavefield, he shown that, for long time reversal windows, there is a saturation regime that limits the SNR. The reason is the following: Once the saturation regime is reached, all the eigenmodes are well resolved. However, A and B are always located at the nodes of some eigenmodes and these eigenfrequencies cannot be transmit or receive in the cavity.

15 mm

3.3. Time-reversal as a temporal correlator '-*• »c-

v.;

More precisely, neglecting the acoustoelectric responses of the transducers and taking into account the modal decomposition of the impulse response AABW on the eigenmodes ipn(x) of the cavity with eigenfrequency co„ we get. AAB(0 = X>»W,,(*)

sin(co„r)

(t > 0).

(11)

This signal is recorded in B and a part A r = [t\; ^] is timereversed and reemitted as AT hAB (-t) = I A(f)'

re [

} 0,

•f

^ # 4 A

Fig. 6. (a) time-reversed signal observed at point A. The observed signal is 210 lis long; (b) time-reversed wavefield observed at different times around point A on a square of 15 x 15 mm. Physica Scripta T90

~~t2' ~h]'

(12)

elsewhere,

So the time-reversed signal in A reads: <j>^{t) =

' dxh^{tR

+

T)/J AB (T)

J2 l/co^MWniB)

J2

l/mm^m{A)^m{B)Im (13) © Physica Scripta 2001

Chaos and Time-Reversed Acoustics with lmn equal to: Imn =

dt sin(o>mT) sin com(z + t)

1 |"2 = -sin(co„0 dT[sin((a»m —

J

<W„)T) +

sin((wm + a)„)x)]

i\

1 ont) I cdr[cos((ft»m + -cos(a>„0

CO„)T) +

sin((a»m +

sin(co„f) C0m — W„

h\B(-t) * hBA(t) = hAA(-t)

[-cos((w m - con)t2) + cos((cum - ain)t\)\

sin(o>„/)

+^ ' 0<

m

•[sin((cuw - aj„)t2) - sin((com - a)„)t[)],

— Q)„

if u)m ^ u)„, \ATcos(co„t),

i f co„

a>„.

(15) Under the assumption that the eigenmodes are not degenerated, then co„ = com 4> n = w, and the second term represents the diagonal elements of the sum over n and m. The first term is only important if the difference am — co„ is small, i.e. for neighboring eigenfrequencies. In the case of a chaotic cavity, next neighbors tend to repulse each other and if the characteristic distance Aco is sufficiently large so that AT » I/Aco, the non-diagonal terms of Imn are negligible compared to the diagonal contributions and one obtains Imn = \5mnATcos(cani)

+ 0(\/co).

a final peak at t = 0. For the same reason, the reversed point B cannot exactly transmitted any waveform in the cavity. Due to the boundaries, a Dirac excitation at B will also give rise to a transmitted signal //BB(0- SO, in the limit of very long time-reversal window we get for a one channel time-reversal experiment tha cavity formula deduced by C. Draeger,

CU„)T)].

The second term of each integral gives a contribution of order l/(co m +co„) < < AT which can be neglected. Thus we obtain

(16)

275

* A B B(0-

(19)

3.4. Time reversal as a spatial correlator As for the multiple scattering medium, focusing properties of the time-reversed wave can be calculated using the spatial correlator approach. If we note AAB(0 the propagation impulse response from point B to an observation point A' (with coordinates r\) different from the source location A the time-reversed signal recreated at A' at time t\ = 0 writes: hA]i(t)hA
0tr('l,O) =

(20)

Thus the directivity pattern of the time-reversed wavefield is given by the cross correlation of the Green functions that can be developed on the eigenmodes of the cavity 0 t r (r,, 0) = V — ^ ( A ) ^ ( r , ) ^ ( B ) . £

—d

(21)

fli-

Note that in a real experiment one has to take into account the limited bandwidth of the transducers, so a spectral function F(co) centered on frequency coc, with bandwidth Aa», must be introduced and we can write Eq. (21) in the form

In the limit AT —*• oo, the time-reversed signal observed in A 0 tr (r,, 0) = by a reversal in B is given by

J2^n(^n(n)^n(B)F(ajn).

