Quantum light sources with configurable lifetime leveraging parity-time symmetry

Domain: physics
Published: 2025-11-27
Editorial outcome: minor_revision

Abstract
Quantum light sources with configurable photon lifetimes are essential for large-scale quantum circuits, enabling applications in programmable quantum computing, quantum communications, and quantum metrology. However, the fundamental trade-off between efficiency and photon lifetime imposes significant challenges on the design of high-performance, large configurable lifetime quantum light sources. Here, we report on such chip-scale quantum light sources by harnessing the unique feature of parity-time (PT) symmetry. The core design centers on employing PT-symmetric coupling between two microresonators of distinct circumferences, enabling a broad range and selective tuning of the intracavity photon density of states. By controlling the alignment between resonators, we achieved an near 20-fold photon lifetime tuning range (7.1±1.5~129.6±1.9 ps), with the shortest lifetimes near the exceptional point (EP). The device generates energy-time entangled photon pairs with 87.1±1.1% interference visibility and a heralded second-order autocorrelation of g(2)H(0) = 0.069±0.001. Our work highlights the potential of PT symmetry for advanced quantum applications, including high-speed communication and programmable quantum computing, quantum coherent tomography, and beyond.

Introduction
On-chip generation of diverse optical quantum states has been widely demonstrated, highlighting its considerable potential for advancing integrated quantum information processing applications1–8. As the field advances, the scalable and configurable generation and manipulation of quantum states across various degrees of freedom become increasingly essential5,6,9–19. Photon lifetime is one of the critical degrees of freedom in optical quantum states, predominantly determined by the band gap of a two-level system or the cavity quality factor (Q factor) of a resonator-based photon emitter20. Photons with long lifetimes, on the order of nanoseconds, typically exhibit relatively narrow linewidths, ranging from a few MHz to several hundred MHz. They are indispensable for aligning with the atomic transition linewidths to enable effective photon-atom interactions, crucial for quantum repeater-based long-distance quantum secure communications21–23. On the other hand, to maximize the rate of quantum communication systems, the lifetime of single-photons τ must be compressed, since the communication bandwidth is proportional to 2π/τ, as governed by Fourier reciprocity13. Short-lived photons are also required in quantum optical coherent tomography24,25. Furthermore, the ability to configure photon lifetime is vital for many applications. Dynamically tunable photon lifetime schemes have been utilized as quantum random number generators10. Moreover, sources with configurable photon lifetime are highly promising for frequency-encoded qubits, a key component in scalable quantum information processing technologies26.

On the integrated photonics platform, research has shown that the control of photon lifetime can be realized by integrated modulation modules to shape spectra of single-photon wave packets9,12–16. However, implementing modulation after photon generation complicates the quantum system, introducing extra losses. In recent years, optical microresonators with spontaneous nonlinear effects have been extensively demonstrated as quantum light sources1–8,27–36. It has been shown that different photon lifetimes can be achieved by setting distinct coupling gaps between the resonator and the bus waveguide20. Still, modifying the lifetime of a photon becomes challenging once the devices of a quantum light source are fabricated. More importantly, the fundamental trade-off between efficiency and bandwidth imposes significant constraints on the design of high-performance configurable photon lifetime quantum light sources. That is, to generate photons with a short lifetime, the duration of light-matter interaction must be reduced (related to an increase of the bandwidth of the cavity), which in return severely deteriorates the generation efficiency35,36. Therefore, it is crucial to find other photon lifetime adjustment schemes that do not sacrifice the efficiency or the photon-pairs generation rate (PGR).

Recent advances in parity-time (PT)-symmetric systems have attracted substantial attention due to their easy implementation yet profound implications in photonics37–41. The non-Hermitian property of PT-symmetric systems, especially at the exceptional point (EP), where the eigenvalues and corresponding eigenvectors coalesce, gives rise to a range of unconventional light-matter interaction phenomena42,43. These properties enable wide applications in areas such as lasing44, enhanced sensing45,46, coherent perfect absorption47, optical frequency combs48,49, and beyond. In addition, we have shown in a recent work that by leveraging PT symmetry and breaking the uniformity of the resonance linewidth of the coupled microcavities, the trade-off between efficiency and bandwidth has been relieved for classical applications such as nonlinear optical signal processing50. This indicated that the trade-off between short photon lifetime and large PGR may be relieved via a similar linewidth-manipulation method. Indeed, integrated resonance-manipulated devices51–54 have been studied for several potential quantum applications, such as spectral-uncorrelated photon-pairs generation51,52 or high-coincidence efficiency photon-pairs sources53. Nevertheless, achieving direct and large-range control of cavity photon lifetime remains elusive.

