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LQO TECHNION—93—QCAP The Many Faces of Chaos CERN l.IBRFIF!IES» GENE/Fl ||l||\\l||l||||\|||l|\\||||||||lllllllllllllllll PBEBIBSEI Department of , Technion—Israel Institute of Technology, 32 000 Haifa, Israel

Abstract — Various approaches to quantum chaos are reviewed and compared. It is not difficult to generate quantum evolutions (unitary mappings of ) which are chaotic. Some of the algorithms achieving this result are quite formal and apparently devoid of physical interest. In the more promising approaches, the following property stands out: a quantum system whose classical analog is chaotic displays hypersensi tivity to small perturbations of its Hamiltonian. The long range evolution of such a system is unpredictable in the presence of small, uncontrollable perturbations. This unpredictability is the hallmark of physical chaos.

INTRODUCTION

The study of classical chaos is a mature and well understood discipline, covering a wide array of physical phenomena. On the other hand, the very existence of quantum chaos was until recently a controversial subject Indeed, the notion of chaos refers to the dynamical behavior as t —> oo, and has a fundamentally asymptotic character. On the other hand, for large enough but still finite times, the , which has undeniable heuristic value when we want to compare classical and quantum systems, fails for any nonlinear system In classical Hamiltonian dynamics, chaos arises because orbits may be unstable and errors in the initial data grow exponentially In quantum , the situation seems to be quite different. The dynamical law is a unitary evolution, 1/xt = U $0. Starting from a slightly different initial state 1,/26 yields 1/Jl = Ud}6, with the same unitary U. It follows that the scalar product of the perturbed and unperturbed states is constant: (1/1,,1%} : (1/JO , 1%}. In other words, small imperfections in the preparation of the initial state d0 uat grow. This elementary argument, however, is not convincing: if, instead of considering isolated classical trajectories, we use Liouville’s equation which describes an ensemble of classical systems, it readily follows from Koopman’s theorem [4] that the overlap of two different Liouville densities is constant in time, just as the overlap of two quantum wave functions, which also describe statistical ensembles. The mere constancy of this overlap therefore does not guarantee the absence of chaos. OCR Output in time, just as the overlap of two quantum wave functions, which also describe statistical ensembles. The mere constancy of this overlap therefore does not guarantee the absence of chaos. There is indeed a very simple way of generating a quantum chaos which closely parallels any type of classical chaos Consider an autonomous obeying the equations of

dmk/dt:V(a:1,...,xN),k k:1,...,N. (1) lf N 2 3, such a system may be chaotic. Irrespective of its physical nature, it is always formally possible to introduce a Hamiltonian,

H:EVk(xl,...,xN)pk, (2) where the pk are new variables, defined to be canonically conjugate to the :1:k. This Hamiltonian obviously gives Eq. (1) as the law of motion. (Note that the Lagrangian, L E Z pk ick ——H, is numerically equal to zero. This is a highly constrained canonical system.) Quantization may then proceed as usual by the introduction of a $(:101, . . . ,:1:N) and the substitution pk —>—i7i3/Oxk. We then still have Eq. (1) as the Heisenberg equation of motion for the operators wk, and since the latter commute (and therefore can be simultaneously diagonalized) any chaos in the solution of the classical equations (1) will be reflected as chaos in the of the expectation values We thus see that there is no formal incompatibility between quantum theory and chaos. The only relevant question is whether chaos can be found in “natural” quantum systems (in particular, in those which are experimentally ) just as we encounter chaos in ordinary planetary systems, or in fluid mechanics. For example, a classical model of the lithium atom is chaotic; l1ow is this fact reflected in the properties of real lithium atoms?

QUANTUM ASPECTS OF CLASSICAL CHAOS

A is not the analog of a point in the classical . The classical analog of a quantum state is a Liouville probability density. lf two Liouville functions are initially concentrated around neighboring points, and have some overlap, that overlap remains constant in time, by virtue of Koopman’s theorem Each one of tl1e Liouville functions may become distorted beyond description, until all phase space appears thoroughly mixed when seen on a coarse scale; yet, the overlap of these functions remains constant. However, the interesting problem is not how a Liouville probability density, which was initially given, will later overlap with the tortuous domain covered by the time evolution of another, initially given Liouville density. The experimentally relevant question is how each one of these time dependent domains overlaps with a fixed domain of phase space. ln a classical chaotic system, the final probability density, seen on a coarse scale, is homogeneous and roughly independent of the initial conditions. This property is called . OCR Output Turning now to quantum theory, we may inquire whether Wigner’s function [6]

