1. INTRODUCTION

Quantum photonic technology exhibits great potential for broad applications, ranging from quantum communication [1] and quantum computing [2] to quantum metrology [3]. Lying in the heart of these applications is the capability of generating high-quality indistinguishable single photons and/or entangled photon pairs. A bright, single-mode, and high-purity integrated single-photon source is essential for a variety of applications, from quantum cryptography [4], quantum teleportation [5], and random number generation [4] to linear optical quantum computing [6]. The broad application potential has excited significant interest around the world to explore various approaches to producing single-photon quantum states [4,7–12].

Current techniques for single-photon generation are primarily based upon either single photons emitted from single-emitter quantum systems, such as single atoms [13], single ions [14], quantum dots [9], and color centers [15], or single photons heralded from photon pairs produced through spontaneous parametric down-conversion (SPDC) or spontaneous four-wave mixing (SFWM) in a nonlinear medium [4,8,11,12]. In the latter approach, as the prepared photon pairs are strongly correlated in time, the detection of one photon can herald the existence of the other [16]. Despite its probabilistic nature, this approach is particularly suitable for diverse quantum functionalities due to its simplicity, versatility, and the flexibility of nonlinear optical processes [4,12].

For a heralded single-photon source, the Klyshko efficiency [17], i.e., the raw detected heralding efficiency, is critical for many practical quantum applications. For example, a Klyshko efficiency greater than 50% is required for efficient linear optical quantum computing in entangled photon-pair sources [18]. For a loophole-free Bell test of the Clauser–Horne–Shimony–Holt inequality, the Klyshko efficiency needs to exceed $2(\sqrt{2}-1)\approx 82.8\%$ for maximally entangled photons [19,20] and $2/3\approx 66.7\%$ for nonmaximally entangled photons [21,22]. Significant efforts have been devoted recently to optimize the heralding efficiency of single-photon sources [20,22–32], and the Klyshko efficiency has been improved to 82.8% at a wavelength of 810 nm [20,23,29] and 75.6% at a telecom wavelength of 1550 nm [22]. However, these photon sources, based on bulk crystals or optical waveguides, generally exhibit broad bandwidths in photon emission spectra. High-quality optical cavities are able to dramatically enhance the spontaneous emission processes and thus generate photons with high efficiency, high purity, and narrow emission spectra, which are of great promise for broad applications, such as quantum communication, quantum frequency conversion, and efficient light–matter quantum interfacing [33–40].

In this paper, we demonstrate an on-chip telecom-band heralded single-photon source on a silicon chip with high quality and high efficiency. Combining the cavity-enhanced SFWM process in a high-$Q$ silicon microdisk resonator, a direct and flexible device-fiber coupling scheme, and, in particular, highly efficient superconducting single-photon detectors [41], we are able to herald single photons with a Klyshko efficiency of 48%. The detected single photons exhibit a conditional self-correlation below 0.05 for detected photons of 0.01 counts per 5 ns gate at a repetition rate of 3 MHz. All these performances are among the best for the chip-scale heralded single-photon sources reported to date [27,42]. In particular, for the first time, to the best of our knowledge, we experimentally verify a quantitative relation between the Klyshko efficiency and high-order photon correlations [43,44] without applying photon-number resolving detectors, which not only helps us understand the photon statistics of the heralding processes, but also provides a simple and accurate approach to verifying the Klyshko efficiency.

