Photon-number-resolving detector (PNRD) is of great importance both in fundamental research of quantum physics and in practical quantum information processing techniques. It serves as a powerful tool not only for the the photon nature study [1, 2], but also for the cryptographic security improvement against photon-number-splitting attacks in quantum key distribution. In particular, PNRDs are requisite in the linear optical quantum computation which, as originally proposed by Knill, Laflamme, and Milburn, relies critically on photon-number resolving of the incident superposition states [3, 4]. Avalanche photodiodes (APDs) are usually used For single-photon detection, where an incident photon excites a single photo-carrier followed by avalanche multiplication to transform the photo-excited carrier to a detectable macroscopic current pulse. In the Geiger-mode operation, the avalanche multiplication is saturated in order to optimize the quantum efficiency. This nevertheless sets undesired immunity to another photo-excited carrier. As a result, photon-number-resolving measurements are disabled in conventional APD-based single-photon detectors. By using multiple detectors, photon-number resolving could be realized but the detection efficiency was decreased [5, 6]. To date, some individual photon detectors have been demonstrated to exhibit interesting photon-number-resolving capability, such as visible light photon counters[7], superconducting optical detectors[8, 9], and field effect transistors with quantum dots[10]. All those breakthroughs have already stimulated vast promising applications [1, 5] although their performance is still limited by the requisite cryogenic operation. In principle, direct photon-number-resolving detection can be achieved by using individual APD detectors operated in the non-saturated avalanche mode rather than the typical Geiger mode. By suppressing the electronic noise in the device down to a sufficiently low level, the single-photon click could be discriminated from the electronic noise with a non-saturated avalanche multiplication, which also allows the multiplication for the second photo-electron. The intrinsic physics governed the non-saturated mode requires further experimental explorations to unveil the essentials that determine the photon-number resolving fidelity and capability. We will explore in this paper the inherent features of the non-saturated avalanche multiplication of single-photon and multi-photon clicks, as well as the multi-photon induced saturation. We will also address the question regarding whether the detection efficiency could be increased by counting the weak avalanche signals previously buried in the noise by using a robust optical noise-cancellation technique.

In our experiment, we used an InGaAs/InP APD in the non-saturated avalanche mode for detecting the near-infrared photon signals. InGaAs/InP APDs typically work in the gated Geiger mode with repetitive pulses above the breakdown voltages [11, 12] to enable and then subsequently quench the APD in order to reduce the dark count [13, 14]. However, the gating pulses typically produce strong charging and discharging pulses on the capacitance of the APD (spike noise) that make it quite difficult to discriminate the weak avalanche signals. In the conventional gated Geiger mode, the avalanche multiplication of the APD is usually set very high in order to achieve efficient discrimination, causing the APD instantaneously saturated. It has been thought that APDs cannot resolve the photon number in a short time interval. Recently, Kardynal and his co-workers changed this cognition. They realized a PNRD based on an In-GaAs/InP APD[15] through a self-differential technique[16]. As the weak avalanche current could be measured with a 21 dB suppression of the spike noise, the InGaAs/InP APD was operated at the non-saturated mode to detect multiple photons. By measuring the weak avalanche current before the APD became saturated, the incident photon number could be distinguished up to 4 according to the amplitude of the avalanche signal [15]. It requires a high suppression of the spike noise to operate the InGaAs/InP APD at the low boundary of the non-saturated mode. In order to plot the whole characteristics of this novel mode, we invented an optical self-differential technique to realize PNRD based on an InGaAs/InP APD. Compared to the electronic self-differential technique[16], the optical method is much easier to achieve because light pulses are immune to the electromagnetic field of the surrounding circuits and optical components are stable and precisely controllable. A 31 dB suppression of the spike noise was obtained, providing a large dynamical multiplication platform to study the avalanche built-up when the InGaAs/InP APD responded to more than one photon.

 

Fig. 1. Set-up of the optical self-balancing single-photon detector. APD: avalanche photodiode; AMP: RF amplifier (×10); LD: distributed feedback laser diode at 1550 nm; EDFA: erbium-doped fiber amplifier; IF: inline fiber filter at 1550 nm with pass bandwidth of 3 nm; PC: fiber polarization controller; PBS: fiber polarization beam splitter; PD_1,2: pin photodiodes.

