FRINGE: a protocol for self-referenced quantum state estimation via photon-number-resolved interferometry

Abstract: We introduce a self-referenced method for quantum-state tomography of light based on photon-number-resolved double-slit interferometry. Two identical copies of the unknown quantum field illuminate laterally displaced slits, guaranteeing perfect spatiotemporal mode matching without a separate local oscillator. In the far-field, detection at transverse position $x$ is associated with a relative slit phase $φ(x)$, and an $N$-photon event projects the detected quantum field onto a state $|N;φ(x)\rangle$. The resulting distribution $P(N,φ)$ is the quantum analogue of a Frequency Resolved Optical Gating (FROG) trace: whereas FROG reconstructs the classical complex spectral field $E(ω)$ from a spectrally resolved second harmonic of a pulse with its delayed self, our measurement reconstructs the Fock-space wavefunction or density matrix from binomially weighted self-interference. The scheme requires no known or mode-matched reference and is compatible with commercially available photon-number-resolving cameras. Beyond conceptual simplicity and automatic mode matching, the FROG analogy permits direct transfer of mature ultrafast-optics methodologies (e.g., mixed-state, ptychographic, and vectorial extensions) into quantum optics, offering a versatile route to tomography of quantum photon states.