  We numerically investigate the electronic structures around a vortex core in
a bilayer superconducting system, with s-wave pairing, Rashba spin-orbit
coupling and Zeeman magnetic field, with use of the quasiclassical Green's
function method. The Bardeen-Cooper-Schrieffer (BCS) phase and the so-called
pair-density wave (PDW) phase appear in the temperature-magnetic-field phase
diagram in a bulk uniform system [Phys. Rev. B 86, 134514 (2012)]. In the low
magnetic field perpendicular to the layers, the zero-energy vortex bound states
in the BCS phase are split by the Zeeman magnetic field. On the other hand, the
PDW state appears in the high magnetic field, and sign of the order parameter
is opposite between the layers. We find that the vortex core suddenly shrinks
and the zero-energy bound states appear by increasing the magnetic field
through the BCS-PDW transition. We discuss the origin of the change in vortex
core structure between the BCS and PDW states by clarifying the relation
between the vortex bound states and the bulk energy spectra. In the high
magnetic field region, the PDW state and vortex bound states are protected by
the spin-orbit coupling. These characteristic behaviors in the PDW state can be
observed by scanning tunneling microscopy/spectroscopy.
