We investigate single-photon scattering in a waveguide quantum electrodynamics system in both the Markovian and the non-Markovian regimes, where a giant $\mathrm{\Lambda }$-type atom is coupled to two waveguides via different atomic transitions at two separated points. The phases of coupling strengths between the giant atom and waveguides are considered. In the Markovian regime, with a proper choice of phases, including the phases accumulated by photons traveling between coupling points and phase difference between two coupled channels, the incident photons can be manipulated to undergo elastic scattering (frequency-preserving) or inelastic scattering (frequency-converting) process. The frequency conversion with unit efficiency can be achieved and the condition of the optimal frequency conversion is identified. Furthermore, we show that asymmetric frequency conversion can also be achieved by phase control. In addition, our numerical simulations demonstrate that the present scheme shows significant robustness under the influence of moderate experimental parameter errors. In the non-Markovian regime, the scattering spectra show more complicated line shapes and become very sensitive to changes in detuning. More interestingly, under specific conditions, non-Markovianity can induce frequency conversion that cannot occur in the Markovian regime. Multiple narrow asymmetric scattering windows are further obtained, which provides more possibilities for studying asymmetric frequency conversion at off-resonant conditions. Our results may have potential applications in designing quantum devices involving giant atoms.

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• Quantum optics
• Quantum optics with artificial atoms
• Waveguides