Cross-phase modulation and entanglement in a compound gradient echo memory

We present a theoretical model for a Kerr-like interaction between two registers of a compound gradient echo memory (GEM). This type of interaction is known to generate cross-phase modulation (XPM) between optical fields, an effect that is limited by the typically small values of nonlinearities in crystals. Here we show that in GEM systems the phase shift increases linearly with the interaction time and quadratically with the strength of the field. Increasing storage (interaction) times would then lead to stronger XPM effects even with fields with very low intensity. This interaction also generates two other effects: entanglement between the registers, which depends on the strength of the interaction and its spatial profile, and an interaction-induced gradient. We show that the latter produces leakage during the storage stage depending on the shape of the stored pulses, an undesirable consequence that can be minimized by carefully designing the temporal profile of the input fields.