Physics – Quantum Physics
Scientific paper
2002-01-25
Nature 412, 712-717 (2001)
Physics
Quantum Physics
Published in Nature 412, 712-717 (2001)
Scientific paper
10.1038/35089017
Heisenberg's principle$^1$ states that the product of uncertainties of position and momentum should be no less than Planck's constant $\hbar$. This is usually taken to imply that phase space structures associated with sub-Planck ($\ll \hbar$) 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' $A \gg \hbar$ develop spotty structure on scales corresponding to sub-Planck $a = \hbar^2 / A \ll \hbar$. 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 perturbations$^2$. 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 environment$^{3-8}$. It may also be relevant in setting limits on sensitivity of Schr\"odinger cats used as detectors.
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