  We study the effect of Markovian environmental noise on the dynamics of a
two-level quantum system which is steered adiabatically by an external driving
field. We express the master equation taking consistently into account all the
contributions to the lowest non-vanishing order in the coupling to the
Markovian environment. We study the master equation numerically and
analytically and we find that, in the adiabatic limit, a zero-temperature
environment does not affect the ground state evolution. As a physical
application, we discuss extensively how the environment affects Cooper pair
pumping. The adiabatic ground state pumping appears to be robust against
environmental noise. In fact, the relaxation due to the environment is required
to avoid the accumulation of small errors from each pumping cycle. We show that
neglecting the non-secular terms in the master equation leads to unphysical
results, such as charge non-conservation. We discuss also a possible way to
control the environmental noise in a realistic physical setup and its influence
on the pumping process.
