  Motivated by recent experiments, we analyze transport of Cooper pairs through
a double-island Josephson qubit. At low bias in a certain range of gate
voltages coherent superpositions of charge states play a crucial role. Analysis
of the evolution of the density matrix allows us to cover a wide range of
parameters, incl. situations with degenerate levels, when dissipation strongly
affects the coherent eigenstates. At high noise levels the so-called Zeno
effect can be observed, which slows down the transport. Our analysis explains
certain features of the I-V curves, in particular the visibility and shape of
resonant peaks and lines.
