We propose a non-Hermitian momentum-lattice model based on an atomic ensemble coupled with spatially modulated optical fields. By introducing modulation to the optical fields, we realize a momentum lattice featuring nonreciprocal couplings, long-range interactions, and highly tunable parameters. In this platform, the Brillouin zone emerges in real space, enabling band braiding in the $E\text{−}x$ space, leading to band structures such as unlinks, unknots, Hopf links, and trefoil knots. The resulting band topology is characterized by the discriminant number in one dimension and homotopy classes in two dimensions. Based on the theory called non-Hermitian absorption spectroscopy, we establish a direct correlation between complex band topological structures and spatially resolved absorption spectra. Interestingly, one can take a photo of the band braiding using an optical camera. Moreover, we propose a protocol to implement a three-band non-Hermitian lattice for exploring non-Abelian topology in a synthetic two-dimensional Brillouin torus and discuss its extension to three dimensions.

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• Geometric & topological phases
• Quantum optics
• Atoms
• Non-Hermitian systems
• Topology