  Rotating wave approximation in a quantum spin system driven by a linearly
polarized alternating magnetic field with quadrupole interaction presents is
investigated in detail in this paper. The conventional way to employ the
rotating wave approximation is to assume the dynamics being restricted in the
reduced Hilbert space. However, when the driving strength is relatively strong
or the driving is off resonant, the leakage from the target resonance subspace
along with the effects from the counter-rotating terms cannot be neglected
anymore. We propose the rotating wave approximation applied in the full Hilbert
space to take the leakage from the target resonance subspace into account. To
include the effects from the counter-rotating terms, we utilize the
counterrotating hybridized rotating wave method in the reduced Hilbert space.
The performance of these rotating wave approximation methods is compared by
estimating the state fidelity as well as the operator fidelity and the results
reveal that different methods may be employed for different practical
circumstances.
