Dynamical quantum ergodicity from energy level statistics

Abstract

Ergodic theory provides a rigorous mathematical description of chaos in classical dynamical systems, including a formal definition of the ergodic hierarchy. How ergodic dynamics is reflected in the energy levels and eigenstates of a quantum system is the central question of quantum chaos, but a rigorous quantum notion of ergodicity remains elusive. Closely related to the classical ergodic hierarchy is a less-known notion of cyclic approximate periodic transformations [see, e.g., I. Cornfield, S. Fomin, and Y. Sinai, Ergodic Theory (Springer-Verlag, New York, 1982)], which maps any “ergodic” dynamical system to a cyclic permutation on a circle and arguably represents the most elementary form of ergodicity. This paper shows that cyclic ergodicity generalizes to quantum dynamical systems, and provides a rigorous observable-independent definition of quantum ergodicity. It implies the ability to construct an orthonormal basis, where quantum dynamics transports any initial basis vector to have a sufficiently large overlap with each of the other basis vectors in a cyclic sequence. It is proven that the basis, maximizing the overlap over all such quantum cyclic permutations, is obtained via the discrete Fourier transform of the energy eigenstates, with overlaps given by specific measures of spectral rigidity. This relates quantum cyclic ergodicity to energy level statistics. The level statistics of Wigner-Dyson random matrices, usually associated with quantum chaos on empirical grounds, is derived as a special case of this general relation. To demonstrate generality, we prove that irrational flows on a 2D torus are classical and quantum cyclic ergodic, with spectral rigidity distinct from Wigner-Dyson. Finally, we motivate a quantum ergodic hierarchy of operators and discuss connections to eigenstate thermalization. This work provides a general framework for transplanting some rigorous concepts of ergodic theory to quantum dynamical systems.