1. Introduction

In recent years, biphoton generation has become one of the essential ways to generate photonic qubits in long-distance quantum communication [1–5]. The biphoton pair is a pair of time-correlated single photons. One can detect the first single photon of a biphoton pair as a trigger to start a quantum operation or process. The first photon acts like a messenger, and it is called the heralding photon. One can then employ the second single photon in the same pair in the operation. The second photon is named the heralded single photon, and it is a practical and convenient qubit in quantum information processing. There are two commonly-used processes for the biphoton generation: spontaneous parametric down-conversion (SPDC) [6–21] and spontaneous four-wave mixing (SFWM) [22–61]. The SPDC process is often employed with nonlinear crystals. The linewidth of single-mode (or multi-mode) SPDC biphotons can be as narrow as a few MHz (or a few hundred kHz in each mode) with the assistance of an optical cavity [12–14,21]. The SFWM process is often applied with laser-cooled atoms or room-temperature or heated atomic vapors. The SFWM biphotons generated from laser-cooled atoms have a linewidth of sub-MHz down to 50 kHz due to low decoherence rates in the system [30,39,40]. However, their generation rate is less than $10^{3}$ pairs/s on average due to the duty cycle in the generation process. The generation rate of the SFWM biphotons using a heated atomic vapor can be $3.7{\times }10^{5}$ pairs/s, while they can still maintain their linewidth below MHz [27–29]. Compared with those using laser-cooled atoms, the production procedure of the biphoton sources using room-temperature or heat atomic vapors is more convenient.

The measurement of the conditional auto-correlation function (CACF) is a way to quantify the single-photon purity of the heralded photons of biphoton pairs [7–17,32–61]. As depicted in Fig. 1(a), the CACF involves the measurement of three-fold coincidence counts. Once the single-photon counting module (SPCM) $D_s$ receives a trigger from the heralding photon, one performs the Hanbury-Brown-Twiss (HBT) measurement of the zero delay time on the heralded photons with a 50/50 beam splitter and two SPCMs $D_{h1}$ and $D_{h2}$. The CACF, $g^{(2)}_{s=1\vert h,h}$, as a function of the time difference between the heralding and heralded photons, $\tau$, is defined by

 

Fig. 1. (a) Schema of the CACF measurement setup, i.e., the three-fold coincidence counts of the zero-delay HBT measurement. Heralding photons were detected by the SPCM $D_s$, whose output triggered or started a measurement event. After passing through a fiber-based 50/50 beam splitter, heralded photons were detected by the SPCMs $D_{h1}$ and $D_{h2}$. (b) Relevant energy levels and transitions in the double-$\Lambda$ SFWM process, where $|1\rangle$, $|2\rangle$, $|3\rangle$, and $|4\rangle$ correspond to the $^{87}$Rb energy levels of $|5S_{1/2}, F=2\rangle$, $|5S_{1/2}, F=1\rangle$, $|5P_{1/2}, F=2\rangle$, and $|5P_{3/2}, F=1,2\rangle$. We applied the pump and coupling fields to generate the signal and probe photons, which were sent to the heralding and heralded channels in (a), respectively.

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In literature, the CACF measurements examined whether a biphoton source can be claimed as the source of heralded single photons. The criterion for the source of heralded single photons is that the CACF value is less than 0.5. In general, a larger value of the cross-correlation function (CCF) between the heralding and heralded photons results in a smaller or equivalently better value of CACF. We give a few examples in the following: In Ref. [48], Podhora et al. reported a CACF of approximately 0.44 at a CCF of about 7.0 with an SFWM biphoton source based on a hot atomic vapor. In Ref. [16], Belhassen et al. achieved a CACF of about 0.1 at a CCF of 44 with an SPDC biphoton source based on an AlGaAs waveguide. In Ref. [17], Seri et al. accomplished a CACF of around 0.05 at a CCF of 45 with an SPDC biphoton source based on a PPLN crystal. In Ref. [51], Davidson et al. achieved a CACF of around 0.035 at a CCF of 90 with an SFWM biphoton source based on a hot atomic vapor. In Ref. [15], Lenhard et al. accomplished a CACF of around 0.007 at a CCF of 280 with an SPDC biphoton source based on a PPKTP crystal. In Ref. [53], Lu et al. reported a CACF of approximately 0.006 at a CCF of about 800 with an SFWM biphoton source based on a silicon microdisk resonator.

A few references reported the formulas of CACF without derivations and under some assumptions or approximations. In Ref. [32], Chou et al. gave an empirical formula to show that the CACF is inversely proportional to the CCF. In Ref. [48], Podhora et al. used the approximation that all photons in the heralding and heralded channels nearly came from the biophotons in the state of $|1_s, 1_h\rangle$. They brought out a formula of the zero-delay CACF, $g^{(2)}_{s=1\vert h,h}(0)$, without derivation, and demonstrated that an expermental value of $g^{(2)}_{s=1\vert h,h}(0)$ is consistent with the formula’s prediction. In Ref. [15], Lenhard et al. showed a formula of $g^{(2)}_{s=1\vert h,h}(0)$ as a function of the signal-to-background ratio without derivation. The formula is only valid when the heralded photons, paired with the heralding photons, are completely uncorrelated with the background or noise photons. In Ref. [16], Belhassen et al. employed an approximation that all photons in the heralding and heralded channels came from biphotons, most of which were in $|1_s, 1_h\rangle$ and few of which were in $|2_s, 2_h\rangle$, and further assumed a low detection efficiency and the condition that a time bin width covered the entire biphoton’s temporal profile. The authors derived a formula that $g^{(2)}_{s=1\vert h,h}(0)$ is approximately equal to twice the product of the mean number of heralded photons and their zero-delay unconditional auto-correlation. They showed that a measured value of $g^{(2)}_{s=1\vert h,h}(0)$ was consistent with the formula’s prediction. To our knowledge, there is neither a systematic study on the CACF nor a derivation for a more general formula of $g^{(2)}_{s=1\vert h,h}(\tau )$ in literature.

