1 Introduction

Photonic quantum states can be characterized by the amplitude and phase for higher photon numbers, and by the photon number distribution for low photon numbers. In the latter case, measurements are always taken using a single-photon detector (SPD). Most SPDs are binary systems that do not take account of photon number, wavelength, phase or position. SPDs with more functions for revealing the states of photons are significant in the applications of quantum physics, such as quantum radar [1], quantum computing [2] and quantum communication. For instance, SPDs capable of resolving photon number may improve the security of practical quantum key distribution (QKD) [3]. SPDs are also expected to produce the information of time, space and intensity of photons in the applications of molecular luminescence imaging and lifetime measurement [4–6]. The SPDs with functions to resolve many quantum states directly can be called as multi-functional SPDs, which is of great concern for both fundamental research and practical applications, especially in infrared wavelength.

Superconductor single photon detector (SSPD) is a new single-photon detector developed in recent decade [7]. It is a promising candidate in many applications for wide-response band, low-noise and high-repetition rate. But traditional SSPDs are also binary detectors. Multi-element SSPDs were first introduced to resolve photon in experiment [8, 9]. Many attempts [10–18] have been made to resolve photon number by parallel or series connected niobium nitride nanowires. But it is a challenging work to give more information of incident photons, such as the space information, wavelength and polarization state simultaneously for SSPDs.

For revealing more information of incident photons, we proposed a multi-functional system based on SSPDs combined with RF combiners without exciting instability as parallel or series connected niobium nitride nanowires chips [15]. The demonstrated system is capable of registering incident photons, resolving photon number and spatial distribution simultaneously by the amplitude of output pulses. At the same time, this system is simple in structure and working stable and inherits the merit of SSPDs.

2 Experimental

The process of fabricating SSPD chips was similar to that described in Refs. [19, 20]. First, we successfully deposited high-quality niobium nitride (NbN) films with a thickness of 5 nm on MgO substrate, and fabricated NbN nanowire based on our deposited NbN films. Electron-beam lithography and reactive-ion etching were used to fabricate the NbN nanowire. For improving the detection efficiency, an optical cavity was fabricated to enhance the photon absorption as Refs. [21, 22]. The measured efficiencies of SSPDs are in the range of 20–30 % with negligible dark counts (less than 10 cps) for 1.5 μm wavelength. The contact pads were also patterned on the chips by means of a sputtering system and optical lithography for circuit connection. All micro-fabrication processes were optimized as Ref. [23].

The fabricated NbN nanowire has a meandering structure covering a 10 μm × 10 μm area, and its width is around 60 nm. The critical temperature is >11 K at the 10 % point of room temperature resistance, and the critical current density is in the range of 5–7 (×106 A/cm2). The patterned nanowire has a square resistance of ~300 Ω/square.

The schematics of proposed system were shown in Fig. 1. The photons were divided into i channels using an optical splitter, as in Fig. 1. Variable delayers were introduced to control the incidence time of photons in a laser pulse so that all elements delivered synchronous output pulses to the RF combiner. Polarization controllers were used to maximize the efficiency of the detectors, because SSPDs are polarization-sensitive detectors [24]. All optical components were connected by single-mode fiber. A picosecond pulsed laser with a wavelength of 1,550 nm, an adjustable repetition rate (1–100 MHz) and a typical pulse width of about 47 ps was used as a photon source in the experiment.

Fig. 1
figure 1

a Schematics of the proposed system and b–d operation process for a typical case (i = 4) of a, where b is optical intensity over time; c gives the output signals of the four elements after detection of the split photons from b; and d is the final output, which theoretically combines the signals given in c

Full size image

The bias circuit of the superconductor nanowires used in this experiment differs in some respects from that of traditional SSPDs. A SourceMeter instrument (Keithley 2400) offering a constant voltage source (typically 2 V) was used to drive multiple superconducting nanowires. Fixed resistors (80–200 kΩ) were connected to the SourceMeter and the superconducting nanowires of SSPD elements through the DC port of a bias tee, supplying the superconducting nanowires with a bias current ranging from 25 to 40 μA. Then, all of the superconducting nanowires were biased at the measured points by choosing appropriate fixed resistors, thereby resulting system efficiency of 3 % at 1,550 nm and negligible dark counts (<1 cps). Many tests verify this bias setting multi-SSPDs system without introducing additional noise or instability, as in the case of traditional SSPDs.

