  We study the stochastic dynamics of a two-level quantum system interacting
with a stochastic magnetic field, and a single frequency electromagnetic field,
with and without making the rotating wave approximation (RWA). The
transformation to the rotating frame does not commute with the stochastic
Hamiltonian if the stochastic field has nonvanishing components in the
transverse direction. Hence, making the RWA modifies the stochastic terms in
the Hamiltonian. Modification of the decay terms is also required in a master
equation approach (i.e., the Liouville--von Neumann density matrix equation)
for describing the dynamics. For isotropic Gaussian white noise, the RWA
dynamics remains Markovian, although the Lindblad terms in the master equation
for the density matrix become time-dependent when the non-commutation of the
RWA transformation and the noise Hamiltonian is properly accounted for. We also
treat Ornstein--Uhlenbeck noise, and find, in contra-distinction to the white
noise case, a significant difference in the dynamics calculated with the RWA
when the non-commutation of the RWA transformation and the noise Hamiltonian is
taken into account. These findings are applicable to the modeling of any open
quantum system coupled to an electromagentic field.