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Fig. 2. (a) APD response to the gating pulse. (b) Avalanche signal waveform from the optical self-balancing single-photon detector

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Figure 1 shows the schematic setup of the optical self-differential photon detector. The In-GaAs/InP APD was gated by negative short pulses with a repetition rate of 25 MHz. The pulse duration was 1 ns and the pulse amplitude was 3.2 V. The operation temperature of the APD was set at -35°C to decrease the dark-count noise. The APD was illuminated by a pulsed laser at 1544 nm with 19 ps pulse duration, which was synchronized at 1/25 of the APD gating frequency. The avalanche multiplication was activated by a photo-excited carrier in the InP layer in a strong electric field, resulting in a macroscopic current pulse between the APD electrodes. At the same time, APD also produced the spike noise in response to each gate as shown in Fig. 2(a). The APD response was magnified by a low-noise RF amplifier to trigger a laser diode at 1550 nm. The response bandwidth of the laser diode was 2.5 GHz, fast enough to transfer the electronic signals to light pulses while keeping the same shape. In this way, all the AC electronic signals were transformed to optical signals, preserving the original information from the APD including the spike noise and the avalanche signals. An erbium-doped fiber amplifier (EDFA) was then used to amplify the optical signal with a gain of about 15 dB. The inline filter was inserted to suppress the amplified spontaneous emission from the EDFA. The polarization controller together with the polarization beam splitter (PBS) composed an adjustable beam splitter to divide the optical pulses into two equal components to a balanced optical detector. The fibers connecting the splitter and the detectors have different lengths to introduce a delay of one gate period between the two components. A fiber stretcher was employed to precisely control the delay between the two components with 0.17-ps resolution. Two conventional photodiodes were used to detect the optical signals from each fiber. The response of the photodiodes exactly replayed the detection signal of the APD. At the output of the balancer, the identical spike noise was subtracted and the avalanche signal emerged as the positive peak followed by a negative peak separated by one repetitive cycle as shown in Fig. 2(b). A maximum spike noise suppression of 31 dB was attained by precisely adjusting the beam splitting ratio and the optical delay. With this optical self-differential photon detector, the spike noise could be much suppressed and the weak avalanche current could be measured, enabling the detector to be operated from the linear mode to the saturated mode.

 

Fig. 3. Peak output voltage distributions at different detection efficiencies.

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We performed photon-number-resolving detection with the optical self-differential photon detector by measuring the avalanche voltage output and analyzing the peak output voltage probabilistic distribution under different avalanche multiplication of the InGaAs/InP APD to investigate the built-up of the avalanche pulse. The augment of the avalanche multiplication was achieved by increasing the bias voltage applied on the APD, leading to the increase of the detection efficiency. The waveforms of the output voltage were measured by an 6-GHz oscilloscope with different average incident photon numbers. Figures 3(a~h) plot the histograms of the output signal peak voltage recorded for different detected average photon numbers.

The single-photon pulses, which were attenuated from a coherent light source in the experiment, exhibited the photon number statistics determined by the Poissonian distribution. The probability of the InGaAs/InP APD peak output signal at a certain voltage could be thus calculated from the probabilistic distribution of different photon-number states according to the Poissonian distribution

where $p(\mu ,n)=\genfrac{}{}{0.1ex}{}{{\mu }^n}{n!}{e}^{-\mu }$ denotes the probability containing n photons in a single pulse with the average detected photon number µ. And ρ(n,V) is the avalanche voltage signal distribution excited by the n-photon-number state. By assuming a Poissonian distribution for the incident photons, we simulated the experimental data as shown by the red lines in Fig. 3, which indicated that the experimental data fitted exactly the Poissonian superposition of photon-number states.

