Sub-Planck structure in phase space and its relevance for quantum decoherence


Heisenberg's principle1 states that the product of uncertainties of position and momentum should be no less than the limit set by Planck's constant, /2. This is usually taken to imply that phase space structures associated with sub-Planck scales () do not exist, or at least that they do not matter. Here I show that this common assumption is false: non-local quantum superpositions (or ‘Schrödinger's cat’ states) that are confined to a phase space volume characterized by the classical action A, much larger than , develop spotty structure on the sub-Planck scale, a = 2/A. Structure saturates on this scale particularly quickly in quantum versions of classically chaotic systems—such as gases that are modelled by chaotic scattering of molecules—because their exponential sensitivity to perturbations2 causes them to be driven into non-local ‘cat’ states. Most importantly, these sub-Planck scales are physically significant: a determines the sensitivity of a quantum system or environment to perturbations. Therefore, this scale controls the effectiveness of decoherence and the selection of preferred pointer states by the environment3,4,5,6,7,8. It will also be relevant in setting limits on the sensitivity of quantum meters.

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Figure 1: Snapshots of the quantum Wigner distribution and of the classical probability density in phase space of an evolving chaotic system.
Figure 2: The compass state, equation (9).
Figure 3: Snapshots of area 2π extracted from Fig. 1a–c.


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This research was supported in part by the National Security Agency. I thank A. Albrecht, N. Balazs, C. Jarzynski, Z. Karkuszewski and J. P. Paz for useful chaotic conversations.

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Correspondence to Wojciech Hubert Zurek.

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Zurek, W. Sub-Planck structure in phase space and its relevance for quantum decoherence. Nature 412, 712–717 (2001).

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