Direct detection of quantum non-Gaussian light from a dispersively coupled single atom.

Many applications in quantum communication, sensing and computation need provable quantum non-Gaussian light. Recently, such light witnessed by a negative Wigner function has been estimated using homodyne tomography from a single atom dispersively coupled to a high-finesse cavity \cite{Hacker2019}. It opens an investigation of quantum non-Gaussian light for many experiments with atoms and solid-state emitters. However, at their early stage, an atom or emitter in a cavity system with different channels to the environment and additional noise are insufficient to produce the negative Wigner function. Moreover, the homodyne detection is frequently challenging for such experiments. We analyse those issues and prove such cavities can be used to emit quantum non-Gaussian light employing single-photon detection in the Hanbury-Brown and Twiss configuration and quantum non-Gaussianity criteria suitable for this measurement. We investigate in details the cases of considerable cavity leakage when the negativity of Wigner function disappears completely. Advantageously, quantum non-Gaussian light can be still conclusively proven for a large set of the cavity parameters at the cost of overall measurement time, even if noise is present.