The Ehrenfest Oscillations in The Level Statistics of Chaotic Quantum Dots

TL;DR

This paper investigates Ehrenfest oscillations in the energy level statistics of chaotic quantum dots, revealing deviations from universal Wigner-Dyson behavior due to Ehrenfest and ergodic times, with distinct oscillation periods in perturbative and nonperturbative parts.

Contribution

It introduces the analysis of Ehrenfest oscillations in the level correlation function of chaotic quantum systems with broken time-reversal symmetry, highlighting their dependence on Ehrenfest time and their impact on spectral statistics.

Findings

Ehrenfest oscillations depend on Ehrenfest time in the intermediate frequency region.

Oscillation periods differ between perturbative and nonperturbative parts of R(ω).

Amplitude of Ehrenfest oscillations in nonperturbative part exceeds that in perturbative part.

Abstract

We study a crossover from classical to quantum picture in the electron energy statistics in a system with broken time-reversal symmetry. The perturbative and nonperturbative parts of the two level correlation function, $R(\omega)$ are analyzed. We find that in the intermediate region, $\Delta\ll\omega\sim t_E^{-1}\ll t_{erg}^{-1}$, where $t_E$ and $t_{erg}$ are the Ehrenfest and ergodic times, respectively, $R(\omega)$ consists of a series of oscillations with the periods depending on $t_E$, deviating from the universal Wigner-Dyson statistics. These Ehrenfest oscillations have the period dependence as $t_E^{-1}$ in the perturbative part. [For systems with time-reversal symmetry, this oscillation in the perturbative part of $R(\omega)$ was studied in an earlier work (I. L. Aleiner and A. I. Larkin, Phys. Rev. E {\bf 55}, R1243 (1997))]. In the nonperturbative part they have the period dependence as $(\Delta^{-1}+\alpha t_E)^{-1}$ with $\alpha$ a universal numerical factor. The amplitude of the leading order Ehrenfest oscillation in the nonperturbative part is larger than that of the perturbative part.