Quantum non-Gaussian Depth of Single-Photon States

We introduce and experimentally explore the concept of the non-Gaussian depth of single-photon states with a positive Wigner function. The depth measures the robustness of a single-photon state against optical losses. The directly witnessed quantum non-Gaussianity withstands significant attenuation, exhibiting a depth of 18 dB, while the nonclassicality remains unchanged. Quantum non-Gaussian depth is an experimentally approachable quantity that is much more robust than the negativity of the Wigner function. Furthermore, we use it to reveal significant differences between otherwise strongly nonclassical single-photon sources.

Figure 1

The classification of quantum state sets used in our discussion, where $\mathsf{a}\subset \mathsf{b}\subset \mathsf{c}\subset \mathsf{d}$. $\mathsf{a}$ is the set of states with negative Wigner function. All states in the set $\mathsf{b}$ are guaranteed to be quantum non-Gaussian. Likewise, all states in the set $\mathsf{c}$ are nonclassical. The set $\mathsf{d}$ contains all states in general. Equivalently, all classical states are contained in the complement $\overline{\mathsf{c}}$ and all Gaussian mixtures are in $\overline{\mathsf{b}}$. The borders of $\mathsf{b}$ and $\mathsf{c}$ therefore represent NC and QNG witnesses. The points and non-solid lines represent various realistic quantum states and their respective paths under attenuation. These quantum states can approach vacuum inside different sets. An ideal single-photon state ($P_0|0⟩⟨0|+P_1|1⟩⟨1|$, red dotted line) is an extremal case of an infinitely robust QNG state. Other realistic states may exhibit infinite NC depth or leave nonclassical states altogether (dashed lines). The green dot-dashed line represents realistic single-photon states with positive Wigner function, as generated experimentally.

Figure 2

The autocorrelation measurement scheme. The heralded state in the signal arm is attenuated, then $p_1$ and $p_{2+}$ are measured [6].

Figure 3

Estimated probabilities of heralded single-photon states, each series representing various attenuations of a particular state. Full diamonds denote the cw SPDC source: orange—coincidence window 2 ns; red—low pump, coincidence window 2 ns. Cyan triangles denote the pulsed SPDC source. Square markers denote the quantum dot: purple squares—above-band excitation, cw pump; blue squares—resonant excitation, pulsed pump. The dot-dashed lines represent theoretical prediction of attenuation from the initial point. The dashed red line is the limit of dark counts for the red attenuation data. There are two lower bounds for $p_{2+}$; the solid black line is a bound for classical states and the blue line is for Gaussian mixtures. Quantum states below these lines are NC or QNG, respectively. Error bars are determined by error propagation from event count errors [6]. Horizontal error bars are smaller than plot points.