Abstract
In this paper, the behavior of the quantum speed up with detuning for an open two-level system embedded in a reservoir at zero temperature within the Janes-Cummings model is studied. We consider two examples of spectral densities in the framework of non-Markovian environments: a high-Q cavity is slightly off-resonant with the transition frequency of the qubit and a two-level system coupled to an anisotropic photonic band gap crystal environment without Markovian or Born approximation. Generally, it is found that there exist a sudden transition from no speedup to speedup at certain critical detuning value, and the physical parameters do not affect the decoherence process for long driving times. Interestingly, we find that the control of the speed of the quantum evolution can be benefit from the combination of the detuning and spectral width density in the first case, while the acceleration of the quantum evolution occurs as the qubit transition frequency is detuned inside the gap for the anisotropic dispersion model. Finally, we clarify how the excited populations and non-Markovianity can be influenced by the detuning to realize the quantum speed limit.