Coherent control in circuit QED

Superconducting charge qubits fabricated inside a transmission line resonator have been used to successfully demonstrate strong interaction of an artificial atom with a single photon [1]. This architecture has also been used to show high-visibility and long coherence time (T$_{1}\sim $7 $\mu $s, T$_{2}\sim $ 500 ns) Rabi oscillations [2] and in the detailed study of measurement-induced dephasing [3]. Here we will discuss protocols to realize one and two-qubit logical gates in circuit QED. These are based on resonant and off-resonant irradiation of the transmission line resonator. First experimental results towards the realization of these gates will be presented. Supported by NSA and ARDA under ARO Contract No. W911NF-05-1-0365 and the NSF under Grants No. ITR-0325580 and No. DMR-0342157. [1] A. Wallraff \textit{et al.}, Nature 431, 162 (2004). [2] A. Wallraff \textit{et al.}, Phys. Rev. Lett., 95, 060501 (2005). [3] D. Schuster \textit{et al.}, Phys. Rev. Lett., 94, 123602 (2005).