  We report the observation of discrete vortex bound states with the energy
levels deviating from the widely believed ratio of 1:3:5 in the vortices of an
iron based superconductor KCa2Fe4As4F2 through scanning tunneling microcopy
(STM). Meanwhile Friedel oscillations of vortex bound states are also observed
for the first time in related vortices. By doing self-consistent calculations
of Bogoliubov-de Gennes equations, we find that at extreme quantum limit, the
superconducting order parameter exhibits a Friedel-like oscillation, which
modifies the energy levels of the vortex bound states and explains why it
deviates from the ratio of 1:3:5. The observed Friedel oscillations of the
bound states can also be roughly interpreted by the theoretical calculations,
however some features at high energies could not be explained. We attribute
this discrepancy to the high energy bound states with the influence of nearby
impurities. Our combined STM measurement and the self-consistent calculations
illustrate a generalized feature of the vortex bound states in type-II
superconductors.
