  The dynamics of a qubit in a structured environment is investigated
theoretically. One point of view of the model is the spin-boson model with a
Lorentz shaped spectral density. An alternative view is a qubit coupled to
harmonic oscillator (HO), which in turn coupled to a Ohmic environment. Two
different methods are applied and compared for this problem. One is a
perturbation method based on a unitary transformation. Since the transformed
hamiltonian is of rotating wave approximation (RWA) form, we call it the
transformed rotating wave approximation (TRWA) method. And the other one is the
numerically exact method of the quasi-adiabatic propagator path-integral
(QUAPI) method. TRWA method can be applied from the first point of view. And
the QUAPI method can applied from both points of views. We find that from the
1st point of view QUAPI only works well for large $\Gamma$. Since the memory
time is too long for the practical evaluation of QUAPI when $\Gamma$ is small.
We call this treatment as QUAPI1. And from the 2nd point of view, QUAPI works
well for small $\Gamma$, since the non-adiabatic effect become more important
as $\Gamma$ increases, one need smaller time-step and more steps to obtain
accurate result which also quickly runs out the computational resources. This
treatment is called QUAPI2. We find that the TRWA method works well for the
whole parameter range of $\Gamma$ and show good agreement with QUAPI1 and
QUAPI2. On the other hand, we find that the decoherence of the qubit can be
reduced with increasing coupling between HO and bath. This result may be
relevant to the design of quantum computer.