In the present work, we experimentally studied the relation between the CACF and CCF under various experimental conditions with a hot-atom SFWM biphoton source. We further developed an analytical formula to express the quantitative connection between the CACF and CCF. The formula is derived based on general theory without using the assumptions or approximations of all photons coming from biphotons, low detection efficiencies, and time bin widths completely covering biphoton’s temporal profiles. It is valid for SFWM and SPDC biphoton sources of all kinds of media with different background-photon types. The predictions from the formula are in good agreement with the experimental data. Thus, using the formula, one can immediately obtain the CACF value from the CCF value without measuring HBT-type three-fold coincidence counts. Our study advanced the knowledge of the biphoton source’s CACF. The formula developed in this work is a convenient tool for examining the single-photon purity or antibunching property of heralded photons.

2. Experimental setup

We produced biphotons, or pairs of heralding and heralded single photons, with a paraffin-coated vapor cell of isotopically enriched $^{87}$Rb atoms via the SFWM process [27–29]. Figure 1(b) shows the relevant energy levels and transitions in the SFWM process. The pump and coupling fields were generated from two external-cavity diode lasers (Toptica DL DLC pro) with wavelengths of about 780 and 795 nm, respectively. Nearly all population was optically pumped to the ground state $|1\rangle$ ($|5S_{1/2},F=2\rangle$) by a donut-shaped laser beam, which is named the hyperfine optical pumping (HOP) field. The donut-shaped beam minimized the influence of the HOP field on the SFWM interaction region [27,47]. The coupling field resonantly drove the transition from $|2\rangle$ ($|5S_{1/2},F=1\rangle$) to $|3\rangle$ ($|5P_{1/2},F=2\rangle$). It had an $e^{-2}$ full width of 1.3$\sim$1.4 mm. The pump field drove the transition from $|1\rangle$ to $|4\rangle$ ($|5P_{3/2},F=1\rangle$ or $|5P_{3/2},F=2\rangle$), and was blue-detuned by 2.00 or 1.84 GHz. It had an $e^{-2}$ full width of 1.2$\sim$1.3 mm. We heated the vapor cell to enhance the atoms’ optical depth (OD). The spectral $e^{-1}$ half width of the Doppler-broadened atomic vapor was approximately 54$\Gamma$, where $\Gamma = 2\pi {\times }6$ MHz is the spontaneous decay rate or angular natural linewidth of $|3\rangle$ and $|4\rangle$. A weak laser field was employed to determine the OD, $\alpha$, and its transmission was given by $\exp (-\alpha )$. The absorption spectrum of this laser field revealed that at the transition frequency from $|5S_{1/2},F=2\rangle$ to $|5P_{1/2},F=2\rangle$, the OD in the SFWM interaction region was around 8.2.

The pump and coupling fields and the signal and probe photons propagated in the same direction, and their beam profiles completely overlapped in the vapor cell. This all-copropagation scheme enables the phase match in the SFWM process and significantly reduces the decoherence rate induced by the Doppler effect [27,28]. The pump and coupling fields were linearly polarized in the orthogonal configuration, while the signal and probe photons have their polarizations normal to the pump’s and coupling’s polarizations, respectively. We utilized polarization filters and etalon filters to attenuate the pump and coupling fields considerably, and hence, their contributions to the SCPMs’ counts were negligible. Details of the polarization and etalon filters, the leakage rates, and the SPCMs’ dark count rates can be found in Ref. [27,28]. The collection efficiencies of the signal and probe photons in this work are 7.6% and 6.3%, respectively. To determine the collection efficiencies, we have considered SPCMs’ quantum efficiencies, optical fibers’ coupling efficiencies, and attenuations due to the polarization and etalon filters and all the other optical components in the signal and probe photons’ propagation paths.

We performed the CACF measurement with the setup depicted in Fig. 1(a). The signal photons were sent to the heralding channel. The probe photons were sent to the heralded channel, which passed through a fiber-based 50/50 beam splitter (BS). Detection of a signal photon by the SPCM $D_s$ initiated a three-fold coincidence count measurement. The count from $D_s$ triggered or started the measurement of a time tagger (IDQ ID900-MASTER). We utilized SPCMs $D_{h1}$ and $D_{h2}$ to detect photons from the two outputs of the BS. The event that one count from $D_{h1}$ and another from $D_{h2}$ arrive within the same time bin is recorded as a three-fold coincidence count by the time tagger. One count from $D_{h1}$ plus another from $D_{h2}$ arriving at different time bins is not considered as a three-fold coincidence count. Hence, the HBT measurement operated at zero delay time. The time tagger recorded not only the three-fold coincidence count, i.e., $N_s(0) N_{h1}(\tau ) N_{h2}(\tau )$ in Eq. (1), but also the two-fold coincidence counts, i.e., $N_s(0) N_{h1}(\tau )$ and $N_s(0) N_{h2}(\tau )$, and the signal photon’s trigger, i.e., $N_s(0)$, where $\tau$ is the time difference between a probe count and a signal trigger. The time bin width in the CACF measurement was set to 2.0 or 4.0 ns, which was about ten times shorter than the temporal FWHM of biphotons. A time bin width shorter than the chosen one affected the measured CACF value very little. Thus, the chosen width is the bin width limit, and the CACF data can directly verify the analytical formula without any convolution or deconvolution process. Under the above condition, we set the time bin width as long as possible to obtain a reasonable signal-to-noise ratio at the accumulation time of 2 hours. Most CACF data had a bin width of 4 ns, and a few data had a temporal width of less than 35 ns had a bin width of 2 ns. The probability of detecting two or more counts from the same SPCM $D_{h1}$ or $D_{h2}$ within the same time bin was tiny, so they were considered as one count for simplicity. The FPGA of the time tagger was programmed to provide histograms of the three-fold and two-fold coincidence counts versus bin number. We used the histograms and the total trigger number to calculate the CACF.