(22)

Thus the summation is limited to a finite number of modes, that is typically in our experiment of the order of some hundreds. As we do not know the exact eigenmode distriThis expression gives a simple interpretation of the residual bution for each chaotic cavity, we cannot evaluate this temporal lobes which are observed experimentally in Fig. expression directly. However, one may use a statistical 6. The time-reversed signal observed at the origin cannot approach and consider the average over different realizbe simply reduced to a Dirac distribution S(t), but is equal, ations, which consists in summing over different cavity even for AT > > 1 /Aco, to another cross-correlation product. realizations. So we replace in Eq. (22) the eigenmodes product by their expectation values {...). We use also a qualitatCAT(t) = A A A ( - 0 * *BB(0 ive argument proposed by Berry [18-20] to characterize (AT) irregular modes in chaotic systems. If chaotic rays support = drhvB(t + T)AAA(T) (18) an irregular mode, it can be considered as a superposition of a large number of plane waves with random direction and phase. This implies that the amplitude of an eigenmode has a Gaussian distribution with (\l/~) =
© Physica Scripta 2001

(17)

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Mathias Fink A similar result has also been observed in the time-reversal experiment conducted in a multiply scattering medium. A clear refocusing has been obtained with only a single array element. (Fig. 4). The focusing process works with broadband pulses (the transducer center frequency is 3.5 MHz with a 50% bandwidth at —6dB). For each individual frequency there is no focusing and the estimate of the spatial correlation is very noisy. However, for a large bandwidth, if we have statistical decorrelation of the wavefields for different frequencies, the time-reversed field is self-averaging.

3.6. Time reversal and backscattering enhancement Coherent interference effects in random media are not only observed in time-reversal experiments. Different experx(mm) iments have shown that coherent interference effects survive Fig. 7. Spatial distribution of the intensity distribution. in random media despite disorder. Especially, it manifests itself in the coherent backscattering effect, which was first observed in optics in 1985 [21], and later in acoustics in 1997 One obtains finally: [22]. It is well explained by constructive interferences between a multiple scattering path and its reciprocal (25) (•Mr,, 0)) = J2^2J^2n I r i ~ ro I /l„)
3.5. Phase conjugation versus time reversal: self averaging in the time domain. This interesting result emphasizes the great interest of time-reversal experiments, compared to phase conjugated experiments. In phase conjugation, one only works with monochromatic waves and not with broadband pulses. For example, if one works only at a frequency a>„ , so that there is only one term in Eq. (22), one cannot refocus a wave on point A. An omnidirectional transducer, located at any position B, working in monochromatic mode, sends a diverging wave in the cavity that has no reason to refocus on point A. The refocusing process works only with broadband pulses, with a large number of eigenmodes in the transducer bandwidth. Here, the averaging process that gives a good estimate of the spatial correlation function is not obtained by summing over different realizations of the cavity, like in Eq. (25), but by a sum over "pseudorealizations" which corresponds to the different modes in the same cavity. This come from the fact that in a chaotic cavity, we may assume a statistical decorrelation of the different eigenmodes. As the number of eigenmodes available in the transducer bandwidth increases, the refocusing quality becomes better and the focal spot pattern becomes closed to the ideal Bessel function. Hence, the signal to noise level should increase as the square-root of the number of modes in the transducer bandwidth. Physica Scripta T90

/(A . B ) =

JA1B (t)dt.