As shown in Fig. 1a, D 1, D 2, … and D i present independent SSPD elements; A 1, A 2, … and A i are low-noise amplifiers; A is a RF power amplifier. The signals amplified by A were recorded using a real-time oscilloscope, and pulse analysis was carried out with a digital counter. First, the output signal of each element was amplified using a low-noise amplifier, which avoid after-pulse or instability for the superconductor nanowire oscillation caused by the reflection of circuit as isolators. Then, the amplified signals were attenuated with a variable RF attenuator. Both low-noise amplifier and RF attenuation were also used to control the amplitude from each SSPD element. For convenience, the combined signal was further amplified using A, an RF power amplifier. The electrical components were connected by coaxial cable.

The process of this operation is presented in (b)–(d) of Fig. 1 for the case i = 4. Using a pulsed laser, the source of photons for detection was a pulsed signal with a very narrow width (~several tens ps). At a low optical intensity, the number of photons in a laser pulse fluctuates and conforms to the Poisson distribution. Figure 1b shows the curve of an optical signal over time in random time interval. The incident photons in Fig. 1b were divided by a splitter and detected by four SSPD elements (D 1, D 2, D 3 and D 4), as in Fig. 1a. The output signals of all of the elements were shown in Fig. 1c and were combined as one channel, as shown in Fig. 1d, which restored the incident shape with photon number information.

3 Results and discussion

3.1 Detection probability

The main approach to prove the detection event of resolving photon number is to analyze the detection probability over incident photon number [7]. According to the theory of quantum optics, the photon number (n) obeys Poisson distribution for ideal coherent light beam. The detection probability for resolving n-photon with an identical efficiency (η) of all detectors was given in Eq. (1) in the system of Fig. 1.

$$P_{n} (\mu ) = \frac{i!}{{\left( {i - n} \right)! \cdot i^{n} }} \cdot \left[ {1 - \exp \left( { - \mu \eta } \right)} \right]^{n}$$
(1)

where n and μ denote the photon number and the mean photon number per pulse, respectively. Numerical calculation of Eq. (1) indicate that at high i and low μ conditions, the detection probability is linear to μ n, where n is the resolved photon number.

To assess n-photon number detection function, the detection probability was analyzed with two SSPD elements (i = 2) in experiment and in theory by Eq. (1).

In theory, there are four possible cases of the photon detection event: (1) D 1 detects and D 2 does not detect the photons; (2) D 1 does not detect and D 2 detects the photons; (3) neither D 1 nor D 2 detects the photons; and (4) both detect the photons. Of these four possibilities, only case 3 did not produce a voltage pulse. Therefore, this system should give three types of voltage pulse.

Figure 2a gives the output signals measured in the two-element system, where ‘D 1 + D 2’ denotes the pulses when both D 1 and D 2 detected photons. The similar shape of the pulses indicates that the repetition rate of the system is close to that of traditional SSPDs. As shown in Fig. 2a, there were three types of pulse, whose amplitudes measured to be 198 ± 5, 250 ± 6 and 441 ± 3 mV. The ranges of the pulse heights were 175–222, 230–270 and 422–461 mV, respectively, the three types of pulse measured over 105 counts. That enables us to distinguish them easily by amplitude across the narrow distribution.

Fig. 2
figure 2

Output voltage pulses of the proposed system when i = 2: a gives the output pulses recorded by a 6-GHz real-time oscilloscope, b is detection probability calculated by the pulse rate measured by a digital counter over optical intensity. The negative pulse in a is caused by the inverse amplifier

Full size image

The pulses with amplitude of 198 mV are produced only D 1 detecting photon(s), and the 250 mV pulses are produced by only D 2 detecting photon(s). The pulses with an amplitude of 441 mV are produced by both D 1 and D 2 detecting photon(s). Therefore, the former two are single-photon detection event, and the last one is two-photon detection.

