Direct characterization of a nonlinear photonic circuit's wave function with laser light

Integrated photonics is a leading platform for quantum technologies including nonclassical state generation \cite{Vergyris:2016-35975:SRP, Solntsev:2014-31007:PRX, Silverstone:2014-104:NPHOT, Solntsev:2016:RPH}, demonstration of quantum computational complexity \cite{Lamitral_NJP2016} and secure quantum communications \cite{Zhang:2014-130501:PRL}. As photonic circuits grow in complexity, full quantum tomography becomes impractical, and therefore an efficient method for their characterization \cite{Lobino:2008-563:SCI, Rahimi-Keshari:2011-13006:NJP} is essential. Here we propose and demonstrate a fast, reliable method for reconstructing the two-photon state produced by an arbitrary quadratically nonlinear optical circuit. By establishing a rigorous correspondence between the generated quantum state and classical sum-frequency generation measurements from laser light, we overcome the limitations of previous approaches for lossy multimode devices \cite{Liscidini:2013-193602:PRL, Helt:2015-1460:OL}. We applied this protocol to a multi-channel nonlinear waveguide network, and measured a 99.28$\pm$0.31\% fidelity between classical and quantum characterization. This technique enables fast and precise evaluation of nonlinear quantum photonic networks, a crucial step towards complex, large-scale, device production.