  Collective modes of bilayered superconducting superlattices (e.g., YBCO) are
investigated within the conserving gauge-invariant ladder diagram approximation
including both the nearest interlayer single electron tunneling and the
Josephson-type Cooper pair tunneling. By calculating the density-density
response function including Coulomb and pairing interactions, we examine the
two collective mode branches corresponding to the in-phase and out-of-phase
charge fluctuations between the two layers in the unit cell. The out-of-phase
collective mode develops a long wavelength plasmon gap whose magnitude depends
on the tunneling strength with the mode dispersions being insensitive to the
specific tunneling mechanism (i.e., single electron or Josephson). We also show
that in the presence of tunneling the oscillator strength of the out-of-phase
mode overwhelms that of the in-phase-mode at $k_{\|} = 0$ and finite $k_z$,
where $k_z$ and $k_{\|}$ are respectively the mode wave vectors perpendicular
and along the layer. We discuss the possible experimental observability of the
phase fluctuation modes in the context of our theoretical results for the mode
dispersion and spectral weight.
