1. Introduction

Single photons constitute an important resource for several quantum optical applications, such as optical quantum computing, randomness generation, metrology and quantum communication [1]. An ideal source would be deterministic, producing single photons on demand. However, heralded single photons sources, which produce single photons at random times accompanied by a heralding signal that announces the photon’s creation, are sufficient for many applications [1–4]. A common way to implement heralded sources is through the process of spontaneous parametric downconversion (SPDC). As shown in Fig. 1(a), SPDC produces photons in pairs, thus the detection of one photon announces the presence of the other.

 

Fig. 1 Scheme to produce heralded single photons using (a) a normal downconversion source and a heralding detector, T₁ and (b) cascaded downconversion, where the output of the first downconversion acts as the pump for the second, heralded with heralding detectors T₁ and T₂. (c) Scheme for the evaluation of the single photon character of the source. The produced heralded photons are sent on a 50:50 beamsplitter with g⁽²⁾ detectors A and B on each end.

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However, the quality of heralded single photons produced by SPDC is limited, since a detection at the heralding detector is not always associated with a single photon. Indeed, there could be no photon present, due to optical losses and detector dark counts, or more than one photon present because of double-pair emission [5]. Moreover, for applications where pure single photons are required, the spectral correlations resulting from SPDC can also be detrimental [6–10].

Many strategies have been proposed to improve the quality of SPDC single photon sources. These include for example various multiplexing schemes, which aim to parallelize single photon production using various degrees of freedom [2, 5, 11–18], and the production of spectrally pure states by controlling the modal structure of the photon pair emission [7–10].

Here, we investigate a different approach to improve the statistics of heralded single photons from SPDC, which is to use photon precertification [19], implemented with cascaded downconversion (CSPDC) as shown in Fig. 1(b). CSPDC has already been demonstrated using separate nonlinear waveguides [20], from an integrated on-chip setup [21] and with a hybrid system using rubidium vapor and a χ⁽²⁾ nonlinear waveguide [22]. Previous works have shown that triplets produced by CPSDC can be entangled in energy-time [23] and in polarization allowing the production of heralded Bell pairs [24], and demonstrated photon precertification [25]. In the context of single-photon heralding, CSPDC is equivalent to pumping a regular SPDC source with heralded single photons, resulting in photon pairs with fundamentally different statistics. At low photon rates, the main benefit is to provide a second heralding signal which can reduce the negative effects of dark counts at the heralding detectors. This approach can potentially result in single photons with a lower heralded second-order correlation function g⁽²⁾ [26], given by

Here we examine the possible advantages of cascaded downconversion for heralding single photons. In other words, given a fixed SPDC-based heralded single photon source and detectors, is it advantageous to pump it using heralded single photons rather than just using a coherent beam from a laser? We address these questions first using an analytic treatment in section 2, followed by a more exact numerical simulation in section 3 based on the detector statistic model introduced by Bussières et al. [28].

2. Analytical model

We can use an approximate analytic treatment to estimate each term in Eq. (1) and thus determine situations where CSPDC can be advantageous. We consider a heralded single photon produced by regular SPDC, as shown in Fig. 1(a). The singles rate at the heralding detector, S₁, and at the g⁽²⁾ detectors, S_A and S_B, are given by

Since our goal is to optimize the $g_{\mathrm{S}}^{(2)}$ value that such a source can produce, we treat the rate of pair creation, which can easily be tuned by changing the intensity of the pump, as a free parameter. Finding the value of N which minimizes $g_{\mathrm{S}}^{(2)}$ and substituting it in Eq. (6) results in an optimal second-order correlation function of

We now consider the second scenario, in which the same SPDC source is pumped with heralded single photons from another SPDC process, as shown in Fig. 1(b). In this case, Eq. (1) becomes

The rate of detected triples (both heralding detectors and either g⁽²⁾ detector) should be dominated by genuine triplets [23], so that the rates are given by

Substituting the values from Eqs. (9)–(11), and performing the same optimization as above yields

The second term in the previous equation comes from the interaction of two pairs in the same coincidence window, and is negligible unless the performance of the g⁽²⁾ detectors is significantly better than that of the heralding detectors.

