Linear microwave response of a charge-type qubit

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