Superluminal pulse propagation and amplification without inversion of microwave radiation via four-wave mixing in superconducting phase quantum circuits

We study the interaction of the microwave fields with an array of superconducting phase quantum circuits. It is shown that the different four-level configurations i.e. cascade, N-type, diamond, Y-type and inverted Y-type systems can be obtained in the superconducting phase quantum circuits by keeping the third order of the Josephson junction potential expansion whereas by dropping the third order term, just the cascade configuration can be established. We study the propagation and amplification of a microwave field in a four-level cascade quantum system, which is realized in an array of superconducting phase quantum circuits. We find that by increasing the microwave pump tones feeding the system, the normal dispersion switches to the anomalous and the gain-assisted superluminal microwave propagation is obtained in an array of many superconducting phase quantum circuits. Moreover, it is demonstrated that the stimulated microwave field is generated via four-wave mixing without any inversion population in the energy levels of the system (amplification without inversion) and the group velocity of the generated pulse can be controlled by the external oscillating magnetic fluxes. We also show that in some special set of parameters, the absorption-free superluminal generated microwave propagation is obtained in superconducting phase quantum circuit system.