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

Heisenberg's principle states that the product of uncertainties of position and momentum should be no less than the limit set by Planck's constant, Planck's over 2pi/2. This is usually taken to imply that phase space structures associated with sub-Planck scales (<<Planck's over 2pi) 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 Planck's over 2pi, develop spotty structure on the sub-Planck scale, a = Planck's over 2pi2/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 perturbations 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 environment. It will also be relevant in setting limits on the sensitivity of quantum meters.