Completely Ergodic Quantum Dynamics in Quasiperiodically Driven Systems

The ergodicity of quantum dynamics with time-independent (or time-periodic) Hamiltonians is often defined through certain statistical properties of (quasi-)energy eigenstates or energies. Such a definition, however, cannot be applied for general time-dependent Hamiltonian dynamics for which eigenstates and eigenenergies may not exist. In this work, we present a new, stronger form of quantum ergodicity, called complete quantum ergodicity (CQE), based on the concept of state design from quantum information theory: CQE requires that the trajectory of any time-evolved wavefunction visits every corner of the Hilbert space uniformly over time. This is a form of ergodicity which is more in line with traditional notions of ergodicity in dynamical systems, in that the time-averaging of a trajectory reproduces an averaging over the space that it moves in. While CQE cannot be attained by time-independent (-periodic) time due to (quasi-)energy conservation, we present explicit solvable examples of CQE achieved by a class of aperiodic, deterministic, and unitary evolution with minimal complexity such as the Fibonacci drive. Using both analytic and numerical techniques, we discuss the implications of our results in the deep thermalization of quantum states and the formation of k-designs in time.