2. RESULTS

A. Characterization of Photon Pairs

Telecom-band photon pair sources have been developed in nonlinear bulk crystals, optical fibers, and nonlinear waveguides based on either SPDC or SFWM [45–52]. A dispersion-engineered high-$Q$ silicon microresonator, either a microring or a microdisk, enables triple cavity resonances with equal frequency spacing, leading to very efficient SFWM [Figs. 1(a)–1(c)]. This type of device has been explored in recent years to generate photon pairs with large photon flux, high pair correlations, and excellent single-mode properties [53–62]. The device employed in this paper is a silicon microdisk with a radius of 4.5 μm and a thickness of 260 nm sitting on a silica pedestal. The device is patterned on a silicon-on-insulator (SOI) wafer by electron beam lithography, with the silicon layer etched by fluorine-based plasma and the silica layer released by hydrofluoric acid [61]. The microdisk resonator exhibits intrinsic optical quality factors around $3.33\times {10}^5$, $5.09\times {10}^5$, and $4.68\times {10}^5$ for three adjacent cavity modes located at 1497.1, 1515.6, and 1534.4 nm, respectively. The free spectral ranges are around 2.437 THz, with a difference of 0.35 GHz. This difference is smaller than the loaded cavity linewidths when critically coupled or over-coupled ($>0.77\mathrm{GHz}$). We pumped the device at the center mode at 1515.6 nm, with 5 ns square pulses at a repetition rate of 3 MHz, to produce signal and idler photon modes symmetrically located around the pump. The produced photon pairs are coupled out of the cavity directly into a tapered single-mode optical fiber and are then delivered to superconducting nanowire single-photon detectors (SNSPDs) to characterize the photon properties [Figs. 1(d) and 2].

 

Fig. 1. Schematics of photon-pair source and heralding single photons. (a) Schematic of a photon source based upon degenerate SFWM in a silicon microresonator. (b) and (c) illustrate frequency and energy diagrams of the process, respectively. (d) Schematic of heralding single photons out of a silicon microresonator through direct tapered-fiber coupling with single-photon detectors.

Download Full Size | PDF

 

Fig. 2. Experimental setup. This setup measures cross correlation, self-correlation, heralding efficiency, and conditional self-correlation at the same time. MZI, Mach–Zehnder interferometer; VOA, Variable optical attenuator; EPG, Electronic pulse generator; Coincidence counter, a multi-channel ps-resolution time-tagger for coincidence counting; SNSPDs, superconducting nanowire single-photon detectors. The inset microscope image shows the silicon microdisk with two forks located symmetrically to the microdisk. In this experiment, the coupling tapered fiber is supported by the input fork (to the right of the device) only and is not touching the device or the output fork.

Download Full Size | PDF

The strong cavity enhancement results in highly efficient photon pair generation with high cross correlations. An example of the photon cross correlation is shown in Fig. 3(b), with a pump power of 11.7 μW dropped into the cavity. Under this moderate pump power, two-photon absorption in silicon is negligible and the device operates in the low gain regime to guarantee the quality of photon pairs. In this regime, the signal-idler cross correlation is given by

 

Fig. 3. (a) Cross-correlation peak values and photon counts per gate at various optical drop powers. The laser is modulated to 5 ns square pulses with a 3 MHz repetition rate. (b) Cross-correlation trace. The lineshape is fitted by the red curve, taking into account the photon lifetimes and the detector timing jitter. The Gaussian response of the detector timing jitters of two detectors is shown in gray. (c) Self-correlation trace. The red curve shows the convolution of the ideal single-mode thermal source and the detector responses.

Download Full Size | PDF

Figure 3(c) shows an example of the experimentally recorded photon self-correlation, which is given by $g_{ss}^{(2)}(\tau )=\frac{C_{ss}(\tau )}{N_s^2}$, where $C_{ss}(\tau )$ represents the coincidence flux density of the signal photon itself measured with a Hanbury-Brown-Twiss-type setup (see Fig. 2). Due to the detector timing jitter, the photon self-correlation is expected to exhibit a peak value of 1.67, instead of 2 [62–64], for an ideal photon source [solid curve in Fig. 3(c)]. The $g_{ss}^{(2)}(\tau )$ shown in Fig. 3(c) exhibits a peak value of 1.60 close to the theoretical value of 1.67. The small mismatch is likely due to the imperfection of the photon sources, such as mixed photon states [64] and noise photons, which can be improved by using a pulse pump with a broader frequency band [65] and using a narrow-band filter for the emitted photons, respectively.