The photons in the heralded channel depicted by Fig. 1(a) consist of biphotons’ probe photons (whose count rate is denoted by $n_p$), fluorescence photons ($n_f$), leakage photons ($n_l$), and SPCMs’ dark counts ($n_d$) [28]. The fluorescence photons were generated by a small population in $|2\rangle$ plus the coupling field’s excitation to $|3\rangle$. The leakage photons came from the leakage of the coupling field. We determined $n_{p}$, $n_f$, $n_l$, and $n_d$ as follows. First, we measured $n_d$ by blocking all the incoming photon fluxes. Next, we determined the leakage rate $n_l$ of the coupling field by blue-detuning it $\sim$1 GHz from resonance, as to make its interaction with the atomic vapor negligible. The increment of the count rate gave $n_l$. Then, the coupling field was tuned to its normal condition, i.e., was set to the transition frequency from $|2\rangle$ to $|3\rangle$. The increment of the count rate gave $n_f$. Finally, we switched on the pump field, and the experiment was in the condition of biphoton generation. The increment of the count rate gave $n_{p}$. During the above procedure, the HOP field was always present. The leakage rate of the pump field in the heralded channel was negligible. As an example, Table 1 in Appendix B shows the ratios ($C_{p}$, $C_f$, $C_l$, and $C_d$) of the four rates to their total rate determined by the above method under a representative experimental condition.

3. Results and discussion

We applied the pump and coupling laser fields to the atomic vapor to generate biphotons, or pairs of the signal and probe photons, with the double-$\Lambda$ transition scheme as shown in Fig. 1(b). In the atomic vapor, the signal photons propagated at the speed of $c$, and the probe photons moved like slow light based on the effect of electromagnetically induced transparency. We utilized the detection of a signal photon to trigger or start the time tagger. The arrival or delay time of the probe photon, which paired with the signal photon producing the trigger, was recorded by the time tagger as a coincidence count. We employed a delay line between the probe SPCM and the time tagger to ensure no missing probe count. For a representative example, Fig. 2(a) shows the number of coincidence counts as a function of the delay time. Not all the coincidence counts came from the paired probe photons. The red line in the figure shows the baseline of around 1500 counts per time bin. The biphoton pairs contributed the counts above the baseline, and the number of the total biphoton counts divided by the data accumulation time gives a detection rate of 5100 pairs/s, corresponding to a generation rate of $1.1{\times }10^6$ pairs/s. The baseline was generated by the dark counts, the leakage of the coupling field, the fluorescence photons induced by the coupling field, and the unpaired probe photons that did not pair with the triggering or heralding signal photons.

 

Fig. 2. (a) The coincidence count (left axis) or the CCF, i.e., $g^{(2)}_{s,h}$ defined by Eq. (2), (right axis) as a function of the delay time, $\tau$, between the heralding and heralded photons. Data were accumulated for 2 minutes with a time bin width of 0.8 ns. Circles connected by gray lines are the experimental data. Red line represents the baseline count contributed by the background photons. The data reveal a detection rate of 5100$\pm 120$ counts/s and a peak value of CCF of 10.6. (b) The CACF, i.e., $g^{(2)}_{s=1\vert h,h}$ defined by Eq. (1), as a function of $\tau$. Data were accumulated for 120 minutes with a time bin width of 4.0 ns. Circles connected by gray lines are the experimental data. The minimum value of CACF is 0.34, and the average value of all the data points of $\tau \geq$ 150 ns is 1.95. Blue line is the theoretical prediction calculated with the formula in Eq. (3), which will be explained in the text. The experimental conditions of (a) and (b) were the same, and the pump and coupling powers ($P_p$ and $P_c$) were 17 and 42 mW, respectively.

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Using the number of coincidence counts in each time bin, we calculated the cross-correlation function (CCF) between the heralding or triggering signal photons and the heralded photons. The CCF is defined by

With the same experimental condition as Fig. 2(a), we measured the conditional auto-correlation function (CACF) of the heralded photons versus the delay time as shown in Fig. 2(b), where the CACF is defined by Eq. (1). The measurement procedure has been described in the third paragraph of Sec. 2. Similar to the measurement of Fig. 2(a), we inserted two delay lines, i.e., two long BNC cables with nearly equal lengths, between the time tagger and the two probe SPCMs. The two arm lengths from the BS to the two detectors ($D_{h1}$ and $D_{h2}$) to the time tagger were nearly equal. Since the uncertainty of the difference between the arrival times via the two arms was significantly less than the time bin width, the delay lines did not affect the measured CCF and CACF values. They enabled the delay time of about 60 ns between the peak (or dip) of the CCF (or CACF) data and the signal photons’ triggers as shown in Figs. 2(a) [or Figs. 2(b)]. As shown by Fig. 2(a), the coincidence counts of $\tau \geq$ 150 ns nearly all came from the background photons. As shown in Fig. 2(b), the average value of the CACF of these background photons is very close to 2.0, and a significant fluctuation outside of the dip of the CACF was due to a low count rate. The bunching-or-antibunching statistics or zero-delay auto-correlation of these background photons is very similar to that of the thermal light. Comparing the CCF and the CACF data, one can see that a more significant value of CCF results in a smaller value of CACF. The maximum or peak CCF of about 10.6 corresponds to the minimum CACF of around 0.34, clearly exhibiting the property of single photons. Furthermore, the blue line in Fig. 2(b) is the theoretical prediction of an analytical formula, which will be described later. The theoretical prediction is in good agreement with the experimental data.

