We study the non-Markovian dynamics of a damped oscillator coupled with areservoir. We present exact formulas for the oscillator's evolution directlyfrom the BCH formula by series expansion with neither Markovian nor rotatingwave approximation (RWA). Based on these, we show the existence of thenon-Markovian feature of the system's evolution for the damped oscillator. Bynumerical simulation we find that the non-Markovian feature exists within awide range of the coupling strength, even when the coupling strength is verysmall. To this problem, prior art results have assumed RWA and the existence ofnon-Markovian feature was found when the system-reservoir coupling is strongenough. However, as we show, given such a strong coupling, the originalHamiltonian without RWA is actually not physical. Therefore, our exact studyhere has thoroughly concluded the issue of non-Markovian feature.
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