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In order to examine whether or not the quantum phase transition of Dicke typeexists in realistic systems, we revisit the model setup of the superconductingcircuit QED from a microscopic many-body perspective based on the BCS theorywith pseudo-spin presentation. By deriving the Dicke model with the correctcharging terms from the minimum coupling principle, it is shown that thecircuit QED system can exhibit superradiant quantum phase transition in thelimit N\rightarrow\infty. The critical point could be reached at easiness byadjusting the extra parameters, the ratio of Josephson capacitance C_{J} togate capacitance C_{g}, as well as the conventional one, the ratio of Josephsonenergy E_{J} to charging energy E_{C}.

Superradiant quantum phase transition in a circuit QED system: a revisit from a fully microscopic point of view