Single-photon sources (SPS) are the fundamental elements usually used as information carriers in many quantum information processes. To date, many types of alternative single photon sources have been developed, e.g., molecule or atom emission,^[1] nitrogen vacancies in diamond,^[2] quantum dots,^[3] and the parametric down-conversion (PDC) sources.^[4] Because of the feasibility and reliability, the PDC sources have been widely adopted in practical implementation of quantum communication, such as long-distance quantum key distribution (QKD),^[5–7] quantum repeater,^[8] and entanglement swapping.^[9]

In the last two decades, PDC sources have been studied extensively, with two typical applications as the heralded single-photon source (HSPS) and the entanglement photon pair source. PDC sources are based on the spontaneous parametric down-conversion (SPDC) process, which possess a probabilistic nature for both the continue-wave (CW) pump^[10,11] and pulsed pump^[12,13] configurations. As a result, the photons emitted from the PDC sources follow a certain statistical distribution such as Poisson or thermal distribution, which can be achieved by properly adjusting the experimental set-up.^[14] Nevertheless, the pulsed pumping regime has more advantages over the CW one, for providing the time information of the emitted photons, which can be very useful in quantum communications.

It is necessary to know the properties of the PDC source in most applications. For instance, when a PDC source is applied to a QKD system, it can generate the decoy-state either actively or passively,^[15–18] dramatically increasing the security of realistic QKD. In those protocols, it is important to know the brightness (i.e., the mean photon number per time unit) and the photon channel transmissions to estimate the key generation rate. Unfortunately, the realistic PDC sources suffer from the lossy channels, inefficient single mode fiber coupling, and non photon-number-resolving detectors. All these factors lead to the assessment of the source brightness being a difficult task. Consequently, efforts have been made to characterize the PDC sources.^[19,20] The method in Ref. [19] is related to the CW pump regime, while the method in Ref. [20] is related to the pulsed one. Both methods are less practical for their complexity. Hence, a more simple and straightforward method is needed to assess the properties of the PDC sources in practical quantum information processing.

Here we propose a model of the pulsed pump PDC source, and derive the analytic formulas for the single and coincidence counting rates, which are the functions of the brightness and the transmittance. The coincidence counting rates include the total, the real, the accidental, and the side peak ones. Then by substituting the measured values into the statistical model of the single and the coincidences counts, one can simply and quickly calculate all the parameters that are needed, such as the brightness and the total efficiency of each photon channel. Moreover, with the measured single and coincidence counts in a Hanbury–Brown–Twiss (HBT) experiment,^[21] one can obtain the second-order autocorrelation function . We experimentally verify our model by using a PPLN based non-degenerate PDC source; with the measured single counting rate and coincidence counting rate, we estimate the brightness and the photon channel transmissions; and then, we predict the real and accidental coincidence counting rates. We finally measure the heralded second-order autocorrelation function, which indicates that our source is very close to the single-photon source.

According to the probabilistic nature of the PDC source, we develop a statistical model to access the properties of the sources. A simple model of the PDC source is shown in Fig. 1. As shown in Fig. 1, one pump photon is focused on a nonlinear crystal (NC). After the spontaneous parametric down-conversion process, a photon pair which includes a signal photon and an idler photon maybe generated simultaneously. Then the photon pair is separated by a dichroic mirror (DM) in the case of collinear generation with non-degenerate wavelengths, and individually collected into fibers by a coupler. The idler photon is directly detected by the single photon detector D₁. While the signal photon is sent into a 50/50 fiber beam splitter (FBS) and detected by two single photon detectors D₂ and D₃, respectively. It should be noted that the FBS is not necessary in determining the source brightness and photon channel transmission, which is only used to perform the HBT experiment and measure the second-order autocorrelation function .

Fig. 1.

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	Fig. 1. (color online) The model of SPDC source.

2.1. Single and coincidences counts

Since all the counts are detected within a time window either in a CW or a pulsed laser scheme, in the following for a convenient description, we suppose a pulsed laser scheme with a repetition rate of f. Assume that the mean photon number of photon pairs emitted per pulse from the crystal is μ₀. Denote the dark count rate of each detector as d_i (i = 1,2,3), and the transmission efficiency of each mode as η_i. Here η_i includes all the optical efficiency, the fiber coupling efficiency, and the efficiency of the detectors. According to our statistical model, the singles and any kind of coincidence can be easily deduced as follows.

