Mode-locked biphoton states, exhibiting unique discrete spectral and temporal multimode behavior within a single spatial mode profile, can be generated by spontaneous parametric down-conversion (SPDC) photons via cavity filtering [1–5], cavity enhancement [6–16], and by spontaneous four-wave mixing in integrated microring resonators [17–21]. The mode-locked behavior in a biphoton frequency comb (BFC) manifests itself as interesting comblike temporal correlations via Hanbury–Brown–Twiss-type interferometry [7–10,14,16,22], which can be naturally exploited by direct joint temporal intensity (JTI) measurements to verify high-dimensional frequency-bin entanglement with complementary joint spectral intensity (JSI) measurements [23]. Such oscillations in the temporal domain can also be used in chip-scale optical quantum information processing [24–27], in time-bin-based quantum communication protocols [28–32], and in quantum networks based on time-resolved correlation measurements [33,34], including multiboson correlation sampling schemes [35,36].

Time-energy entanglement, first formulated by Franson using a pair of unbalanced Mach–Zehnder interferometers (MZIs) [37], has proven to be quintessential from fundamental and practical perspectives in the quantum optics community. For example, the Franson interference is important for testing quantum nature in a loophole-free test [38] for high-quality long-distance time-energy entanglement distribution [39–44] and for securing time-bin-based high-dimensional quantum key distribution (QKD) against collective Gaussian attacks [29,45,46]. This distinct mode-locked feature of BFC can be revealed nonlocally via a Franson interferometry, leading to discrete time-energy-entangled Franson interference recurrences, spaced by the round-trip time of the cavity [2–5]. The cavity geometry, the reflectivity of the mirrors, finesse F, and the free spectral range (FSR) of the cavity determine the JSI and the JTI of the generated mode-locked BFC states. Indeed, the cavity serves as a tool to tailor the parameters of the two-photon state via the two-photon mode-locked process, for instance, to probe the time-energy entanglement via Franson interferometry using off-the-shelf telecom compatible filters [2–5] or to match the bandwidth of the SPDC photons to a specific atomic transition line for quantum memory storage [11–13,15,47–52]. Moreover, to the best of our knowledge, in all of the previous demonstrations for cavity-filtered mode-locked BFCs [1–5], the impact of the key figure of merit, the cavity finesse F on both mode-locked temporal correlations, and Franson interference recurrences remain unexplored.

In this paper, we utilize mode-locked singly filtered BFCs based on the cavity-filtered scheme with a Fabry–Perot (FP) cavity that is flexible and robust for experimental implementations. We examine the role of cavity finesse F in the mode-locked signal-idler temporal cross correlations as well as in the time-energy entanglement via Franson interferometry with our experimental demonstrations and theoretical descriptions. First, we observe the mode-locked temporal oscillations of a singly filtered BFC using our second-order cross-correlation function measurements with a cavity FSR of 5.03 GHz and a bandwidth of 457 MHz. Such direct JTI measurements using commercial single-photon detectors signify the existence of spectral phase coherence in the cavity-filtered BFC scheme for the first time. Second, we measure and quantify the falloff trend of the time-energy entanglement in our singly filtered BFCs based on 15.15 GHz and 5.03 GHz cavities via Franson interferometry, spanning up to six time bins with the highest central time-bin accidental-subtracted visibility over 97% for both cavities. Such JSI measurements are consistent and complementary with our JTI measurements. We find out the cavity finesse F plays a central role towards optimum temporal correlations and Franson interference recurrence in our singly filtered mode-locked BFCs by comparing different cavity configurations. For the first time, via a higher cavity finesse F of 45.92 in a 15.11 GHz BFC, we measured a Franson recurrence visibility of 69.86% with an approximately 3.13-fold improvement of Franson visibility compared to the Franson visibility with cavity finesse F of 11.14 at the sixth time bin, approaching the quantum-classical limit of 70.7%. Moreover, for the 16th time bin in the same BFC, there is a potential approximately 24-fold improvement of the Franson visibility for a cavity finesse F of 45.92 compared to the Franson visibility with a cavity finesse F of 11.14. Near-optimum Franson interference recurrence and a time-bin Schmidt number KT near 16 effective modes in similar FSR fiber cavity are predicted with a cavity finesse F of 200. Our results pave the way for producing high-quality mode-locked temporal correlations for time-resolved correlation-based quantum networks and can be helpful in pushing the limit of the Hilbert space dimensionality in the temporal domain [3], for high-quality long-distance high-dimensional time-energy entanglement distributions [4,53,54] and on-chip optical quantum information processing.

