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arXiv:quant-ph/0412159 · Physical Review Letters

Emergence of Chaos in Quantum Systems Far from the Classical Limit

Salman Habib, Kurt A. Jacobs, Kosuke Shizume

Published Jan 10, 2006• 65 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.96.010403

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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.
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.
A fundamental requirement for the emergence of classical behavior from an underlying quantum description is that certain observed quantum systems make a transition to chaotic dynamics as their action is increased relative to $\ensuremath{\hbar}$. While experiments have demonstrated some aspects of this transition, the emergence of quantum trajectories with a positive Lyapunov exponent has never been observed directly. Here, we remove a major obstacle to achieving this goal by showing that, for the Duffing oscillator, the transition to a positive Lyapunov exponent can be resolved clearly from observed trajectories even with measurement efficiencies as low as 20%. We also find that the positive Lyapunov exponent is robust to highly mixed, low-purity states and to variations in the parameters of the system.
Continuous observation of a quantum system yields a measurement record that faithfully reproduces the classically predicted trajectory provided that the measurement is sufficiently strong to localize the state in phase space but weak enough that quantum backaction noise is negligible. We investigate the conditions under which classical dynamics emerges, via a continuous position measurement, for a particle moving in a harmonic well with its position coupled to internal spin. As a consequence of this coupling, we find that classical dynamics emerges only when the position and spin actions are both large compared to $\ensuremath{\hbar}$. These conditions are quantified by placing bounds on the size of the covariance matrix which describes the delocalized quantum coherence over extended regions of phase space. From this result, it follows that a mixed quantum-classical regime (where one subsystem can be treated classically and the other not) does not exist for a continuously observed spin-12\frac{1}{2}21​ particle. When the conditions for classicality are satisfied (in the large-spin limit), the quantum trajectories reproduce both the classical periodic orbits as well as the classically chaotic phase space regions. As a quantitative test of this convergence, we compute the largest Lyapunov exponent directly from the measured quantum trajectories and show that it agrees with the classical value.
This thesis makes a unique contribution to the field of quantum chaos by theoretically demonstrating the effect that measurement has on the emergence of chaos from the quantum world and demonstrating a means to control the onset of chaos in the quantum system using adaptive measurements. Here we investigate how the choice of the continuous measurement strategy for an open quantum system affects the emergence of chaos in the transition from the quantum limit to the classical limit when the system is dissipative. We consider two models in our research. The Duffing oscillator is classically chaotic and also dissipative( ie. an open quantum system), whereas the driven top is classically a closed system; adding dissipation via continuous measurement therefore changes the behaviour in from the classical limit. The first half of this thesis presents the investigation of a dissipative system whose classical limit is chaotic. We explore the emergence of chaos from the open quantum system that is continuously monitored and investigate the dependence on the choice of monitoring by changing a single parameter in a homodyne measurement scheme, effectively changing the information gained by the measurement. We show that the emergence of chaos in the regime where quantum effects are still present can be determined solely by changing the measurement parameter. This is a result of the interplay between the quantum interference effects induced by the nonlinear dynamics and the localisation and decoherence that occurs due to the measurement. We also investigate the case where the classical limit is regular for the Duffing oscillator, and demonstrate the semiclassical effect of chaos induced by the measurement back-action. A certain choice of measurement leads to a noise which drives the system to large spread in the dimensionless position enabling a non-classical transition mechanism that is classically forbidden, inducing chaos in the system. These results are verified by the numerical calculation of the maximal Lyapunov exponent in the quantum regime. The second part of this thesis investigates the possibility of controlling the degree of chaos with quantum control. We design an effective control scheme to control the degree of chaos using the measurement dependency of the state. We propose an adaptive measurement scheme which changes the homodyne measurement angle in real time depending on the direction of the state's interference fringes in phase space. This is done using the knowledge gained by the measurement signal. We show that this control scheme can enhance or suppress chaos. By enhancing the degree of chaos we are also able to push the onset of chaos further into the quantum regime than was possible before. By suppressing chaos we generate highly non-classical states and regular motion. The feasibility of experimentally realising this control technique is discussed in detail. The final section of this thesis considers a chaotic system that is not dissipative in the classical limit: the driven top. We investigate the effect that opening the quantum system to decoherence has on the degree of chaos when we continuously measure the system. We demonstrate that the presence of decoherence suppresses the chaos and alters the dynamics of the quantum system. This is seen to worsen as the strength of the measurement is increased unless a particular measurement is chosen that perfectly cancels out the decoherence resulting in the Hamiltonian evolution in addition to noise from the measurement. These results are verified by the separation time between classical and quantum dynamics.
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.