  Coherent microwave-to-optical conversion is crucial for transferring quantum
information generated in the microwave domain to optical frequencies, where
propagation losses can be minimised. Among the various physical platforms that
have realized coherent microwave-to-optical transduction, those that use atoms
as transducers have shown rapid progress in recent years. In this paper we
report an experimental demonstration of coherent microwave-to-optical
conversion that maps a microwave signal to a large, tunable 550(30) MHz range
of optical frequencies using room-temperature $^{87}$Rb atoms. The
inhomogeneous Doppler broadening of the atomic vapor advantageously supports
the tunability of an input microwave channel to any optical frequency channel
within the Doppler width, along with simultaneous conversion of a multi-channel
input microwave field to corresponding optical channels. In addition, we
demonstrate phase-correlated amplitude control of select channels, resulting in
complete extinction of one of the channels, providing an analog to a frequency
domain beam splitter across five orders of magnitude in frequency. With
frequency-division multiplexing capability, multi-channel conversion, and
amplitude control of frequency channels, neutral atomic systems may be
effective quantum processors for quantum information encoded in frequency-bin
qubits.