B. Heralding Efficiency

Heralding efficiency is a critical parameter for a heralded single-photon source, which characterizes the conditional probability of heralding single photons out of photon pairs generated from nonlinear optical process. In this experiment, we use the signal photons at 1497.1 nm to herald the idler photons at 1534.4 nm. The SFWM process always produces the signal and idler photon pairs simultaneously inside the cavity [Fig. 1(a)]. However, after the photon pairs are created, the idler photons have to survive a few processes to be successfully heralded, as illustrated by Fig. 1(d). First, the idler photons have to survive the loss inside the optical cavity before being extracted out into the tapered optical fiber. Second, the photons coupled into the tapered optical fiber need to survive the propagation loss inside the tapered region of the optical fiber. The photons then propagate through the optical components and are detected by single-photon detectors. The Klyshko efficiency, at region B in Fig. 1(d), is therefore given by

 

Fig. 4. Improving heralding efficiency by increasing the taper coupling rate. (a) The red and blue curves are theoretical predictions of ${\eta }_K$ for the over-coupled and under-coupled conditions, respectively. Experimental data points are from various fiber-taper coupling conditions denoted by different colors, with corresponding normalized transmission traces shown in (b). (b) The normalized cavity transmission traces in various taper coupling conditions, with the theoretical fitting in black. $T_c$ represents the transmissivity dip. ${\eta }_e$ stands for the photon extraction efficiency following ${\eta }_e=(1\pm \sqrt{T_c})/2$. The loaded optical qualities are around $1.03\times {10}^4$, $2.43\times {10}^4$, $1.90\times {10}^5$, and $4.06\times {10}^5$, from top to bottom.

Download Full Size | PDF

The photon extraction efficiency and the tapered fiber efficiency contribute to the preparation efficiency as ${\eta }_P={\eta }_e{\eta }_{tf}$. ${\eta }_e$ represents the probability of coupling photons generated inside the cavity out into the tapered optical fiber. The photons are coupled out with an external coupling rate of ${\mathrm{\Gamma }}_e$ after surviving the intrinsic photon decay/loss rate of ${\mathrm{\Gamma }}_0$ inside the device, which results in an extraction efficiency of ${\mathrm{\Gamma }}_e/({\mathrm{\Gamma }}_e+{\mathrm{\Gamma }}_0)$. Although ${\mathrm{\Gamma }}_0$ is a constant value characterizing the material and scattering loss inside the device, ${\mathrm{\Gamma }}_e$ can be tuned by the fiber-device distance to change the photon extraction efficiency. This efficiency can be quantified by the transmissivity at the center of the resonance $T_c$, given by ${\eta }_e=(1\pm \sqrt{T_c})/2$, where the plus and minus signs correspond to the over-coupled and under-coupled conditions, respectively (see Supplement 1 for details). As shown in Fig. 4(b), the device coupling can be tuned from under-coupling with a photon extraction efficiency of 13.2% to well over-coupling with a photon extraction efficiency up to 97.8%, such that nearly all photons are extracted out of the device into the delivery fiber. Consequently, the device is flexibly controlled to achieve a high efficiency of heralding single photons. Another factor for preparation efficiency is ${\eta }_{tf}$, the transmissivity of the photons through the tapered region of the coupling fiber. This efficiency is related to tapered-fiber transmittance as ${\eta }_{tf}=\sqrt{T_{tf}}$, because only the loss of the tapered-fiber after the cavity affects heralding efficiency (Supplement 1). The tapered fiber used in this experiment has a high transmittance of $T_{tf}=83.0\%$, which leads to an efficiency of ${\eta }_{tf}=91.1\%$. As a result, the overall efficiency of preparing the single photons inside the single-mode fiber is given by ${\eta }_P=\sqrt{T_{tf}}(1\pm \sqrt{T_c})/2$, represented by the curves in Fig. 4(a).