Neglecting the second term, Eq. (7) and Eq. (12) can be combined to show that the cascaded approach is advantageous if

Therefore, for given figures of merit H_AB and H₁, an approach using cascaded downconversion will be advantageous as long as the probability of a secondary downconversion is sufficiently high, as shown in Fig. 2.

 

Fig. 2 An example of the minimal g⁽²⁾ for identical detectors. The necessity for high conversion efficiency in the SPDC crystal becomes less pronounced as the detector figure of merit increases. In this case, for superconducting nanowire detectors, which can have figures of merit above 10⁹ [1], secondary downconversion requires an efficiency of 10⁻⁸ or better for CSPDC to be advantageous. This criterion is easily met by efficient SPDC crystals such as lithium niobate waveguides [20].

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3. Numerical modeling of detector behavior

For a more exact modeling of detector behavior, we use an adapted version of the bucket-detector matrix formalism introduced in Bussières et al. [28]. The photons are modeled to follow the Poissonian statistics expected for distinguishable photon triplets [31], reflecting the fact that current experimental realizations of CPSDC have displayed high spectral correlations [23]. The case of indistinguishable single photons [32] can also be modeled by this method, although at low photon pair rates this does not meaningfully impact results.

3.1. Modeling the g⁽²⁾ of the output photons

Considering that the g⁽²⁾ detectors are only needed to measure the g⁽²⁾, the properties of these detectors are not intrinsic to the source. As such, we can consider the output of the source using perfect detectors that do not have any dark counts. In the limit where d_AB → 0, H_AB → ∞and therefore Eq. (13) is trivially met if P > 0. In such a case, only multiple-photon events will contribute to the T_1AB and F terms for SPDC and CSPDC respectively.

Analytically, Eqs. (7) and (11) become

Therefore, the improvement in the g⁽²⁾ for the CSPDC case is given by

The results of the simulation are shown in Fig. 3 for a typical SPDC crystal and detector performance. We notice that in this case, where the performance of the g⁽²⁾ detectors is ignored, CSPDC provides a dramatic improvement over SPDC. While the maximal advantage over SPDC is only achieved at impractically low rates, at 0.01 cps the heralded g⁽²⁾ is already lowered by two orders of magnitude.

 

Fig. 3 With ideal g⁽²⁾ detectors, we find a substantial improvement of the minimal g⁽²⁾. In this example, η₁ = η₂ = 0.7, d₁ = d₂ = 10 s⁻¹, W = 2 ns and P = 10⁻⁶.

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3.2. Modeling the g⁽²⁾ with identical detectors

In practice, the g⁽²⁾ detectors will not be perfect, which will significantly impact the measured heralded g⁽²⁾. Therefore, we now consider a more realistic approach where all the detectors have similar performance. Here H_AB = H₁ = H₂ = H, and Eq. (13) becomes

Finally the improvement becomes

Considering that 0 < η < 1 we have 1.25 < f (η) < 1.91, which gives a maximal improvement factor of 2.91 for the minimum g⁽²⁾ achievable. This is confirmed by the simulation results, shown in Fig. 4. While the improvement in the optimal g⁽²⁾ is not as substantial as the last case considered, it can be reached while maintaining practical pair production rates. Significantly, CSPDC can be advantageous even for equal photon production rates. Furthermore, the minimal value of the heralded g⁽²⁾ is achieved on a broad range of photon production rates. A CSPDC source can therefore remain at the optimal g⁽²⁾ despite orders of magnitude of change in pump power.

 

Fig. 4 A typical example of the g⁽²⁾ as a function of pump rate assuming η = 0.7, P = 10⁻⁶, W = 5 ns and d = 20 s⁻¹ for SPDC and CSPDC. CSPDC maintains the minimal g⁽²⁾ on an extended range of heralded single photon production rates.