WM, P) = (rh)`N / z>(q — rl q + r) €2"’""’” dr, (3) which is the quantum analog of a Liouville density, is also subject to mixing. The answer is negative: Wl (q, p) has a much smoother time evolution than a Liouville function In particular, it can never develop contorted substructures on scales smaller than 71. Therefore, Wigner’s function VV(q, p) does not possess the mixing property, as defined above. In general, it is found empirically, by numerical simulations, that tends to suppress the appearance of chaos. Quantum wave packets may remain localized, even when classical orbits are strongly chaotic, because the breakup of KAM (Kolmogorov-Arnol’d—Moser) surfaces starts in limited regions, and the remnants of these surfaces effectively act as barriers to quantum wave packet motion while permitting extensive classical flow A similar phenomenon appea.rs in simple models where the Hamiltonian includes a time~dependent perturbation In these models, which may have a single degree of freedom, the physical system is prepared in a state involving only one, or at most a few energy levels of the unperturbed Hamiltonian. One then finds that the time evolution of the quantum system involves only a few more neighboring energy levels, so that the energy remains "localized” in a narrow domain, even though no vestige remains of the KAM manifolds, the corresponding classical evolution is chaotic, and the classical energy increases without bound in a diffusive way [10]. As a consequence of this energy "localization,” the quantum motion is almost periodic and the initial state recurs repeatedly [11], as it would for a ti1ne—independent Hamiltonian with a point spectrum. The peculiarity here is not the recurrence itself, which is similar to that in a Poincaré cycle [12], and is completed only after an inordinately long time, but the fact that the quantum state after an arbitrarily long time can be computed accurately with a finite amount of work. This suggests a curious paradox [13]. Rather than computing a classically chaotic by numeri cal integration of Hamilton’s equations, we could quantize in the standard way the classical Hamiltonian (with an arbitrarily low, but finite value of li) and then integrate the Schrodinger equation in order to follow the motion of a small wave packet. For example, we could integrate the evolution of the solar system for trillions of years, by assuming that the Sun and the planets, and all their moons, and all the asteroids, are point particles with constant and exactly known masses, and by replacing these classical points by Gaussian wave packets of optimum size. Could this be a less complex task (for t —+ oo) than the direct integration of Hamilton’s equations? Unfortunately, nothing can be gained by this subterfuge, because the initial quantum state from which we obtain the final wave packet is not itself a small wave packet, but is likely to be spread throughout all the accessible phase space. lf we want a genuine quantum simulation of a classical orbit of total duration t, we must start from a wave packet of size Aq ~ e’L‘(Aq)nna; and Ap ~ e“L’(Ap)nm;, where L is the . Then, for a given value of the final uncertainty, the initial Aq Ap OCR Output must behave as exp(—2Lt), and this requires using a fake h which decreases as exp(—2Lt). As a consequence, we must take into account an increasing number of states, because the smaller Ti, the larger the density of energy levels for a given energy. A detailed analysis [13] then shows that the number of terms needed in the expansion into energy eigenstates increases exponentially with t, with at least the same Lyapunov exponent as for the classical problem. We thus see that if we attach to the word “chaos” the meaning that the computational of a dynamical evolution increases faster than the actual duration of the motion (so that long range predictions are impossible, except statistically), then a genuine quantum system, with fixed Ti and a discrete energy spectrum, is never chaotic. On the other hand, if we want the correspondence principle to hold for a classically chaotic system, all we have to do is to use a fake value of h which decreases as e`2L’, where t is the total duration of the motion. It is crucial to specify which limit is taken first, t -—> oo or Ti —> 0 [14]. For fixed Fi, the correspondence principle is expected to break down after a finite time, of the order of log(S/Ti)/L, where S S: fpdq of the classical orbit.

UNITARY MAPS

An explicit model in which genuine quantum chaos occurs with properties similar to those of classical chaos is the configurational quantum cat map [15]. A charged particle moves in a square in the my plane, with periodic boundary conditions. It is subjected to time—periodic electromagnetic kicks (the electric field vector lies in the my plane, and the magnetic one is perpendicular to it). lt is found that the quasi-energy spectrum is absolutely continuous, and every value is countably infinite degenerate. The expectation value of the energy grows at an exponential rate, and a wave packet, initially concentrated in a small region of configuration space, gets exponentially stretched and folded, and is quickly distributed over the entire square. lt is even possible to have quantum chaos in a finite dimensional Hilbert space. Consider a unitary evolution, 1/2(t) = U(t) ¢, where the elements of the unitary matrix U(t) have a chaotic evolution, and