Quantum-classical correspondence in integrable systems  

作　　者：Yiqiang Zhao Biao Wu 

机构地区：[1]International Center for Quantum Materials,School of Physics,Peking University,Beijing 100871,China [2]Collaborative Innovation Center of Quantum Matter,Beijing 100871,China [3]Wilczek Quantum Center,School of Physics and Astronomy, Shanghai Jiao Tong University,Shanghai 200240,China

出　　处：《Science China(Physics,Mechanics & Astronomy)》2019年第9期95-103,共9页中国科学：物理学、力学、天文学（英文版）

基　　金：supported by the National Key Research and Development Program of China (Grant Nos. 2017YFA0303302, and 2018YFA0305602);the National Natural Science Foundation of China (Grant Nos. 11334001, and 11429402)

摘　　要：We find that the quantum-classical correspondence in integrable systems is characterized by two time scales. One is the Ehrenfest time below which the system is classical;the other is the quantum revival time beyond which the system is fully quantum. In between, the quantum system can be well approximated by classical ensemble distribution in phase space. These results can be summarized in a diagram which we call Ehrenfest diagram. We derive an analytical expression for Ehrenfest time, which is proportional to h-1/2. According to our formula, the Ehrenfest time for the solar-earth system is about 1026 times of the age of the solar system. We also find an analytical expression for the quantum revival time, which is proportional to h-1. Both time scales involve ω(I), the classical frequency as a function of classical action. Our results are numerically illustrated with two simple integrable models. In addition, we show that similar results exist for Bose gases, where 1/N serves as an effective Planck constant.We find that the quantum-classical correspondence in integrable systems is characterized by two time scales. One is the Ehrenfest time below which the system is classical; the other is the quantum revival time beyond which the system is fully quantum. In between, the quantum system can be well approximated by classical ensemble distribution in phase space. These results can be summarized in a diagram which we call Ehrenfest diagram. We derive an analytical expression for Ehrenfest time, which is proportional to h-1/2. According to our formula, the Ehrenfest time for the solar-earth system is about 1026 times of the age of the solar system. We also find an analytical expression for the quantum revival time, which is proportional to h-1. Both time scales involve ω（I）, the classical frequency as a function of classical action. Our results are numerically illustrated with two simple integrable models. In addition, we show that similar results exist for Bose gases, where 1/N serves as an effective Planck constant.

关 键 词：quantum-classical CORRESPONDENCE Ehrenfest TIME QUANTUM REVIVAL TIME INTEGRABLE systems 

分 类 号：N[自然科学总论]