(26)

and plotted on Fig 7. The intensity is stronger at the source point (B = A) than the background intensity (A ^ B) by a factor about 2 as expected by reciprocity arguments. A comparison of Eq. (26) and Eq. (20) shows that the field intensity /(A, B) is nothing else than the amplitude of the wave observed at the source position A at time 0 after a TR process from point B. Hence the coherent backscattering effect answers the following question: Where is the best point to time reverse a wave in a chaotic cavity with only one point transducer1] The answer is that the peak of the time reversed wave is twice higher if the initial source point A and the time reversal point B are at the same location. C. Draeger has also shown that another interest for choosing point A as the time-reversed point is that the residual background noise around the focal spot is also weaker than in the case A ^ B [16]. 4. Conclusion In this paper, we have shown how chaotic ray dynamics in random media and in cavities enhances resolution in time-reversal experiments. Multi-pathing makes the effective size of the TRM much larger than its physical size. This 'Q Physica Scripta 2001

Chaos and Time-Reversed Acoustics study proves the feasibility of acoustic time-reversal in media with chaotic ray dynamics. Paradoxically, chaotic dynamics is useful, as it reduces the number of channels needed to insure an accurate time-reversal experiment. This result can be easily extended to electromagnetic propagation through random and chaotic media. Perhaps one of the most spectacular application of time-reversed technologies will be, in the future, wireless communication, where the multipathing may enhance the amount of information that can be transmitted between antenna arrays in complex scattering environments like cities. From quantum chaos to wireless communications, time reversal symmetry plays an important role and, contrary to long-held beliefs, chaotic scattering of acoustic or microwave signals may strongly enhance the amount of independent information that can be transmitted from an antenna array to a volume of arbitrary shape. References 1. Fink, M., Prada. C , Wu. F. and Casscreau, D.. IEEE Ultrasonics Symposium Proceedings, Montreal, Vol 1. pp. 681-686, (1989). 2. Fink, M.. IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 39. 555 (1992). 3. Fink, M., Physics Today 50. 34 (1997).

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4. Jackson, D. R. and Dowling, D. R., J. Acoust. Soc. Am. 89, 171 (1991). 5. Cassereau. D. and Fink, M., IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 39, 579 (1992). 6. de Rosny, J. and Fink. M., to be published. 7. Cassereau, D. and Fink, M., J Acoust. Soc Am. 96, 3145 (1994). 8. Kino. G. S.. "Acoustics Waves", Prentice Hall, Signal Processing Series (1987). 9. Derode, A., Roux, P. and Fink, M„ Phys. Rev. Lett. 75, 4206 (1995). 10. Snieder, R. and Scales. J., Phys. Rev. E 58, 5668 (1998). 11. Derode, A.. Tourin, A. and Fink. M., J. Acoust. Soc. Am. (2000). 12. Derode, A. Tourin, A. and Fink. M., J. Appl. Phys. 85, 6343 (1999). 13. Roux, P., Roman, B. and Fink, M., Appl. Phys. Lett. 70, 1811 (1997). 14. Kupperman, W. A. et ai, J. Acoust. Soc. Am. 103, 25 (1998). 15. Draeger, C. and Fink, M„ Phys. Rev. Lett. 79, 407 (1997). 16. Draeger, C. and Fink. M.. J. Acoust. Soc. Am. 105, 618 (1999). 17. Draeger, C. and Fink, M„ J. Acoust. Soc. Am. 105. 611 (1999). 18. Berry. M. V., in "Les Houches 1981 -chaotic behaviour of deterministic systems". (North Holland, Amsterdam, 1983), pp. 171 271. 19. McDonald, S. W. and Kaufman. A. N., Phys. Rev. A 37, 3067 (1988). 20. Weaver. R. and Burkhardt, J. J.. Acoust. Soc. Am. 96. 3186 (1994). 21. van Albada. M. P. and Lagendijk, A., Phys. Rev. Lett. 55, 2692 (1985); Wolf, P. E. and Maret, G.. Phys. Rev. Lett. 55, 2696 (1985). 22. Tourin, A., Derode, A., Roux. P., Van Tiggelen, B. A. and Fink. M., Phys. Rev. Lett. 79, 3637 (1997). 23. de Rosny. J., Tourin, A. and Fink, M.. Phys. Rev. Lett. 84, 1693 (2000). 24. Weaver, R. L. and Lobkis, O. I., Phys. Rev. Lett. 84, 4942 (2000).