Figure 2c shows the detection probability over the mean photon number per laser pulse. As shown in Fig. 2c, the slopes are 1.02 ± 0.06 and 1.01 ± 0.03 for the pulses from D 1 to D 2, respectively, which conforms to the linear relationship (n ~ 1), known to be characteristic of the single-photon regime. The fitted slope for pulses D 1 + D 2, which corresponds to the power of μ, is 2.01 ± 0.06, indicating two photons detection (n ~ 2) as in Fig. 2b. The experimental result is quite consistent with the square relation concluded from Eq. (1). Therefore, the resolved photon number with our proposal is consistent with theoretical analysis.

3.2 Resolving photon number

As the evidenced capability of resolving photon number, we used this proposal to resolve photon number up to 4 in experiment.

An i = 4 system, as shown in Fig. 1, was implemented in this experiment, and the resulting output pulses are given in Fig. 3, where we adjust the pulse amplitude for all elements to be around 160 mV recorded by the oscilloscope. Then, our system degenerated, becoming similar to the system described in Ref. [15]. There were four types of voltage pulse, with amplitudes of ~160, 320, 480 and 640 mV, as shown in Fig. 3.

Fig. 3
figure 3

Output pulse for photon number resolution with identical amplitude of output pulse for all elements

Full size image

Therefore, the pulse with amplitude of 640 mV implies four-photon detection, and the other three amplitudes correspond to three photons, two photons and single photon. In practice, the photon number can be calibrated from the measured values as described in theoretical terms in Ref. [25]. In this case, this system is similar to other SPDs [9, 10, 12 and 26] who implemented the function of resolving photon number using pulse amplitude.

Experiments were also implemented to test the response band, repetition rate and signal-to-noise ratio. As the pulse shape in Fig. 3, all detection events have the similar recovery time, producing similar repetition rate for this system. Besides, the noise level was not increased when the RF signal was combined. These features are important for practical application with multi-functional SPDs.

3.3 Resolving spatial distribution

Furthermore, our proposal makes it possible to give the spatial distribution of an incident optical signal. The photons from different space were coupled to different fibers of the proposed system. For the practical application of resolving spatial distribution of this propose, the optical splitter in Fig. 1 was not needed.

As in the previous analysis, the amplitude of the output pulse can be taken as the sum of the combination of element amplitudes. In this session, the amplitude of each elements was adjusted to take a geometric series distribution, such as (1, 2, 4, …, 2i), and the sum of their combination was (0, 1, 2, 3, …, 2i+1 − 1), which corresponds to the amplitudes in all detection cases. As a result, the amplitude was unique for all cases of photon spatial distribution. It is possible to prove the uniqueness of the sum corresponding to a combination be mean of the inductive method in mathematics. Thus, we distinguish all photon distribution by the final amplitude in the proposed system.

In this session, we introduced three elements (i = 3) to the system, as in Fig. 1. To observe the phenomenon of resolving photon spatial distribution, variable attenuators (0–20 dB) were introduced after the splitter shown in Fig. 1 to imitate an optical signal with a practical space distribution of photons over the elements in this session.

As illustrated in Fig. 4, there were seven types of pulse with different amplitudes. The amplitudes of the three individual elements were measured at ~60, 120 and 240 mV, and the other pulses were the combinations of all elements. Obviously, the amplitude difference of adjacent pulses was larger than the noise level, which enabled us in practice to differentiate pulse types by amplitude.

Fig. 4
figure 4

a Output pulses with geometric series amplitudes and the possible states of detectors; b and c number distribution and spatial distribution of photons concluded from a. The coordinate ‘1’ of x-axis in c denotes the position at D 1, and so on

Full size image

In Fig. 4, the red rectangles indicate that the element detected the photons, and the green rectangles that it did not. The figure shows that pulse amplitude carries information about the photon number and spatial distribution. For example, pulse D 1 + D 2 in Fig. 4a corresponds to the detection of two photons. At the same time, the amplitude also indicates the spatial distribution of the photons detected. For example, the pulse with amplitude ~360 mV in Fig. 4a implies that detectors D 2 and D3 registered photons, whereas D 1 detected none in the time interval.

