  We analyze tunneling of Cooper pairs across voltage biased asymmetric
single-Cooper-pair transistors. Also tunneling of Cooper pairs across two
capacitively coupled Cooper-pair boxes is considered, when the capacitive
coupling and Cooper pair tunneling are provided by a small Josephson junction
between the islands. The theoretical analysis is done at subgap voltages, where
the current-voltage characteristics depend strongly on the macroscopic
eigenstates of the island(s) and their coupling to the dissipative environment.
As the environment we use an impedance which satisfies Re[Z]<<R_Q and a few
LC-oscillators in series with Z. The numerically calculated I-V curves are
compared with experiments where the quantum states of mesoscopic SQUIDs are
probed with inelastic Cooper pair tunneling. The main features of the observed
I-V data are reproduced. Especially, we find traces of band structure in the
higher excited states of the Cooper-pair boxes as well as traces of multiphoton
processes between two Cooper-pair boxes in the regime of large Josephson
coupling.
