  One of the major difficulties in realizing a high-dimensional frequency
converter for conventional optical vortex (COV) stems from the difference in
ring diameter of COV modes with different topological charge numbers l. Here,
we implement a high-dimensional frequency convertor for perfect optical vortex
(POV) modes with invariant size through the four-wave mixing (FWM) process by
utilizing Bessel-Gaussian beams instead of Laguerre-Gaussian beams. The
measured conversion efficiency from 1530 nm to 795 nm is independent of l at
least in subspace of {-6,...,6}, and the achieved conversion fidelities for
two-dimensional (2D) superposed POV states exceed 97%. We further realize the
frequency conversion of 3D, 5D and 7D superposition states with fidelities as
high as 96.70%, 89.16% and 88.68%, respectively. The reported scheme is
implemented in hot atomic vapor, it's also compatible with the cold atomic
system and may find applications in high-capacity and long-distance quantum
communication.
