## Abstract

The potential data rate of a quantum network is limited by both the entangled photon source (EPS) and quantum memories. While an EPS whose bandwidth matches with broadband quantum memories (BBQMs) can take full advantage of the data rate limit. The EPS with GHz bandwidth is usually obtained by filtering a much broader EPS signal that is generated through spontaneous parametric down-conversion (SPDC), but this method has obvious drawbacks, e.g., large space requirements, high losses, and relatively low spectral brightness. Here we present a simple and compact method to generate a single-longitudinal-mode sub-GHz-bandwidth time-energy EPS using a type-II SPDC in a submillimeter-length Fabry-Pérot cavity. The proposed photon source offers superior figures of merit: the maximum coincidence to accidental coincidence ratio is approximately 1800, the detected pair flux ranges up to 42500 pairs per second, and the source has a high Klyshko efficiency of 25%. This source offers a very potential way to boost the performance of broadband quantum memories and high-speed quantum networks.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

As the key elements of the quantum networks, EPS and quantum memory both determine the data rate of the network. EPSs can be obtained from SPDC [1,2] and spontaneous four wave mixing (SFWM) [3,4], the bandwidth of photons can reach terahertz or even more. While BBQMs with relative narrow bandwidth represent the fundamental building blocks of data-rates limit for quantum networks [5–7]. These BBQMs are realized based on atomic frequency combs (AFCs) in an erbium-doped fiber [8] and Raman scattering (RS) in a warm cesium atomic vapor [9] or cold atomic ensemble [10]. BBQMs that operate with GHz bandwidth have broken the limited data rates of megahertz [8,9]. In the existing demonstrations of BBQMs for photonic states, the stored photons with GHz bandwidth are usually obtained by direct filtering photons with much broader spectra that are generated by SPDC in nonlinear crystals [8]; this is because the typical emission bandwidth of a photon pair in SPDC is too broad (∼THz) to enable effective coupling to a GHz-bandwidth quantum memory. This direct filtering method typically suffers from drawbacks such as large space requirements, high transmission losses and relatively low spectral brightness. Cavity-enhanced SPDC is an appropriate technique that enables enhancement of the spectral brightness and narrowing of the bandwidths of the photons to be performed simultaneously [11]. Earlier works focused on the generation of narrow-bandwidth photon pairs (NBPPs) with bandwidths of less than 100 MHz [12–15]. These photon sources are based on free-space cavities, which generally have relatively small free spectral ranges (FSRs, which are usually less than GHz) and therefore are unsuitable for preparation of a GHz-bandwidth photon pair. A compact method was later developed to generate single longitudinal mode (SLM) NBPPs with a monolithic cavity [16–20]. A single photon source based on a monolithic lithium niobate whispering gallery mode resonator was also reported [21]. In a monolithic cavity, photons can be generated directly in the SLM because of the cluster effect inside the cavity when the cavity is designed with appropriate finesse [19,20]. These cavities are too long to generate GHz-bandwidth photon pairs when the SLM operating conditions are met. That will not take full advantage of the BBQMs’ data rates. Additionally, to generate and collect photons efficiently, strict mode matching is required for both the pump beam and the generated photon pairs.

Time-energy entanglements in the telecom band are very useful in long-distance quantum communication for insensitivity to polarization disturbances in the fibers. Considerable research progress in the development of telecom band time-energy entangled sources has been demonstrated based on a variety of platforms, including periodically poled KTiOPO_{4} (PPKTP) [22,23] and periodically poled lithium niobate (PPLN) crystals and waveguides [24,25]. To demonstrate time-energy entanglement, the time delay of the interferometers should be much larger than the coherence times of the single photons [26,27]. So that the construction of the phase-stable interferometer will also benefit from the suitable-bandwidth SLM EPS. Although major progress has been made in the generation of time-energy EPSs, a sub-GHz bandwidth SLM time-energy entangled source based on a tiny monolithic cavity has not been reported to date.

