Time-reversible and fully time-resolved ultra-narrowband biphoton frequency combs

Time-reversibility has proven to be ubiquitous in various branches of physics, including classical^1–15 and quantum systems.^16–27 For example, time-reversal techniques were exploited to optimize the efficiency of storage and retrieval of photons in atomic ensembles^3,4,10,17,19 and coherent perfect absorbers,^8,14 to realize near-unity absorption efficiency using microwave photons^21–23 and photonic crystals,⁷ and to realize superabsorption.¹⁵ Furthermore, time-reversal techniques have found many applications, such as strong focusing,^5,9 holography,¹² transfer of quantum state between distant optical¹⁸ and microwave cavities,²⁴ and quantum communication.²⁵ Among the time-reversible systems, the absorption of single photons by atomic ensembles is a fundamentally interesting problem and has important applications in quantum science because it forms the essential element of a quantum repeater, the quantum memories.^28,29 The standard atomic quantum memory protocols include electromagnetically induced transparency (EIT),^30,31 controlled reversible inhomogeneous broadening (CRIB),^32,33 gradient echo memory (GEM),¹⁷ and atomic frequency comb (AFC).^34,35 It is known that the quantum sources have to be resonant with atoms and exhibit narrow spectra that match the bandwidth of these quantum memories (typically between 0.1 and 100 MHz).^28,29,36 Nevertheless, the typical bandwidth of spontaneous parametric downconversion (SPDC)^29,37 and spontaneous four-wave mixing³⁸ spans a range of 0.1 to 100 THz. There are several orders of magnitude difference between quantum source and atomic quantum memories. An effective approach to address this challenge is to employ cavity-enhanced^26,39–57 or filtered schemes^58–62 to narrow the output spectrum of SPDC or through four-wave mixing on chip.^63–65 However, all previous SPDC and four-wave mixing sources have either lacked time-reversibility^{39–42,44–47,50–60} or are not compatible with the bandwidth of atomic memories.^{26,40–43,45,49,50,54,61,62} While the time-reversal waveforms of a singly resonant photon-pair source have been demonstrated,²⁶ the result only shows time-reversed waveforms of a single-frequency mode without multimode temporal oscillations.

In this study, we present the demonstration of a time-reversed, fully time-resolved, ultra-narrowband telecommunication biphoton source utilizing single-sided biphoton frequency combs (BFCs) across a massive number of over 5000 frequency modes. Our approach involves realizing reversing temporal oscillations in the biphoton source by a planar fiber Fabry–Pérot cavity (FFPC). This FFPC is effectively exchanged between the signal and idler channels within the single-sided BFCs. We directly observe fully time-resolved and time-reversed second-order cross correlation and joint temporal intensity (JTI) from our biphoton sources without any temporal broadening from the commercial single-photon detectors. The average measured cavity free-spectral range (FSR) and the linewidth of exponential-decay and time-reversed single-sided BFCs are $≈$42.66 MHz, $≈$4.60 MHz, $≈$42.67 MHz, and $≈$4.60 MHz, respectively. The time-reversibility and fully time-resolved temporal oscillations in our source can be useful for improving the quantum storage efficiency in atomic memories and completely utilizing the temporal information in multimode BFCs.

Subsequently, we conducted Hanbury Brown and Twiss (HBT) interference measurements to validate the generation of single photons using our time-reversed single-sided BFCs. We verified this for each cavity round trip, and we measured minimum heralded g⁽²⁾(0) = 0.013 ± 0.002 and g⁽²⁾(0) = 0.016 ± 0.002 for the zeroth round trip with our time-reversible singly filtered BFCs. The measured minimum heralded g⁽²⁾(0) for the first to fourth cavity round trips is all smaller than 0.5. We also observe minimum heralded g⁽²⁾(0) = 0.204 ± 0.002 and g⁽²⁾(0) = 0.212 ± 0.002 to further validate the multimode emission and preservation of single-photon generation in our source. Our biphoton source operates within the telecommunication band, and it exhibits the narrowest average cavity FSR of $≈$42.66 MHz and linewidth of $≈$4.60 MHz. Hence, our source is directly compatible with chip-scale AFC quantum memories^66,67 without conversion of quantum frequency.⁶⁸ By fully harnessing the potential of time-reversed single-sided BFCs in the frequency domain, our sources are expected to have an average number of 5786 and 5722 frequency modes, which can be useful for denser time–frequency quantum information processing and time–frequency multiplexed quantum storage toward long-distance quantum networks.

