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arXiv:quant-ph/9805068

Does quantum chaos exist? (A quantum Lyapunov exponents approach.)

W. A. Majewski

Published May 22, 1998• 6 citations
arXivHTMLPDF

Abstract

We shortly review the progress in the domain of deterministic chaos for quantum dynamical systems. With the appropriately extended definition of quantum Lyapunov exponent we analyze various quantum dynamical maps. It is argued that, within Quantum Mechanics, irregular evolution for properly chosen observables can coexist with regular and predictable evolution of states.

DOI
10.48550/arxiv.quant-ph/9805068

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We shortly review the progress in the domain of deterministic chaos for quantum dynamical systems. With the appropriately extended definition of quantum Lyapunov exponent we analyze various quantum dynamical maps. It is argued that, within Quantum Mechanics, irregular evolution for properly chosen observables can coexist with regular and predictable evolution of states.
The relationship between chaos and quantum mechanics has been somewhat uneasy--even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes--such as chaos--that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time-series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next-generation experiments.
The dynamical status of isolated quantum systems is unclear as conventional measures fail to detect chaos in such systems. However, when quantum systems are subjected to observation--as all experimental systems must be--their dynamics is no longer linear and, in the appropriate limit(s), the evolution of expectation values, conditioned on the observations, closely approaches the behavior of classical trajectories. Here we show, by analyzing a specific example, that microscopic continuously observed quantum systems, even far from any classical limit, can have a positive Lyapunov exponent, and thus be truly chaotic.
Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.
We propose a new diagnostic for quantum chaos. We show that the time evolution of complexity for a particular type of target state can provide equivalent information about the classical Lyapunov exponent and scrambling time as out-of-time-order correlators. Moreover, for systems that can be switched from a regular to unstable (chaotic) regime by a tuning of the coupling constant of the interaction Hamiltonian, we find that the complexity defines a new time scale. We interpret this time scale as recording when the system makes the transition from regular to chaotic behavior.
We show that there is genuine chaos in quantum dynamics by introducing a physical distance between two quantum states. Qualitatively different from existing distances for quantum states, for example, the Fubini-Study distance, the physical distance between two mutually orthogonal quantum states, can be very small. As a result, two quantum states, which are initially very close by physical distance, can diverge from each other during the ensuing quantum dynamical evolution. We are able to use physical distance to define the quantum Lyapunov exponent and the quantum chaos measure. The latter leads to a quantum analog of the classical Poincaré section, which maps out the regions where quantum dynamics is regular and the regions where it is chaotic. Three different systems-a kicked rotor, the three-site Bose-Hubbard model, and the spin-1/2 XXZ model-are used to illustrate our results.
The agenda of dissipative quantum chaos is to create a toolbox that would allow us to categorize open quantum systems into "chaotic" and "regular" ones. Two approaches to this categorization have been proposed recently. One of them is based on the spectral properties of generators of open quantum evolution. The other one utilizes the concept of Lyapunov exponents to analyze quantum trajectories obtained by unraveling this evolution. By using two quantum models, we relate the two approaches and try to understand whether there is an agreement between the corresponding categorizations. Our answer is affirmative.