The pump (and coupling) field’s intensity or its power, $P_p$ (and $P_c$), at the fixed beam size affects the CCF and consequently influences the CACF. A higher $P_c$ results in a larger peak value of CCF, i.e., Max$[g^{(2)}_{s,h}(\tau )]$, or signal-to-background ratio (SBR) [27]. On the other hand, a higher $P_p$ results in a smaller Max$[g^{(2)}_{s,h}(\tau )]$ or SBR [28]. Consequently, either a higher $P_c$ or a lower $P_p$ produces a smaller minimum value of CACF, i.e., Min$[g^{(2)}_{s=1\vert h,h}(\tau )]$, or better single-photon purity. To show the universal relation between Max$[g^{(2)}_{s,h}(\tau )]$ and Min$[g^{(2)}_{s=1\vert h,h}(\tau )]$, we varied $P_c$ and $P_p$ and measured the CCF and CACF. Figure 3 shows Min$[g^{(2)}_{s=1\vert h,h}(\tau )]$ versus Max$[g^{(2)}_{s,h}(\tau )]$, and both were measured at the same experimental condition. The accumulation time of each CCF data was 2 minutes to achieve a good signal-to-noise ratio like the one shown in Fig. 2(a). However, the accumulation time of each CACF data was 2 hours to achieve a similar signal-to-noise ratio like the one shown in Fig. 2(b). We obtained the error bar or standard deviation of each Max$[g^{(2)}_{s,h}(\tau )]$ with a couple of the CCF measurements right before and right after the CACF, and that of each Min$[g^{(2)}_{s=1\vert h,h}(\tau )]$ with the estimations of time bin uncertainty and shot noise. The black lines in the figure are the theoretical predictions of Eq. (3), which will be illustrated in the next paragraph. The consistency between the theoretical predictions and experimental data is satisfactory.

 

Fig. 3. The minimum value of CACF, i.e., Min$[g^{(2)}_{s=1\vert h,h}(\tau )]$, versus the peak value of CCF, i.e., Max$[g^{(2)}_{s,h}(\tau )]$, both of which were measured at the same experimental condition. Green diamonds are the data measured at $P_p =$ 8$\sim$34 mW (horizontally from right to left) and $P_c = 17$ mW, red circles represent those at $P_p = 17$ mW and $P_c =$ 17$\sim$45 mW (horizontally from left to right), and blue squares are those at $P_p = 34$ mW and $P_c =$ 17$\sim$48 mW (horizontally from left to right). Upper and lower black lines are the predictions calculated with Eq. (3) at $g^{(2)}_{p_0,b}(0) =$ $g^{(2)}_{b,b}(0) =$ 2.00 and 1.95.

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Upon the detection of the heralding photon, i.e., the signal photon that produced a trigger of the time tagger, the CACF of the heralded photon that can consist of the probe photon paired with this signal photon and the background photon is defined by Eq. (1). Since there is a universal relation between the CCF and CACF as demonstrated by Fig. 3, we derived an analytical formula in Appendix A given by

We plot the blue line in Fig. 2(b) with the formula in Eq. (3). Note that the peak (or dip) of the CCF (or CACF) data appeared at the delay time of about 60 ns as shown in Fig. 2(a) [or Fig. 2(b)]. Since the theoretical prediction of the CACF was calculated from the CCF data, this delay time in the CCF data automatically appeared in the CACF prediction. To obtain the blue line, we used the values of $g^{(2)}_{p_0,b}(0)$ and $g^{(2)}_{b,b}(0)$ listed in Table 1. All the values of $C_{p}$, $C_f$, $C_l$, and $C_d$ in Table 1 were measured under the same experimental condition as Fig. 2(b), where $C_{p}$, $C_f$, $C_l$, and $C_d$ represent the ratios or percentages of the biphoton’s probe, fluorescence, leakage, and dark-count rates to the total photon rate in the heralded channel, respectively. With the four percentages, we then employed the formulas in Eqs. (17) and (18) to obtain the values of $g^{(2)}_{p_0,b}(0)$ and $g^{(2)}_{b,b}(0)$. A large fraction of the heralded photons were the probe photons from the biphotons and the fluorescence photons induced by the coupling field due to the population in state $|2\rangle$ [28]. They are the photons of thermal light, and their zero-delay auto-correlation should be 2. A small fraction of the heralded photons came from the leakage photons of the coupling field and the dark counts of the SPCMs [28]. Their zero-delay auto-correlation should be 1. Consequently, the zero-delay auto-correlation of the heralded photons should be very close to 2, as demonstrated by Fig. 3 in Ref. [28].

We further measured the unconditional auto-correlation function (UCAF) versus the delay time of the heralded photons, i.e., $g^{(2)}_{h,h}(\tau )$, under the same experimental condition as Fig. 2, which is given by

The detection setup in the measurement merely consisted of the heralded channel without the heralding channel in Fig. 1(a). Since the photon counts from $D_{h1}$ and those from $D_{h2}$ have no preferential order in time, the data of $g^{(2)}_{h,h}(\tau )$ are symmetric around $\tau = 0$ as shown in Fig. 4. Nearly at $\tau = 0$, $g^{(2)}_{h,h}$ had a maximum value of about 1.98, inferring that the heralded photons were mostly thermal light. At a large positive or negative $\tau$, the value $g^{(2)}_{h,h}$ asymptotically approached to 1, indicating that the photon detected by $D_{h1}$ and those by $D_{h2}$ became uncorrelated. Considering the linewidth of the two-photon transition formed by the probe photons and coupling field, the temporal FWHM of $g^{(2)}_{h,h}(\tau )$ is reasonable. Thus, the data of $g^{(2)}_{h,h}(\tau )$ in Fig. 4 are convincing. In theory, the value of $g^{(2)}_{h,h}(0)$ is equal to that of $g^{(2)}_{b,b}(0)$. The measured $g^{(2)}_{h,h}(0)$ in Fig. 4 confirmed the value of $g^{(2)}_{b,b}(0)$ in Table 1, implying that the formulas in Eqs. (17) and (18) are valid. With the values of $g^{(2)}_{p_0,b}(0)$ and $g^{(2)}_{b,b}(0)$ given in Table 1, the theoretical prediction in Fig. 2(b) has no adjustable parameter. Hence, one can use Eq. (3) to obtain the CACF without the HBT-type three-fold coincidence measurement.