Here, we present a simple and compact method to generate an SLM sub-GHz bandwidth time-energy EPS via type-II SPDC in a submillimeter-length Fabry-Pérot (FP) cavity. We fabricated a tiny FP cavity based on a type-II quasi-phase matching (QPM) PPKTP crystal. The cavity length is 0.92 mm, which has a relatively large FSR of approximately 100 GHz. The cavity is formed by direct coating of the planar end faces of the crystal and thus does not require precision mode matching for the pump beam, mode matching can be achieved for beam waist ranges from 50μm to 350μm. We use a homemade temperature controller to stabilize the central frequency of the emitted photons of the monolithic PPKTP FP cavity. After a general theoretical analysis is performed to determine the conditions for SLM operation in such a tiny monolithic cavity, we perform experiments to characterize the parameters of the GHz-bandwidth photon pair source. A maximum coincidence-to-accidental coincidence count ratio (CAR) of 1800 at a pump power of 50 mW can be obtained, and the detected photon flux ranges up to 4.25 × 10^{4} pairs per second with CAR of 300. A Klyshko efficiency or symmetric heralding efficiency of 25% is obtained, which represents the highest value achieved in cavity-enhanced photon pair generation to date. The measured cross-time correlation between the photon pair yields bandwidths of 546 MHz and 735 MHz for the two orthogonally polarized photons. An estimated spectral brightness of 2.636 ± 0.056 (s·mW·MHz)^{−1} is obtained. The SLM of the photons is verified by performing a single-photon Michelson interference experiment.

## 2. Results

#### 2.1. Principles and experimental setups

A type-II QPM SPDC should satisfy the conditions for conservation of both energy (${\omega _p} = {\omega _s} + {\omega _i}$) and momentum ($\Delta k = {k_p} - {k_s} - {k_i} - 2\pi /\Lambda \approx 0$), where ${\omega _x}(x = p,s,i)$ and ${k_x}(x = p,s,i)$ are the frequency and the wave-vector of the pump, the signal and the idler, respectively; $\Lambda $ is the poling period of the crystal.

The photon pair generated in a single-pass SPDC can be expressed as [20]:

Therefore, the joint spectral intensity for cavity-enhanced SPDC can be expressed as:

Based on the equations above, we can simulate the conditions for SLM operation based on the parameters of our own cavity. The simulated results are shown in Fig. 1.

The simulations indicate that we can obtain SLM-photon emission from a tiny cavity with appropriate finesse. For the 0.92 mm FP cavity, the FSR ($FS{R_{y,z}} = c/2{n_{y,z}}L$, where *n _{y}* and

*n*are the refractive indexes of along the

_{z}*y*and

*z*principal axes of the crystal) values for the

*y*and

*z*axes are 93.61 GHz [Fig. 1(a)] and 89.42 GHz [Fig. 1(b)], respectively (the

*y*and

*z*polarization modes are assumed to be the signal and idler photons, respectively). The cluster space is defined as the frequency difference between two nearby frequency modes that can be resonant with the cavity simultaneously and can be expressed as $\Delta {\Omega _c} = ({FS{R_y} \times FS{R_z}} )/({FS{R_y} - FS{R_z}} )$ [19]; the cluster space for our current experiment is 1997.8 GHz. This cluster space is greater than half of the SPDC bandwidth of the 0.92 mm bulk crystal (the full width at half maximum is 3239 GHz). The numbers of cavity modes contained in one cluster space for the signal and idler photons can be expressed as ${N_s} = {{{n_y}} / {({n_z} - {n_y})}},{N_i} = {{{n_z}} / {({n_z} - {n_y})}}$, and the difference between the numbers should be 1. For SLM operation, the difference of the FSRs of the signal and idler photons should be greater than the spectral widths of the single resonances. In the situation presented here, $\Delta {\nu _{FSR}}\textrm{ = }FS{R_y} - FS{R_z} > 1/2(\Delta {\nu _y} + \Delta {\nu _z})$, where $\Delta {\nu _{y,z}} = FS{R_{y,z}}/{F_{y,z}}$ is the resonance bandwidth for the

*y*and

*z*polarizations, and ${F_{y,z}}$ is the finesse for these two polarizations. If we assume that ${F_y} = {F_z}$, then the finesse of the cavity should be greater than 22. In the simulations, when ${F_y} = {F_z}\textrm{ = }50$ [Fig. 1(c)], it could be seen that in addition to the dominant frequency mode, there will still be some other frequency components in the signal spectrum. When we used the coating parameters in our experiments (${F_y} = 171,{F_z}\textrm{ = }122$), these other frequency components could be neglected [Fig. 1(d)].