In the experiment, the Klyshko efficiency is measured as the conditional probability of heralding an idler photon when a signal photon is already registered, i.e., ${\eta }_K=P(i|s)=N_{is}/N_s$, where $N_{is}$ and $N_s$ are the signal-idler coincidence flux and the signal photon flux registered in the experiment. As shown in Fig. 2, the signal channel is split into two channels, $s_1$ and $s_2$, which can herald idler photons individually. Since the extraction efficiency, optical loss, and detection efficiency in these signal channels do not affect the idler photons to be heralded, the heralding efficiency is identical for these separate signal channels,

As shown in Fig. 3, due to the detector timing jitters, the signal-idler coincidence counts spread over a certain time window, outside of which accidental coincidence counts dominate. As a result, a proper integrating time ${\tau }_{\text{int}}$ is required to obtain the correct signal-idler coincidence counts, $N_{is}={\int }_{-{\tau }_{\text{int}}/2}^{+{\tau }_{\text{int}}/2}{C}_{si}(\tau )\mathrm{d}\tau$. Detailed analysis shows that an integrating time of 1 ns, which is about three times the full width of the half-maximum of $g_{si}^{(2)}(\tau )$, is able to collect nearly all the correlated photon pairs, while excluding most of the accidental coincidence counts. A detailed discussion on the integrating time can be found in Supplement 1. Figure 4(a) shows the experimentally recorded Klyshko efficiency and the corresponding preparation efficiency with different device-coupling conditions. It shows clearly that we are able to achieve a Klyshko efficiency up to 48%, which is comparable to the highest values for telecom-band photon sources reported to date [22,31,32]. The experimental observations of heralding efficiencies agree closely with the theoretical prediction based on Eq. (2) [red and blue curves in Fig. 4(a)]. The slight discrepancy between the experimental data and theoretical prediction, especially at the extremely overcoupled conditions, is likely from the imperfections in the coupling ideality [67], the laser locking of shallow resonances, and/or the photon source, which requires further investigation.

There is a tradeoff between the heralding efficiency and the power efficiency in the microresonator photon source [68], where the power efficiency evaluates the amount of photon pair fluxes under certain levels of optical input power. To generate photons at the same flux with a higher heralding efficiency, the input power needs to be increased due to the decreased loaded optical quality at the over-coupled condition. A straightforward solution for the power efficiency issue is to improve the intrinsic optical qualities of the device and then work in the over-coupled condition. Moreover, the tradeoff can be solved by designing a waveguide to be critically coupled by the pump mode for power efficiency and overly coupled by the signal and idler modes for heralding efficiency.

C. Photon Antibunching

The heralded single photons not only show high Klyshko efficiency at the over-coupled condition of the microresonator, but also have a great single-photon property at the same condition. One important characteristic of single photons is the photon antibunching property [8,12], which is measured by conditional self-correlation [16,69]. We performed the experiment on the signal photons by splitting them into two channels and recording the coincidence counts among signal channel 1, signal channel 2, and the idler channel (see Fig. 2). The conditional self-correlation is thus given by Refs. [16,69],

 

Fig. 5. Power dependence traces of preparation efficiency (${\eta }_{smf}$, red) and conditional self-correlation ($g_c^{(2)}$, blue). The empty circles estimate the Klyshko efficiency, taking into account Raman noise and two-photon cases.

Download Full Size | PDF

Figure 5 shows that the recorded heralding efficiency drops by a certain amount with a decrease in the pumping power. This is due to the increased impact of noise photons [70], which mainly come from Raman noise produced in the delivery fiber [61] (with a length of $\sim 5\mathrm{m}$). As Raman noise depends linearly on the optical power while the SFWM depends on it quadratically, the impact of Raman noise increases at low pumping levels [70]. On the other hand, the heralding efficiency decreases slightly at high pumping levels, which is due to the multi-photon effects [71]. The probability of generating multi photons within the integrating time increases with the increased pumping level. Because the photon coherence time is shorter than the integrating time, the multi-photon effects can be mitigated by using faster detectors and thus narrower integrating times. Both Raman noise and multi-photon effects can be obtained by measuring noise photons and photon flux, as shown in the red circles in Fig. 5. The prediction agrees in trend with the experimental data. The mismatch of a few percentages could be due to the imperfection of the source or the uncertainty in the measuring setup and requires further study.