 

Fig. 4. The unconditonal auto-correlation function of the heralded photons, i.e., $g^{(2)}_{h,h}$, as a function of the delay time $\tau$. The experimental condition here is the same as that in Fig. 2. The time bin width was 4.8 ns. Circles are the experimental data. Red line is the best fit, which reveals $g^{(2)}_{h,h}(0) =$ 1.98.

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Let’s consider the unconditional or free-running count rates of the four kinds of photons in the heralded channel in the experiment. First, various powers of $P_p$ and $P_c$ gave the probe photon rates of ($3{\sim }13){\times }10^4$ counts/s. Next, the HOP field mentioned in the first paragraph of Sec. 2. emptied most of the population in state $|2\rangle$. The fluorescence photon rate was ($6{\sim }17){\times }10^3$ counts/s, and a higher $P_c$ resulted in a higher rate. Furthermore, since we installed the polarization and etalon filters to block the strong coupling light [28], the leakage rate of the coupling field was merely 200$\sim$600 counts/s. Finally, the sum of the dark count rates of the SPCMs $D_{h1}$ and $D_{h2}$ was about 300 counts/s. According to the above rates, most of the photons were the bunching thermal light, and a few of them, including the SPCMs’ dark counts, behaved like the randomly bunching light. The values of $g^{(2)}_{p_0,b}(0)$ and $g^{(2)}_{b,b}(0)$ of the data points in Fig. 3 were all above 1.95. Hence, we use $g^{(2)}_{p_0,b}(0) = g^{(2)}_{b,b}(0) =$ 2.00 (upper limit) and 1.95 (lower limit) in the formula of Eq. (3) to plot the black lines in the figure.

To further test the formula in Eq. (3), we intentionally increased the leakage of the coupling light by degrading the polarization filter. A low pump power was utilized to have a low biphoton generation rate. About 68% of the photons in the heralded channel were the leakage photons, around 26% were the biphotons’ probe photons, and approximately 6% were the fluorescence photons, and dark counts were negligible. Using Eqs. (17) and (18), we obtained $g^{(2)}_{p_0,b}(0) =$ 1.32 and $g^{(2)}_{b,b}(0) = 1.10$, which was confirmed by the measured $g^{(2)}_{h,h}(0)$ of 1.09. Utilizing Eq. (3) and the measured peak CCF of 4.1, we predicted a CACF minimum of 0.55. The measured CACF minimum was 0.52$\pm$0.06. While the coherent or randomly bunching photons are the majority of the photons in the heralded channel, the consistency between the theoretical prediction and experimental data of the CACF is also satisfactory.

From the practical point of view, a biphoton source with a high generation and a narrow linewidth or, equivalently, a long temporal width is desirable. So, the spectral brightness, i.e., the generation rate per linewidth, is an important figure of merit of biphotons. It is linearly proportional to the product of the biphoton detection rate, $R_d$, and the temporal full width at half maximum, $\tau _p$. In the double-$\Lambda$ SFWM biphoton source, $\tau _p$ mostly depends on the coupling power $P_c$ [27,62], and $R_d$ mainly depends on the pump power $P_p$ [28,62]. At various powers of $P_p$ and $P_c$, Fig. 5 shows the peak value of CCF, i.e., Max$[g^{(2)}_{s,h}(\tau )]$, as a function of $R_d \tau _p$. For the following reason, we fitted the data with the $y = 1 + A/x$ function, where $A$ is the fitting parameter. The peak CCF equals the sum of 1 and SBR as demonstrated by Fig. 2(a). When the heralded photons are mostly the biphotons’ probe photons, the SBR is inversely proportional to the spectral brightness [28]. In Fig. 5, the consistency between the experimental data and best fit is satisfactory, suggesting a universal relation between the spectral brightness and the peak CCF and a higher spectral brightness associated with a smaller peak CCF. In other words, a higher spectral brightness results in a larger minimum CACF, i.e., Min$[g^{(2)}_{s=1\vert h,h}(\tau )]$, or a worse single-photon purity. Nevertheless, the parameter $A$ depends on the experimental condition and the measurement method. How to make $A$ as large as possible to achieve a high spectral brightness at a reasonably good single-photon purity certainly deserves more effort.

 

Fig. 5. The peak value of CCF, i.e., Max$[g^{(2)}_{s,h}(\tau )]$, as a function of the product of detection rate ($R_d$) and temporal full width at half maximum ($t_p$). Green diamonds are the data measured at $P_p =$ 8$\sim$34 mW (horizontally from left to right) and $P_c = 17$ mW, red circles represent those at $P_p = 17$ mW and $P_c =$ 17$\sim$45 mW (horizontally from right to left), and blue squares are those at $P_p = 34$ mW and $P_c =$ 17$\sim$48 mW (horizontally from right to left). Black line is the best fit of all the data. The fitting function is given by $y = 1 +A/x$, where $A$ is the fitting parameter.

