  Employing a sixth order expression for the differential time evolution
operator based on the Magnus expansion (ME), we conducted quantum dynamics
calculations of a $\Lambda$-system driven by two sinusoidal time dependent
fields. For a closed system dynamics, we confirmed the equivalence of the
dynamics in the Hilbert space and the Liouville space numerically. We also
conducted open system quantum dynamics calculation by generalizing the ME to
the non-Hermitian dynamics in the Liouville space for the case where the
effects of photonic bath are represented by Lindblad operators. In both cases,
the accuracy of the rotating wave approximation (RWA) was assessed. We found
significant errors of RWA during initial stages of the dynamics for
representative cases where electromagnetically induced transparency or coherent
population trapping can be observed. The presence of bath for open system
quantum dynamics reduces the errors of RWA, but significant errors for
off-diagonal elements of the density operator can still be seen. We also found
that approaches to steady state limits of exact dynamics are slower than those
for RWA. These results demonstrate the utility of the ME as a general and
reliable tool for closed and open system quantum dynamics for time dependent
Hamiltonians, and expose potential issues of drawing conclusions based solely
on RWA.
