Superposition-preserving photon-number amplifier

We analyze a pure-state photon-number amplifier (PNA), i.e., a device that performs the action [Formula Presented] where [Formula Presented] is a positive integer. We derive expressions for the fidelity of a finite internal state PNA and demonstrate that the amplifier fidelity, i.e., the similarity between the obtained and the desired output state can be made to approach unity. Finally we outline a systematic procedure to find the Hamiltonian for any finite Hilbert-space PNA and compute the Hamiltonian for a simple but nontrivial PNA.