Quasi-localization dynamics in a Fibonacci quantum rotor

We analyze the dynamics of a quantum kicked rotor (QKR) driven with a binary Fibonacci sequence of two distinct drive amplitudes. While the dynamics at low drive frequencies is found to be diffusive, a long-lived pre-ergodic regime emerges in the other limit. Further, the dynamics in this pre-ergodic regime can be associated with the onset of a dynamical quasi-localization, similar to the dynamical localization observed in a regular QKR. We establish that this peculiar behavior arises due to the presence of localized eigenstates of an approximately conserved effective Hamiltonian, which drives the evolution at Fibonacci instants. However, the effective Hamiltonian picture does not persist indefinitely and the dynamics eventually becomes ergodic after asymptotically long times.