In this work, we demonstrate a lifetime-configurable quantum light source by leveraging a dual-ring coupled PT-symmetric system. The PT symmetry introduces an extra degree of freedom through unbalanced losses to control the decay rates of microresonators, making it naturally suitable for adjusting the lifetime of photons. The maximum loss contrast near and away from the EP enables orders of magnitude control of the photon lifetime. Meanwhile, the PGR has maintained a moderately high value thanks to the undegraded pump enhancement via the selective linewidth manipulation technique. Experimentally, we demonstrate a near 20-fold variation in photon lifetime, from 7.1±1.5 to 129.6±1.9 ps. Energy-time entangled photon pairs are generated, exhibiting a typical raw two-photon interference visibility of 87.1±1.1%. A typical heralded second-order autocorrelation g(2)H(0) of 0.069±0.001 is also obtained.

Results
System design and principle of operation
We demonstrate the configurable photon lifetime quantum light source using spontaneous four-wave mixing (SFWM) in a dual-ring coupled PT-symmetric system, as sketched in Fig. 1a. The main ring (red) with a radius R1 is coupled to the auxiliary ring (blue) with a radius R2. The intrinsic decay of the main and auxiliary ring is γ1 and γ2, respectively. The coupling strength between two rings is κ. The thermal tuning module is fabricated upon the auxiliary ring. The auxiliary ring is coupled to the bus waveguide with a coupling decay rate γc, resulting in a total decay rate γ2 + γc. When γc ≫ γ1,2, the decay rates of the two coupled rings become severely unbalanced, and a passive PT-symmetric system is thus formed47,50,55. Given that the radius ratio is set to R1 = 2R2, the free spectral ranges (FSRs) of the two rings (denoted as FSR1 and FSR2, respectively) differ by a factor of 2, as depicted by the intracavity photon density of state (DOS) in Fig. 1b. A thermal tuning module is fabricated upon the auxiliary ring to adjust the detuning between the two rings. When a voltage is applied to the thermal tuning electrode, the resonance of the auxiliary ring gradually aligns with that of the main ring, consequently reducing the Q factors of the corresponding resonances of the main ring (see Fig. 1c(ii) and d). The resonances influenced by this adjustment are designated as the signal and idler resonances, whereas the central resonance, which remains unchanged due to the FSR difference of the two rings, functions as the pump resonance (see Fig. 1d(iii)). The purpose of this design is twofold. On one hand, the symmetrically reduced Q factors of the signal and idler resonances result in shorter lifetimes of paired photons. On the other hand, the unaffected high-Q pump resonance ensures high SFWM efficiency, thereby maintaining a high PGR. Physically, this phenomenon arises because SFWM primarily occurs in the main ring, where the pump light is predominantly confined50, while the generated photons are emitted through the bus waveguide. The photon lifetime of the emitted photons is largely determined by the coupling decay introduced by the auxiliary ring.

Without loss of generality, we consider in the following the evolution of signal resonance since the signal and idler resonances act similarly. According to temporal coupled-mode theory (TCMT), the non-Hermitian Hamiltonian of the system reads41,50

H = [ω1 - i(γ1/2) κ; κ ω2 - i(γ2/2 + γc)] (1)

where ω1, ω2 are the resonant frequencies of the main ring and the auxiliary ring, respectively. Define δω = ω2 - ω1 as the detuning between the two rings. It is worth noting that the Hamiltonian is defined in a lossy PT-symmetric configuration by factoring out an average loss (γ1 + γ2 + γc)/4, in which the unbalanced losses compose an equivalent gain-loss mechanism in this passive PT-symmetric system47,55. Solving the complex eigenfrequencies of the Hamiltonian when two rings are aligned, i.e., under the condition of δω = 0, they can be expressed as41,50

ω± = ω1 - i(γ1 + γ2 + γc)/4 ± 1/4 * sqrt(16κ^2 - (γ2 + γc - γ1)^2) (2)

Discussion simplifies when the intrinsic losses of the two rings are identical, i.e., γ1 = γ2, which can be approximated in our design and fabricated devices. The EP occurs at the degeneracy point of complex eigenfrequencies, i.e., κ = γc/4 (detailed analyses are given in Supplementary Note 1).

