  We investigate the electric quadrupole interaction of an alkali-metal atom
with guided light in the fundamental and higher-order modes of a vacuum-clad
ultrathin optical fiber. We calculate the quadrupole Rabi frequency, the
quadrupole oscillator strength, and their enhancement factors. In the example
of a rubidium-87 atom, we study the dependencies of the quadrupole Rabi
frequency on the quantum numbers of the transition, the mode type, the phase
circulation direction, the propagation direction, the orientation of the
quantization axis, the position of the atom, and the fiber radius. We find that
the root-mean-square (rms) quadrupole Rabi frequency reduces quickly but the
quadrupole oscillator strength varies slowly with increasing radial distance.
We show that the enhancement factors of the rms Rabi frequency and the
oscillator strength do not depend on any characteristics of the internal atomic
states except for the atomic transition frequency. The enhancement factor of
the oscillator strength can be significant even when the atom is far away from
the fiber. We show that, in the case where the atom is positioned on the fiber
surface, the oscillator strength for the quasicircularly polarized fundamental
mode HE$_{11}$ has a local minimum at the fiber radius $a\simeq 107$ nm, and is
larger than that for quasicircularly polarized higher-order hybrid modes, TE
modes, and TM modes in the region $a<498.2$ nm.