The experimental setup is illustrated schematically in Fig. 2(a). A monolithic cavity is formed by coating the end face of a type-II PPKTP crystal. The front face is coated with an anti-reflection (AR, T>99%) coating at 775 nm and a highly reflective (HR, R>99.9%) coating at 1550 nm, while the rear face is coated with an AR coating at 775 nm and a partially-reflective (PR, R=96%) coating at 1550 nm; this forms an FP cavity that is resonant at 1550 nm. The crystal has dimensions of 1 mm×2 mm×0.92 mm and is periodically poled with a periodicity of 46.2 μm to obtain quasi-phase matching (QPM) for SPDC with a pump beam at 775 nm and the signal and idler photons at 1550 nm. The crystal’s temperature is controlled using a homemade temperature controller with stability of ±1 mK. The pump beam propagates along the *x* axis. A 775 nm continuous wave laser (TA Pro, Toptica; bandwidth <1 MHz) is used as the pump. The collimated Gaussian beam’s waist is modified from 1 mm to 200μm by using a group of lenses to ensure that the photons generated by SPDC are resonant with the cavity. A dichroic mirror (DM) is used here to verify the cavity’s properties via a photoelectric detector and an oscilloscope (OSC) (see 4.2. Appendix B for details). After the PPKTP crystal, an infrared lens is used to collect the photons with high efficiency. The signal and idler photons are then divided using a polarizing beam splitter (PBS) and collected, while the pump beam is removed by filters (FELH 1400/1000, Thorlabs). The signal and idler photons are subsequently sent to all-fiber-based unbalanced Michelson interferometers (UMIs, where each UMI consists of a fiber beam splitter and two Faraday rotation mirrors) to demonstrate the time-energy entanglement. Two superconductor nanowire single-photon detectors (SNSPDs, Scontel) and a coincidence device (Timeharp-260, PicoQuant) are used to perform the single-photon and two-photon correlation measurements.

#### 2.2. Characterization of the FP cavity and correlation measurement

We first characterize the FP cavity using an infrared laser source (CTL-1550, Toptica). The detailed experimental setup is shown in 4.2. Appendix B. The results obtained are shown in Figs. 2(b)–2(e). According to the SLM operating conditions, the FSR difference between the two modes is 4.19 GHz, which is much greater than the average bandwidth of the signal and idler photons of 0.64 GHz. The measured results thus fit the SLM operating theory well. By estimating the single mode excitation probability based on the method introduced in Ref. [28], the mode excitation probability of our source is about 0.92.

Next, we perform correlation measurements. First, we measure the single counts and the coincidences at various pump powers without the UMIs [and obtain results as shown in Fig. 3(a)], where the maximum photon flux detected is 4.25 × 10^{4} at a pump power of 550 mW. The average Klyshko efficiency is approximately (25.00 ± 0.16)%, and the dependence of average Klyshko efficiency on the pump power is also measured, with results as shown in Fig. 3(b). The dependence of Klyshko efficiency or symmetric heralding efficiency with pump power can be found in Ref. [29–32], for relative low pair generation rate, the dark noise of the system dominates in the measurement, therefore the Klyshko efficiencies increases very fast with pump power. When the pair generation rate is larger than dark noise of the system (in the pump power range where the multi-photon coincidence events can be ignored), the Klyshko efficiencies will increase proportional with pump power. We then calculate the CAR [as shown in Fig. 3(c)] using the equation $CAR = {{{R_c}} / {{R_{ac}}}}$, where the maximal value is approximately 1800 at the pump power of 50 mW. Time-window in the CAR value measurement is 6.4 ns that can ensure all the effective coincidence counts in one time-bin. Here, ${R_c}$ is the coincidence count and ${R_{ac}}$is the accidental coincidence count. The model for fitting the curve can be found in Ref. [33].