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Physica Scripta. T90, 278-282, 2001

Single-Mode Delay Time Statistics for Scattering by a Chaotic Cavity K. J. H. van Bemmel, H. Schomerus and C. W. J. Beenakker Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands Received September 1, 2000

PACS Ref: 05.45.Mt, 42.25.Dd, 42.25.Hz

P = 1 (2) indicates the presence (absence) of time-reversal symmetry. We assume a single polarization for simplicity, We investigate the low-frequency dynamics for transmission or reflection of a as in the microwave experiments in a planar cavity [2]. wave by a cavity with chaotic scattering. We compute the probability distribution of the phase derivative /dw of the scattered wave amplitude, The dynamical observable is the correlator p of an element known as the single-mode delay time. In the case of a cavity connected of the scattering matrix S(co) at two nearby frequencies, Abstract

to two single-mode waveguides we find a marked distinction between detection in transmission and in reflection: The distribution P(4>') vanishes for negative (j>' in the first case but not in the second case.

1. Introduction

p = Snm(co +1 Sco)S*nm(co - \ Sco).

(1)

The indices n and m indicate the detected outgoing mode and the incident mode, respectively. The single-mode delay time (j)' is defined by [3-5]

Microwave cavities have proven to be a good experimental testing ground for theories of chaotic scattering [1]. Much work has been done on static scattering properties, but recently dynamic features have been measured as well [2]. A key dynamical observable, introduced by Genack and coworkers [3-5], is the frequency derivative (j)r = d(j)/dco with / = \S (a>)\2 the intensity of the scattered wave in nm of the phase of the wave amplitude measured in a single mode n for unit incident intensity in mode m. If we write speckle of the transmitted or reflected wave. Because one the scattering amplitude S = v7e'* in terms of amplitude nm speckle corresponds to one element of the scattering matrix, and phase, then (j)' = dcp/dco. We will investigate the distriand because <$>' has the dimension of time, this quantity isbution of (j>' in an ensemble of chaotic cavities having slightly called the single-channel or single-mode delay time. It is different shape, at a given mean frequency interval A a linear superposition of the Wigner-Smith delay times intro- between the cavity modes. For notational convenience, duced in nuclear physics [6,7]. we choose units of time such that 2n/A = 1. The probability distribution of the Wigner-Smith delay The single-mode delay times are linearly related to the times for scattering by a chaotic cavity is known [8]. The Wigner-Smith [6,7] delay times x\, 12,..., TJV, which are the purpose of this paper is to derive from that the distribution eigenvalues of the matrix P(4>') of the single-mode delay time. The calculation follows closely our previous calculation of P(') for reflection from a disordered waveguide in the localized regime [9]. The g = - i S f t ^ = C / t d i a g ( T , , . . . , T ) E / . (3) JV absence of localization in a chaotic cavity is a significant dco simplification. For a small number of modes TV connecting the cavity to the outside we can calculate P{<j)') exactly, while To see this, we first expand the scattering matrix linearly in for A f » l we can use perturbation theory in 1/TV. The large-TV distribution has the same form as that following from diffusion theory in a disordered waveguide [4,5], but S(co±\8co)= VTU±\ida)VTdiag(T:u...,TN)U. (4) for small TV the distribution is qualitatively different. In particular, there is a marked distinction between the distriSince S is symmetric for /? = 1, one then has V — U. For bution in transmission and in reflection. jS = 2, V and U are statistically independent. Combination 2. Formulation of the problem The geometry studied is shown schematically in Fig. 1. It consists of an TV-mode waveguide connected at one end to a chaotic cavity. Reflections at the connection between waveguide and cavity are neglected (ideal impedance matching). The TV modes may be divided among different waveguides, for example, TV = 2 could refer to two single-mode waveguides. The cavity may contain a ferrimagnetic element as in Refs. [10,11], in which case time-reversal symmetry is broken. The symmetry index Physica Scripta T90

\

-<sXV»

.

