Photonic Reservoir Engineering via 2D $\Lambda$-Type Atomic Arrays in Waveguide QED

Electromagnetically induced transparency (EIT) in $\Lambda$-type atomic systems underpins quantum technologies such as high-fidelity memory and nonlinear optics, but conventional setups face intrinsic limitations. Standard geometries of one-dimensional atomic chains coupled to waveguides allow only a single bright superradiant channel, while subradiant modes remain weakly accessible, limiting control over collective radiative behavior and dark-state pathways. This leads to unwanted inelastic processes, degrading memory fidelity and reducing nonlinear photon generation efficiency. Here, we propose two two-dimensional (2D) atomic lattice geometries coupled to a photonic crystal waveguide, namely Zigzag and Orthogonal structures. In the Zigzag model, engineered collective super- and subradiant modes produce a flattened EIT window, broadening the transmission bandwidth and suppressing unwanted scattering to enhance quantum memory fidelity. In the Orthogonal model, four-wave mixing (FWM) intensity is amplified by up to six orders of magnitude relative to a conventional one-dimensional $\Lambda$-type EIT chain with identical $\Gamma_{1D}$, $\Omega_c$, and probe intensity, with localized idler photons forming well-defined spectral modes. These results demonstrate a versatile route to engineer structured photonic reservoirs for on-demand photon generation, high-fidelity quantum storage, and enhanced nonlinear optical processes.