Single counting rate For an n-photon pair state generated from the PDC source, the yield of the detector D_i for each photon pair state can be written as

Then the counting rate of detector D_i can be expressed as

where is the photon number distribution.

Total coincidence counting rate In our model, one should measure the total coincidence between detectors D₁ and D_j (j = 2,3), which includes the real coincidence and the accidental coincidence. As is known, the coincidence is obtained when the two detectors click simultaneously within the same time window. The yield of the detector for each pulse is already given by Eq. (1), then we can easily obtain the total coincidence counting rate

Real coincidence counting rates The real coincidence comes from the case that the idler and one path of the signal photon from the same n-photon pair are simultaneously detected, its yield is given by

where “rc” indicates the real coincidence, and “q” shows that the photon pair participates in the coincidence. Furthermore, the counting rate of real coincidence can be obtained in the same way

Accidental coincidence counting rate As for the accidental coincidence, it can be divided into four classes: (i) the click events of detectors D₁ and D_j are both from the dark counts; (ii) D₁ click is caused by the dark counts, and D_j is caused by the PDC photons; (iii) D_j click is caused by the dark counts, and D₁ is caused by the PDC photons; (iv) the detected photons of D₁ and D_j come from different PDC photon pairs. Therefore, the accidental coincidence can be given as

Side peak coincidence counting rates If the relative delay between two paths is adjusted to around (), the coincidence can still be measured, which is called the side peak coincidence. This coincidence comes from two independent pulses, therefore the corresponding expression is

From the above deduction, we can arrive at three conclusions. First, the singles, the total and side peak coincidence counting rates can be measured directly from an experiment, we can use the measured counting values to assess the brightness and efficiency. Second, it has been proven through Mathematica that the sum of the real and accidental coincidence counting rates is exactly equal to the total coincidence, i.e., , which in turn validates our model. Third, the accidental and side peak coincidences are not the same thing.

2.2. Brightness and photon channel transmission

Note that our statistical model can apply to any certain photon number distribution of the PDC source. If we have a prior knowledge of the photon number distribution, then we can determine the brightness (μ₀) and photon channel transmission (η_i) by just measuring the single and total coincidence counting rates. Now we take the typical Poisson and thermal distributions for examples.

The Poisson distribution can be expressed as , therefore the expressions (2) and (3) can be further simplified as

The thermal distribution is , then expressions (2) and (3) can be given by

Finally, after substituting the measured dark count rate d_i, and with the measured single and total coincidence counting rates, μ₀ and η_i can be numerically solved.

Particularly if the condition is satisfied, and the dark count rate can be neglected, for both distributions we can obtain the further simplified results

The μ₀ and η_i can also be simply obtained. Equation (10) indicates that the singles and total coincidence are proportional to μ₀, while the side peak coincidence is in proportion to μ²₀.

2.3. The second-order autocorrelation function

The second-order autocorrelation function can be found via an HBT experiment by measuring the three-path coincidence using three detectors (D₁, D₂, and D₃), as shown in Fig. 1. For any photon number distribution, the second-order autocorrelation function^[22] has always been expressed as

and the heralded second-order autocorrelation function^[23] can be obtained by

where C₂₃ is the coincidence counting rate of D₂ and D₃, and C₁₂₃ is the coincidence counting rate of D₁, D₂, and D₃. Here, we also take the Poisson and thermal distributions for examples to discuss the distribution of the second-order autocorrelation function.

For the Poisson distribution, when the delay of the two path (τ) is 0, the coincidence between D₂ and D₃ is given by

where t is the transmittance of the FBS, , and . Then is estimated by

This result indicates that is independent of the brightness.

For the thermal distribution, the coincidence between D₂ and D₃ when τ = 0 can be obtained in a similar way

When neglecting the dark count rate, we can obtain an approximated result

Therefore the second-order autocorrelation function for the thermal distribution is

With , we can obtain the following relationship:

From the above derivation, we can easily see that implies a perfect single photon source. However, due to the disturbance from external experiment conditions, we usually regard it as achieving an HSPS when . Alternatively, describes a classical source (for Poisson and for thermal ).

