Generation of superposition coherent states of microwave fields via dissipation of a superconducting qubit with broken inversion symmetry

We present an efficient approach for dissipative generation of quantum superposition states in the microwave resonator, which is coupled to a superconducting qubit. We investigate the situation where the inversion symmetry of potential energy of the qubit is broken, and the strong two-photon nonlinear coupling between qubit and resonator can be realized via the transverse and longitudinal couplings. According to the two-photon dissipation and driving process, the dissipation of a qubit can be utilized to steer the microwave field into a quantum superposition of distinct coherent states, i.e., the Schr\"odinger cat state. In addition, we also extend the method to produce an entangled coherent state of two spatially separated resonators. Our scheme is based on quantum reservoir engineering and turns detrimental noise into a resource, which makes it feasible in experimental implementation. The present result may have potential applications in the field of quantum information processing with circuit QED systems.