The working principle of our experiment is illustrated in Fig. 1(a). We used singly filtered mode-locked BFC sources where the SPDC source is a fiber-coupled type-II phase-matched periodically poled KTiOPO4 (ppKTP) waveguide of fiber-coupling efficiency of ≈50% per facet, driven by a self-injection-locked continuous-wave laser at 658 nm wavelength [55]. The singly filtered BFCs are generated by sending the signal photon of the orthogonally polarized frequency-degenerate photon pairs at 1316 nm with a 245 GHz FWHM phase-matching bandwidth to three fiber FP cavities with 5.03 GHz, 15.11 GHz, and 15.15 GHz FSR, respectively. The idler photon is untouched. The cavity FWHM bandwidths are 0.46 GHz, 0.329 GHz, and 1.36 GHz, respectively. The generated singly filtered mode-locked BFCs state is then directed to a stabilized Franson interferometer for nonlocal interference measurements. After the Franson interferometer, coincidences are recorded with two commercial superconducting nanowire single-photon detectors (SNSPDs, ≈85% detection efficiency and root-mean-square timing jitter ≈55ps, PhotonSpot, Inc.). We first measure the quantum signature of Franson interference, the three temporally separated correlation peaks as presented in the inset of Fig. 1(a). We observe clear single-sided temporal second-order cross-correlation structures for all Franson correlation peaks. Details of the experimental setup for singly filtered mode-locked BFC sources can be found in Fig. 5 of Appendix A.

Our Franson interferometry is composed of two unbalanced MZIs with the path-length difference measured to be ΔT=4.84ns, supporting the probing of time-energy entanglement [37]. We use a thermal heater in the long path of arm1 to sweep and fine-tune the relative phase shift ΔT1 between the two MZIs to probe the time-energy entanglement of our singly filtered BFCs. In addition, a motorized stage delay in arm2 with ΔT2 up to 360 ps is used to select different time bins, and temperature controllers with long-term temperature stability of ≈1mK are implemented to stabilize our Franson interferometry.

Here, we analyze the role of cavity finesse F in the model of mode-locked singly filtered BFC, focusing on temporal second-order cross-correlation function and Franson interference recurrences based on our experimental setup. The mode-locked signal-idler temporal second-order cross-correlation function of singly filtered BFC can be written as [4,8,14] KS′I′cross(τ)=∫−∞∞dω2πiNSsinc(ωtcoh/2)e−4ω2ln(2)/Δω2eiω(τ−tcoh/2)1+i1πFsin(πω/ΔΩ)e−iπω/ΔΩ.(1)

Here, τ=tsignal−tidler; NS is the source brightness; tcoh=3.62ps is the coherence time of our SPDC source; and Δω/2π=225GHz is the FWHM bandwidth of the Gaussian bandpass filter (BPF). The key figure-of-merit cavity finesse F is given by F=ΔΩ2Δωc=πΔωcΔTcavity=πarcsin(1−1−lrt21−lrt4)=λQ2L,(2)where ΔΩ is the FSR of the cavity; Δωc is the cavity bandwidth; ΔTcavity is cavity round-trip time; lrt is the total round-trip optical losses including scattering losses, clipping losses, absorption and transmission losses; λ is the cavity mode wavelength; Q is the cavity quality factor; and L is the cavity optical length. For a linear planar FP cavity, the cavity finesse F is proportional to cavity quality factor Q.

