Accurate Detection of Arbitrary Photon Statistics

We report a measurement workflow free of systematic errors consisting of a reconfigurable photon-number-resolving detector, custom electronic circuitry, and faithful data-processing algorithm. We achieve an unprecedented accurate measurement of various photon-number distributions going beyond the number of detection channels with an average fidelity of 0.998, where the error is primarily caused by the sources themselves. Mean numbers of photons cover values up to 20 and faithful autocorrelation measurements range from $g^{(2)}=6\times {10}^{-3}$ to 2. We successfully detect chaotic, classical, nonclassical, non-Gaussian, and negative-Wigner-function light. Our results open new paths for optical technologies by providing full access to the photon-number information without the necessity of detector tomography.

Figure 1

The autocorrelation $g^{(2)}$ evaluated from the measured photon statistics (solid marker) and the corresponding ideal statistics (empty marker) of various optical signals with mean photon number $⟨n⟩$ [14]. Shown are the coherent states with $g^{(2)}=1$ (blue triangle up), thermal states (also termed chaotic light) with $g^{(2)}=2$ (red circle), $M_{\mathrm{th}}$-mode thermal states with $M_{\mathrm{th}}=1$, 2, 4, 10 (violet triangle down), and $m$-photon-subtracted thermal states for $m=0$, 1, 2, 3 (black circle). The cases of $M_{\mathrm{th}}=1$ and $m=0$ coincide with the thermal state. Furthermore, the emission from a cluster of $N_p$ single-photon emitters is shown for $N_p=1$,…,9 with $g^{(2)}=1-1/N_p$.

Figure 2

(a) Experimental setup of the PNRD based on a discrete optical network with full reconfigurability and continuous tunability of splitting ratios, pulse-height spectrum of the analog output of the detector, and scheme of photon-statistics retrieval. Measured (blue bars) and the corresponding theoretical photon statistics (green dots) for (b) thermal state with $⟨n⟩=4.93(4)$, (c) two-photon-subtracted thermal state, (d) single-photon, and (e) heralded nine-photon state that emulates emission from a cluster of single-photon emitters. (Inset) Wigner function evaluated from the measured statistics. Note that the data agree with theory even beyond the number of channels of the PNRD (ten).

Figure 3

A numerical analysis of EME total variation distance $\mathrm{\Delta }$. With more measurement runs $R$, the statistical error in the data is lower and the EME result approaches the true photon statistics despite the limited number of channels $M=10$. Here shown are coherent state with $⟨n⟩=10$ (blue triangle up), thermal state with $⟨n⟩=5$ (red circle), $N_p$-photon cluster with $N_p=1$ (green square), and $N_p=9$ (green rhombus) for a single value of $\lambda =10^{-3}$. The gray area illustrates the observed scaling ($0.25/\sqrt{R}–14/\sqrt{R}$.)