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We study the propagation and amplification of a microwave field in afour-level cascade quantum system which is realized in a superconducting phasequantum circuit. It is shown that by increasing the microwave pump tonesfeeding the system, the normal dispersion switches to the anomalous and thegain-assisted superluminal microwave propagation is obtained in this system.Moreover, it is demonstrated that the stimulated microwave field is generatedvia four-wave mixing without any inversion population in the energy levels ofthe system (amplification without inversion) and the group velocity of thegenerated pulse can be controlled by the external oscillating magnetic fluxes.We also show that in some special set of parameters, the absorption-freesuperluminal generated microwave propagation is obtained in superconductingphase quantum circuit system.

Gain-assisted superluminal microwave pulse propagation via four-wave mixing in superconducting phase quantum circuits