  Light is an excellent medium for both classical and quantum information
transmission due to its speed, manipulability, and abundant degrees of freedom
into which to encode information. Recently, space-division multiplexing has
gained attention as a means to substantially increase the rate of information
transfer by utilizing sets of infinite-dimensional propagation eigenmodes such
as the Laguerre-Gaussian 'donut' modes. Encoding in these high-dimensional
spaces necessitates devices capable of manipulating photonic degrees of freedom
with high efficiency. In this work, we demonstrate controlling the optical
susceptibility of an atomic sample can be used as powerful tool for
manipulating the degrees of freedom of light that passes through the sample.
Utilizing this tool, we demonstrate photonic mode conversion between two
Laguerre-Gaussian modes of a twisted optical cavity with high efficiency. We
spatiotemporally modulate the optical susceptibility of an atomic sample that
sits at the cavity waist using an auxiliary Stark-shifting beam, in effect
creating a mode-coupling optic that converts modes of orbital angular momentum
$l=3\rightarrow l=0$. The internal conversion efficiency saturates near unity
as a function of the atom number and modulation beam intensity, finding
application in topological few-body state preparation, quantum communication,
and potential development as a flexible tabletop device.
