TY - JOUR

T1 - Out-of-time-order correlators in quantum mechanics

AU - Hashimoto, Koji

AU - Murata, Keiju

AU - Yoshii, Ryosuke

N1 - Funding Information:
Open Access, ©c The Authors. Article funded by SCOAP3.
Funding Information:
We would like to thank Taro Kimura, Michikazu Kobayashi, Yasusada Nambu, Makoto Negoro, and Takahiro Sagawa for valuable discussion and comments. The numerical calculations were partially carried out using the TSC-computer of Topological Science in Keio university. The work of K.H. was supported in part by JSPS KAKENHI Grant Numbers 15H03658, 15K13483. The work of K.M. was supported by JSPS KAKENHI Grant Number 15K17658. The work of R.Y. was supported by the MEXT-Supported Program for the Strategic Research Foundation at Private Universities Topological Science (Grant No. S1511006).

PY - 2017/10/1

Y1 - 2017/10/1

N2 - The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time, which could be the reason of the difficulty for the numerical analyses to exhibit the exponential growth.

AB - The out-of-time-order correlator (OTOC) is considered as a measure of quantum chaos. We formulate how to calculate the OTOC for quantum mechanics with a general Hamiltonian. We demonstrate explicit calculations of OTOCs for a harmonic oscillator, a particle in a one-dimensional box, a circle billiard and stadium billiards. For the first two cases, OTOCs are periodic in time because of their commensurable energy spectra. For the circle and stadium billiards, they are not recursive but saturate to constant values which are linear in temperature. Although the stadium billiard is a typical example of the classical chaos, an expected exponential growth of the OTOC is not found. We also discuss the classical limit of the OTOC. Analysis of a time evolution of a wavepacket in a box shows that the OTOC can deviate from its classical value at a time much earlier than the Ehrenfest time, which could be the reason of the difficulty for the numerical analyses to exhibit the exponential growth.

KW - AdS-CFT Correspondence

KW - Black Holes

KW - Models of Quantum Gravity

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U2 - 10.1007/JHEP10(2017)138

DO - 10.1007/JHEP10(2017)138

M3 - Article

AN - SCOPUS:85032278508

VL - 2017

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

SN - 1126-6708

IS - 10

M1 - 138

ER -