Observation of the quantum capacitance in a Cooper-pair transistor

The effective capacitance of a single Cooper-pair box (SCB) can be defined as the second derivative of its energy with respect to gate voltage. This capacitance has two parts, the geometric capacitance, $C_{geom}$, and the quantum capacitance, $C_Q$. $C_Q$ is due to the anti-level crossing caused by the Josephson coupling energy $E_J$, and depends parametrically on the gate voltage. This capacitance, which is dual to the Josephson inductance, can be substantially larger than $C_{geom} $ as well as negative. To detect $C_Q$, we have measured the in- phase and out-of-phase rf-signal reflected from a Cooper-pair transistor (CPT), embedded in a resonant circuit. Under suitable biasing conditions the CPT acts as a SCB in series with a capacitance. It can be shown that the imaginary part of the reflected signal depends linearly on $C_Q$, and we can thus measure $C_Q$ directly from the reflected signal as a function of the gate voltage. The measured data agrees well with the theoretical prediction assuming that the system is in the ground state. We can extract the ratio $E_J/E_C$ for each of the two junctions in the CPT, where $E_C$ is the charging energy of the CPT.