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Exponential sensitivity and chaos in quantum systems.

Poincare was the first to appreciate that exponential sensitivity in nonlinear systems can lead to exceedingly complicated dynamical behavior [1—4]. For lack of illustrative examples and "visual aids, " however, the importance of his work was not immediately appreciated by the scientific community. This situation changed dramatically with the advent of powerful and fast computers and nowadays Poincare's ideas are counted among the foundations of the theory of nonlinear dynamics and chaos. By now chaotic phenomena have been identified in nearly every branch of physics. Surprisingly, complicated systems are not necessary for chaos to emerge. In fact, chaos can be found in the simplest dynamical systems. Well known examples are the driven pendulum [5], the double pendulum [6], and the hydrogen atom in a strong magnetic [7] or microwave field [8]. With chaos in classical dynamical systems well established, the question arises whether quantum systems are able to display exponential sensitivity and chaos. It was argued long ago [9,10] that quantum chaos is strictly impossible in autonomous finite quantum systems which exhibit a discrete energy spectrum. Moreover, a peculiar quantum suppression effect, first discovered by Casati and collaborators in the dynamics of the kicked rotor [11],seems to suppress exponential instabilities and chaos in periodically perturbed quantum systems [8,11— 14]. Fishman, Grempel, and Prange [15] showed that this effect is related to Anderson localization [16], a destructive phase interference effect which can occur in strongly disordered solids. As a consequence of these "negative" results, a shift in focus occurred in quantum chaos research. Instead of looking for exponential sensitivity, the main stream of quantum chaos research is currently concentrated on identifying the fingerprints of classical chaos in quantum systems. This type of research is usually identified as "quantum chaology" a term coined and justified by Berry [17]. Today, quantum chaology is by far the most fruitful branch of quantum chaos research and produced a host of new insight into the classical/quantum correspondence of classically chaotic systems. This does not mean that exponential sensitivity was never identified in quantum systems. Fox and Eidson, e.g. , presented a laser model which exhibits exponential sensitivity and chaos [18]. The same is true for a model of nuclear collective motion discussed by Bulgac [19]. Even models in solid state physics [20,21] or models coupling a quantum system dynamically with its boundary [22] are known to exhibit chaos. All these models, however, share a common drawback. Every one of these models introduces a classical approximation to some (but not all) of its quantum degrees of freedom. Therefore, they are best described by the term "half-classical. " In the laser-chaos model, e.g. , it is the radiation field which is treated classically [18] and in the model for chaos in nuclei it is the collective degrees of freedom which are treated classically [19]. The separation into a classical and a quantum system is justified in these models on the basis of arguments analogous to the Born-Oppenheimer approximation well known in molecular physics [23, 24]. Therefore, exponential sensitivity and chaos can be observed in the classical as well as in the quantum subsystems of half-classical quantum systems over the time period of validity of the classical approximation. Another interesting approach to the quantum chaos problem was recently discussed by Ford et al. [25]. Using algorithmic complexity theory [25,26] they showed explicitly that the quantized version of Arnol'd's cat map (see below) is not chaotic. Summarizing the present state of research it is fair to say that no fully quantized system has ever been produced which would show exponential sensitivity and chaos. It is the purpose of this Letter to present the first such system. It is based on one of the paradigms of quantum mechanics, the Stern-Gerlach apparatus (see, e.g. , Ref. [27]), and can thus be studied experimentally. Consider a spin-precession apparatus X which consists of a co11inear sequence of two homogeneous magnetic field sections A and 8, respectively (see Fig. 1 for a case with two magnets). The magnets are arranged according to the recursive rule