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Observation of quantum capacitance in the Cooper-pair transistor

Prior to 2005, the qubits fabricated and studied at Chalmers incorporated a radio-frequency single-electron transistor (RF-SET), which functioned as an electrometer to measure the charge of the single Cooper-pair box (SCB) two-level system. For this type of projective measurement, one must necessarily move away from the optimal point (charge degeneracy) to perform a charge measurement. Another drawback of this design concerns scalability. In a multiple qubit system, each of the qubits of this type requires its own RF-SET.

In addition to the DC line for the SCB voltage gate, one needs two DC lines (voltage gate and bias), and a radio-frequency coax for operation of the RF-SET. One must also synchronize the SCB gate and RF-SET gate to cancel cross capacitances. Our newly designed readout scheme utilizes the effective capacitance of the SCB by placing the SCB in a resonant LC circuit. In this way, one can combine the two-level system and measurement apparatus into a single device. Furthermore, unlike measuring charge, the effective capacitance readout is most sensitive at the charge degeneracy---the optimal point for qubit operation.

The effective capacitance of a single Cooper-pair box can be defined as the derivative of the induced charge with respect to gate voltage and has two parts, the geometric capacitance and the quantum capacitance. The latter is due to the level anti-crossing caused by the Josephson coupling and is dual to the Josephson inductance. It depends parametrically on the gate voltage and its magnitude may be substantially larger than. Calculations show that such methods are capable of single-shot readout of the quantum state.

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