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We present analytic solutions to the evolution in generalized tight-bindingmodels, which consider complex first-neighbor couplings with equal amplitudeand arbitrary phases. Our findings provide a powerful tool for efficientlycalculating expectation values and correlations within the system, which areotherwise difficult to compute numerically. We apply our results to relevantexamples in quantum light manipulation using N-port linear couplers, describingthe evolution of single(multi)-mode squeezing, single photon added (subtracted)Gaussian states, and second-order site-to-site photon correlations.Significantly, our analytic results outperform standard numerical calculations.Our study paves the way for a comprehensive mathematical framework describingthe spatial evolution of quantum states across a wide range of physical systemsgoverned by the tight-binding model.

Analytic Evolution for Complex Coupled Tight-Binding Models: Applications to Quantum Light Manipulation