By using Eqs. (1) and (2), we obtain the Franson interference recurrence envelopes with maximum and minimum coincidence counts as follows: $$\begin{gathered} CS′I′max(τ)=ηS′ηI′16∫0Tgdτ[(|KS′I′(cross)(τ)|2)+|KS′I′(cross)(τ−nΔTcavity)|2 \\ +2|KS′I′(cross)*(τ)KS′I′(cross)(τ−nΔTcavity)|2], \end{gathered}$$(3)and $$\begin{gathered} CS′I′min(τ)=ηS′ηI′16∫0Tgdτ[(|KS′I′(cross)(τ)|2)+|KS′I′(cross)(τ−nΔTcavity)|2 \\ −2|KS′I′(cross)*(τ)KS′I′(cross)(τ−nΔTcavity)|2], \end{gathered}$$(4)where ηS′,ηI′ include all losses encountered in our measurement setup and Tg is the coincidence counting window of 2 ns. Subsequently we can obtain the Franson interference recurrence visibilities at the nth time bin in a singly filtered BFC to be Vn=exp(−nπ/F).(5)

Equations (1) and (5) allow us to evaluate the impact of cavity finesse F on the signal-idler temporal second-order cross-correlation function and visibilities for the Franson interference recurrence numerically, enabling the comparison with our experimental measurements of singly filtered mode-locked BFCs.

Next, in Fig. 1(b), by using experimental setup in Fig. 5 of Appendix A, we measure the JTI of our 5.03 GHz FSR singly filtered BFC using large-alphabet encoding [29]. We sift the coincidence data into frames of duration d×Tbin, composed of d time bins of duration Tbin. We choose Tbin to be 16 ps to enhance the resolution of temporal correlations. This d of 128 shows that there are ∼4 clean cross correlations from a* to d* in our 5.03 GHz singly filtered BFC. The red arrow indicates both the cross sections of measured JTI [also shown in the inset of Fig. 1(b)], and the decay direction of singly filtered temporal waveform. The inset of Fig. 1(b) shows the corresponding g(2) measurements of the signal-idler temporal second-order cross-correlation function for the same singly filtered BFC. Using a 5.03 GHz FSR fiber cavity and commercial SNSPDs, we observe a clear comblike structure of the temporal correlation, resulting from the mode-locking process of our singly filtered BFC source. The time interval trt is measured to be ≈200ps, which matches well with a cavity FSR of 5.03 GHz (≈198.8ps). Cavity damping rate is extracted to be ≈458.7MHz from our measurements with the exponential decay of the mode-locked oscillations envelope, and this value closely matches the cavity bandwidth of 457 MHz [8,9]. Our result in the inset of Fig. 1(b) is consistent with our results in Fig. 1(b). This is the first experimental measurement of signal-idler temporal second-order cross-correlation function that probes the JTI in cavity-filtered BFC scheme using only commercial detectors, certifying the existence of phase coherence for our multimode quantum source. Such JTI measurements, when paired with JSI measurements [example JSI measurements for the same singly filtered BFC are shown in Fig. 1(c), where we scan multiple frequency bins within the phase-matching bandwidth of our biphoton source using a pair of tunable frequency filters], reveal genuine high-dimensional frequency-bin entanglement via providing mutual information of biphoton phase coherence [23].

We next perform Franson interference measurements for the 5.03 GHz FSR singly filtered BFC. Figure 1(d) shows the optimized maximum Franson interference of the central time bin in a 5.03 GHz FSR cavity. For this measurement, we sweep the thermal heater in arm1 to tune the relative phase-shift ΔT1 between the two MZIs, whereas, fixing the motorized stage position ΔT2 at the center of the optimum Franson interference. By postselecting coincidence events occurring within a 2 ns wide coincidence time window, while changing the phases of the interferometers (ΔT1), we obtain a maximum visibility of 97.79% for the 5.03 GHz FSR singly filtered BFC, after accidental coincidence subtraction. The Franson visibility is calculated based on (Cmax−Cmin)/(Cmax+Cmin), where Cmax is the maximum interference fringe coincidence counts and Cmin is the minimum interference fringe coincidence counts.