To experimentally investigate this approach, we utilized the following experimental setup in Fig. 1(a). For the pumping source, we utilize an external cavity diode laser (ECDL) operating at a wavelength of 780 nm. The entangled photons are produced through a type-II ppKTP waveguide. To eliminate the residual pump photons, we employ a long-pass filter (LPF) and a bandpass filter (BPF) with a 95% passband transmission (specifically, the Semrock NIR01-1570/3 filter). The signal and idler photons are then separated using a polarizing beam splitter (PBS). To reverse the temporal oscillations of singly filtered BFCs, we selectively pass either the signal photons or the idler photons through a FFPC. Figure 1(b) displays the fully time-resolved temporal waveforms of an exponential-decay single-sided BFC. For the exponential-decay single-sided BFC, the average cavity FSR is measured to be $≈$23.442 ns, which is corresponding to $≈$42.66 MHz. The average cavity linewidth is measured to be $≈$4.60 MHz, deriving from the theoretical fitting (black solid and blue dashed lines). In Fig. 1(c), we present the completely resolved temporal waveforms of a time-reversed single-sided BFC; its average cavity FSR is measured to be $≈$23.438 ns, which corresponds to $≈$42.67 MHz. The cavity linewidth is measured to be $≈$4.60 MHz, deriving from the theoretical fitting (black solid and red dashed lines). We note that the slight difference between the average cavity FSRs (⁠$≈$42.66 and $≈$42.67 MHz) for exponential-decay and time-reversed single-sided BFCs comes from the fitting error. We extract the average cavity FSR from measuring the temporal spacing between correlation peaks, which corresponds to the average of the spacing between every peak. Hence, here and throughout this work, we represent all our results with the average cavity FSR and linewidth, and the frequency mode number reported is calculated based on the average FWHM bandwidth of our SPDC source and the average cavity FSR. In Figs. 1(b) and 1(c), the inset illustrates the measured pump dependencies of single counts for signal and idler photons. The signal photons are represented by the blue symbols, while the red symbols represent the idler photons. These measurements were obtained from the central correlation peak in our time-reversible single-sided BFCs. In both single-sided BFCs, it is observed that the unfiltered idler (signal) photons exhibit higher single counts compared to the filtered signal (idler) photons. This discrepancy can be attributed to the filtering loss incurred by a FFPC. To ensure stable photon-counting measurements, we must regulate the temperature of the FFPC due to its ultra-narrow cavity FSR (corresponds to a cavity length of 2.04 m). Temperature control of the FFPC is performed through our custom-made double-temperature shielding layers, providing a temperature control stability of ∼1.6 MHz (⁠$≈$1 mK) over 24-h. Other detailed information of ultra-narrow FFPC used in this work is provided in the supplementary material.

Then, we proceed to measure the photon statistics of the exponential-decay single-sided BFC, as predicted in Eq. (2). The results of our measurements are presented in Fig. 2. It is important to note that in practical temporal measurements, the sharp peaks of the cross correlation function often appear broader or not resolvable due to the limited temporal resolution of the single-photon detectors.^{26,39,40,42–54,58–61} In our study, we utilize a cavity with an average FSR of $≈$42.66 MHz, enabling us to completely resolve the discretized JTI of the single-sided BFC without any temporal broadening. This result is performed using only commercial superconducting nanowire single-photon detectors (SNSPDs) with an approximate detection efficiency of 85% and a root-mean-square timing jitter of about 55 ps (Photon Spot, Inc.).

