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A Lagrangian formalism is used to derive the Hamiltonian for a $\lambda$/4resonator shunted by a current-biased Josephson junction. The eigenstates andthe quantum dynamics of the system are analyzed numerically, and we show thatthe system can function as an efficient detector of weak incident microwavefields.

Quantized resonator field coupled to a current-biased Josephson junction in circuit QED