  We consider an atomic frequency comb based quantum memory inside an
asymmetric optical cavity. In this configuration it is possible to absorb the
input light completely in a system with an effective optical depth of one,
provided that the absorption per cavity round trip exactly matches the
transmission of the coupling mirror ("impedance matching"). We show that the
impedance matching results in a readout efficiency only limited by irreversible
atomic dephasing, whose effect can be made very small in systems with large
inhomogeneous broadening. Our proposal opens up an attractive route towards
quantum memories with close to unit efficiency.
