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Phase sensitive amplifiers (PSA), contrary to usual phase insensitiveamplifiers (PIA), are in principle capable to achieve noiseless amplification,i.e. exhibit a quantum-limited noise figure (NF) of 0 dB. When implementedusing four-wave mixing (FWM) in a nonlinear fiber, extra waves can be generatedby undesired FWM processes, which may introduce extra input ports for vacuumfluctuations, thus potentially degrading the NF. In this situation, we givehere a general analytical quantum derivation of the PSA NF, valid for anarbitrary number of nonlinearly coupled modes. This expression is usable assoon as a linear input-output relation can be found for the annihilation andcreation operators of the involved modes. It predicts that the noise leveldepends on the number of interacting waves. We illustrate the usefulness ofthis expression in the case of six waves, corresponding to four interactingquantum modes. In this example the signal NF is degraded by 0.4 dB, compared to10 dB obtained for PIA operation of the same scheme.

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