Due to the mode-locking process of our singly filtered BFC, the two-photon mode-locked oscillations in the temporal domain will result in discretized time-energy entanglement. Indeed, by using both 5.03 GHz and 15.15 GHz FSR singly filtered BFCs, we measure the Franson interference recurrences over two and six time bins, respectively, as presented in Fig. 2. For a 15.15 GHz FSR singly filtered BFC, we observe the Franson interference fringes visibilities to be 98.97%, 74.32%, 54.37%, 40.56%, 31.50%, and 22.32%, after subtracting the accidental coincidence counts as shown in Figs. 2(a)–2(f) (see Fig. 6 in Appendix B for the measured quantum signatures of Franson interferences for the 15.15 GHz FSR singly filtered BFC). Then, in Fig. 2(g), we compare our experimental results with theoretical modeling. In the insets of Fig. 2(g), we observe Franson interference fringes visibilities to be 97.79% and 72.32%, after accidental coincidence subtraction for the 5.03 GHz FSR singly filtered BFC. The number of measurable time bins is limited by both the FSR of the cavity and the motorized stage position ΔT2 in arm2 of our Franson interferometer. Besides, we also measure the noninteger cavity round-trip-time Franson interferences for the center time bin in both singly filtered BFCs. This is illustrated in Fig. 7 of Appendix B with an absence of the Franson interference fringes, which confirms the discretization of singly filtered BFCs due to mode-locking process. These JSI measurements are consistent with our JTI measurements in Fig. 1(b) of ≈200ps cavity round-trip time, confirming the mode-locked phase coherence behavior of our singly filtered BFC.

Interestingly, we find that although these two cavities have different FSRs and cavity bandwidths, they have similar cavity finesse F (F5.03GHz=10.93 and F15.15GHz=11.14), resulting in the close trend of the recurrence visibilities in the Franson interference, illustrated by bidirectional black arrows in Fig. 2(g) between experiment and theory. This indicates that the cavity finesse F plays a key role in determining the quality of time-energy entanglement over multiple time bins. By increasing this figure of merit, the falloff of Franson interference recurrence visibilities over multiple cavity round-trip times can be improved.

Figure 3 presents the modeling and measurement of the normalized signal-idler temporal second-order cross-correlation function and Franson interference recurrence visibilities for the two cavities. To better resolve the mode-locked oscillations, we set the FWHM timing jitter to be 20 ps, relevant to current state-of-the-art low-jitter superconducting single-photon detectors [56]. In Figs. 3(a) and 3(b), we plot the signal-idler temporal second-order cross-correlation function of 5.03 GHz [experiment is presented in Fig. 1(b)] and 15.15 GHz FSR cavities with different cavity finesses F for comparison. Both results show that when the cavity finesse F increases to 30 in cavities under the same FSR, the cavity damping rate becomes smaller, resulting in an increased probability to detect signal photons at multiples of the cavity round-trip time trt after idler photons arrive at the single-photon detector. We observe a similar trend for Franson interference recurrence visibilities (measurements results are given in Fig. 2) as shown in Figs. 3(c) and 3(d). There is a clear improvement of modeled Franson visibilities when the cavity finesse F is ≈3 times larger than our experimental results in Fig. 2. For the 5.03 GHz FSR singly filtered BFC, we find that a small cavity finesse F of 30 can result in all the Franson interference recurrence visibilities to be higher than the quantum-classical limit of 70.7% [37,57] in the 1 ns temporal range. For the 15.15 GHz FSR singly filtered BFC, the Franson visibility at the 16th time bin with cavity finesse F of 30 is ≈14-fold higher than that with cavity finesse F of 11.14. Due to the increasing probability of detecting mode-locked temporal oscillations at the single-photon detector with a cavity finesse F of 30, the Franson interference recurrence visibilities increase correspondingly. Here, via using a 45.92 cavity finesse F with a 15.11 GHz FSR singly filtered BFC, we experimentally demonstrate improved Franson interference recurrence. Figure 3(e) shows the Franson recurrence interference fringes with six discrete time bins measured in a 45.92 cavity finesse F singly filtered BFC. The experimentally measured (theoretical) visibilities for the six time bins (in red) for the 15.11 GHz FSR singly filtered BFC are 98.76% (100%), 92.69% (93.39%), 86.45% (87.21%), 80.64% (81.45%), 75.01% (76.06%), and 69.86% (71.03%), respectively. All measurements are performed with a coincidence window of 2 ns, and the reported visibilities are after subtracting accidental coincidences. This exemplifies that the cavity finesse F is a key parameter for optimizing both the temporal correlations and the Franson interference recurrence of the mode-locked biphoton states.