(

Fig. 1. Sketch of a chaotic cavity coupled to N propagating modes via one or more waveguides. The shape of the cavity is the quartered Sinai billiard used in recent microwave experiments [2], © Physica Scripta 2001

Single-Mode Delay Time Statistics for Scattering by a Chaotic Cavity The result is

of Eqs. (1), (2), and (4) leads to [9] T

iV

i2 ±i

r> Hi

279

T u v

iii

ut = £/,„,, v,- = Vin.

(14)

(5)

dfi oc sin 2y dy j | da,-.

(6)

The joint distribution function Eq. (7). For /? = 1 one has

The distribution of the Wigner-Smith delay times for a chaotic cavity was calculated in Ref. [8]. It is a Laguerre ensemble in the rates ju, = 1/T„

P(T+,T_)

follows from

/»(t+,T_)=1L|T_|(^.-Ti)-4

(15)

X eXp(-T + (T^ - T i ) - ' ) 0 ( T + - |T_|),

P{li,,..., nN) <x f ] \ii, - ft/ Y\ niN'2 exp(-\Pn k )8Qi k ).

(7)

while for /? = 2

The step function 0(x) = 1 for x > 0 and 0(x) = 0 for x < 0. i > ( T , T _ ) = l T 2 _ ( ^ - T 2 J - 6 + It follows from Eq. (7) that (5Z,-T,-> = 1, a result that was (16) x exp(-2T + (i5. - T I ) - 1 ) ^ - |T-I). known previously [12]. To calculate the joint distribution P(I, <j>') from Eq. (5), we also need the distribution of the coefficients u, and v,. This F i r s t w e c ° n s i d e r t h e c a s * P=\n^m. Because of the V follows from the Wigner conjecture [13], proven in Ref. [8], f ^ ;\™e h a s | v i ' = !"2J a n d , M f J ? 1 ' T h e r e " fore a = 0 and cf>' = %+, so 4>' is independent of/. Integration according to which the matrices U and V are uniformly dis. of Eq (15) over T results m tributed in the unitary group. The calculation for small ' " N is now a straightforward integration, see Section 3. A (17) For large N we can use perturbation theory, see Section 4. P{#) = H'-\<j>' + 2' + 2 ) e x p ( - 2 / 0 W ) Because of the uniform distribution of U and V, independent of the r,'s, we can evaluate the average of <j>' directly I n t h i s c a s e < as w e l 1 a s i n t h e c a s e N=V> c a n t a k e o n o n l y positive values, but this is atypical, as we will see shortly. for any N, From Eqs. (10) and (12) we find the relation between I and the parametrization of U, (8) N' 7 = sin2 2y sin2(a3 — ai — 1x2). (18) We define the rescaled variable '/(<£') = N
-1),

(19)

3. Small number of modes For N = 1 there is no difference between the Wigner-Smith delay time jand the single-mode delay time. In that case 7 = 1 and (/>' = ') = c^'-2~m

exV(-\p/4>')6{4>').

in agreement with Refs. [16,17]. For the case N = 2, ft = 1, n = m we use that u\ = v\, u2 = v2 and obtain the parametrization

(9) / = 1

(20)

sin 2y sin (04 — a\ — a 2 ),

The normalization coefficient cp equals (2n)~1/2 for /? = 1 a = (cos2y)/7. (21) and 1 for j? = 2. Now we turn to the case N = 2. By writing out the summation in Eq. (5) for 7 and 4>', one obtains (/>' = T + + ar_ The distribution P(I, a) resulting from the measure (14) is with T± = j(x\ ± T 2 ) and />(/, a ) = ^ - / 1 / 2 ( l - /)~ 1/2 (1 - 7a 2 r 1 / 2 0(/)0(l - 7)0(1 - 7a 2 ). 2n / = | M l | 2 | v 1 | 2 - h | M 2 | 2 | v 2 | 2 + MlW *v 1 v*+ M >2vtv 2 , (10) (22)

« = (\ui\2\vl\2-\u2\2\v2\2)/I.