Furthermore, the accurate distribution over space and the photon number of the incident photons can be estimated from Fig. 4a. The pulse number with different amplitudes shown in this figure was measured with a digital counter. The photon number distribution of the incident optical pulse is then shown in Fig. 4b. It should be noted that the system is unable to resolve the photon number when that number is larger than the element number. However, this problem can be resolved by adding more detection elements. At the same time, the spatial distribution can be ascertained from Fig. 4c. The results in Fig. 4c indicate the probability of photons hitting positions D 1, D 2 and D 3. The proposed system features the multi-functional ability to detect photons, register incident photons and resolve the photon number and spatial distribution.

3.4 Extension, time jitter and applications

3.4.1 Extension to more elements

Many applications demand to resolve the incident optical signal with more photons or with more pixels.

However, the system should have a good signal-to-noise ratio and a large dynamic range for practical reasons. In the system we have presented, the signal-to-noise ratio is ~1.5 for the purpose of discrimination. The dynamic range of amplitude is 60–300 mV. Therefore, the number of elements is limited to three (60, 120 and 240 mV) by the low-noise amplifier (gain ≈ 12 dB). It would be feasible to introduce commercially available low-noise amplifiers with a gain of 40 dB instead of 12 dB to this experiment at the same noise level. This would raise the maximum amplitude to several V, increasing the number of possible elements to i = 8 for enhanced dynamic range. The system with more than eight elements can also be achieved by multi-series amplifiers and more RF combiners with tree structure in theory.

3.4.2 Time jitter

Time jitter, the undesired deviation of the output signal, is a critical parameter of SPD. In fact, the jitter was always obtained by the parameter FWHM from Gaussian fitting with the sample space of the deviation between output signal time and standard time.

In theory, the jitter of our proposal was obtained from the new sample space as Eq. (2) composed by the samples from many sets. While the average arriving time (μ l ) was set to be identical by delayers as Fig. 1, the new sample space should be less than the maximum one.

$$f = \mathop \sum \limits_{l = 1}^{l = i} \frac{1}{{\sqrt {2\pi } \sigma_{l} }}\exp \left[ { - \frac{{\left( {t - \mu_{l} } \right)^{2} }}{{2\sigma_{l}^{2} }}} \right]$$
(2)

while the σ l in Eq. (2) was identical for all sample space, the new space composed by all samples from all sets should have same Poison distribution. Therefore, the combined signal should have same time jitter in theory.

Experimental results measured by the system in session 3.1 were ~52 ps for the combined signal, 57 and ~55 ps for both detectors. These jitters include the chips jitter, optical component jitter and electronics jitter as our previous results [27]. Considering experiment error, the measured jitters are reasonable according to the theory analysis above. It is apparent that the time jitter of this proposed system was not degenerate from the single SSPD as the theoretical analysis.

3.4.3 Practical applications

Currently, this proposed system only produced a probability, not practical quantum states like other SPDs with multi-pixel technology over time or space. Therefore, the measured value, such as photons number, should be calibrated as Ref. [25].

Ideally, the direct measurement of quantum states with our proposal requires high efficiency as Eq. (1). SSPD with high efficiency closing to 100 % is prospective for the high intrinsic quantum efficiency [28, 29]. The main limitation of system efficiency is technology problem. We believe that it is promising to produce unite-efficiency with SSPD for energy gap (several meV) of Cooper pairs much less than photons energy. It will be practical to be applied in application modern quantum optics and quantum communication with this proposal. Currently, it is also worthy of many applications.

4 Conclusions

In summary, these experiments have demonstrated the effectiveness of multi-functional SPDs based on SSPDs assisted by a RF power combiner. All open mode of SSPD enable us to detection the incident signal whose arriving time is unknown, and not need synchronous signal or additional circuit to control the gate mode of detectors as APD system. The photon number up to four was resolved experimentally, and our system was capable of giving the spatial distribution over three pixels by the amplitudes of output pulses. The practical applications of readout circuits and bias setting have also been discussed in detail. The experimental results also show that this proposal is feasible, stable and cost-effective, and thus has potential applications in the fields of QKD, quantum computing, ultra-weak light imaging, etc.