Consider a common situation that occurs in coupled ring-systems when the coupling between two rings is strong and ring-bus waveguide coupling loss is weak, i.e., κ > γ1,2 ≫ γc10. The system exhibits a typical mode splitting at δω = 0, as mapped in Fig. 1c(i). In contrast, the loss of the coupling decay γc in our design is much larger than the intrinsic losses γ1,2. When the coupling coefficient is set to be κ ≈ γc/4, the system operates near the EP, as mapped in Fig. 1c(ii). A merged single resonance with a significantly low-Q factor is formed at δω = 0, which is naturally suitable for generating short-lived photons.

The photon lifetime adjustment process is achieved by changing the detuning between two rings using a thermal tuning module, as illustrated in Fig. 1a, b and d. Initially, the auxiliary ring with resonant frequency ω2 is misaligned with the main ring ω1. The signal and idler resonances can be regarded as resonances from a single ring under quasi-critical coupling with a linewidth of Δω = 2γ when detuning is large, e.g., δω ≫ FSR1/2 (the high-Q pump resonance functions in the same way. Detailed analyses refer to Supplementary Note 2)50,56. Therefore, the signal and idler resonances are in high-Q state in this case, as shown in Fig. 1d(i), and the photon lifetime is calculated by20,57,58.

τhigh-Q = 1/Δω = 1/(2γ) (3)

As the voltage applied to the thermal tuning electrode of the auxiliary ring gradually increases, the resonance of the auxiliary ring approaches that of the main ring (Fig. 1d(ii)) until they are completely aligned (δω = 0), as shown in Fig. 1d(iii). In case (iii), the signal and idler resonances operate near the EP59, with a linewidth of 2Im(ω+) ≈ γc, according to Eq. (2). At this point, the photon lifetime is defined as the inverse of the mode linewidth by

τlow-Q = 1/γc (4)

It can be derived from Eqs. (3) and (4) that the maximum achievable contrast of the photon lifetime is γc/2γ1. Thus, increasing the coupling decay rate of the auxiliary and the bus-waveguide or reducing the intrinsic loss of the main ring can enhance the contrast ratio in the tunability of the photon lifetime.

Device characterizations
We design the device on a low-loss integrated silicon nitride-on-insulator (SNOI) platform to experimentally demonstrate the configurable photon lifetime quantum light source. The radius of the main ring is designed as R1 = 228 μm (with a FSR close to 100 GHz to match the International Telecommunication Union’s (ITU) dense wavelength division multiplexer (DWDM) channels), and the radius of the auxiliary ring is R2 = 114 μm. The transmission spectra from 1534 nm to 1540 nm of the system under different voltages, i.e., 0 V and 6.6 V, are plotted in Fig. 2a, b, respectively (for details of spectra characterizations, see Methods and Supplementary Note 3). Under the 0 V bias, the two rings are misaligned, and the overall transmission spectrum exhibits a series of quasi-critical-coupled high-Q resonances from the main ring (FSR1 = 0.78 nm). Between each two high-Q resonances, there are auxiliary ring resonances (yellow lines) with a relatively low extinction ratio (FSR2 = 1.56 nm) due to the auxiliary ring being in a state of over-coupling with the bus waveguide. Under the bias of 6.6 V, the resonances of the main ring and the auxiliary ring are aligned, resulting in a distribution of alternating low-Q and high-Q resonances in the transmission spectrum. Resonances that are three FSR away from the pump resonance on both sides are selected as the signal and idler resonances to avoid imperfect pump rejection filtering, which might cause large noise photon counts. To be concrete, the resonance centred at 1536.9 nm (C51 DWDM channel) is chosen as the pump resonance. The resonances centred at 1534.6 nm (C48) and 1539.3 nm (C54) are chosen as the signal and idler resonances, respectively, as shown by the dashed areas in Fig. 2a, b.

