Quasilocalization 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 preergodic regime emerges in the other limit. Furthermore, the dynamics in this preergodic regime can be associated with the onset of a dynamical quasilocalization, 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.

Figure 1

(a) The kinetic energy of a FQKR grows diffusively at lower drive frequencies and dynamically quasilocalizes at high frequency ($\tau <0.1$). The inset shows that the fluctuations in the quasilocalized (preergodic) regime are evenly spread about the mean kinetic energy calculated from the perturbation analysis (black dashed line) using Eq. (8). (b) The quasilocalization is destroyed, and an ergodic behavior is found to emerge at very long times. This can be seen from the linear growth of the kinetic energy with unit slope for $N>{10}^5$. The black dashed line with unit slope is provided for visual reference. Furthermore, the time after which localization is destroyed progressively increases as the frequency (${\tau }^{-1}$) is increased. The kick amplitudes chosen for the plot are ${\mathcal{K}}_1=10, {\mathcal{K}}_2=12$, and the initial state is chosen to be the angular momentum eigenstate $|{\psi }_0\rangle =|l=0\rangle$.

Figure 2

The normalized expansion coefficients defined in Eq. (7) at (a) stroboscopic instants $N$ and (b) Fibonacci instants $\mathcal{N}$.

Figure 3

Typical eigenstates of the effective Fibonacci Hamiltonian $H_{\mathrm{Fi}}$, peaked around different values of $l$, for ${\mathcal{K}}_1=10, {\mathcal{K}}_2=12$, and $\tau =0.01$. The eigenstates are found to be localized in the angular momentum space.

Figure 4

(a) Evolution of the kinetic energy observed at stroboscopic instants $N$ for a regular rotor, ${\mathcal{K}}_2=15$. The energy dynamically localizes for all driving frequencies, barring resonance conditions, which are not considered in this paper. (b) and (c) Typical eigenstates of the Floquet propagator $U_F$ localized around different values of $l$, defined in Eq. (A2), with $\mathcal{K}=15$ and kick frequency $\tau =1$ (b) and $\tau =0.001$ (c).

Figure 5

(a) Evolution of the kinetic energy observed at stroboscopic instants $N$ when the rotor is driven with (a) biperiodic and (b) aperiodic binary sequences and the kick amplitudes are chosen as ${\mathcal{K}}_1=10$ and ${\mathcal{K}}_2=12$. The energy saturates for all driving frequencies in the case of the biperiodic sequence, while it grows diffusively in the other case.

Figure 6

Evolution of the kinetic energy for the FQKR with different choices of $l_0$. The kick amplitudes are chosen to be ${\mathcal{K}}_1=10$ and ${\mathcal{K}}_2=12$.