Memory-based Probabilistic Noiseless Amplification of Coherent states

We analytically show that probabilistic amplification of a weak coherent state stored inside an atomic medium can be achieved via detection of coherently scattered photons. We show that this is because of collective excitations created among atoms in the ensemble. We describe the physics of the amplification and identify the failure events, which occur during the amplification process. The amplification is realized by coherently mapping a weak coherent state in an ensemble of $\mathrm{\Lambda }$-level atoms followed by detection of multiple Raman scattered photons, conditionally projecting the coherent state into an amplified state upon retrieval.