The Ehrenfest Oscillations in The Level Statistics of Chaotic Quantum Dots

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.