Multiple single-photon generations in three-level atoms coupled to cavity with non-Markovian effects

In this paper, we show how to generate the multiple single-photon wavepackets of arbitrary temporal shape from an optical cavity coupled with $N$ three-level atoms driven by a driving field in the non-Markovian regime. We derive an exact analytical expression of the optimal driving field for generating such wavepackets, which depends on two detunings of the cavity and driving field with respect to the three-level atoms. The cavity we used consists of two mirrors facing each other, where one is perfect and the other exists the dissipation (one-sided cavity), which couples with the corresponding non-Markovian input-output fields. If the first single-photon wavepacket generated by the Markovian system is the same as the non-Markovian case, the Markovian system cannot generate the same multiple single-photon wavepackets as the non-Markovian one when the spectral widths of the other environments taking values different from the spectral width of the first environment, while setting the equal spectral widths for the different environments can generate this. The generated multiple different single-photon wavepackets are not independent of each other, which satisfy certain relations with non-Markovian spectral parameters. We analyse the transition from Markovian to non-Markovian regimes and compare the differences between them, where the cavity interacts simultaneously with the multiple non-Markovian environments. Finally, we extend the above results to a general non-Markovian quantum network involving many cavities coupled with driven three-level atoms.