Photonic quantum information processing using the frequency continuous-variable of single photons

The celebrated Hong--Ou--Mandel effect illustrates the richness of two-photon interferometry. In this work, we demonstrate that this extends to the realm of time-frequency interferometry. Taking advantage of the mathematical analogy which can be drawn between the frequency and quadrature degrees of freedom of light when there is a single photon in each auxiliary mode, we consider the equivalent of the Hong--Ou--Mandel effect in the frequency domain. In this setting, the $n$-Fock state becomes equivalent to a single-photon state with a spectral wave function given by the $n^{th}$ Hermite--Gauss function and destructive interference corresponds to vanishing probability of detecting single photons with an order one Hermite--Gauss spectral profile. This compelling analogy motivates us to propose an interferometric strategy that uses a frequency-engineered two-photon state to achieve enhanced phase precision that scales inversely with the number of modes. Finally, we generalise the Gaussian Boson Sampling model to time-frequency degrees of freedom of single photons.