  We study the dynamics of a two-level system described by a slowly varying
Hamiltonian and weakly coupled to the Ohmic environment. We follow the
Bloch--Redfield perturbative approach to include the effect of the environment
on qubit evolution and take into account modification of the spectrum and
matrix elements of qubit transitions due to time-dependence of the Hamiltonian.
This formalism is applied to two problems. (1) We consider a qubit, or a
spin-1/2, in a rotating magnetic field. We show that once the rotation starts,
the spin has a component perpendicular to the rotation plane of the field that
initially wiggles and eventually settles to the value proportional to the
product of angular rotation velocity of the field and the Berry curvature. (2)
We re-examine the Landau--Zener transition for a system coupled to environment
at arbitrary temperature. We show that as temperature increases, the thermal
excitation and relaxation become leading processes responsible for transition
between states of the system. We also apply the Lindblad master equations to
these two problems and compare results with those obtained from the
Bloch--Redfield equations.
