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

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