JQI-QuICS-CMTC Seminar

Date
Fri, Sep 8, 2017 4:15 pm - 5:15 pm
Location
PSC 2136

Description

Speaker: Efim Rozenbaum

Speaker Affiliation: JQI and CMTC

Title: Lyapunov Exponent and Out-of-Time-Ordered Correlator's Growth Rate in a Chaotic System

Abstract: It was proposed recently that the out-of-time-ordered four-point correlator (OTOC) may serve as a useful characteristic of quantum-chaotic behavior, because in the semi-classical limit, hbar -> 0, its rate of exponential growth resembles the classical Lyapunov exponent. Here, we calculate the four-point correlator, C(t), for the classical and quantum kicked rotor -- a textbook driven chaotic system -- and compare its growth rate at initial times with the standard definition of the classical Lyapunov exponent. Using both quantum and classical arguments, we show that the OTOC's growth rate and the Lyapunov exponent are in general distinct quantities, corresponding to the logarithm of phase-space averaged divergence rate of classical trajectories and to the phase-space average of the logarithm, respectively. The difference appears to be more pronounced in the regime of low kicking strength K, where no classical chaos exists globally. In this case, the Lyapunov exponent quickly decreases as K -> 0, while the OTOC's growth rate may decrease much slower showing higher sensitivity to small chaotic islands in the phase space. We also show that the quantum correlator as a function of time exhibits a clear singularity at the Ehrenfest time t_E: transitioning from a time-independent value of t^-1 ln[C(t)] at t < t_E to its monotonous decrease with time at t>t_E. We note that the underlying physics here is the same as in the theory of weak (dynamical) localization [Aleiner and Larkin, Phys. Rev. B 54, 14423 (1996); Tian, Kamenev, and Larkin, Phys. Rev. Lett. 93, 124101 (2004)] and is due to a delay in the onset of quantum interference effects, which occur sharply at a time of the order of the Ehrenfest time.

*Snacks and drinks will be served at 4 pm*