The spectral brightness of the photon pair is calculated using the formula ${R_{est}} = {{{R_{\det }}} / {({\alpha _s}{\alpha _i})}}$[15], where *R _{est}* is the estimated photon pair production rate,

*R*is the detected pair rate, and ${\alpha _{s,i}}$ include all efficiencies such as the fiber collection efficiency and the transmittance of the filters and the detector efficiencies, respectively, for the signal and idler photons. Next, we provide estimates of the overall detection efficiencies for the signal and idler photons. The transmittances of the two filters (i.e., FELH 1000 and FELH 1400) are 97% and 99%, respectively. The total fiber transmission loss is 2.5 dB (1 dB and 1.5 dB for the signal and the idler, respectively). The detection efficiency of our SNSPD is approximately 60%. The other losses, which are caused by the infrared lens and the PBS, are estimated to be 1%. By considering the fiber coupling efficiency of 55% and all the losses noted above, the estimated emission spectral brightness is 2.636 (s·mW·MHz)

_{det}^{−1}.

We employ second-order cross correlation function (SCCF) to confirm the SLM status of our source, while stimulated emission tomography (SET) can also be applied to characterize quantum-correlated photon pairs [34]. The SCCF for type-II double resonant SPDC is given in Refs. [15,35]. The measured SCCF is shown in Fig. 3(d) as black squares. The SCCF measurements were performed with 25 ps coincidence resolution and 30 s coincidence times at a pump power of 300 mW with a CAR of 500; The red solid line represents the results of numerical simulation (please see 4.1. Appendix A for theoretical details of SCCF). The full width at half maximum (FWHM) value of the SCCF is approximately 0.412 ns. Equation (5) shows that the correlation function has oscillatory damping with a multi-peak comb-like shape, where the damping rate of the peak values is equal to the cavity damping rate. The envelope of these peaks decays as a function of ${e^{ - 2\pi {\gamma _{s,i}}|\tau |}}$. In our cavity, ${\gamma _s}$ and ${\gamma _i}$ are 546 MHz and 735 MHz, respectively. Without consideration of the responses of the detection and coincidence systems, the FWHM of the second-order cross correlation time is defined as ${T_{FWHM}} = 1.39/2\pi \gamma$, where $\gamma$ is the geometric mean of ${\gamma _s}$ and ${\gamma _i}$. Therefore, we should obtain an FWHM of 0.349 ns. In the experiments, the time jitter of the detection and coincidence systems is approximately 60 ps, which results in broadening of the time cross-correlation curve to 0.412 ns [15,36]. The simulation fits well with measured data. And that is the first part in confirming the SLM status of the EPS.

#### 2.3. Characterization of time-energy entanglement and SLM status

We then use two UMIs to characterize the time-energy entanglement. When both the signal and idler photons pass through the long arm and the short arm of two identical UMIs, these cases are indistinguishable and the photonic state can then be expressed as $|\Phi \rangle = 1/\sqrt 2 (|{SS} \rangle + |{LL} \rangle )$[37]. We can measure the coincidences by tuning the UMI phase of the signal photons while the phase of the idler photons remains fixed. The coincidence results are shown in Fig. 4. The visibilities are 87.1 ± 0.7% (88.6 ± 0.7%) and 89.2 ± 0.6% (90.7 ± 0.6%) without (with) subtraction of the accidental coincidence counts for idle UMI phases of 0 and π/4, respectively.