Cavity finesse F tailors the intrinsic joint biphoton properties of mode-locked singly filtered BFCs as shown earlier in the Franson interference recurrence and signal-idler temporal second-order cross-correlation function. For the above signal-idler temporal second-order cross-correlation functions and Franson interference recurrence, cavity finesse F plays a key role because it is proportional to the number of round trips before a single photon leaves the cavity or is lost via dissipation. Indeed, from Eq. (2), for our linear planar FP cavity, the cavity finesse F is proportional to cavity quality factor Q. Therefore, increasing the cavity finesse F is equivalent to increasing the cavity Q factor where photons can travel more round trips in the cavity, whereas, storing more energy before they are lost in dissipation or leave the cavity. This results in a lower cavity damping rate with the same FSR, therefore, increasing the detection probability in signal-idler temporal second-order cross-correlation function after multiple cavity round-trip times trt. This flattens the time-energy entangled Franson recurrence interferences. In our case, the design of FP cavity mostly depends on our measurement setups. The FSR decides the frequency-bin spacing, time interval trt, and Franson revival periods, whereas, our motorized stage delay in arm2 of Franson interferometry sets the measurable Franson interference recurrence. Therefore, in order to access multiple time and frequency bins of our singly filtered BFCs, we choose to use 5 GHz and 15 GHz FSR cavities in this paper. In our prior works, we implemented a larger cavity FSR for denser Franson revival interferences, whereas, having less frequency bins in our BFC [3,4]. The bandwidth of the cavity, which is related to its finesse F and FSR, can be chosen to achieve higher detection rate in signal-idler temporal second-order cross-correlation function and to approach flattened Franson revival interferences.

In Fig. 4, we present the temporal second-order cross-correlation peak values and Franson interference recurrences for the 6th and 16th time bin in 15 GHz FSR singly filtered BFCs for higher cavity finesse F. For mode-locked oscillations in the temporal domain, we plot signal-idler temporal second-order cross-correlation peak values versus different cavity finesse F of 11.14, 30, 50, 100, and 200 in a 15.15 GHz FSR cavity with an FWHM timing jitter of 20 ps as shown in Fig. 4(a). It can be seen that by using a moderate cavity finesse F of 200 for both the 6th and the 16th time bins, signal photons are more likely to be detected after idler photons are counted at the detector, resulting in higher temporal correlation peak values. This improved JTI may be useful for temporal-multiplexing using singly filtered BFCs. In Fig. 4(b), we illustrate the effect of cavity finesse F on Franson interference recurrences and time-bin Schmidt number KT with our experimental results of a cavity finesse F of 11.14 and a higher cavity finesse F of 45.92 superimposed (orange stars).

The improvement of Franson interference recurrence is described in Eq. (5). The measured sixth time-bin Franson visibility for a 15.11 GHz FSR cavity with a higher finesse F of 45.92 is 69.86% (71.03% for theoretical visibility) after subtracting accidental coincidences, which is very close to the visibility of quantum-classical limit of 70.7% (the black dotted line). Here, by using a cavity with higher finesse F of 45.92, we observe an experimental ≈3.13-fold improvement of the Franson visibility. All the other measurement data in 45.92 cavity finesse F singly filtered BFC are presented in Fig. 3(e). Besides, for the 16th time bin in the same BFC, there is a potential ≈24-fold improvement of the Franson visibility for a cavity finesse F of 45.92 compared to the Franson visibility with a cavity finesse F of 11.14. The light green and yellow bars represent the theoretical time-bin Schmidt number KT from the Franson recurrence visibilities up to the 6th and 16th time bins, respectively. Here, we define time-bin Schmidt number KT analog to the frequency-bin Schmidt number from literature [3,4,18,19]. In the time domain, we have a Schmidt number given by KT=(∑λn2)−1, where ∑λn=1 with {λn} being that domain’s Schmidt mode eigenvalues. Then, due to the difficulty in measuring joint temporal amplitude (JTA), we use pure state assumption [3,4] such that JTA equals the square root of the binned JTI. By sampling the temporal wave function of our singly filtered BFC, at τ=nΔTcavity, we can obtain its binned JTI as |Ψ(nΔT)|2=exp(−2nΔΩΔTcavity)∑n=0Nexp(−2nΔΩΔTcavity). Combining the JTI of our singly filtered BFC and Eq. (5), we can obtain time-bin Schmidt mode eigenvalues as λn=e−πn/F∑m=0Me−πm/F, for 0≤n≤M, where M+1 is the number of Franson recurrence interference time bins. Therefore, given Franson recurrence interference visibility up to the 6th (experiment) and 16th (theory) time bins for 15 GHz singly filtered BFCs, we can extract their corresponding time-bin Schmidt number KT. We note that this method can also be used for extracting the time-bin Schmidt number using Hong–Ou–Mandel revival interferences [2,3].

