Josephson junctions are quantum mechanical devices with sinusoidal nonlinearity and their use as amplifiers of non-classical microwaves in cavities at low temperatures, is discussed. Self-pumped amplifiers as well as externally pumped three- and four-photon amplifiers are considered, and a quantum theory that describes their operation is developed. It is shown that the sinusoidal nonlinearity can be expanded into a series of terms, the first few of which are familiar from quantum optical amplifiers that use nonlinear optical materials, while the others are higher order corrections.
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