Low-frequency anomalies in dynamic localization

Quantum mechanical spreading of a particle hopping on tight binding lattices can be suppressed by the application of an external ac force, leading to periodic wave packet reconstruction. Such a phenomenon, referred to as dynamic localization (DL), occurs for certain magic values of the ratio $\Gamma=F_0/ \omega$ between the amplitude $F_0$ and frequency $\omega$ of the ac force. It is generally believed that in the low-frequency limit ($\omega \rightarrow 0$) DL can be achieved for an infinitesimally small value of the force $F_0$, i.e. at finite values of $\Gamma$. Such a normal behavior is found in homogeneous lattices as well as in inhomogeneous lattices of Glauber-Fock type. Here we introduce a tight-binding lattice model with inhomogeneous hopping rates, referred to as pseudo Glauber-Fock lattice, which shows DL but fails to reproduce the normal low-frequency behavior of homogeneous and Glauber-Fock lattices. In pseudo Glauber-Fock lattices, DL can be exactly realized, however at the DL condition the force amplitude $F_0$ remains finite as $\omega \rightarrow 0$. Such an anomalous behavior is explained in terms of a $\mathcal{PT}$ symmetry breaking transition of an associated two-level non-Hermitian Hamiltonian that effectively describes the dynamics of the Hermitian lattice model.