  We show how population inversion (PI) occurs in a two-level system (TLS)
while measuring its charge using a capacitively coupled superconducting
single-electron transistor (SSET), biased in the subgap regime, where the
current through the SSET is carried by different cycles involving tunneling of
both Cooper pairs and quasiparticles. The PI is directly associated with the
resonant nature of the Cooper-pair tunneling. We also show how the SSET may
strongly relax the TLS, although there is negligible current flowing through
the SSET, i.e. it is turned off. The calculation of the quantum back-action
noise is based on a real-time Keldysh approach. We specifically discuss the
case of a Cooper Pair Box qubit with the SSET capacitively coupled as read-out
device.