In Fig. 2(a), we record the coincidence counts of our exponential-decay single-sided BFC; then, we sift the coincidence data into frames of duration d × T_bin, comprised of d time-bins of duration T_bin. We choose T_bin to be 100 ps to enhance the resolution of temporal correlations. This d of 1400 shows that there are six clean second-order cross correlation peaks, from (a^∗) to (f^∗) in our $≈$42.66 MHz single-sided BFC. Each cross correlation peak represents the joint detection of biphotons corresponding to 0 to 5 cavity round trips. The red arrow indicates both the cross sections of measured JTI [also shown in Fig. 2(b)] and the direction of single-sided temporal waveforms. The inset of Fig. 2(a) is the example measured pump dependencies of coincidence counts from the central correlation peak in our exponential-decay singly filtered BFC. In Fig. 2(b), we show the measured temporal second-order cross correlation functions from the cross sections of JTI in Fig. 2(a). For Fig. 2(b), we normalize all the results with respect to the values of central correlation peak (a^∗), whose peak value is measured to be 68.114. Then, the normalized second-order cross correlation peak values are as follows: 1, 0.563, 0.349, 0.179, 0.121, and 0.075, respectively. The FWHM of the six correlation peaks is consistent with the root-mean-square timing jitter of our commercial SNSPDs. This observation suggests that the JTI of our biphoton source is fully time-resolved. We want to emphasize that there are alternative approaches for achieving full resolution of the JTI in multimode biphoton sources. One such method is the utilization of ultrafast coincidence counting techniques,⁷⁰ time-lens,⁷¹ or low-jitter SNSPDs.⁷² These methods typically require either complex setups or specially designed SNSPDs.

In consistent with Eq. (3), in Fig. 3(a), we experimentally observe a time-reversed single-sided BFC, with six reversed second-order cross correlation peaks in the measured JTI. With its $≈$42.67 MHz cavity FSR, we measure the fully time-resolved and reversed JTI via our single-sided BFC. The photon coincidence data are recorded and sifted into frames of duration d × T_bin. To be consistent with measurements in Fig. 2, we choose T_bin to be 100 ps and the d be 1400, and we show that there are six clean cross correlation peaks, from (a^∗) to (f^∗) in our $≈$42.67 MHz time-reversed single-sided BFC. The blue arrow indicates both the cross sections of measured JTI [also shown in Fig. 3(b)] and the direction of single-sided temporal waveforms. Interestingly, we can clearly observe the reversal in the direction of single-sided temporal oscillations, due to the asymmetric temporal waveform of the single-sided BFC. Our results in Figs. 1–3 are consistent with Eqs. (2) and (3). The inset of Fig. 3(a) is the example measured pump dependencies of coincidence counts from the central correlation peak in our time-reversed BFC. From Figs. 1–3, we measure and confirm that both the ratio of the single and coincidence counts is similar in our time-reversible singly filtered BFCs. Figure 3(b) shows the measured temporal second-order cross correlation functions from the cross sections of JTI in Fig. 3(a) for all six peaks. For Fig. 3(b), we normalize all the results with respect to the values of central correlation peak (a^∗), whose peak value is measured to be 73.420. Then, the normalized second-order cross correlation peak values are as follows: 1, 0.534, 0.320, 0.180, 0.120, and 0.076, respectively. In summary, in our time-reversible single-sided BFCs, the average cavity FSR ΔT is $≈$23.438 ns and the average effective temporal resolution of SNSPDs T is $≈$155.35 ps; therefore, we achieve an average ratio T/ΔT ≈ 0.0066, enabling us to experimentally fully resolve the JTI of our multimode biphoton sources using only commercial SNSPDs.

Next, we conducted measurements to examine the anti-bunching behavior of our time-reversed ultra-narrowband single-sided BFCs. In Fig. 4(a), we perform the HBT interference measurements via an exponential-decay singly filtered BFC. To access g⁽²⁾(0), we direct the idler photons to an HBT interferometer, heralded by the signal photons. The g⁽²⁾(0) value is measured from recording the threefold coincidence counts. Owing to the average cavity FSR (⁠$≈$23.442 ns) and the FWHM of each cross correlation peak (average $≈$155.6 ps), the accidental coincidence outside the correlated photons becomes dominant, leading to an increase in the value of g⁽²⁾(0). To address this, we adopt a coincidence window of 1 ns around each peak maximum.⁵¹ This choice ensures that the coincidence window remains within the boundaries of the individual coincidence peaks, effectively excluding events that do not originate from our single-sided BFCs. Through the implementation of this approach, we are able to accumulate a sufficient number of coincidence counts, which provides us with reliable single-photon statistics. The minimum heralded g⁽²⁾(0) = 0.013 ± 0.002 is measured for the zeroth round trip at 0.25 mW pump power at a heralding rate of 20 kHz. For the first, second, third, and fourth cavity round trips, the minimum heralded g⁽²⁾(0) is measured to be 0.092 ± 0.002, 0.164 ± 0.002, 0.208 ± 0.003, and 0.465 ± 0.003 with 0.3, 0.34, 0.4, and 0.45 mW, respectively. At low pump powers, the observed g⁽²⁾(0) values for all cavity round trips remain below 0.5. This indicates single-photon state generation from our exponential-decay single-sided BFC. The value of heralded g⁽²⁾(0) exhibits proportionality to the pump power, which can be attributed to the Poisson statistics. The increase in minimum heralded g⁽²⁾(0) for multiple cavity round trips vs the zeroth cavity round trip is a result of the decreasing signal-to-noise ratio observed for those specific cross correlation peaks.