The diminished visibilities occur for two main reasons: first, the asymmetry-based losses between the two UMIs will reduce the visibility; and second, a small proportion of the photon pairs in the other longitudinal modes will also reduce the visibility. It has been demonstrated that when the interference visibility is greater than 71%, the Bell inequality can be violated [38,39]; the Bell S parameter is related to the two visibilities by $S = \sqrt 2 (V1 + V2)$, where *V1* and *V2* are the visibilities at phases 0 and π/4, respectively; in the present case, the estimated S parameter is 2.49 ± 0.02 (2.54 ± 0.02) without (with) subtraction of the accidental coincidence counts. The details of the UMI phase tuning are presented in 4.3. Appendix C.

When the photon frequency mode contains only one longitudinal mode, the reduction in the interference visibility will occur much more slowly than for a multi-longitudinal mode photon in an unbalanced interferometer [14]. Therefore, we can verify whether the photon output from the cavity contains an SLM by performing a Michelson interference experiment. As shown in Fig. 5(a), the horizontally polarized photons (i.e., the signal photons) are directed into a free-space Michelson interferometer and the output is coupled into a SNSPD to perform coincidence measurements with the idler photons. At different values of Δ*x* (Δ*x*=2Δ*L*, where Δ*L* is the path difference between the two arms), we can calculate the visibility based on the maximum and minimum values of the coincidence; the results are presented in Fig. 5(b).

For a Lorentzian shape for the cavity spectrum, the dependence of the visibilities with respect to Δ*x* can be expressed as shown in Eq. (9) (please see 4.4. Appendix D for details). From the fitting results, we obtain $\Delta {\nu _N}$ of 568.9 MHz, which fits well with the results of the cavity status measurements (546 MHz, with a deviation of approximately 4.19%). Combined with the simulation of the SCCF, we can confirm the SLM status of our EPS.

#### 2.3. Comparison of parameters between our source and other reported results

We compared the performance of our photon pair source with that of other reported works based on bulk or integrated guided-wave cavities; the results are listed in Table 1. These results demonstrate that the average Klyshko efficiency of our source is the highest among the various cavity-enhanced photon sources reported to date. The high average Klyshko efficiency was achieved for several reasons. First, appropriate selection of the lenses for the pump beam, signal and idler photons means that a relatively high collection efficiency is obtained for collection of the photon pair in the single-mode fiber; second, low filtering losses are achieved for the signal and idler photons using high transmittance filters; and third, SNSPDs are used in the experiments, which means that the detector has relatively high detection efficiency and a low dark count when compared with an InGaAs single photon detector. The CAR of our source is also the highest achieved for cavity-enhanced SPDC to date. The spectral brightness of our source is relatively low for three main reasons: first, our crystal is much shorter in length (approximately one order of magnitude shorter) than other previously reported sources; second, the poling period at the C-band (46.2 μm) is much longer than that near 800 nm (approximately 10 μm). Therefore, the effective number of periods at the fixed crystal length is much smaller because the photon pair generation rate is proportional to the square of the effective number of periods [40]; as a result, the spectral brightness would be approximately 21 times smaller than the sources reported at wavelengths near 800 nm when the same crystal length is used; third, the photon generation rate is proportional to pump power density, therefore a smaller beam waist would results in a higher photon pair generation rate when the pump power is the same [40], the beam waist used in the present work is about 200 μm, which is larger than most cavity enhanced SPDC sources. Although the spectral brightness is relatively low when compared with the other reported sources, the total detected photon pair rate is the highest reported to date for cavity-enhanced SLM photon pair sources. The total photon pair flux can be increased further with use of higher pump powers, because watt-level lasers operating at around 775 nm are easily obtained. To solve this problem for the proposed setup, we can fabricate the cavity to be coated for resonance with both the pump wavelength and the down-converted photon pair wavelength simultaneously; the pump power requirements can then be reduced significantly because the pump power is enhanced as a result of the resonance effect, for example, if both end faces are 95% reflection-coated for 775 nm, there would be a reduction in the pump power of approximately 20 times when the intra-cavity pump power is the same.

