  We present an experimental demonstration of converting a microwave field to
an optical field via frequency mixing in a cloud of cold $^{87}$Rb atoms, where
the microwave field strongly couples to an electric dipole transition between
Rydberg states. We show that the conversion allows the phase information of the
microwave field to be coherently transferred to the optical field. With the
current energy level scheme and experimental geometry, we achieve a photon
conversion efficiency of ~0.3% at low microwave intensities and a broad
conversion bandwidth of more than 4 MHz. Theoretical simulations agree well
with the experimental data, and indicate that near-unit efficiency is possible
in future experiments.
