This study explores the versatile framework of time-frequency variables in the context of quantum information, with a specific focus on the use of single photons as a paradigmatic system. In our first analysis, we discern the contributions of intensity and spectral resources, unveiling their impacts on the precision of parameter estimation as the number of probes increases. Remarkably, we demonstrate the potential for quadratic scaling using quantum mode correlations and derive mathematical expressions for the optimal states. In a second investigation, inspired by the state achieving the Heisenberg limit, we introduce a novel method for encoding GKP (Gottesman, Kitaev, and Preskill) states in our paradigmatic system. We analyze the scalability of error detection and correction with the total photon number. We demonstrate that these codes can effectively correct displacements in time-frequency phase space and photon losses. This research highlights the unifying potential of time-frequency variables as a platform for diverse quantum applications.
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