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Ehrenfest-Time Effects in Mesoscopic Systems

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Semiclassical Approach to Mesoscopic Systems

Part of the book series: Springer Tracts in Modern Physics ((STMP,volume 245))

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Abstract

This chapter is devoted to the most prominent universal effect in ballistic systems beyond RMT that can be accessed by semiclassical methods: the effect of a non-zero Ehrenfest-time. The Ehrenfest-time provides a separation between the timescales when the dynamics of an initially spatially localised wave packet can be described by classical mechanics on the one hand and when it is dominated by wave interference on the other hand, for an illustration of this transition see Fig. 1.1. For an estimate we consider two points inside the wave packet a distance \(\lambda_F\) apart and calculate the time until they feel the effect of the boundary, i.e. until they are a distance of the order of the system size \({\mathcal L}\) apart taking into account the possible exponential separation of neighbouring trajectories in chaotic systems.

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Notes

  1. 1.

    We will often refer to this calculation as field-theoretical approach to describe Ehrenfest-time effects.

  2. 2.

    Using the estimate that \(1/\lambda\) is of the order of the free flight time, \(\lambda\approx p/(m{\mathcal L}),\) we obtain \(2m\lambda\hbar/p^2\approx\lambda_F/{\mathcal L}\) and by this consistency with Eq. (4.1).

  3. 3.

    The reason for the special choice of c in Eq. (4.3) can be best seen within the configuration-space approach introduced in Chap. 2. One therefore needs the monodromy matrix element \(M_{12}\approx(m\lambda)^{-1}\exp(\lambda t/2)\) determining the position difference of two orbits at time t, given by W, as a function of the initial angle difference \(\epsilon\) and the approximation \(\lambda\approx p/(m{\mathcal L}).\) This yields \(t_{\rm enc}\approx2/\lambda\ln\left[W/({\mathcal L}\epsilon)\right]\) in the configuration-space approach. For a detailed derivation see [13]. Comparing then the constants c in the configuration and in the phase-space approach one finally arrives at Eq. (4.3).

  4. 4.

    This condition for the minimal loop duration is equivalent to the one in the last section. Note here that setting \(W={\mathcal L}\) in \(t_{\rm enc}\approx2/\lambda\ln\left[W/({\mathcal L}\epsilon)\right]\) obtained in the last footnote yields the same expression for \(t_{\rm enc}\) as setting \(c=1\) in Eq. (2.45) as done in the last section.

  5. 5.

    As the calculation in [12] shows, this contribution is actually only non-zero due to taking into account the encounter fringes shown in Fig. 4.10.

  6. 6.

    For a detailed explanation how such contributions increasing with increasing Ehrenfest-time arise see Sect. 4.5.2.

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  1. Institut für Theoretische Physik, Universität Regensburg, Regensburg, Germany

    Daniel Waltner

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Waltner, D. (2012). Ehrenfest-Time Effects in Mesoscopic Systems. In: Semiclassical Approach to Mesoscopic Systems. Springer Tracts in Modern Physics, vol 245. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24528-2_4

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  • DOI: https://doi.org/10.1007/978-3-642-24528-2_4

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