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We consider the single photon transistor in coupled cavity system ofresonators interacting with multilevel superconducting artificial atomsimultaneously. Effective single mode transformation is used for thediagonalization of the the hamiltonian and impedance matching in terms of thenormal modes. Storage and transmission of the incident field are described bythe interactions between the cavities controlling the atomic transitions oflowest lying states. Rabi splitting of vacuum induced multiphoton transitionsis considered in input/output relations by the quadrature operators in theabsence of the input field. Second order coherence functions are employed toinvestigate the photon blockade and localization-delocalization transitions ofcavity fields in oscillating regime of photon states described by the thepopulation imbalance. Refection and transmission of cavity output fields areinvestigated in the presence of the multilevel transitions. Accumulation andfiring of the reflected and transmitted fields are used to investigate thesynchronization of the bunching spike train of transmitted field and populationimbalance of cavity fields. In the presence of single photon gate field, gainenhancement is explained for transmitted regime.

On-Chip Photonic Transistor based on the Spike Synchronization in Circuit QED