Linear microwave response of a charge-type qubit

Using Bloch-type equations with phenomenological energy relaxation and deph asing terms, we investigate analytically the linear response of a supercond ucting charge qubit (single-Cooper-pair box) to a microwave field within a two-band model. Coherence of the macroscopic quantum states of the autonomo us device is due to the competition of irradiation and dephasing. An extra qubit impedance arises from this quantum effect which can be expressed in t erms of quantum resistance unit, quality and detuning. The theoretical resu lts may support the design of microwave experiments with charge-type qubits . Especially, we present a proposal for determining the dephasing time by m eans of reflection measurements, anticipating a high quality of the extra ( quantum) impedance.