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.

Figures (4)

• Figure 1: Schematic illustration of (a) Frequency-Resolved Optical Gating (FROG) for reconstructing ultrashort laser pulses and (b) Fock-Resolved Interferometry for Number-Gated quantum-state estimation (FRINGE). In FROG, $E(t)$ and $E(t-\tau)$ are mixed in a nonlinear crystal, and the resulting spectrogram enables reconstruction of $E(t)$. In FRINGE, a quantum state interferes with a displaced (or delayed) copy of itself and the photon-number distribution is measured in the far field. The two experiments are mathematically analogous, allowing decades of advances from ultrafast optics to be translated to quantum optics.
• Figure 2: Reconstruction of single-slit Fock coefficients from interferometric number–gated data. (a) Magnitudes $|c_n|$ and (b) phases $\arg c_n$ of a single-slit squeezed state $D(\alpha)S(\zeta)\ket{0}$, recovered from the true distribution $P(N,\phi)$. Blue circles: ground truth $c_n$ coefficients of the squeezed–coherent state. Orange rectangle: FRINGE-reconstructed $c_n$ coefficients; the gauge is fixed by $c_0>0$ and $\arg c_1=0$. (c) Synthetic FRINGE data generated from the ground truth squeezed–coherent single-slit state. (d)$P(N,\phi)$ prediction from the reconstructed single-slit coefficients $\{c_n\}$ obtained by our FRINGE procedure. The overlap across $n$ demonstrates accurate recovery of both amplitudes and relative phases.
• Figure 3: Interferometric number-phase maps $P(N,\phi)$ in the Young double-slit geometry (Figure \ref{['fig:FROGvsFRINGE']}b). (a) Ideal trace $P_{\rm true}(K,\phi)$ synthesized from a random quantum state. (b) Detected trace $P_{\det}(N,\phi)$ after applying a finite quantum efficiency of $\eta=0.5$, which binomially thins the photon number statistics. (c) Exact reconstruction of the lossless trace $P_{\rm true}(K,\phi)$ obtained from panel (b) by applying the closed-form inverse \ref{['eq:coef-inverse-final']}.
• Figure 4: FRINGE reconstruction of a mixed state. (a)–(b) Real parts of the true and reconstructed density matrices $\Re(\rho_{\rm true})$ and $\Re(\rho_{\rm rec})$; (c)–(d) imaginary parts $\Im(\rho_{\rm true})$ and $\Im(\rho_{\rm rec})$. Data were simulated for a mixture of two coherent states in a truncated Fock space ($d=8$). The FRINGE trace $P(N,\phi)$ was generated from Equations \ref{['eq:balanced-fringe']}. The density matrix was then recovered by nonlinear least squares over the parameterization $\rho=T T^\dagger/Tr(TT^\dagger)$, with a small diagonal anchor extracted from the trace.