(11)

To find the joint distribution P(I, a) we parametrize U in terms of 4 independent angles, U =

cosyexp(—iaj) sinyexp(—iai — ia2) — sin y exp(—ia3 + ia2) cos y exp(—ia3) ) • (12)

with a,- e (0, 27i) and y e (0, rc/2). The invariant measure dpi oc |Detg|dyr],da, in the unitary group follows from the metric tensor g, defined by Tr dUdtf

= 'JTgijdxjdxj, 'V

Physica Scripta 2001

{*,-} = {y, «i. «2, «3}-

(13)

The joint distribution of / and 4>' = 2f takes the form /»(/, ') = / Jo

r

dt_ dz + 7 > (T + , T.

0'/2

H'-^ir1)-

-T+\

1 (23)

The distribution of 7 following from integration of P(J,oc) over a is given by Eq. (19) with 7 -> 1 - 7 , as it should. The integrations over T + , T _ , and 7, needed to obtain T*^') can be evaluated numerically, see Fig. 2. Notice that P(') has a tail towards negative values of cj>'. Physica Scripta T90

280

K. J. H. van Bemmel, H. Schomerus and C.W. J. Beenakker

N = l N = 2,n ^ m N = 2,n = m

-0.5 0 0.5

1 1.5

-0.5 0 0.5

2 2.5

1 1.5

2 2.5

i' Fig. 2. Distribution of the single-mode delay time in the case of preserved Fig. 3. Same as in Fig. 2, for broken time-reversal symmetry (fi = 2)/The time-reversal symmetry (/? = 1). The curves for N = 1, 2 follow from Eqs. (9), curves for N = 1, 2 and for N » 1 follow from Eqs. (9), (23), and (36). There (17), and (23). The curve for N » 1_follows from Eq. (36), and is the same for is no difference between n = m and n-£m for any N. The large TV-result for P = 2 is the same as for /) = 1. n = m and n # m. The delay time <j>' = (j> l(<j>') is rescaled such that the mean is 1 for all curves. Data points are the result of a Monte Carlo calculation in the Laguerre ensemble (with N = 400, n / m representing the large-JV limit).

For N = 2, fi = 2 it doesn't matter whether n and m are equal or not. Parametrization of both U and V leads to / = (1 - xi)(l - X2) + X\X2 + 2yf{\ - X l ) ( l

eigenvalues, = \(qBx +p±

X2)X\X2COSf],

y/lpqBx + q2B2 + p2^j,

(30)

(24) a — (1 — x\ — xi)/I,

(31)

(25) ** = 2>.rM

with a measure d/j. oc dxidx2drj and x\, x2 e (0,1), r\ e (0, n). Performing the inverse Fourier transforms and returning to The joint distribution P{I, a) is now given by the variables ' and / leads to P(I, a) = i/'/ 2 0(7)0(l - 1)0(1 - lot2). (26)

P(I) = 9(1)6(1 -1).

(N3I/nY/lexp(-NI)

P(I, (j)') =

Integration over a leads to [16,17] (27)

^(B2-B2ri2^{-NI%^yT).

