  One of the uncertainties in nuclear physics is whether a phase transition
between hadronic nuclear matter to quark matter exists in supranuclear matter
equations of state. Such a feature can be probed via gravitational-wave signals
from binary neutron star inspirals that contain information of the induced
tides. The dynamical part of the tides is caused by the resonance of pulsation
modes of stars, which causes a shift in the gravitational-wave phase. In this
paper, we investigate the dynamical tides of the interfacial mode ($i$-mode) of
spherical degree $l=2$, a non-radial mode caused by an interface associated
with a quark-hadron phase transition inside a hybrid star. In particular, we
focus on hybrid stars with a crystalline quark matter core and a fluid hadronic
envelope. We find that the resonant frequency of such $i$-modes typically
ranges from 300Hz to 1500Hz, and the frequency increases as the shear modulus
of the quark core increases. We next estimate the detectability of such a mode
with existing and future gravitational-wave events from the inspiral waveform
with a Fisher analysis. We find that GW170817 and GW190425 have the potential
to detect the $i$-mode if the quark-hadron phase transition occurs at
sufficiently low pressure and the shear modulus of the quark matter phase is
large enough. We also find that the third-generation gravitational-wave
detectors can further probe the $i$-mode with intermediate transition pressure.
This finding opens a new, interesting direction for probing the existence of
quark core inside a neutron star.