In the whole region of the non-saturated avalanche operation, the distributions of the peak voltages showed a series of maxima and shoulders in Gaussian shapes, which were originated from the avalanche currents induced by different number of photons. In this region, the detection efficiency of the InGaAs/InP APD monotonically increased with the avalanche multiplication. When the detection efficiency was as low as 1%, the gap between n-photon and (n+1)-photon (n≥1) peaks was about 40 mV, which was very close to the widths of 1- and 2-photon peaks (30 and 42 mV), and less than the widths of n-photon peaks (n≥3). This gap was insufficient to distinctly resolve the photon number, since there existed considerable overlapping of different photon-number peaks as shown in Figs. 3(a, b). Decreasing the avalanche multiplication to a detection efficiency below 1% could reduce the gap and width of the photon-number peaks. It nevertheless caused much more overlapping between different photon-number peaks and thus impaired to resolve photon numbers. At the detection efficiency of 10% and 20% (dark-count probabilities were 7.2×10^-6 and 2.6×10^-5/per pulse, respectively) as shown in Figs. 3(c~f), the gaps between 1- and 2-photon peaks were 108 and 140 mV while the widths of them were 70 and 90 mV, respectively. The gap between the photon-number peaks increased faster than the corresponding width as the detection efficiency increased from 10% to 20%. Figures 3(c~f) clearly indicate that up to 4 photons could be distinctly resolved. As the multiplication gain increased further, the multi-photon gap increased slowly since the avalanche gain was close to the saturation region. The gaps between n- and (n+1)-photon peaks (n>1) were almost zero when the detection efficiency was 36%. As a result, the peak distributions showed no difference when the incident photon number was more than 1 as shown in Figs. 3(g, h), and the APD could only distinguish 0, 1 and (n>1) photons. It is interesting to note that the avalanche peak voltage was less than 960 mV for all the measurements in various cases, indicating the avalanche saturation. Efficient optical spike-cancellation enabled distinct discrimination of avalanche signals well below saturation, and the optimum photon-number resolving could be achieved with the detection efficiency around 20% as shown in Figs. 3(e, f).

The maxima of the photon-number peaks at different detection efficiencies are plotted in Fig. 4. The error-bars show the corresponding peak distribution width. The width of n-photon peak was √n(n>1) scaled to the 1-photon peak, which was caused by the statistical fluctuation. The width of 1-photon peaks was determined by the avalanche multiplication, and the excess noise derived from the statistical nature of the avalanche multiplication of the InGaAs/InP APD, similar to the result with Si-APDs [17]. When the detection efficiency was 1%, the maxima of n-photon peaks were centered around 63.5, 107.0, 140.0, 177.5, 213.0, and 248.0 mV, corresponding to the avalanche current pulses induced by 1, 2, 3, 4, 5, and 6 photons, respectively. The maxima peak voltage values are obtained by fitting the curves in Fig. 3. The peak amplitudes showed approximatively linear increase with the photon numbers. But the slope of the curve was small, and the gaps were close to the width. As a result, the photon-number peaks were not well-separated. This sets the low boundary for the InGaAs/InP APD to resolve photon numbers. When the detection efficiency was increased to 10% and 20%, the maxima of the n-photon (n=1~6) peaks were centered around (144.1, 252.0, 347.5, 426.5, 488, and 550.0 mV) and (224.0, 364.0, 460.0, 552.5, 629.0, and 690.0 mV), respectively. The width of n-photon peak at 10% was also √n scaled to the 1-photon peak, while the width of n-photon peak at 20% was 0.926√n scaled to the 1-photon peak. We recognized that at the detection efficiency of 20% the avalanche multiplication was evidenced to get saturated at multi-photon clicks. In this region, the photon-number peak increased with the photon number at relatively large slopes, allowing to distinguish the photon numbers before avalanche became saturated at about 6~7 photons. When the detection efficiency increased to 36%, all the peaks reached the maximum amplitude of 960 mV, the saturation effect appeared obviously and the peak output voltage was independent of the incident photon number more than 2. This sets the upper boundary for the InGaAs/InP APD to resolve photon numbers.

 

Fig. 4. Peak output voltage for different photon numbers.

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Figure 4 clearly shows that the InGaAs/InP APD operated from approximatively linear mode to the saturated mode, with the detection efficiency from 1% to 36%. We called this region as sub-saturated mode. The photon-number resolving performance of the InGaAs/InP APD is mainly dependent on two factors. One was the distribution width of the peaks, which was determined by the avalanche multiplication, and the statistical nature of the avalanche multiplication. The other one was the multi-photon avalanche saturation, which limited the maximum resolvable photon number.

In conclusion, the optical self-differential technique provided an efficient technique for the spike cancellation of the InGaAs/InP APD, allowing the sensitive detection on the tiny avalanche current change induced by different numbers of incident photons. The high suppression of the spike noise enabled us to analyze the whole region of the sub-saturated mode. In this region, the peak output voltage of the detector was proportional to the incident photon number. Finally, a photon-number-resolving detection was achieved with the detection efficiency as high as 36%.

This work was funded in part by National Natural Science Fund of China (10525416), National Key Project for Basic Research (2006CB921105), Key Project Sponsored by the National Education Ministry of China (108058), Projects from Shanghai Science and Technology Commission (8530708200 and 08dz1400703).