Generalised expression of the noise figure of phase sensitive amplifiers for an arbitrary number of modes

Phase sensitive amplifiers (PSA), contrary to usual phase insensitive amplifiers (PIA), are in principle capable to achieve noiseless amplification, i.e. exhibit a quantum-limited noise figure (NF) of 0 db. When implemented using four-wave mixing (FWM) in a non-linear fibre, extra waves can be generated by undesired FWM processes, which may introduce extra input ports for vacuum fluctuations, thus potentially degrading the NF. In this situation, we give here a general analytical quantum derivation of the PSA NF, valid for an arbitrary number of nonlinearly coupled modes. This expression is usable as soon as a linear input-output relation can be found for the annihilation and creation operators of the involved modes. It predicts that the noise level depends on the number of interacting waves. We illustrate the usefulness of this expression in the case of six waves, corresponding to four interacting quantum modes. In this example the signal NF is degraded by 0.4 db, compared to 10 db obtained for PIA operation of the same scheme.

Automated summary

Phase sensitive amplifiers have the potential to achieve noiseless amplification with a quantum-limited noise figure of 0 db, but undesired FWM processes in a non-linear fiber can introduce extra waves and potentially degrade the NF. A general analytical quantum derivation of the PSA NF is provided here for an arbitrary number of nonlinearly coupled modes, showing that the noise level depends on the number of interacting waves. Illustrative example with six waves demonstrates a 0.4 db degradation in signal NF compared to PIA operation with a 10 db noise figure.