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The processing of information encoded in frequency combs or spectral lattices has multiple applications for both classical and quantum states of light ranging from communications to spectroscopy. There is a strong interest in all-optical approaches for ultra-fast processing on integrated platforms. Here, we develop a concept and demonstrate experimentally all-optical flexible spectral comb reshaping in a nonlinear waveguide for two novel applications. First, we reveal that the evolution of an optical spectral comb can emulate wave dynamics in multi-dimensional lattices, which is a nontrivial generalization of previous theoretical proposals. In our experiment, a discrete signal spectrum is modulated by stronger pumps co-propagating in a nonlinear fiber with Kerr-type nonlinearity. Four-wave mixing Bragg scattering then induces coupling between many spectral lines, including nonlocal couplings between spectral lines which are further apart. We find that such a configuration can be exactly mapped to wave dynamics in complex multi-dimensional lattices, and as a representative example we realize a tube of triangular lattice. Importantly, the nontrivial phase of complex-valued couplings can give rise to synthetic gauge fields, and we directly measure corresponding asymmetric spectral reshaping. Our approach is scalable to higher-dimensional synthetic lattices. Second, we show that such a lattice with nonlocal couplings can enable the full reconstruction of the input spectra, including information on the phase and coherence, with a single-shot spectral intensity measurement. We demonstrate the reconstruction of input states composed of four frequency channels. Remarkably, the coherent nature of nonlinearly induced couplings is applicable for quantum states with spectral encoding.
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