Department of Physics, Indian Institute of Technology Madras, Chennai 600036, India.
Phys Rev E. 2019 Jan;99(1-1):012201. doi: 10.1103/PhysRevE.99.012201.
The out-of-time-ordered correlator (OTOC) is a measure of quantum chaos that is being vigorously investigated. Analytically accessible simple models that have long been studied in other contexts could provide insights into such measures. This paper investigates the OTOC in the quantum bakers map which is the quantum version of a simple and exactly solvable model of deterministic chaos that caricatures the action of kneading dough. Exact solutions based on the semiquantum approximation are derived that tracks very well the correlators until the Ehrenfest time. The growth occurs, surprisingly, at the exponential rate of the classical Lyapunov exponent which is half of that expected semiclassically. This exponential growth is modulated by slowly changing coefficients. Beyond this time, saturation occurs at a value close to that of random matrices. Using projectors for observables naturally leads to truncations of the unitary time-t propagator and the growth of their singular values is shown to be intimately related to the growth of the out-of-time-ordered correlators.
过时有序相关器（OTOC）是一种正在被积极研究的量子混沌度量。在其他背景下长期研究的、具有解析解的简单模型可以为这些度量提供一些启示。本文研究了量子面包师映射中的 OTOC，它是简单且可精确求解的确定性混沌模型的量子版本，它模拟了揉面团的作用。基于半量子近似的精确解被推导出来，这些解可以很好地跟踪相关器，直到 Ehrenfest 时间。令人惊讶的是，这种增长以经典 Lyapunov 指数的指数率发生，而经典 Lyapunov 指数是半经典的一半。这种指数增长被缓慢变化的系数所调制。在此时间之后，饱和发生在接近随机矩阵的值。使用用于可观测量的投影器自然会导致幺正时间传播子的截断，并且它们奇异值的增长被证明与过时有序相关器的增长密切相关。
