We propose and analyze a general scheme to engineer programmable and nonreciprocal interactions in an array of atoms coupled to a photonic waveguide. By subjecting each atom to a local harmonic frequency modulation with a defined phase ${\theta }_i$, we demonstrate the emergence of effective long-range couplings whose parameters are dynamically controllable. For atoms coupled to a conventional waveguide, the modulation dresses the atomic operators, leading to effective interactions governed by the Bessel function ${\mathcal{J}}_0$ whose argument depends on the phase differences $\mathrm{\Delta }{\theta }_{ij}$. For atoms coupled to a photonic crystal waveguide near its band edge, where the density of states is sharply varying, the modulation can be tuned to selectively activate the ${\mathcal{J}}_1^2$ term, which simultaneously scales both the coherent and dissipative components of the complex coupling strength ${\mathrm{\Gamma }}_{ij}$. Crucially, the spatial profile of the local modulation phases imprints a factor $exp\left(\mathrm{i}\mathrm{\Delta }{\theta }_{ij}\right)$ onto the waveguide-mediated coupling strength ${\mathrm{\Gamma }}_{ij}$, enabling the synthesis of nonreciprocal quantum interactions. Our approach thus provides a flexible toolbox for simulating a wide range of non-Hermitian Hamiltonians on waveguide-QED platforms.

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• Quantum optics