Heisenberg's principle1 states that the product of uncertainties of
position and momentum should be no less than Planck's constant β. This is
usually taken to imply that phase space structures associated with sub-Planck
(βͺβ) scales do not exist, or, at the very least, that they do not
matter. I show that this deeply ingrained prejudice is false: Non-local
"Schr\"odinger cat" states of quantum systems confined to phase space volume
characterized by `the classical action' Aβ«β develop spotty structure
on scales corresponding to sub-Planck a=β2/Aβͺβ. Such
structures arise especially quickly in quantum versions of classically chaotic
systems (such as gases, modelled by chaotic scattering of molecules), that are
driven into nonlocal Schr\"odinger cat -- like superpositions by the quantum
manifestations of the exponential sensitivity to perturbations2. Most
importantly, these sub-Planck scales are physically significant: a determines
sensitivity of a quantum system (or of a quantum environment) to perturbations.
Therefore sub-Planck a controls the effectiveness of decoherence and
einselection caused by the environment3β8. It may also be relevant in
setting limits on sensitivity of Schr\"odinger cats used as detectors.Comment: Published in Nature 412, 712-717 (2001