Generating the Schrodinger cat state in a nanomechanical resonator coupled to a charge qubit

We propose a scheme for generating the Schr\“{o}dinger cat state based on geometric operations by a nanomechanical resonator coupled to a superconducting charge qubit. The charge qubit, driven by two strong classical fields, interacts with a high-frequency phonon mode of the nanomechanical resonator. During the operation, the charge qubit undergoes no real transitions, while the phonon mode of the nanomechanical resonator is displaced along different paths in the phase space, dependent on the states of the charge qubit, which yields the Schr\“{o}dinger cat state. The robustness of the scheme is justified by considering noise from environment, and the feasibility of the scheme is discussed.