Quantum Chaos and Physical Distance Between Quantum States


It has been long argued and accepted that there is no true quantum chaos due to the linearity of the Schrödinger equation. Berry even suggested to change quantum chaos, a confusing oxymoron,  to quantum chaology as the study of quantum systems whose classical counterparts exhibit chaos. We show that there is genuine quantum chaos despite that quantum dynamics is linear. This is revealed by introducing a physical distance between two quantum states. Qualitatively different from existing distances for quantum states, for example, the Fubini-Study distance, the physical distance between two mutually orthogonal quantum states can be very small.  As a result, two quantum states, which are initially very close by physical distance, can diverge from each other during the ensuing quantum dynamical evolution. This kind of sensitivity to initial conditions is reduced to the one in classical dynamics in the semiclassical limit. In the end, we discuss how nonlinear Newton’s equations of motion emerge out of the linear Schrödinger equation. 

Reference: Wang, Wang and Wu, PRE 103, 042209 (2021)