D. Klyshko Efficiency and its Verification

Photon-number-resolving detectors allow unique access to additional high-order information that can be used to characterize nonclassical photon states [72]. In single-photon detectors without photon-number-resolving abilities, however, access to high-order correlation is limited and the Klyshko efficiency is generally measured by Eq. (3). In practice, the accuracy of the Klyshko efficiency depends on the integrating time and accidental coincidence background (see Supplement 1 for more details) and there is no existing method to verify the Klyshko efficiency. It turns out the Klyshko efficiency can also be obtained fairly accurately from the high-order photon correlation measured by the three-detector setup that characterizes the photon antibunching in the previous section (Fig. 2). Notice that, interestingly, although the photons of the two signal channels are correlated [Fig. 3(c)], the heralding processes/events of the idler photons by these two signal photons are independent of each other. Thus, the probability that both signal channels fail to herald idler photons is the multiplication of the probabilities that an individual signal channel fails to herald an idler photon, given by

 

Fig. 6. (a) Experimental data of self-correlation counts, self-correlation background counts, and triple-coincidence counts shown in blue, gray, and red, respectively. (b) The ratio of triple-coincidence counts to self-correlation counts has a constant value of $P(i|s_1{s}_2)=72\%$. The Klyshko efficiency can be extracted from this value by $P(i|s_1{s}_2)=1-{(1-{\eta }_K)}^2$.

Download Full Size | PDF

There have been discussions about the origin of triple-coincidence counts regarding whether they are the leakage of the cross correlation or the self-correlation [43,44]. We believe that the triple-coincidence counts are functions of three time variables ($t_{s_1}$, $t_{s_2}$, and $t_i$) and thus can be related to cross correlation or self-correlation depending on which time variable is integrated. Our results indicate that triple-coincidence counts are closely related to the self-correlation counts when $t_i$ is integrated. Further, the ratio of triple-coincidence counts to self-correlation counts, $P(i|s_1{s}_2)$, can be used to verify the Klyshko efficiency quantitatively. $P(i|s_1{s}_2)$ not only preserves information about the heralding efficiency, including optical loss and detection efficiency, but also contains temporal information that can better characterize correlated photon sources in the presence of background noise.

3. CONCLUSION

We study the properties of heralded single photons produced by spontaneous four-wave mixing in a high-$Q$ silicon microresonator. We investigate the heralding efficiency of the cavity-enhanced photon source theoretically and experimentally. To improve the heralding efficiency, we over-couple the resonant cavity by a tapered optical fiber, where the photon extraction efficiency is above 97%. The Klyshko efficiency is thus improved to 48%, the highest value in cavity-enhanced photon sources reported to date [33–39,53,54,56–63]. These efficiencies are comparable to the highest values for on-chip heralded single-photon photon sources [27,42]. The Klyshko efficiency can be further improved by increasing the fiber-taper transmittance [73], decreasing the loss in optical components, and using detectors with higher detection efficiencies [74]. We also study the power dependence of Klyshko efficiency and the photon antibunching, with Raman noise and two-photon effects analyzed [70,71]. Moreoever, the Klyshko efficiency is found to be related to the ratio of triple-coincidence counts to self-correlation counts for the first time to our knowledge. This relation originates from the independence of the heralding processes and applies universally to heralded single-photon sources based upon nonlinear optical processes. Although the experiment is carried out in fiber-coupled microresonators, the analysis of the Klyshko/preparation efficiency can be generalized to Fabry–Perot cavities [33–39], prism-coupled resonators [63], and waveguide-coupled resonators [53–59]. The developed on-chip cavity-enhanced single-photon source with high heralding efficiencies and strong photon antibunching, as well as CMOS compatibility, has great potential in various quantum photonics applications [1–6].