The Q factors of all resonances are fitted and marked by the green (high-Q) and orange (low-Q) circles (the right axis). Before and after thermal tuning, the Q factors of the resonances in the main ring that coupled to the auxiliary ring exhibit a significant reduction, whereas the unaffected resonances keep a similar high-Q factor under quasi-critical coupling. According to TCMT, the system parameters are extracted by the fittings: γ1 = γ2 = 3.0 GHz, γc = 146.8 GHz, κ = 45.5 GHz. It can be seen that κ ≈ γc/4 is approximately satisfied, indicating that the system operates near the EP.

Correlated photon-pairs generation
Next, we characterize the photon lifetime of the correlated photon pairs generated from the lifetime configurable device. Figure 3a plots the experimental setups for photon-pairs generation. The pump light from a continuous wave (CW) laser is passed through an erbium-doped fibre amplifier (EDFA), a variable optical attenuator (VOA), and a polarization controller (PC), and then into a 90/10 beam splitter (BS). 10% of the pump enters a power metre (PM) to monitor the input power entering the chip. The remaining 90% of the pump is filtered out using a high-isolation (with an isolation of ≥120 dB) band-pass DWDM at the C51 channel and then coupled into the chip. The band-pass DWDM is employed here to filter out noise photons at the signal and idler wavelengths of the preceding system. The second band-stop DWDM is set after the generation of correlated photon pairs as the pump rejection filter. Figure 3b gives the test system of correlated photon pairs. A tunable bandwidth filter (TBF) is applied to separate the signal and idler wavelengths, and then send the photons to the superconducting nanowire single-photon detectors (SNSPDs, with a detection efficiency of about 90% and a dark count rate of about 30 Hz). The output signals of SNSPDs are further connected to a time-to-digital converter (TDC) to record the coincidence events.

The coincidence counts (CCs) of generated photon pairs under the bias voltages of 0 V and 6.6 V are recorded in Figs. 4a, b, respectively. A 1000-time Monte Carlo simulation is used to fit the coincidence count histograms32. A composite distribution of exponential and Gaussian functions is applied to numerically examine the lineshapes60,61, since the timing jitters of the detection systems are included in the coincidence peaks. The total standard deviations of the two counting histograms are w1 = 153.7 ± 1.3 ps and w2 = 83.0 ± 1.0 ps, respectively. Accordingly, the fittings giving the lifetimes τ of long and short-lived photons are 129.6 ± 1.9 ps and 7.1 ± 1.5 ps, respectively, and the timing jitter of the detection system σ is 82.8 ± 0.9 ps. The uncertainties are extracted from the Monte Carlo simulations. An near 20-fold photon lifetime adjusting contrast is achieved, matching the Q factor variations in spectral tests given earlier (detailed analyses of the extraction of photon lifetime are given in Supplementary Note 4). From TCMT, the expected photon lifetimes before and after the alignment are τhigh-Q = 167.7 ps and τlow-Q = 6.5 ps, respectively, according to the relation between decay rate and photon lifetime (Eqs. (3) and (4)). Here, the photon lifetime predicted for long-lived photons deviates from the experimental results. This discrepancy arises because, in the long-lifetime regime, the additional losses introduced by the auxiliary ring to the main ring are neglected in TCMT. Nevertheless, the expressions derived from the TCMT can be used as a simple model to roughly approximate the tunable range of the photon lifetime.

Next, Fig. 4c, d record the peak values of CCs with respect to the on-chip pump powers, from 0.25 mW to 14.12 mW. The CC rates scale with the square of the pump power, aligning with the theoretical predictions for the PGR in SFWM35. The error bars are derived from Poissonian photon-counting statistics28,32. The corresponding coincidence-to-accidental ratios (CARs) are then recorded in Fig. 4e, f. Under the 0 V bias, the maximum CAR reaches 31, while under the 6.6 V bias, the CAR reaches 170.

It is noteworthy that the coincidence count rate increases surprisingly, even though the Q factors of signal and idler resonances are decreased when comparing the states before and after the rings are aligned. This can be attributed to the over-coupling of signal and idler resonances with an enhanced photon escape probability, which can be effectively modelled by taking into account the different Q factors of evolved three resonances in the SFWM process35,36,62–64. Theoretical calculation reveals an on-chip PGR of high-Q case (0 V bias) of 85 kHz (under 14.12 mW on-chip pump); and a higher PGR of 140 kHz is expected after the linewidth manipulation (6.6 V bias), agreeing well with the experimental results in Fig. 4c, d (~1.65 times larger PGR in the linewidth-manipulated case), excluded the system losses and detection efficiency of SNSPDs (see Methods and more details in Supplementary Note 4).