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For a given set of detectors, the g⁽²⁾ is minimized by reducing the pump power to control the pair production rate and is optimized when the pair production rate is low enough that effects from dark counts becomes dominant. Since CSPDC decreases the false heralding rates by requiring two heralds, the minimal value of g⁽²⁾ achievable is lower in cases where Eq. (17) is fulfilled. As shown in Fig. 4, for typical parameters, this is achievable with commercially available detectors. For example, a shallow junction silicon single-photon avalanche diode (SPAD) detector has a figure of merit above 10⁸ and superconducting nanowire detectors can achieve figures of merit about 10⁹ [1]. Materials such as periodically poled lithium niobate (PPLN) in optical waveguides can reach conversion efficiencies of 10⁻⁶ [20, 33]. Using these elements, CSPDC would be advantageous regardless of Klyshko efficiency.

As a concrete example, we consider the use of detectors with a figure of merit of 6 × 10⁶ [34] and a PPLN waveguide with conversion efficiency 4 × 10⁻⁶ and coupling efficiency from the waveguide to the detector of 40% [35]. In such a case, a minimal $g_{\mathrm{S},\min }^{(2)}$ of 1.6 × 10⁻⁶ could be achieved with regular SPDC. This would require pumping the crystal with 0.38 nW and would result in 1.9 × 10³ heralded single photons per second. If the setup was then converted to use CSPDC by adding a second crystal of similar efficiency, heralded single photons could be produced at the same rate with an improved g⁽²⁾ of 1.3 × 10⁻⁶. This would require a pump power of 0.74 mW. The heralded g⁽²⁾ could be reduced further to $g_{\mathrm{C},\min }^{(2)}=8,4\times {10}^{-7}$ by pumping the setup with 6.4 µW, although this would reduce the rates to 16 heralded single photons per second.

4. Conclusion

Our results confirm that CSPDC allows for a lower heralded g⁽²⁾ than normal SPDC. This maximal improvement can be achieved using typical detectors. Moreover, compared to a normal SPDC source operated to produce an optimal g⁽²⁾, CSPDC can lead to a lower g⁽²⁾ at equal rates. We therefore expect that most current implementations of heralded single photons using downconversion could produce a lower g⁽²⁾ if they were integrated to a cascade.

In cases where CSPDC is advantageous, the optimal range of the photon production rate is also larger, spanning multiple orders of magnitude, meaning that the value of the g⁽²⁾ is resistant to even large fluctuations in pump beam intensity. This characteristic is exemplified in Fig. 4.

While CSPDC provides a maximal improvement to the g⁽²⁾ at low rates, this could in principle be overcome with multiplexed schemes which increase the production rates without lowering the number purity of heralded photons [1]. Nevertheless, we expect our process to be most useful in cases where g⁽²⁾ purity is paramount but where rates are either not critical or limited by other factors, such as detectors or switching electronics. This could be the case for some implementations of linear optical quantum computing, as the trade-off between rates and number purity is highly dependent on the particulars of the scheme [3], although the theory so far has focused on sources with higher values of g⁽²⁾. Our scheme could also be useful for the characterization of single-photon detectors, which can be done with heralded single photons but is currently hindered partly by multi-photon events [4].

An important consideration for heralded single photons is the spectral purity of photons, which would be limited by the intrinsic spectral correlations of cascaded downconversion experiments performed to date [23]. However, some schemes are able to control these correlations. For example, we expect that that this challenge could be addressed by using recently published schemes such as spectral multiplexing with feed-forward control [17], by using the SPDC in a type-II waveguide where the crystal’s length is controlled [36] or by manipulating tripartite frequency correlation of the produced triplets pre-heralding [32].

The method used in the simulations can be extended further, by considering, for example, different arrangement of detectors using CSPDC. We also believe that the method extends to more complex heralded single photon source schemes, and for schemes using single-photons as pumps for nonlinear processes.