Full-frequency voltage noise spectral density of a single-electron transistor

We calculate the full-frequency spectral density of voltage fluctuations in a single-electron transistor (SET), used as an electrometer biased above the Coulomb threshold so that the current through the SET is carried by sequential tunneling events. We consider both a normal-state SET and a superconducting SET. The whole spectrum, from low-frequency telegraph noise to quantum noise at frequencies comparable to the SET charging energy ${(E}_C/ħ)$ to high-frequency Nyquist noise, is described. We take the energy exchange between the SET and the measured system into account using a real-time diagrammatic Keldysh technique. The voltage fluctuations determine the backaction of the SET on the measured system, and we specifically discuss the case of superconducting charge qubit read-out and measuring the so-called Coulomb staircase of a single Cooper-pair box.