  The two-level quantum system (qubit) in a precessing magnetic field and in
contact with a heat bath is investigated. The exact reduced dynamics for the
qubit in question is obtained. We apply the approach based on the block
operator matrices theory, in contrast with the standard methods provided by the
theory of the open quantum systems. We also present the solution of the Riccati
operator equation associated with the Hamiltonian of the system. Next, we study
the adiabatic approximation for the system under consideration using quantum
fidelity as a way to measure validity of the adiabatic theory. We find that in
the weak coupling domain the standard condition that ensures adiabatic behavior
of the spin in the magnetic field also guarantees its adiabatic evolution in
the open system variant of this model. Therefore, we provide the explicit
example of the open quantum system that satisfies the adiabatic approximation
firstly formulated for the closed quantum systems.
