  We use Hanbury-Brown-Twiss interferometry (HBTI) to study various quantum
phases of hard core bosons (HCBs) and ideal fermions confined in a
one-dimensional quasi-periodic (QP) potential. For HCBs, the QP potential
induces a cascade of Mott-like band-insulator phases in the extended regime, in
addition to the Mott insulator, Bose glass, and superfluid phases. At critical
filling factors, the appearance of these insulating phases is heralded by a
peak to dip transition in the interferogram, which reflects the fermionic
aspect of HCBs. On the other hand, ideal fermions in the extended phase display
various complexities of incommensurate structures such as devil's staircases
and Arnold tongues. In the localized phase, the HCB and the fermion
correlations are identical except for the sign of the peaks. Finally, we
demonstrate that HBTI provides an effective method to distinguish Mott and
glassy phases.
