Ehrenfest oscillations in the level statistics of chaotic quantum dots

We study the crossover from a 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(ω) are analyzed. We find that in the intermediate region Δ≪ω∼t_E^-1≪t_erg^-1 , where t_E and t_erg are the Ehrenfest and ergodic times, respectively, R(ω) 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(ω) was studied in an earlier work [I. L. Aleiner and A. I. Larkin, Phys. Rev. E, 55, R1243 (1997)]). In the nonperturbative part they have the period dependence as (Δ^-1+αt_E)^-1 with α a universal numerical factor. The amplitude of the leading order Ehrenfest oscillation in the nonperturbative part is larger than that of the perturbative part.