Attenuated laser pulses are often employed in place for single photons in order to test the efficiency of the elements of a quantum network. In this work we analyse theoretically the dynamics of storage of an attenuated light pulse (where the pulse intensity is at the single photon level) propagating along a transmission line and impinging on the mirror of a high finesse cavity. Storage is realised by the controlled transfer of the photonic excitations into a metastable state of an atom confined inside the cavity and occurs via a Raman transition with a suitably tailored laser pulse, which drives the atom and minimizes reflection at the cavity mirror. We determine the storage efficiency of the weak coherent pulse which is reached by protocols optimized for single-photon storage. We determine the figures of merit and we identify the conditions on an arbitrary pulse for which the storage dynamics approaches the one of a single photon. Our formalism can be extended to arbitrary types of input pulses and to quantum memories composed by spin ensembles, and serves as a basis for identifying the optimal protocols for storage and readout.
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