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Quantum signatures of chaos in a kicked top

Abstract

Chaotic behaviour is ubiquitous and plays an important part in most fields of science. In classical physics, chaos is characterized by hypersensitivity of the time evolution of a system to initial conditions. Quantum mechanics does not permit a similar definition owing in part to the uncertainty principle, and in part to the Schrödinger equation, which preserves the overlap between quantum states. This fundamental disconnect poses a challenge to quantum–classical correspondence1, and has motivated a long-standing search for quantum signatures of classical chaos2,3. Here we present the experimental realization of a common paradigm for quantum chaos—the quantum kicked top2,4— and the observation directly in quantum phase space of dynamics that have a chaotic classical counterpart. Our system is based on the combined electronic and nuclear spin of a single atom and is therefore deep in the quantum regime; nevertheless, we find good correspondence between the quantum dynamics and classical phase space structures. Because chaos is inherently a dynamical phenomenon, special significance attaches to dynamical signatures such as sensitivity to perturbation1,5 or the generation of entropy6 and entanglement7,8, for which only indirect evidence has been available9,10,11. We observe clear differences in the sensitivity to perturbation in chaotic versus regular, non-chaotic regimes, and present experimental evidence for dynamical entanglement as a signature of chaos.

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Figure 1: Stroboscopic phase space plot for a classical kicked top.
Figure 2: Quantum phase space (Husimi) distributions for a quantum kicked top.
Figure 3: Sensitivity to perturbation as a quantum signature of chaos.
Figure 4: Entanglement as a quantum signature of chaos.

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Acknowledgements

We thank I. H. Deutsch and P. Jacquod for discussions. This work was supported by the National Science Foundation (grant no. 0653631) and the Office of Naval Research (grant no. N00014-05-1-420). S.G. was supported by an NSERC Discovery grant.

Author Contributions All authors contributed extensively to this work.

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Correspondence to P. S. Jessen.

Supplementary information

Supplementary Information

This file contains Supplementary Data, Supplementary Figures 1-6 with Legends, Supplementary Table 1 and Supplementary References. (PDF 2890 kb)

Supplementary Video 1 (a)

This video shows experimentally measured data for the evolving quantum phase space distribution, for an initial state centered on the large regular island in the Fy<0 hemisphere and is equivalent to Supplementary Figure 2. (MP4 472 kb)

Supplementary Video 1 (b)

This video contains theoretical model predictions for the evolving quantum phase space distribution, for an initial state centered on the large regular island in the Fy<0 hemisphere and can be compared to Supplementary Video 1(a) and Supplementary Figure 2. (MP4 477 kb)

Supplementary Video 2 (a)

This video shows experimentally measured data for the evolving quantum phase space distribution, for an initial state centered on the lower of the pair of islands in the Fy>0 hemisphere and is equivalent to Figure 2A and Supplementary Figure 3. (MP4 489 kb)

Supplementary Video 2 (b)

This video contains theoretical model predictions for the evolving quantum phase space distribution, for an initial state centered on the lower of the pair of islands in the Fy>0 hemisphere and can be compared to Supplementary Video 2(a), Figure 2A and Supplemental Figure 3. (MP4 470 kb)

Supplementary Video 3 (a)

This video shows experimentally measured data for the evolving quantum phase space distribution, for an initial state localized in the sea of chaos in the Fy>0 hemisphere. And is equivalent to Figure 2B and Supplementary Figure 4. (MP4 484 kb)

Supplementary Video 3 (b)

This video shows the evolving quantum phase space distribution, for an initial state localized in the sea of chaos in the Fy>0 hemisphere and can be compared to Supplementary Video3(a), Figure 2B and Supplemental Figure 4. (MP4 471 kb)

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Chaudhury, S., Smith, A., Anderson, B. et al. Quantum signatures of chaos in a kicked top. Nature 461, 768–771 (2009). https://doi.org/10.1038/nature08396

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