In Fig. 4(b), we present a summary of the HBT measurement results obtained using a time-reversed single-sided BFC. In this experimental configuration, we directed the unfiltered signal photons into an HBT auto-correlation measurement setup, which was heralded by the filtered idler photons. We use the same coincidence window of 1 ns around each correlation peak as in Fig. 4(a). The minimum heralded g⁽²⁾(0) = 0.016 ± 0.002 is measured for the zeroth round trip at 0.28 mW pump power with a heralding rate at 24 kHz. For the first, second, third, and fourth cavity round trips, the minimum heralded g⁽²⁾(0) is measured to be 0.115 ± 0.002, 0.207 ± 0.002, 0.247 ± 0.003, and 0.496 ± 0.003 with 0.33, 0.38, 0.46, and 0.49 mW, respectively. Our results are consistent with the exponential-decay singly filtered BFC in Fig. 4(a). For low pump powers, the heralded g⁽²⁾(0) values for all cavity round trips are smaller than 0.5, and heralded g⁽²⁾(0) is proportional to the pump power. The increase in minimum heralded g⁽²⁾(0) for multiple cavity round trips is due to the same rationale for the exponential-decay single-sided BFC. To further confirm the presence of multimode emission and single-photon anti-bunching characteristics in our time-reversible single-sided BFCs, we further obtain minimum heralded g⁽²⁾(0) = 0.204 ± 0.002, and g⁽²⁾(0) = 0.212 ± 0.002 by using a larger coincidence window of 30 ns to include the zeroth and first cavity round trips for both exponential-decay and time-reversed BFCs (see the supplementary material for more details). Our results indicate that the single-photon quantum nature is preserved in our time-reversed single-sided BFCs.

After demonstrating completely resolvable and reversible of temporal envelopes in our single-sided BFCs, we proceed to compare our ultra-narrowband biphoton source with other studies in the literature. In Fig. 5, we present a summary of the cavity FSR and linewidth of selected cavity-enhanced SPDC sources and filtered BFCs, compared to our time-reversed ultra-narrowband biphoton source. The selected cavity-enhanced SPDCs and post-filtered BFCs include singly resonant,^{26,40,43,48,49} singly filtered,⁶² doubly resonant,^{39,42,44,45,47,50,52–55,57} doubly filtered,⁶⁰ and triply resonant^46,51 configurations. The FSR is expressed in GHz, while the linewidth is measured in MHz. These parameters characterize the multimode quantum sources and highlight the improvements achieved in terms of FSR and linewidth in our time-reversed fully time-resolved biphoton source.

The typical bandwidth of EIT-based quantum memory ranges from 0.1 to 100 MHz,^30,31 and that of CRIB/GEM and AFC-based quantum memories lies between several MHz to 100 MHz.^{17,28,29,32–36} Furthermore, the time-reversal characteristic of our biphoton sources may also be exploited to optimize the efficiency of storage and retrieval of photons in atomic quantum memories.^3,4,10,17,19 Hence, our time-reversed ultra-narrowband biphoton source can be compatible with these atomic quantum memory protocols. The present work has the narrowest FSR in a biphoton source, with an expected number of frequency modes reaching 5786 and 5722 (see the supplementary material for additional information). Furthermore, it has the longest coherence time of up to 217 ns among all the previously reported single-sided configurations. These results highlight the advancements of our biphoton sources made in terms of FSR, frequency mode count, and coherence time. These fully time-resolved multi-temporal modes can also be utilized for high-dimensional quantum information processing^54,55 or temporal-multiplexed quantum storage.⁶⁸ In addition, with the advancement of narrower bandwidth telecom frequency filters, or by using a cavity with a larger FSR,^60–62 it is also possible to implement frequency-multiplexed quantum storage⁵⁴ with dense frequency modes in our BFC sources.

