Quantum amplitude amplification by phase noise

I consider a system in which an atom in a superposition of momentum eigenstates of a periodic potential is subject to periodic random phase jumps in that potential. If the j-th state in superposition is "marked" by giving the (j-1)-th state a phase flip, I show that for the correct amount of noise, a ∼6 dB amplification of the marked state probability amplitude occurs, with no significant amplification in other states.