  An efficient multi-mode quantum memory is a crucial resource for
long-distance quantum communication based on quantum repeaters. We propose a
quantum memory based on spectral shaping of an inhomogeneously broadened
optical transition into an atomic frequency comb (AFC). The spectral width of
the AFC allows efficient storage of multiple temporal modes, without the need
to increase the absorption depth of the storage material, in contrast to
previously known quantum memories. Efficient readout is possible thanks to
rephasing of the atomic dipoles due to the AFC structure. Long-time storage and
on-demand readout is achieved by use of spin-states in a lambda-type
configuration. We show that an AFC quantum memory realized in solids doped with
rare-earth-metal ions could store hundreds of modes or more with close to unit
efficiency, for material parameters achievable today.
