  We study the directional dependence of the coupling between a
nanofiber-guided light field and a two-level atom with an electric quadrupole
transition. We examine the situation where the atom lies on the fiber
transverse axis $x$, the quantization axis for the atomic internal states is
the other orthogonal transverse axis $y$, the atomic upper and lower levels are
the magnetic sublevels $M'$ and $M$ of hyperfine-structure levels of an
alkali-metal atom, and the field is in a quasilinearly polarized fundamental
guided mode HE$_{11}$ with the polarization $\xi=x$ or $y$. We find that the
absolute value of the quadrupole Rabi frequency depends on the propagation
direction of the light field in the cases of ($M'-M=\pm1$, $\xi=y$) and
($M'-M=\pm2$, $\xi=x$). We show that the directional dependence of the coupling
leads to the directional dependence of spontaneous emission into guided modes.
We find that the directional dependence of the atom-field coupling in the case
of quadrupole transitions is not entirely due to spin-orbit coupling of light:
there are some other contributions resulting from the gradient of the spatial
phase factor of the field.