## 3. Conclusion

In conclusion, we experimentally realized a sub-GHz bandwidth SLM photon pair source using a type-II SPDC process in a short FP cavity based on the cluster effect. The maximum CAR, maximum detected photon pair flux and average Klyshko efficiency achieved are outperform most of cavity-enhanced SPDC results. The source presented here can be used directly in fiber-based broadband QMs based on AFC. Though the demonstrated sub-GHz source’s bandwidth is less than GHz and the bandwidth of BBQMs cannot be fully utilized, the scheme shows potential to generate photons which have the suitable bandwidth to take full use of the data rates of BBQMs to a certain degree. By changing the poling period and the coating of the crystal, the scheme can also generate photons at different frequencies and may be directly used for a broadband Raman quantum memory. The presented work provides a potential roadmap toward the generation of a GHz bandwidth photon pair source using a minimized type-II cavity-enhanced SPDC, which has great potential for high-speed quantum communication applications.

## 4. Appendix

## 4.1. Appendix A: Details of the second-order cross correlation function

The second-order cross correlation function for type-II double resonant SPDC is given as [15,33]

## 4.2. Appendix B: Characterization of the status of the cavity

Before we performed the time-energy entanglement measurements, we characterized the properties of the cavity through the second harmonic generation (SHG) process. The details are shown in Fig. 6(a). We used an infrared laser (Toptica, CTL 1550 nm) as the SHG pump beam. We first calibrated the infrared laser, as shown in Fig. 6(b). The wavelength of the infrared laser was calibrated by using an optical spectrum analyzer (YOKOGAWA, AQ6370D), the uncertainty of the calibration is ±5pm. The laser’s output wavelength shows a good linear relationship with the piezoelectric transducer (PZT) voltage. The quarter- and half-wave plates are used to adjust the polarization of the light into the horizontal and vertical directions to measure the transmission spectrum of the cavity. Through fine tuning of the PZT voltage, i.e., tuning of the pump wavelength, we measured the FSR and the bandwidth using a photodiode (PD) and an oscilloscope (OSC). The results of these measurements are shown in the experimental results section. We then adjusted the polarization of the light to 45° with respect to the *y* axis to satisfy the SHG conditions. The beam was directed into the crystal via a group of lenses and a charge-coupled device (CCD) detector was used to acquire the SHG beam, as shown in Fig. 6(a) at the bottom left.

## 4.3. Appendix C: How the UMI phase is tuned

In the fiber UMI, the thermal coefficient of the fiber at 1550 nm is $\frac{{dn}}{{dt}} = 0.811 \times {10^{ - 5}}/^\circ C$ and the fiber length difference of the UMI is 1.022 m, which corresponds to ${L_d} = c\Delta t/2n$ for a 10 ns time delay. The temperature for one tuning period is $\Delta T = \lambda /(2{L_d}\frac{{dn}}{{dT}}) = 0.094K$. In the experiment, the temperature tuning periods and the phase were calibrated using a stable narrow bandwidth laser source. The laser used for calibration the phase of UMI is a continuously tunable narrow bandwidth diode laser (DLC CTL-1550, Toptica), which has output power greater than 30 mW, less than 10kHz line width at 5μs integration time and tuning range of 1520-1630 nm. The phases of the UMIs can remain unchanged for hours because of their strong thermal and acoustic isolation from the surrounding environment. In the Franson interference measurements, when the coincidence counts are at their maximal and minimal values, the three-bin coincidence histogram is as shown in Fig. 7.

## 4.4. Appendix D: Dependence of visibilities in a Michelson interference with optical path length difference

Consider a Lorentzian shape for the cavity spectrum that can be expressed as:

*c*is the speed of light in a vacuum.

*Δx*is the optical path difference. If we assume a constant value of

*R*as the background for the coincidence, the visibility can then be expressed as:

_{dc}*R*can be expressed as shown in Eq. (9). We evaluate

_{dc}*R*to be 1/30, which corresponds to a coincidence background of approximately 50 in the experiment.

_{dc}## Funding

Anhui Initiative in Quantum Information Technologies (AHY020200); National Natural Science Foundation of China (11934013, 61435011, 61525504).

## Disclosures

The authors declare no conflicts of interest.

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