Silvestrov P G, Tworzydło J, Beenakker C W J
Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Feb;67(2 Pt 2):025204. doi: 10.1103/PhysRevE.67.025204. Epub 2003 Feb 26.
We reexamine the problem of the "Loschmidt echo," that measures the sensitivity to perturbation of quantum-chaotic dynamics. The overlap squared M(t) of two wave packets evolving under slightly different Hamiltonian is shown to have the double-exponential initial decay proportional to exp(-constant x e(2lambda(0)t)) in the main part of the phase space. The coefficient lambda(0) is the self-averaging Lyapunov exponent. The average decay (-)M proportional to e(-lambda(1)t) is single exponential with a different coefficient lambda(1). The volume of phase space that contributes to (-)M vanishes in the classical limit variant Planck-->0 for times less than the Ehrenfest time tau(E)=1/2lambda0(-1)|ln Planck|. It is only after the Ehrenfest time that the average decay is representative for a typical initial condition.
我们重新审视“洛施密特回波”问题，它用于衡量量子混沌动力学对微扰的敏感性。在稍微不同的哈密顿量下演化的两个波包的重叠平方(M(t))，在相空间的主要部分显示出与(\exp(-常数\times e^{2\lambda_0t}))成比例的双指数初始衰减。系数(\lambda_0)是自平均李雅普诺夫指数。平均衰减(\langle M\rangle)与(e^{-\lambda_1t})成比例，是具有不同系数(\lambda_1)的单指数形式。对于小于埃伦费斯特时间(\tau_E = \frac{1}{{2\lambda_0^{-1}}|\ln\hbar|})的时间，对(\langle M\rangle)有贡献的相空间体积在经典极限(\hbar\to0)时消失。只有在埃伦费斯特时间之后，平均衰减才代表典型的初始条件。
