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Quantum Localization

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Book cover Quantum Signatures of Chaos

Part of the book series: Springer Series in Synergetics ((SSSYN))

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Abstract

This chapter will focus mainly on the kicked rotator that displays global classical chaos in its cylindrical phase space for sufficiently strong kicking. The chaotic behavior takes the form of “rapid” quasi-random jumps of the phase variable around the cylinder and “slow” diffusion of the conjugate angular momentum p along the cylinder.

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Notes

  1. 1.

    We are indebted to H.J. Sommers for showing this proof to us.

  2. 2.

    Here and in the following 0+ denotes an arbitrarily small positive number.

  3. 3.

    While ħτ∕I for the kicked rotator naturally arises as a dimensionless measure of Planck’s constant, that role will be played by 1∕j for the kicked top. Therefore, it is convenient to set ħ = 1 in this section.

  4. 4.

    The reader will pardon the sloppiness of denoting the coherent state by |Θϕ〉 rather than |jΘϕ〉.

  5. 5.

    The top does not, in general, display the phenomenon of quantum localization under conditions of classical chaos; from this fact it may be understood that the localization length is larger than 2j + 1; see, however, the subsequent section.

  6. 6.

    Equation 72 on p. 3818 of Ref. [48] contains typos and should be read as our Eq. (8.8.71).

  7. 7.

    The “global” spectral average possible here as it was in Sect. 7.2 saves us from having to use the superanalytic Hubbard-Stratonovich transformation.

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Authors and Affiliations

  1. Universität Duisburg-Essen, Essen, Germany

    Fritz Haake

  2. School of Mathematical Science, University of Nottingham, Nottingham, UK

    Sven Gnutzmann

  3. Center for Theoretical Physics, Polish Academy of Sciences, Warszawa, Poland

    Marek Kuś

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  1. Fritz Haake
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  2. Sven Gnutzmann
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  3. Marek Kuś
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Haake, F., Gnutzmann, S., Kuś, M. (2018). Quantum Localization. In: Quantum Signatures of Chaos. Springer Series in Synergetics. Springer, Cham. https://doi.org/10.1007/978-3-319-97580-1_8

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