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Non-Markovian dynamics of open quantum systems without rotating wave approximation

We study the non-Markovian dynamics of a damped oscillator coupled with a reservoir. We present exact formulas for the oscillator's evolution directly from the BCH formula by series expansion with neither Markovian nor rotating wave approximation (RWA). Based on these, we show the existence of the non-Markovian feature of the system's evolution for the damped oscillator. By numerical simulation we find that the non-Markovian feature exists within a wide range of the coupling strength, even when the coupling strength is very small. To this problem, prior art results have assumed RWA and the existence of non-Markovian feature was found when the system-reservoir coupling is strong enough. However, as we show, given such a strong coupling, the original Hamiltonian without RWA is actually not physical. Therefore, our exact study here has thoroughly concluded the issue of non-Markovian feature.