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4. Conclusion

The conditional auto-correlation function (CACF), i.e., $g^{(2)}_{s=1\vert h,h}(\tau )$ defined by Eq. (1), and the cross-correlation function (CCF), i.e., $g^{(2)}_{s,h}(\tau )$ defined by Eq. (2), are the essential properties of biphotons, where $\tau$ is the delay time between the heralding and heralded photons. We systematically studied the relation between the CCF and CACF with a hot-atom SFWM biphoton source. In the SFWM process, we employed the pump and coupling laser fields to generate the biphotons from the atoms. Each biphoton is a pair of single photons, called signal and probe, detected in the heralding and heralded channels, respectively. Photons in the heralding channel were nearly all the signal photons, and those in the heralded channel consisted of the probe photons paired with the heralding signal photons and the background photons. Sources of the background photons were the unpaired probe photons, fluorescence photons, leakage photons, and dark counts. The background photons affect the single-photon purity or CACF of the heralded photons.

Since the pump and coupling powers are two of the most influential parameters in the SFWM process, we varied them and measured the CCF and CACF. The CCF (or CACF) is the histogram of the two-fold (or three-fold HBT-type) coincidence count measurement. The experimental data of CACF versus CCF under the same experimental condition reveal a universal relation regardless of the pump and coupling powers. We further derived an analytical formula for the relation. The formula links the CCF, the zero-delay cross-correlation between the paired probe and background photons, $g^{(2)}_{p_0,b}(0)$, and the zero-delay auto-correlation of the background photons, $g^{(2)}_{b,b}(0)$, to the CACF as shown by Eq. (3). Since the derivation is based on general theory, the formula is also valid for the SPDC biphoton sources and various background photon compositions. We were able to infer the values of $g^{(2)}_{p_0,b}(0)$ and $g^{(2)}_{b,b}(0)$ in Refs. [15,16], and found that the relations between the maximum or average CCF and the minimum or average CACF of their SPDC biphoton sources approximately agreed with the formula. In this work, the theoretical predictions of CACF calculated from the CCF data based on the formula are consistent with the CACF data. Since we determined $g^{(2)}_{p_0,b}(0)$ and $g^{(2)}_{b,b}(0)$ by measuring the percentages of probe photons, fluorescence photons, leakage photons, and dark counts in the heralded channel, there was no adjustable parameter in the theoretical prediction. The consistency suggests that one can use Eq. (3) and the CCF to obtain the CACF without the time-consuming three-fold HBT-type coincidence count measurement. One can also understand how the single-photon purity limits spectral brightness by utilizing Eq. (3) and the data shown in Fig. 5.

To our knowledge, there is neither a systematic study on the CACF nor a derivation for a more general formula of the CACF in literature. Our experimental and theoretical studies advanced the knowledge of the connection between the CACF and CCF in all kinds of biphoton sources. The analytical formula developed in this work provides a convenient and valuable tool for examining the single-photon purity or antibunching property of heralded photons.

Appendix A: Derivation of the formula for the conditional auto-correlation function of heralded photons

We illustrate the derivation of the CACF formula shown in Eq. (3) in this appendix. In Fig. 1(a), once a signal photon in the heralding channel made a trigger to start a measuement of the three-fold coincidence count, photons in the heralded channel could consist of the probe photon paired with the signal photon and all the other photons, i.e., the background photons. Hence, the counts in the SPCMs $D_{h1}$ and $D_{h2}$ are expressed by $N_{h1} = N_{p_0 1}+N_{b 1}$ and $N_{h2} = N_{p_0 2}+N_{b 2}$, where the subscript of $p_0$ represents the paired probe photon, that of $b$ represents the backgroud photons, and those of 1 and 2 indicate $D_{h1}$ and $D_{h2}$, respectively. Then, the definiton of the CACF shown in Eq. (1) is rewritten as

(5)$$g^{(2)}_{s=1\vert h,h}(\tau) = \frac{ \langle N_s(0) \rangle \langle N_s(0) [N_{p_0 1}(\tau)+N_{b1}(\tau)] [N_{p_0 2}(\tau) +N_{b2}(\tau)] \rangle } {\langle N_s(0) N_{h1}(\tau) \rangle \langle N_s(0) N_{h2}(\tau) \rangle},$$

where $\tau$ is the difference between the detection time of the photons in heralded channel and that of the trigger made by the signal photon in the heralding channel.

In the denominator of Eq. (5), $\langle N_s(0) N_{h1}(\tau ) \rangle$ and $\langle N_s(0) N_{h2}(\tau ) \rangle$ relate to the CCF, $g^{(2)}_{s,h}(\tau )$, which is defined by Eq. (2). Since the beam splitter (BS) in Fig. 1(a) does not change the statistics of the correlation between the photons, $g^{(2)}_{s,h}(\tau )$ = $g^{(2)}_{s,h1}(\tau )$ = $g^{(2)}_{s,h2}(\tau )$. For simplicity without degrading generality, the 50/50 BS made $\langle N_{h1} \rangle = \langle N_{h2} \rangle = \langle N_{h} \rangle /2$. The denominator of Eq. (5) is given by

(6)$$\langle N_s(0) N_{h1}(\tau) \rangle \langle N_s(0) N_{h2}(\tau) \rangle = [g^{(2)}_{s,h}(\tau)]^2 \frac{\langle N_s(0) \rangle^2 \langle N_{h}(\tau) \rangle^2}{4}.$$

Therefore, the means of two-fold coincidence count on the left-hand side have been converted to the expression consisting of cross-correlation function and mean photon numbers on the right-hand side.

