Heiss W.D. (ed.) Quantum dots.. a doorway to - tiera.ru

162 C.W.J. Beenakker
〈ρ(E)〉 δ/2
1
0.5
RMT
BS
0
0 0.5 1 1.5
E/ET Fig. 18. Comparison of the smoothed density of states in a chaotic Andreev billiard
as it follows from RMT ((49), with a hard gap) and as it follows from the
Bohr-Sommerfeld (BS) rule ((83), without a hard gap). These are the two limiting
distributions when the Ehrenfest time τE is, respectively, much smaller or much
larger than the mean dwell time τdwell
the Bohr-Sommerfeld form (83). We investigate this crossover in the following
section.
8 Quantum-To-Classical Crossover
8.1 Thouless Versus Ehrenfest
According to Ehrenfest’s theorem, the propagation of a quantum mechanical
wave packet is described for short times by classical equations of motion.
The time scale at which this correspondence between quantum and classical
dynamics breaks down in a chaotic system is called the Ehrenfest time τE [72]. 5
As explained in Fig. 19, it depends logarithmically on Planck’s constant: τE =
α −1 ln(Scl/h), with Scl the characteristic classical action of the dynamical
system and α the Lyapunov exponent.
This logarithmic h-dependence distinguishes the Ehrenfest time from other
characteristic time scales of a chaotic system, which are either h-independent
(dwell time, ergodic time) or algebraically dependent on h (Heisenberg time ∝
1/δ). That the quasiclassical theory of superconductivity breaks down on time
scales greater than τE was noticed already in 1968 by Larkin and Ovchinnikov
[74].
The choice of Scl depends on the physical quantity which one is studying.
For the density of states of the Andreev billiard (area A, opening of width
W ≪ A 1/2 , range of transverse momenta ∆p � pF inside the constriction)
5 The name “Ehrenfest time” was coined in [73].