  We propose a superconducting circuit-atom hybrid, where the Rabi oscillation
of single excess Cooper pair in the island is stabilized via the common
atomic-clock technique. The noise in the superconducting circuit is mapped onto
the voltage source which biases the Cooper-pair box via an inductor and a gate
capacitor. The fast fluctuations of the gate charge are significantly
suppressed by an inductor-capacitor resonator, leading to a
long-relaxation-time Rabi oscillation. More importantly, the residual
low-frequency fluctuations are further reduced by using the general
feedback-control method, in which the voltage bias is stabilized via
continuously measuring the dc-Stark-shift-induced atomic Ramsey signal. The
stability and coherence time of the resulting charge-qubit Rabi oscillation are
both enhanced. The principal structure of this Cooper-pair-box oscillator is
studied in detail.