In the numerator of Eq. (5), $\langle N_s(0) [N_{p_0 1}(\tau )+N_{b1}(\tau )] [N_{p_0 2}(\tau ) +N_{b2}(\tau )] \rangle$ is split into the four terms: $\langle N_s(0) N_{p_0 1}(\tau ) N_{p_0 2}(\tau ) \rangle$, $\langle N_s(0) N_{b1}(\tau ) N_{b2}(\tau ) \rangle$, $\langle N_s(0) N_{p_0 1}(\tau ) N_{b2}(\tau ) \rangle$, and $\langle N_s(0) N_{p_0 2}(\tau )$ $N_{b1}(\tau ) \rangle$. The first term is essential zero due to the single-photon nature of the paired probe photons, i.e.,

(7)$$\langle N_s(0) N_{p_0 1}(\tau) N_{p_0 2}(\tau) \rangle = 0.$$

Next, since the signal photon and the backgound photons are uncorrelated, the second term $\langle N_s(0) N_{b1}(\tau ) N_{b2}(\tau ) \rangle$ is equal to the product of $\langle N_s(0) \rangle$ and $\langle N_{b1}(\tau ) N_{b2}(\tau ) \rangle$. Furthermore, the definition of the zero-delay auto-correlation between the background photons, $g^{(2)}_{b,b}(0)$, is given by

(8)$$g^{(2)}_{b,b}(0) \equiv \frac{\langle N_{b1}(0) N_{b2}(0) \rangle} {\langle N_{b1}(0) \rangle \langle N_{b2}(0) \rangle}.$$

Using $\langle N_{b1} \rangle = \langle N_{b2} \rangle = \langle N_{b} \rangle /2$ due to the BS, we obtain

(9)$$\langle N_s(0) N_{b1}(\tau) N_{b2}(\tau) \rangle = \langle N_s(0) \rangle g^{(2)}_{b,b}(0) \frac{\langle N_{b}(\tau) \rangle^2}{4}.$$

Finally, the values of the third and fourth terms $\langle N_s(0) N_{p_0 1}(\tau ) N_{b2}(\tau ) \rangle$ and $\langle N_s(0) N_{p_0 2}(\tau ) N_{b1}(\tau ) \rangle$ are the same due to the symmetry of the 50/50 BS. Moreover, the definition of the zero-delay cross-correlation between the paired probe photons and the background photons, $g^{(2)}_{p_0,b}(0)$, is given by

(10)$$g^{(2)}_{p_0,b}(0) \equiv \frac{\langle N_{p_0}(0) N_{b}(0) \rangle} {\langle N_{p_0}(0) \rangle \langle N_{b}(0) \rangle} = \frac{\langle N_{p_0}(\tau) N_{b}(\tau) \rangle} {\langle N_{p_0}(\tau) \rangle \langle N_{b}(\tau) \rangle},$$

and its value does not depend on the measurement time at $0$ or $\tau$. Since the signal photon and the backgound photons are uncorrelated, the statistics of the zero-delay correlation between the paired probe photon and the background photons is not affected by the signal photon, i.e.,

(11)$$\frac{\langle N_s(0) N_{p_0 1}(\tau) N_{b2}(\tau) \rangle} {\langle N_s(0) N_{p_0 1} (\tau) \rangle \langle N_{b2}(\tau) \rangle} = \frac{\langle N_{p_0 1}(\tau) N_{b2}(\tau) \rangle} {\langle N_{p_0 1} (\tau) \rangle \langle N_{b2}(\tau) \rangle} = g^{(2)}_{p_0,b}(0),$$

(12)$$\frac{\langle N_s(0) N_{p_0 2}(\tau) N_{b1}(\tau) \rangle} {\langle N_s(0) N_{p_0 2} (\tau) \rangle \langle N_{b1}(\tau) \rangle} = \frac{\langle N_{p_0 2}(\tau) N_{b1}(\tau) \rangle} {\langle N_{p_0 2} (\tau) \rangle \langle N_{b1}(\tau) \rangle} = g^{(2)}_{p_0,b}(0).$$

Based on the above two equations, and $\langle N_{b1} \rangle = \langle N_{b2} \rangle = \langle N_{b} \rangle /2$ as well as $\langle N_{p_0 1} \rangle = \langle N_{p_0 2} \rangle = \langle N_{p_0} \rangle /2$, the third and fourth terms become

(13)$$\begin{aligned} \langle N_s(0) N_{p_0 1}(\tau) N_{b2}(\tau) \rangle \!\!\!&=\!\!\! \langle N_s(0) N_{p_0 2}(\tau) N_{b1}(\tau) \rangle\\ &=\!\!\! g^{(2)}_{p_0,b}(0) \frac{\langle N_s(0) N_{p_0} (\tau) \rangle \langle N_{b}(\tau) \rangle}{4}\\ &=\!\!\! g^{(2)}_{p_0,b}(0) \frac{[g^{(2)}_{s,h}(\tau) \langle N_h(\tau) \rangle - \langle N_b(\tau) \rangle] \langle N_s(0) \rangle \langle N_b(\tau) \rangle}{4}. \end{aligned}$$

To obtain the last equation in the above, we utilize $\langle N_s(0) N_{p_0} (\tau ) \rangle = \langle N_s(0) [N_h(\tau )-N_b(\tau )] \rangle = \langle N_s(0) N_h(\tau ) \rangle - \langle N_s(0) \rangle \langle N_b(\tau ) \rangle$ and the definiton of $g^{(2)}_{s,h}(\tau )$ shown in Eq. (2). According to Eqs. (7), (9), and (13), the numerator of Eq. (5) is written as

(14)$$\begin{aligned} &\langle N_s(0) \rangle \langle N_s(0)[N_{p_0 1}(\tau)+N_{b1}(\tau)] [N_{p_0 2}(\tau) +N_{b2}(\tau)] \rangle\\ & = g^{(2)}_{b,b}(0) \frac{\langle N_s(0) \rangle^2 \langle N_{b}(\tau) \rangle^2}{4} +2 g^{(2)}_{p_0,b}(0) \frac{[g^{(2)}_{s,h}(\tau) \langle N_h(\tau) \rangle - \langle N_b(\tau) \rangle] \langle N_s(0) \rangle^2 \langle N_b(\tau) \rangle}{4}. \end{aligned}$$

Therefore, the mean of the three-fold coincidence count on the left-hand side has been converted to the expression consisting of cross-correlation functions and mean photon numbers on the right-hand side.

