Universal compressive tomography in the time-frequency domain

We implement a compressive quantum state tomography capable of reconstructing any arbitrary low-rank spectral-temporal optical signal with extremely few measurement settings and without any \emph{ad hoc} assumptions about the initially unknown signal. This is carried out with a quantum pulse gate, a device that flexibly implements projections onto arbitrary user-specified optical modes. We present conclusive experimental results for both temporal pulsed modes and frequency bins, which showcase the versatility of our randomized compressive method and thereby introduce a universal optical reconstruction framework to these platforms.