The distribution P(I, <$>') follows upon insertion of Eqs. (16) (32) and (26) into Eq. (23). Numerical integration yields the distribution P(4>') plotted in Fig. 3 As in the previous case, there The averages over M, and T,- still have to be performed. is a tail towards negative (j)f. Up to now the derivation is the same as for the disordered waveguide in the localized regime [9], the only difference being the different distribution of the Wigner-Smith delay 4. Large number of modes times T,-. The absence of localization in a chaotic cavity We now calculate the joint distribution P(I, <j)') for N » 1. greatly simplifies the subsequent calculation in our present First the case n ^ m will be considered, when there is no dis- case. While in the localized waveguide anomalously large tinction between /? = 1 and /? = 2. In the large TV-limit the T,'S lead to large fluctuations in B\ and B , in the chaotic 2 vectors « and v become uncorrelated and their elements cavity the term y? NI2 in Eq. (7) suppresses large delay times. k become independent Gaussian numbers with zero mean Fluctuations in Bk are smaller than the mean by a factor and variance 1/N. We first average over v, following \/s/N. For i V » l we may therefore replace Bk in Eq.(32) Ref. [9]. We introduce the weighted delay time W = I'. by (B ). k The Fourier transform of P(I, W) is given by y(Pi 7 -\-iqW)). The average over v is a Gaussian p(t) = (%2i <5(T - T,-)) of the delay times. It is given by [8] integration, that gives ^ , g r ) = <det(l-i///iV)- 1 ), H = pu*uT +\q(u*uT + u*uJ),

(28) (29)

where w, = «,T ( . The matrix H has only two nonzero Physica Scripta T90

P(T)

N 2nx2

. / ( T + - T)(T - T - ) , T±

3±V8 N

'

(33)

for T inside the interval (T_, T+). The density is zero outside this interval. The resulting averages are (B\) = TV-1 and © Physica Scripta 2001

Single-Mode Delay Time Statistics for Scattering by a Chaotic Cavity (B2) - 2N~2, which leads to />(/, h = (N3I/7t)l/2 e x p ( - 7 w [ l + (4>r - 1)2])6>(/).

(34)

(Recall that $ = <£'/<<£'> = N<j>'.) Integration over $' or / gives P(I) = /Vexp(-/V7)0(/),

(35)

P(f) = ^[l+^'-l) 2 ]~ 3 / 2 .

(36)

This distribution of / and
(37)

The joint distribution P(C0, C\) has the Fourier transform

281

imaginary part of the off-diagonal elements are independently Gaussian distributed with zero mean and unit variance. The real diagonal elements are independently Gaussian distributed with zero mean and variance 2. We diagonalize H, order the eigenvalues from large to small, and multiply the n-th normalized eigenvector by a random phase factor eM", with a„ chosen uniformly from (0,2n). The resulting matrix of eigenvectors is uniformly distributed in the unitary group. The Laguerre ensemble (7) for the rates /i, = 1/T, can be generated by a random matrix of the Wishart type [18,19]. Consider a N x (2N — 1 + 2//?) matrix X, where X is real for p = 1 and complex for ft = 2. (The matrix X is neither symmetric nor Hermitian.) The matrix elements are Gaussian distributed with zero mean and variance (\xnm\2} = 1- The joint probability distribution of the eigenvalues of the matrix XX^ is then given by Eq. (7). The results of our numerical check are included in Figs. 2 and 3. The large-Af limit is represented by N — 400, n ^ m. The analytical curves agree well with the numerical data.

x(Po,Puqo,q\) = (exp(i/>oReC0 + i<7oImC0 + i/jjReCi + i^iImCi)). (38)

6. Conclusion

We have investigated the statistics of the single-mode delay time (/>' for chaotic scattering. For a large number N of X(po, Pi,qo,qi)= (exp(-L)), (39) scattering channels the distribution has the same form as for diffusive scattering [4,5], but for small N the distribution ^ = l E / l n [ l +N-2(p0+plTd2+N-2(q0 + qlTi)2]. (40) is different. The case N = 2 is of particular interest, representing a cavity connected to two single-mode Fluctuations in Z, are smaller than the average by a factor waveguides. For preserved time-reversal symmetry and 1/iV. We may therefore approximate (exp(—L)) =» detection in transmission (ft — 1, n ^ rri), we find that <j>' exp(-L). Because N~2(p0 + £IT,) 2 + N~2(q0 + q^j)2 is of can take on only positive values, similarly to the Wignerorder 1/7V, we may expand the logarithm in Eq. (40). Smith delay times. In contrast, for detection in reflection The average follows upon integration with the density (33), (or for broken time-reversal symmetry) the distribution acquires a tail towards negative '. These theoretical IT\ Po + go ,PJ + 9i , PoPi + °oai ,AU predictions are amenable to experimental test in the {L) + + (41) microwave cavities of current interest [2].