Deep red dotted lines are the experimentally measured sixth time-bin Schmidt number KT of 4.82 (4.92 from theory) for a 11.14 cavity finesse F, and the sixth time-bin Schmidt number KT of 5.91 (5.92 from theory) for a 45.92 cavity finesse F, respectively. As cavity finesse F increases to 200 for the 16th time bin, the time-bin Schmidt number KT is equal to ≈16 effective modes, which is because the Franson revival visibility approaches the optimum limit of 100%. As illustrated in Fig. 4(b), we found out the time-bin Schmidt number KT for a cavity finesse F of 200 is 15.92, and that for a cavity finesse F of 11.14 is 5.99. Hence, the effective temporal Hilbert space dimensionality, thus, potentially increases from 35 to 253. In Fig. 4, we choose to use cavities with FSR of 15 GHz because this FSR is closed to maximum FSR in cavity-enhanced SPDC sources [8,14,16,58–63]. Furthermore, this 15 GHz FSR cavity is compatible with the current fiber Bragg grating filters in optical communications for frequency-domain projection measurements [3]. Here, we note that our schemes are robust and flexible without requiring a stabilization system nor customized cavity design compared to cavity-enhanced SPDC sources [8,14,16,64–68]. Besides, in Fig. 8 of Appendix C, we also illustrate the effect of cavity finesse F on the temporal second-order cross-correlation function and Franson interference visibilities for a 50 GHz FSR singly filtered BFC. We note that the signal-idler temporal second-order cross-correlation function measurements are limited by the detection jitter of commercial single-photon detectors for 50 GHz FSR cavity, and they can be resolved by the Franson interference measurements experimentally accessible with current interferometric technology [2–5,69].

We have shown experimentally and theoretically that cavity finesse F acted on joint biphoton properties of mode-locked singly filtered BFCs, manifesting in the signal-idler temporal second-order cross-correlation function and Franson interference recurrence. We measured the similar trend of time-energy entanglement of 5.03 GHz and 15.15 GHz FSR mode-locked singly filtered BFCs via Franson interferometry with interference fringes observed over six and two time bins, and with optimum central time-bin accidental-subtracted visibility of 98.97% and 97.79%, respectively. Increasing the cavity finesse F from small to moderate values led to a lower cavity damping rate with the same FSR, increasing detection probability in the signal-idler temporal second-order cross-correlation function after multiple cavity round-trip times trt and flattening the visibility decay of the Franson interference recurrence in our singly filtered mode-locked BFCs. Via a higher cavity finesse F 45.92 singly filtered BFC, we experimentally present a ≈3.13-fold improvement of Franson visibility compared to the Franson visibility with cavity finesse F of 11.14 at the sixth time bin, close to the quantum-classical limit of 70.7%. Furthermore, for the 16th time bin in the same BFC, there was a potential ≈24-fold improvement of the Franson visibility for a cavity finesse F of 45.92 compared to the Franson visibility with a cavity finesse F of 11.14. By using a cavity finesse F of 200, we predicted a time-bin Schmidt number KT near 16 effective modes, which was coming from the near-optimum Franson interference recurrence for all the time bins. Our results paved the way for optimum Franson interference recurrence towards scaling up the time-bin Schmidt number KT and the Hilbert space dimensionality in the temporal domain, and for high-quality long-distance high-dimensional time-energy entanglement distribution and QKD.

Acknowledgment. The authors acknowledge discussions with Z. Xie, P. Fan, J. H. Shapiro, F. N. C. Wong, and discussions on the superconducting single-photon detectors with V. Anant, B. Korzh, A. Mueller, M. Spiropulu, and M. Shaw. K.-C.C. and X.C. developed the idea, performed the measurements, and conducted data analysis. K.-C.C. and M.C.S. contributed to the theory and numerical modeling. K.-C.C., X.C., and C.W.W. prepared the paper. All authors contributed to the discussion and revision of the paper.