Ultimate conversion efficiency bound for the forward double-$Λ$ atom-light coupling scheme

Abstract: We show that for the two widely used configurations of the double-$\Lambda$ atom-light coupling scheme, one where the control fields are applied in the same $\Lambda$-subsystem and another where they applied in different $\Lambda$-subsystems, the forward propagation of the probe and signal fields is described by the same set of equations. We then use optimal control theory to find the spatially-dependent optimal control fields which maximize the conversion efficiency from the probe to the signal field, for a given optical density. The present work is expected to find application in the implementation of efficient frequency and orbital angular momentum conversion devices for quantum information processing, as well as to be useful to many other applications using the double-$\Lambda$ atom-light coupling scheme.