Quantum Chaos And Physical Distance Between Quantum States

We show that there is genuine chaos in quantum dynamics 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. We are able to use physical distance to define the quantum Lyapunov exponent and the quantum chaos measure. The latter leads to a quantum analog of the classical Poincare section, which maps out the regions where quantum dynamics is regular and the regions where it is chaotic. Three different systems-a kicked rotor, the three-site Bose-Hubbard model, and the spin-1/2 XXZ model-are used to illustrate our results.