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arXiv:chao-dyn/9702012

How long does the quantum chaos last?

G. Casati, Boris V. Chirikov, O. V. Zhirov

Published Feb 18, 1997• 1 citation
arXivHTMLPDF

Abstract

The main purpose of this Comment is to point out that besides the short time scale of quantum chaos, confirmed once more in recent paper by Alicki et al, there is generally another time scale t_R, which is much longer and on which a partial quantum-classical correspondence persists. Namely, the quantum diffusion closely follows the classical one even though the former is dynamically stable. The absence of the long scale t_R in model studied by Alicki et al. is a result of a special choice for one model's parameters value.

DOI
10.48550/arxiv.chao-dyn/9702012

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The main purpose of this Comment is to point out that besides the short time scale of quantum chaos, confirmed once more in recent paper by Alicki et al, there is generally another time scale t_R, which is much longer and on which a partial quantum-classical correspondence persists. Namely, the quantum diffusion closely follows the classical one even though the former is dynamically stable. The absence of the long scale t_R in model studied by Alicki et al. is a result of a special choice for one model's parameters value.
The threshold for quantum-classical correspondence is D∼ℏ43D \sim \hbar^{\frac43}D∼ℏ34​
In chaotic quantum systems, an initially localized quantum state can deviate strongly from the corresponding classical phase-space distribution after the Ehrenfest time tE∼log⁡(ℏ−1)t_{\mathrm{E}} \sim \log(\hbar^{-1})tE​∼log(ℏ−1), even in the limit ℏ→0\hbar \to 0ℏ→0. Decoherence by the environment is often invoked to explain the persistence of the quantum-classical correspondence at longer timescales. Recent rigorous results for Lindblad dynamics with phase-space diffusion strength DDD show that quantum and classical evolutions remain close for times that are exponentially longer than the Ehrenfest time whenever D≫ℏ43D \gg \hbar^{\frac43}D≫ℏ34​, in units set by the classical Hamiltonian. At the same time, some heuristic arguments have suggested the weaker condition D≫ℏ2D \gg \hbar^{2}D≫ℏ2 always suffices. Here we construct an explicit Lindbladian that demonstrates that the scaling D∼ℏ43D \sim \hbar^{\frac43}D∼ℏ34​ is indeed the threshold for quantum-classical correspondence beyond the Ehrenfest time. Our example uses a smooth time-dependent Hamiltonian and linear Lindblad operators generating homogeneous isotropic diffusion. It exhibits an ℏ\hbarℏ-independent quantum-classical discrepancy at the Ehrenfest time whenever D≪ℏ43D \ll \hbar^{\frac43}D≪ℏ34​, even for ℏ\hbarℏ-independent "macroscopic" smooth observables.
We investigate how decoherence affects the short-time separation between quantum and classical dynamics for classically chaotic systems, within the framework of a specific model. For a wide range of parameters, the distance between the corresponding phase-space distributions depends on a single parameter $\ensuremath{\chi}$ that relates an effective Planck constant ${\ensuremath{\hbar}}_{\mathrm{eff}}$, the Lyapunov coefficient, and the diffusion constant. This distance peaks at a time that depends logarithmically on ${\ensuremath{\hbar}}_{\mathrm{eff}}$, in agreement with previous estimations of the separation time for Hamiltonian systems. However, for $\ensuremath{\chi}\ensuremath{\lesssim}1$, the separation remains small, going down with ${\ensuremath{\hbar}}_{\mathrm{eff}}^{2}$, so the concept of separation time loses its meaning.
Recently, the phenomenon of quantum-classical correspondence breakdown was uncovered in optomechanics, where in the classical regime the system exhibits chaos but in the corresponding quantum regime the motion is regular - there appears to be no signature of classical chaos whatsoever in the corresponding quantum system, generating a paradox. We find that transient chaos, besides being a physically meaningful phenomenon by itself, provides a resolution. Using the method of quantum state diffusion to simulate the system dynamics subject to continuous homodyne detection, we uncover transient chaos associated with quantum trajectories. The transient behavior is consistent with chaos in the classical limit, while the long term evolution of the quantum system is regular. Transient chaos thus serves as a bridge for the quantum-classical transition (QCT). Strikingly, as the system transitions from the quantum to the classical regime, the average chaotic transient lifetime increases dramatically (faster than the Ehrenfest time characterizing the QCT for isolated quantum systems). We develop a physical theory to explain the scaling law.
The quantum dynamics of a classically chaotic model are studied in the approach to the macroscopic limit. The quantum predictions are compared and contrasted with the classical predictions of both Newtonian and Liouville mechanics. The time-domain scaling of the optimal quantum-classical correspondence is analyzed in detail in the case of both classical theories. In both cases the correspondence for observable quantities is shown to break down on a time-scale that increases very slowly (logarithmically) with increasing system size. In the case of quantum-Liouville correspondence such a short time-scale does not imply a breakdown of correspondence since the largest quantum-Liouville differences reached on this time-scale decrease rapidly (as an inverse power) with increasing system size. Therefore the statistical properties of chaotic dynamics are, as expected, well described by quantum theory in the macroscopic limit. In contrast, the largest quantum-Newtonian differences reached on the log time-scale actually increase in proportion to the system size. Since the invariance properties of the Hamiltonian impose functional constraints on the time-varying chaotic coordinates, it is possible to show, moreover, that if the quantum predictions are believed to describe the coordinates of individual chaotic systems, then they also predict macroscopic violations of any kinematic or dynamic constants of the motion. These results for chaotic systems indicate that a valid description of the time-varying properties of individual macroscopic bodies is not available within the standard interpretive framework of quantum theory.
We study observation entropy (OE) for the quantum kicked top model, whose classical counterpart possesses different phases: regular, mixed, or chaotic, depending on the strength of the kicking parameter. We show that OE grows logarithmically with coarse-graining length beyond a critical value in the regular phase, while OE growth is much faster in the chaotic regime. In the dynamics, we demonstrate that the short-time growth rate of OE acts as a measure of the chaoticity in the system, and we compare our results with out-of-time-ordered correlators (OTOC). Moreover, we show that in the deep quantum regime, the results obtained from OE are much more robust compared to OTOC results. Finally, we also investigate the long-time behavior of OE to distinguish between saddle-point scrambling and true chaos, where the former shows large persistent fluctuations compared to the latter.