  Atomic frequency comb (AFC) quantum memories are a promising technology for
quantum repeater networks because they enable multi-mode, high-fidelity storage
of photons with on-demand retrieval. The optimization of the retrieval
efficiency of an AFC memory is important because it strongly impacts the
entanglement generation rate in quantum networks. Despite initial theoretical
analyses and recent experimental demonstrations, a rigorous proof of the
universally optimal configuration for the highest AFC retrieval efficiency has
not been presented. In this paper we offer a simple analytical proof which
shows that the optimized square-tooth AFC provides the highest retrieval
efficiency among all possible comb tooth shapes, under the physical constraint
of maximal optical depth of an atomic ensemble. The optimality still holds even
when the non-zero background absorption and the finite homogeneous broadening
of atoms are considered. Our proof provides experimentalists with rigorous
arguments how to create optimal AFC under realistic experimental conditions.
Finally, we also identify other functional optimization problems where our
proof technique is applicable, thus proving the optimality of the square
function in more general scenarios.
