  We numerically study the electronic structure of a single vortex in two
dimensional superconducting bilayer systems within the range of the mean-field
theory. The lack of local inversion symmetry in the system is taken into
account through the layer dependent Rashba spin-orbit coupling. The spatial
profiles of the pair potential and the local quasiparticle density of states
are calculated in the clean spin-singlet superconductor on the basis of the
quasiclassical theory. In particular, we discuss the characteristic core
structure in the pair-density wave state, which is spatially modulated exotic
superconducting phase in a high magnetic field.