A schematic of our experimental setup is displayed in Fig. 2. A 76 MHz repetition rate, picosecond, mode-locked Ti:sapphire laser at 898 nm is frequency doubled into 449 nm light through a β- (BBO) crystal. It is then used to pump a periodically poled LiNbO₃ (PPLN) crystal, generating nondegenerated parametric down-conversion photon pairs centered at 633 nm and 1545 nm, respectively. The generated photon pairs are separated by a dichroic mirror (DM). Photons at 1545 nm are further split into two paths by a fiber beam splitter (FBS). Then each path is individually coupled into a fiber and filtered by a tunable bandpass filter with full width half maximum of 3 nm. The photons in the two paths are finally sent into home-made super-conducting nanowire single-photon detectors (SNSPD) operating at 2.15 K with a 55% detection efficiency and a dark count rate of 16 Hz. Moreover, two polarization controllers (PC) are placed in front of the SNSPDs to maximize the counting rates. Photons at 633 nm are coupled into a fiber and sent into a silicon avalanche photo diode (SAPD). We use a time-to-digital converter (TDC) to collect the signals from all detectors. The start signal of TDC comes from the clock of the Ti:sapphire laser, which is frequency divided into 380 kHz. The time window is set to 3 ns.

Fig. 2.

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	Fig. 2. (color online) The experimental setup of SPDC source.

First, we performed the HBT experiment, measuring the single counting rates of D₂ and D₃ and their coincidence counting rate C₂₃. Here, we only discuss the Poisson distribution, is calculated to be using Eq. (11). This result demonstrates that the photon number distribution follows the Poisson distribution. The single counts and the total coincidences were measured with pump powers set to , , , , and , respectively. The measured R₁, and are displayed in Fig. 3(a). and are shown in Fig. 3(b). By using our model for the Poisson distribution, we can calculate the mean photon number μ₀ and photon channel transmissions , which are demonstrated in Figs. 4(a) and 4(b), respectively.

Fig. 3.

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	Fig. 3. (a) The measured single count rate of D1, total coincidences between D1, D2 and D1, D3 for different pump power. (b) Side peak coincidences between D1, D2 and D1, D3 for different pump power.

Fig. 4.

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	Fig. 4. (color online) (a) The mean photon number per pulse generated by the crystal with increasing pump power. (b) The photon channel transmission with increasing pump power.

From Fig. 3(a), we can see that the experimental data of R₁, , and can be well linearly fitted, which is in agreement with Eq. (10). Similarly, figure 3(b) shows that the experimental data of and can perfectly match the quadric curve described by Eq. (10). It should be emphasized that all the fittings rely on the condition ; in our experiment, they are estimated to be , , , together with the value of μ₀, the condition can be surely satisfied.

Figure 4(a) demonstrates a proportionality between the pump power and the generated mean photon number μ₀. We can figure out from Fig. 4(b) that the calculated photon channel transmissions do not change with the pump power, which manifests a reliability of our model.

With the estimated values of μ₀ and , substituting the Poisson distribution into Eqs. (5) and (9), we can predict the real and accidental coincidences. The expressions are

Fig. 5.

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	Fig. 5. (color online) (a) The predicted real coincidences between D1, D2 and D1, D3. (b) The predicted accidental coincidences between D1, D2 and D1, D3.

By measuring the single counting rates R₁ and the coincidence counting rates C₁₂, C₁₃, C₁₂₃ together with Eq. (12), we can obtain the heralded second-order autocorrelation function at different pump powers. Figure 6 shows as a function of the pump power. When the mean photon number is kept less than 0.1 as in our experiment, the probability of single photon is much larger than that of multi-photon. What is more, after being heralded, the vacuum pulse can be largely eliminated, and so the single photon becomes the dominant part in the source. Therefore, the second-order autocorrelation function measured under the condition that the idler detector clicks can be very close to that of a single-photon source.

Fig. 6.

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	Fig. 6. The heralded second-order autocorrelation function measured at different pump powers.

We proposed a simple statistical model to characterize a PDC source, which provides the means to estimate the brightness, photon number transmissions, and the second-order autocorrelation function by measuring the single counts and total coincidence counts. Moreover, with the estimated photon channel transmission and mean photon number per pulse, one can predict the real and accidental coincidence counts. We constructed a PDC source and characterized its properties by using our method, the experimental results confirmed our statistical model. This analysis method is simple and less time consuming in accessing a PDC source, which we believe will have a wide application in the field of quantum communication.