  We study the Nambu-Goldstone (NG) modes associated with spontaneous breaking
of the continuous time-translation symmetry. To discuss a quantum time-crystal
with the spontaneously-broken continuous time-translation symmetry, we
introduce the van der Pol type nonlinear-friction to open quantum systems. By
considering small fluctuations around a time-periodic mean-field solution, we
show that a gapless collective mode necessarily appears; this is nothing but
the NG mode associated with a time crystal. We show that its dispersion
relation becomes $\omega=-iC\bm p^2$. We also show that noncommutative breaking
of the time-translation and U(1) symmetries results in mixing of the NG modes,
and the (typically) propagating NG mode appears, whose dispersion relation
becomes $\omega=(\pm C_1-iC_2)\bm p^2$.
