Quantum mechanics of phase-sensitive amplification in a fiber

In a recent paper [C.J. McKinstrie, S. Radic, Opt. Express 12 (2004) 4973], four schemes were described, which produce phase-sensitive amplification in fibers: Degenerate scalar and vector four-wave mixing (FWM), and phase conjugation (PC) combined with frequency conversion (FC), in either order. In this paper, the basic quantum mechanics of these schemes are discussed. For each scheme, formulas are derived for the field-quadrature and photon-number variances. The effects of polarization misalignment on vector FWM, and the effects of unbalanced conversion on PC combined with FC, are studied. Moderate amounts of these imperfections do not degrade significantly the performances of the aforementioned schemes.