Controllable generation of two-mode-entangled states in two-resonator circuit QED with a single gap-tunable superconducting qubit

We study controllable generation of two-mode-entangled states in a circuit QED setup, which consists of two spatially separated superconducting transmission line resonators and a single gap-tunable superconducting qubit. Two sharp coupling sidebands are induced when the artificial atom is suitably driven by a bichromatic microwave field. The two resonators can have squeezing-type interactions with the qubit via the coupling sidebands. If the two resonators are not degenerate, we show that the two resonators can be cooled down into the two-mode squeezed vacuum via dissipation of the qubit. The generation of the two-mode squeezed state is based on a dissipative state-engineering process, which explores the energy relaxation of the qubit as a resource. Moreover, the scheme does not need both the specific preparation of the initial state and the designed special dynamical process of the system. If the resonators are degenerate, we show that entangled coherent states of the resonators can be generated by use of the unitary dynamical evolution process of the system and the state-projection measurement. Moreover, macro entangled coherent states of the resonators with huge photons can in principle be created if the resonators and the qubit have sufficiently long lifetimes. The present scheme has two remarkable features: (1) only a single qubit is used in the generation of the two-mode squeezed state; and (2) the ultrastrong coupling condition and initializing the resonators in coherent states are not required. These make the present scheme more simple and feasible in experimental implementation.