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Recently, waveguide lattices with non-uniform tunneling amplitudes have beenexplored due to their myriad tunable properties, many of which arise from theextended nature of their eigenstates in the absence of disorder. Here, weinvestigate the dynamics, localization, and parity- andtime-reversal-($\mP\mT$) symmetry breaking in lattices that support onlylocalized eigenstates in the disorder-free limit. We propose three families oftunneling profiles that lead to qualitatively different single-particle timeevolution, and show that the effects of weak disorder contain signatures of thelocalized or extended nature of clean-lattice eigenstates. We show that inlattices with localized eigenstates, the signatures of $\mP\mT$-symmetrybreaking are acutely sensitive to the initial form of the wave packet. Ourresults suggest that waveguide lattices with localized eigenstates will exhibita wide array of phenomena that are absent in traditional systems.

Dynamics, disorder effects, and PT-symmetry breaking in waveguide lattices with localized eigenstates