  Phasonic degrees of freedom are unique to quasiperiodic structures, and play
a central role in poorly-understood properties of quasicrystals from excitation
spectra to wavefunction statistics to electronic transport. However, phasons
are challenging to access dynamically in the solid state due to their complex
long-range character and the effects of disorder and strain. We report phasonic
spectroscopy of a quantum gas in a one-dimensional quasicrystalline optical
lattice. We observe that strong phasonic driving produces a nonperturbative
high-harmonic plateau strikingly different from the effects of standard dipolar
driving. Tuning the potential from crystalline to quasicrystalline, we identify
spectroscopic signatures of quasiperiodicity and interactions and map the
emergence of a multifractal energy spectrum, opening a path to direct imaging
of the Hofstadter butterfly.
