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, ℏ/2. This is usually taken to imply that phase space structures associated with sub-Planck scales (≪ℏ) 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 ℏ, develop spotty structure on the sub-Planck scale, a = ℏ2/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.
Subscribe to Journal
Get full journal access for 1 year
only $3.83 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Heisenberg, W. Über den anschaulichen Inhalt der quantentheoretischen Kinematik and Mechanik. Z. Phys. 43, 172–198 (1927); The physical content of quantum kinematics and mechanics (Engl. Trans.) in Quantum Theory and Measurement (eds Wheeler, J. A. & Zurek, W. H.) (Princeton Univ. Press, Princeton, 1983).
Zurek, W. H. Decoherence, chaos, quantum-classical correspondence, and the algorithmic arrow of time. Phys. Script. T76, 186–198 (1998).
Zurek, W. H. Pointer basis of a quantum apparatus: Into what mixture does the wavepacket collapse? Phys. Rev. D 24, 1516–1524 (1981).
Zurek, W. H. Environment-induced superselection rules. Phys. Rev. D 26, 1862–1880 (1982).
Joos, E. & Zeh, H. D. The emergence of classical properties through the interaction with the environment. Z. Phys. B 59, 229 (1985).
Zurek, W. H. Decoherence and the transition from quantum to classical. Phys. Today 44, 36–46 (1991).
Giulini, D., Joos, E., Kiefer, C., Kupsch, J., Stamatescu, L.-O. & Zeh, H. D. Decoherence and the Appearance of a Classical World in Quantum Theory (Springer, Berlin, 1996).
Zurek, W. H. Decoherence, einselection, and the quantum origin of the classical. Rev. Mod. Phys. (in the press); also as preprint (quant-ph 010527) at 〈http://xxx.lanl.gov〉 (2001).
Haake, F. Quantum Signatures of Chaos (Springer, Berlin, 1991).
Casati, G. & Chrikov, B. Quantum Chaos (Cambridge Univ. Press, Cambridge, 1995).
Hillery, M., O'Connell, R. F., Scully, M. O. & Wigner, E. P. Distribution functions in physics: Fundamentals. Phys. Rep. 106, 121–167 (1984).
Berry, M. V. & Balazs, N. L. Evolution of semiclassical quantum states in phase space. J. Phys. A 12, 625–642 (1979).
Korsch, H. J. & Berry, M. V. Evolution of Wigner's phase-space density under a nonintegrable quantum map. Physica D 3, 627–636 (1981).
Zurek, W. H. & Paz, J. P. Decoherence, chaos, and the Second Law. Phys. Rev. Lett. 72, 2508–2511 (1994).
Berman, G. P. & Zaslavsky, G. M. Condition of stochasticity in quantum non-linear systems. Physica (Amsterdam) 91A, 450 (1978).
Habib, S., Shizume, K. & Zurek, W. H. Decoherence, chaos, and the correspondence principle. Phys. Rev. Lett. 80, 4361 (1998).
Caldeira, A. O. & Leggett, A. J. Path-integral approach to quantum Brownian motion. Physica 121A, 587–616 (1983).
Paz, J. P. & Zurek, W. H. in Les Houches Lectures Session LXXII (eds Kaiser, R., Westbrook, C. and David, F.) 533–614 (Springer, Berlin, 2001).
Braun, D., Haake, F. & Strunz, W. A. Universality of decoherence. Phys. Rev. Lett. 86, 2913–2917 (2001).
Hannay, J. H. & Berry, M. V. Quantization of linear maps on a torus—Fresnel diffraction by a periodic grating. Physica 1D, 267–290 (1980).
Caves, C. in Physical Origins of Time Asymmetry (eds Halliwell, J. J., Pérez-Mercader, J. & Zurek, W. H.) 47–77 (Cambridge Univ. Press, Cambridge, 1993).
Miller, P. A. & Sarkar, S. Signatures of chaos in the entanglement of two coupled quantum kicked tops. Phys. Rev. E 60, 1542 (1999).
Braginsky, V. B. & Khalili, F. Y. Quantum nondemolition measurements: the route from toys to tools. Rev. Mod. Phys. 95, 703–711 (1996).
Karkuszewski, Z., Zakrzewski, J. & Zurek, W. H. Breakdown of correspondence in chaotic systems: Ehrenfest versus localization times. Preprint quant-ph/0010011 at 〈http://xxx.lanl.gov〉 (2000).
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.
About this article
Cite this article
Zurek, W. Sub-Planck structure in phase space and its relevance for quantum decoherence. Nature 412, 712–717 (2001). https://doi.org/10.1038/35089017
Optics Letters (2020)
Time-evolution of nonlinear optomechanical systems: interplay of mechanical squeezing and non-Gaussianity
Journal of Physics A: Mathematical and Theoretical (2020)
International Journal of Theoretical Physics (2020)
Physical Review Letters (2019)
New Journal of Physics (2019)