  Quantum memories can be used not only for the storage of quantum information,
but also for substantial manipulation of ensembles of quantum states.
Therefore, the speed of such manipulation and the ability to write and retrieve
the signals of relatively short duration becomes important. Previously there
have been considered the limits on efficiency of the cavity-enhanced atomic
Raman memories for the signals whose duration is not much larger than the
cavity field lifetime, that is, beyond the bad cavity limit. We investigate in
this work the four-wave mixing noise that arises by the retrieval of the
relatively short signals from the cavity-assisted memories, thus complementing
recent considerations by other authors, who mainly concentrated on the limit of
large cavity decay rate. The four-wave mixing noise is commonly recognized as
an important factor, able to prevent achieving a high memories quality in a
variety of the atomic, solid state etc. implementations.
  The side-band noise sources (with respect to the quantized signal, supported
by the cavity) play important role in the four-wave mixing. We propose an
approach that allows one to account for the side-band quantum noise sources of
different physical origin in the cavity-assisted atomic memories using a
unified theoretical framework, based on a two-band spectral filtering of the
noise sources. We demonstrate that in such spectrally-selective memories the
side-band atomic noise sources essentially contribute to the four-wave mixing
noise of the retrieved signal on a par with the side-band quantized field
entering the cavity.
