We investigate the single-photon scattering spectra in a giant-molecule waveguide quantum electrodynamics system, where a giant molecule consisting of two coupled giant atoms is connected to two waveguides via two separate sites, respectively. The phases of coupling strengths between the giant molecule and waveguides are considered. Using a real-space method, we obtain the exact analytical expressions for the single-photon scattering coefficients, which are valid both in the Markovian and non-Markovian regimes. It is shown that the transmission spectra strongly depend on phases, including the phases accumulated by photons traveling between coupling points and the phase difference between two coupled channels of the giant-molecule upper (lower) waveguide. Under the Markovian limit, we find that by selecting appropriate accumulated phases and local phase difference, targeted routing and asymmetrical scattering of photons can be achieved without breaking the symmetry in the coupling configuration between the giant molecule and waveguides. Moreover, all the optimal phase combinations for implementing single-photon routing have been summarized. In the non-Markovian regime, the scattering spectra show more complicated line shapes with multiple staggered peaks and dips. More interestingly, the scattering spectrum exhibits periodic changes during the transition from the Markovian to the non-Markovian regime, which makes one obtain a means to control photon routing in addition to phases. Our results may be useful in controlling and manipulating light-matter interactions involving giant atoms at the single-photon level.

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• Quantum optics
• Waveguide arrays
• Markovian processes
• Non-Markovian processes