Averaging over u we find

=-^r ^^ —A^—•

Inverse Fourier transformation gives TV4 /'(Co, C,) = —-2exp(-iV|Co| 2 - i ^ 3 | C , | 2 + /V2Re C0C*). (2n) (42) The resulting distribution of (j>' and / is P(I, <j>') = (7V 3 //27r) 1/2 exp(-iAr/[l + ($' -

We thank P. W Brouwer and M. Patra for valuable advice. This research was supported by the "Nederlandse organisatie voor Wetenschappelijk Onderzoek" (NWO) and by the "Stichting voor Fundamenteel Onderzoek der Materie" (FOM).

l)2Jje(I). (43)

It is the same as the distribution (34) for nj^m, apart from the rescaling of / by a factor of 2 as a result of coherent backscattering. The distribution (36) of ' for N » 1 is included in Figs. 2 and 3 for comparison with the small ./V-results. 5. Numerical check We can check our analytical calculations by performing a Monte Carlo average over the Laguerre ensemble for the T,'S and the unitary group for the w,'s and v,-'s. For the average over the unitary group we generate a large number of complex Hermitian N x N matrices H. The real and © Physica Scripta 2001

Acknowledgements

References 1. Stockmann, H.-J., "Quantum Chaos — An Introduction", (Cambridge University, Cambridge, 1999). 2. Persson, E., Rotter, I., Stockmann, H.-J. and Barth, M., Phys. Rev. Lett. 85, 2478 (2000). 3. Sebbah, P., Legrand, O. and Genack, A. Z., Phys. Rev. E 59, 2406 (1999). 4. Genack, A. Z., Sebbah, P., Stoytchev, M. and van Tiggelen, B. A., Phys. Rev. Lett. 82, 715 (1999). 5. van Tiggelen, B. A., Sebbah, P., Stoytchev, M. and Genack, A. Z., Phys. Rev. E 59, 7166 (1999). 6. Wigner, E. P., Phys. Rev. 98, 145 (1955). 7. Smith, F. T., Phys. Rev. 118, 349 (I960). 8. Brouwer, P. W., Frahm, K. M. and Beenakker, C. W. J., Phys. Rev. Lett. 78, 4737 (1997); Waves Random Media 9, 91 (1999). 9. Schomerus, H., van Bemmel, K. J. H. and Beenakker, C. W. J., condmat/0004049. Physica Scripta T90

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K. J. H. van Bemmel, H. Schomerus and C.W J. Beenakker

10. So, P., Anlage, S. M., Ott, E. and Oerter, R. N., Phys. Rev. Lett. 74, 2662 (1995). 11. Stoffregen, U., Stein, J., Stockmann, H.-J., Kus, M. and Haake, F., Phys. Rev. Lett. 74, 2666 (1995). 12. Lyuboshitz, V. L., Phys. Lett. B 72, 41 (1977). 13. Wigner, E. P., Ann. Math. 53, 36 (1951); 55, 7 (1952). 14. Fyodorov, Y. V. and Sommers, H.-J., Phys. Rev. Lett. 76, 4709 (1996).

Physica Scripta T90

15. Gopar, V. A., Mello, P. A. and Biittiker, M., Phys. Rev. Lett. 77, 3005 (1996). 16. Baranger, H. U. and Mello, P. A., Phys. Rev. Lett. 73, 142 (1994). 17. Jalabert, R. A., Pichard, J.-L. and Beenakker, C. W. J., Europhys. Lett. 27, 255 (1994). 18. Edelman, A., Linear Algebra Appl. 159, 55 (1991). 19. Baker, T. H„ Forrester, P. J. and Pearce, P. A., J. Phys. A 31, 6087 (1998).

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