# Quantum Calibration of Photon-Number-Resolving Detectors Based on Multi-Pixel Photon Counters
## 3 Results
### Photon Number Distribution Histogram  
Figure 3a,b demonstrate the statistics of MPPC response at the wavelength of 791 nm when the incident average photon number per pulse was about 5.40 and 54.01, respectively. For each averaged incident photon number, we collected 50,000 pulses and recorded the peak amplitude of each pulse in order to construct the histogram. The bin width in statistics is 0.002 V. The peak amplitude histogram of coherent states with different photon number verifies high sensitivity of MPPC and its photon-number-resolving capability. Bounded by the minimum of the voltage distribution, counts of the same peak are added up. The photon number distribution is in accordance with Poisson distribution shown in the insets when the incident photons are relatively low. The red curve is fitted to the experimental data in accordance with a Poisson distribution.  
The probability of detecting \(i\) -photons per pulse is governed by the Poisson distribution:  
\[y=\frac{e^{i}j}{\mathrm{i}\Gamma}, \tag{9}\]  
where \(j\) is detected average photon number per pulse. By fitting the histograms in the insets of Figure 3a,b, we obtained \(j=0.33\) photons per pulse when the incident photon flux was 5.40 photons per pulse, and \(j=3.59\) photons per pulse when the incident photon flux was 54.01 photons per pulse, respectively. The \(j\) divided by the averaged incident photons is the detection efficiency of the detector. The detection efficiency could be roughly deduced to about 6 % at the wavelength of 791 nm. The detection efficiency of the detector at 523 nm was fitted to be about 25 % in the same way.  
Figure 2: Schematic diagram of the experimental setup for MPPC calibration experiment. M1,2: silver mirrors; VA: variable attenuator; FM: flip mirror; ATTN: attenuator with fixed attenuation; BP: band-pass filter; L: focusing lenses (\(f=50\) mm); MPPC: multi-pixel photon counter; OSC: oscilloscope. The pulsed light source is the mode-locked laser at either 791 nm or 523 nm.  
At different incident average photon number, the coefficient of determination is different. The adjusted coefficient of determination \(\overline{R}^{2}\) is used to evaluate the fitting quality, it is computed as \(\overline{R}^{2}=1-\frac{RSS/df_{first}}{TSS/df_{total}}\), in which \(RSS\) and \(TSS\) are the residual sum of squares and total sum of squares, respectively. The closer the value of \(\overline{R}^{2}\) is to 1, the higher the coefficient of determination is. When the incident photons are relatively low, \(\overline{R}^{2}\) is very close to 1, but at large average incident photon number, the coefficient of determination will decrease. We obtain \(\overline{R}^{2}\) is about 0.999 and 0.989 for the incident average photon number per pulse of 5.40 and 54.01, respectively. Therefore, the Poisson fitting could not describe the performance of MPPC completely.  
We repeated the experiment at 523 nm by switching the light source to the sum-frequency mixing output beam. As shown in Figure 3c,d, the detector voltage amplitude histogram of MPPC response at the wavelength of 523 nm when the incident average photon number per pulse is about 1.28 and 12.84, respectively. Similarly, we could obtain \(\overline{R}^{2}\) was about 0.999 and 0.993, respectively. The \(\overline{R}^{2}\) value decreased at larger average photon number again.  
As the average number of incident photon per pulse increases, the possibility that two photons impinging on the same MPPC pixel will increase. As a result, photon-number-resolving capability of MPPC will decrease. In addition, the cross talk will also influence the photon number distribution detected by MPPC [40]. The detected photon number distribution doesn't obey the Poisson distribution anymore. Therefore, the detection efficiency could not represent very well the performance of MPPC any more at larger detected average photon number per pulse.