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The dynamics of a qubit under the decoherence of a two level fluctuator (TLF)in addition to its coupling to a bosonic bath is investigated theoretically.Two different methods are applied and compared for this problem. One is aperturbation method based on a unitary transformation. With the merit of ourunitary transformation, non-adiabatic effect can be taken into accountefficiently. And the other one is the numerically exact method, namely thequasi-adiabatic path-integral (QUAPI) propagator technique. We find that theanalytical method works well for a wide parameter range and show good agreementwith QUAPI. On the other hand, The enhancement and the reduction of quantumdecoherence of the qubit is checked with varying bath temperature $T$ andTLF-bath coupling.

The dynamics of a qubit in a spin-boson environment: a comparison between analytical and numerical method