Here, we also provide detailed discussions regarding the applicability and limitations of our method for potential applications. The post-filtered scheme has the flexibility to utilize cavities with different FSRs and linewidths. Typically, a smaller cavity FSR is better to encode more frequency modes given the fixed bandwidth of spontaneous parametric downconversion (SPDC) sources and to fully resolve temporal oscillations as performed in this work, but when considering the existing minimum bandwidth of telecom tunable filters (usually on the order of few GHz), this also limits the cavity FSR we can choose for frequency-bin encoding applications. Meanwhile, a larger cavity FSR (such as 50 GHz) is directly compatible with off-the-shelf multiplexing components; hence, it is more straightforward to implement frequency-bin encoding in this case; however, given the typical temporal resolution of single-photon detectors (on the order of dozens to hundreds of ps), such a temporal multimode feature cannot be detected easily. As for the cavity linewidth, given the fixed cavity FSR, a smaller linewidth corresponds to flatten temporal cross correlation and longer coherence time, while a larger linewidth leads to the opposite but with less filtering loss. Overall, we can summarize that our post-filtering schemes are robust and flexible without requiring a stabilization system nor a customized cavity design as required in cavity-enhanced SPDC sources, but the cavity-enhanced schemes can achieve higher source brightness without filtering. Finally, we note that the decreasing signal-to-noise ratio for temporal second-order cross correlation measurements (and JTI measurements) with multiple cavity round trips can be improved in the future by employing a cavity with higher finesse F, as confirmed in our prior work.⁷³ 

In this work, we have demonstrated a time-reversible, fully time-resolved ultra-narrowband telecommunication biphoton source using single-sided BFCs. With the ultra-narrowband single-sided BFCs, we realized the reversed and fully time-resolved multimode oscillations without using frequency filters or specially designed cavities. We experimentally observe time-reversed and fully time-resolved second-order cross correlation and JTI measurements via our single-sided BFCs using only commercial single-photon detectors. Here, we point out that the time-reversibility of our source could be useful for enhancing the efficiency of quantum storage in atomic memories. The completely resolved temporal oscillations in our source provide a possible route to harness the temporal information in single-sided BFCs fully. Then, we verified that the multimode emission and single-photon state generation are preserved in our biphoton sources with HBT measurements. To the best of our knowledge, our biphoton source has the narrowest average FSR of $≈$42.66 MHz and a linewidth of $≈$4.6 MHz, resulting in a dense frequency mode number of 5786. In addition, our source exhibits the longest coherence time of 217 ns among all the previously reported singly configurations. These characteristics of our presented sources highlight the advancements in FSR, frequency mode count, and coherence time in the biphoton sources. Our time-reversed and fully time-resolved ultra-narrowband telecommunication biphoton source can be helpful for advanced quantum information processing tasks, such as high-dimensional time–frequency entanglement, and practical quantum applications, such as efficient multimode quantum storage toward long-distance and large-scale quantum networks.

The supplementary material contains information on the joint temporal intensity of exponential-decay and time-reversed single-sided BFCs. It also contains detailed information about FFPC used in this work, data of the heralded second-order auto-correlation function for time-reversible single-sided BFCs using a coincidence window of 30 ns, and the measured frequency spectrum of SPDC and the modeled frequency spectrum for time-reversible single-sided BFCs.

The authors thank Vikas Anant for discussions on the superconducting nanowire single-photon detectors and Justin Caram for the help on the heralded auto-correlation measurements using a HydraHarp 400. This study was supported by the National Science Foundation under Award Nos. 1741707 (EFRI ACQUIRE), 1919355, 1936375 (QII-TAQS), and 2137984 (QuIC-TAQS) and the Army Research Office Multidisciplinary University Research Initiative (Grant No. W911NF-21-2-0214).

The authors have no conflicts to disclose.

K.-C.C. and X.C. contributed equally to this paper.

K.-C.C. developed the idea and conducted the measurements. K.-C.C. and X.C. performed data analysis. K.-C.C., M.C.S., and X.C. contributed to the theory and numerical modeling. X.C. and C.W.W. supported and discussed the studies. K.-C.C., X.C., and C.W.W. prepared the manuscript. C.W.W. supervised the project. All authors contributed to the discussion and revision of the manuscript.

Kai-Chi Chang: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Xiang Cheng: Investigation (equal); Methodology (equal). Murat Can Sarihan: Investigation (equal); Methodology (equal); Software (equal). Chee Wei Wong: Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.