From shaken quasicrystals to laser-driven quantum Hall matter

External modulation can alter the properties of a static system and give rise to unconventional phases of matter. In particular, the interplay between external modulation and localization can lead to exotic and tunable transport. We discuss recent theoretical and experimental advances in this area using ultracold atoms in quasiperiodic lattices. A 1D driven quasicrystal can be mapped via the Harper-Hofstadter mapping to a 2D electron gas at a high magnetic field illuminated by arbitrarily polarized light. By tuning the amplitudes and polarization of the light that drives this integer quantum Hall system in the higher dimension, we experimentally and theoretically reveal a tessellated phase diagram featuring a nested duality-protected pattern of metal-insulator transitions, and generate anomalous critical transport that is neither ballistic nor localized and can be traced back to single-particle multifractality in the spectrum.