Dynamical quantum ergodicity from energy level statistics

Amit Vikram, Victor Galitski

Ergodic theory provides a rigorous mathematical description of classical dynamical systems, including a formal definition of the ergodic hierarchy consisting of merely ergodic, weakly-, strongly-, and K-mixing systems. Closely related to this 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 notion of ergodicity. This paper shows that cyclic ergodicity generalizes to quantum dynamical systems, which is proposed here as the basic rigorous definition of quantum ergodicity. It implies the ability to construct an orthonormal basis, where quantum dynamics transports an initial basis vector to all other basis vectors one by one, while minimizing the error in the overlap between the time-evolved initial state and a given basis state with a certain precision. It is proven that the basis, optimizing the error over all cyclic permutations, is obtained via the discrete Fourier transform of the energy eigenstates. This relates quantum cyclic ergodicity to level statistics. We then show that Wigner-Dyson level statistics implies quantum cyclic ergodicity, but that the reverse is not necessarily true. For the latter, we study an irrational flow on a 2D torus and argue that both classical and quantum flows are cyclic ergodic, while the level statistics is non-universal. We use the cyclic construction to motivate a quantum ergodic hierarchy of operators and argue that under the additional assumption of Poincare recurrences, cyclic ergodicity is a necessary condition for such operators to satisfy eigenstate thermalization. This work provides a general framework for transplanting some rigorous results of ergodic theory to quantum dynamical systems.