  We present analytic solutions to the evolution in generalized tight-binding
models, which consider complex first-neighbor couplings with equal amplitude
and arbitrary phases. Our findings provide a powerful tool for efficiently
calculating expectation values and correlations within the system, which are
otherwise difficult to compute numerically. We apply our results to relevant
examples in quantum light manipulation using N-port linear couplers, describing
the 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 describing
the spatial evolution of quantum states across a wide range of physical systems
governed by the tight-binding model.