We now replace the denominator and numerator of Eq. (5) by the right-hand sides of Eqs. (6) and (14) to obtain

(15)$$g^{(2)}_{s=1\vert h,h}(\tau) = \frac{2 g^{(2)}_{p_0,b}(0) [g^{(2)}_{s,h}(\tau) \langle N_h(\tau) \rangle - \langle N_b(\tau) \rangle] \langle N_b(\tau) \rangle + g^{(2)}_{b,b}(0) \langle N_{b}(\tau) \rangle^2 } {[g^{(2)}_{s,h}(\tau)]^2 \langle N_{h}(\tau) \rangle^2 }.$$

Except the probe photon paired with the signal photon that starts a measurement of the three-fold coincidence count, the background photons can come from all the other photons in the heralded channel. Therefore, $\langle N_b(\tau ) \rangle \approx \langle N_h(\tau ) \rangle$ is very reasonable, and it is the only approximation used in the entire derivation. Substituting $\langle N_b(\tau ) \rangle$ for $\langle N_h(\tau ) \rangle$ in the above equation, we have derived a formula for the prediciton of the CACF which is given by

(16)$$g^{(2)}_{s=1\vert h,h}(\tau) = \frac{2 g^{(2)}_{p_0,b}(0) [g^{(2)}_{s,h}(\tau) - 1] + g^{(2)}_{b,b}(0)} {[g^{(2)}_{s,h}(\tau)]^2}.$$

Appendix B: Determinations of the probe-background and background-background correlations at the zero delay time

The source of background photons are all photons in the heraleded channel except the one paired with the signal photon, which triggered or started a three-fold coincidence count measurement. Thus, the background photons consist of the probe photons, the fluorescence photons, the leakage photons, and the dark photons, i.e., the SPCM’s dark counts. The percentages of these four kinds of photons are denoted by $C_p$, $C_f$, $C_l$, and $C_d$, respectively. The value of the zero-delay cross-correlation between the paired probe photons and the background photons, $g^{(2)}_{p_0,b}(0)$, and that of the zero-delay background-background auto-correlation, $g^{(2)}_{b,b}(0)$, can be obtained from the following formulas:

(17)$$g^{(2)}_{p_0,b}(0) = C_{p} g^{(2)}_{p_0,p} + C_f g^{(2)}_{p_0,f} + C_l g^{(2)}_{p_0,l} + C_d g^{(2)}_{p_0,d} = 2 C_{p} + 2 C_f + C_l + C_d,$$

(18)$$\begin{aligned} g^{(2)}_{b,b}(0) &= C_{p}^2 g^{(2)}_{p,p} + C_f^2 g^{(2)}_{f,f} + C_l^2 g^{(2)}_{l,l} + C_d^2 g^{(2)}_{d,d}\\ &+ 2 [C_{p} C_f g^{(2)}_{p,f} + C_{p} C_l g^{(2)}_{p,l} + C_{p} C_d g^{(2)}_{p,d} + C_f C_l g^{(2)}_{f,l} + C_f C_d g^{(2)}_{f,d} + C_l C_d g^{(2)}_{l,d}]\\ &= 2 C_{p}^2 + 2 C_f^2 + C_l^2+ C_d^2 + 2 [2 C_{p} C_f + C_{p} C_l + C_{p} C_d + C_f C_l + C_f C_d + C_l C_d], \end{aligned}$$

where $g^{(2)}_{p_0,i}$ (where $i = p, f, l, d$) is the zero-delay correlation between the paired probe photons and the background photons, $g^{(2)}_{i,i}$ (where $i = p, f, l, d$) is the zero-delay auto-correlation of the background photons, and $g^{(2)}_{i,j}$ (where $i, j = p, f, l, d$ and $i \neq j$) is the zero-delay cross-correlation between any two kinds of the background photons. Since the paired and unpaired probe photons were generated via the same process, $g^{(2)}_{p_0,i} = g^{(2)}_{p,i}$. The values of $g^{(2)}_{i,i}$ and $g^{(2)}_{i,j}$ are readily known based on the statistics natures of photons and the correlations between them. We experimentally determined $C_{p}$, $C_f$, $C_l$, and $C_d$ with the method illustrated in the last paragraph of Sec. 2. Table 1 shows the measured values of $C_{p}$, $C_f$, $C_l$, and $C_d$ under the experimental condition of Fig. 2, and also displays $g^{(2)}_{p_0,b}(0)$ and $g^{(2)}_{b,b}(0)$ calculated from Eqs. (17) and (18). We used the values of $g^{(2)}_{p_0,b}(0)$ and $g^{(2)}_{b,b}(0)$ and Eq. (3) to plot the blue line in Fig. 2(b), which is in good agreement with the experimental data of CACF.

Table 1. Values of $g^{(2)}_{p_0,b}(0)$ and $g^{(2)}_{b,b}(0)$ calculated from Eqs. (17) and (18) based on the measured parameters of $C_p$, $C_f$, $C_l$, and $C_d$. The experimental condition in the measurement was the same as that of Fig. 2.

View Table

Funding

National Science and Technology Council (110-2639-M-007-001-ASP, 111-2639-M-007-001-ASP, 112-2112-M-007-020-MY3, 112-2119-M-007-007).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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