Energy-time entanglement and heralded single-photon generation
We perform the energy-time entanglement of the short-lived correlated photon pairs generated by CW pumping. The quantum entanglement can be characterized through a Franson-type two-photon interference experiment, by violating Bell’s inequality65,66. As shown in Fig. 3c, we measure two-photon interference fringes using a folded Franson interferometer configuration implemented via an unbalanced Michelson interferometer (UMI)28,32,67,68. This arrangement can be regarded as a folded version of the standard Franson interferometer, in which the two unbalanced interferometers for the signal and idler photons are effectively combined into a single interferometer, employing the symmetry of the original configuration. Figure 5a, b give the CCs of destructive and constructive two-photon interference, respectively. The single-side count rates of signal and idler photons are recorded in the upper panel of Fig. 5c, which are near constants (about 40.9 kHz and 39.6 kHz on average, respectively), indicating that single-photon interference does not occur. The measured interference fringe with respect to the phase is further shown in the lower panel of Fig. 5c. The green circles are the coincidence events in 20 seconds, and the dashed green curve is the fit using a 1000-time Monte Carlo method28,32. The interference visibility is calculated as 87.1 ± 1.1% (without subtracting the accidental coincidence counts), which exceeds 70.7%, indicating effective energy-time entanglement violating the Bell inequality65,66,69.

Lastly, the heralded single photons can be obtained based on the photons generated in pairs. The heralded second-order autocorrelation g(2)H(τ) is measured by a Hanbury Brown–Twiss (HBT) interferometer70, as shown in Fig. 3d. In the experiment, the signal photons are detected directly by SNSPD, while the idler photons are detected with a delay time after passing through the 50:50 beam splitter, and the threefold coincidence events are recorded by TDC. The g(2)H(τ) of heralded single photons (idler) and their heralding photons (signal) is given in Fig. 6. The zero-delay value g(2)H(0) is measured to be 0.069 ± 0.001 with a heralding rate of 256 kHz under an 11 mW on-chip pump, indicating that it can be used as a short-lifetime heralded single-photon source.

Methods
Chip fabrication
The chip is fabricated by LIGENTEC (Switzerland) using a standard thick silicon nitride waveguide process. The waveguide dimensions are 1000 nm × 800 nm (width × height), supporting the fundamental mode only.

Spectra characterizations
A tunable continuous wave (CW) laser with an output power of −15 dBm is scanned from 1520 nm to 1580 nm to characterize the spectra of the coupled ring system. The fibre-to-chip coupling loss is estimated to be 4 dB per facet. A complete analysis of spectra and Q factor fittings can be found in Supplementary Note 3.

Coincidence measurements
For coincidence measurement between the signal and idler photons, we use a high-resolution time-to-digital convertor (TDC) with a time resolution of 1 ps. The timing jitters of signal and idler channels are tested to be σ1 = 68.4 ± 0.3 ps and σ2 = 49.1 ± 0.3 ps, and the total timing jitter in coincidence measurement can be calculated as σtest = sqrt(σ1^2 + σ2^2) = 84.1 ps (see Supplementary Note 4 for detailed test), which is very close to the fittings of coincidence histograms. The CAR values in Fig. 4c, d are calculated by CAR = (C - A)/A, where C and A are the overall and accidental coincidence counts obtained from the peak and background of the coincidence counting spectra.

Photon-pairs generation rate
We theoretically calculate that a PGR = 85 kHz in the high-Q case (0 V bias), and a PGR = 140 kHz in the linewidth-manipulated case (6.6 V bias), respectively, under 14.12 mW pump power, using the following method35,36,62,63

PGR = 8 * ω * v_g^3 * γ^2 * L * P^2 * Q_p^4 * Q_s=i^3 / (e_p^4 * e_s=i^2)

where ω is the resonance frequency, v_g is the group velocity, γ is the nonlinear coefficient, P is the pump power, L = 2πR1 is the circumference of the main ring, respectively. Q_p is the loaded Q factor of the pump resonance, Q_s=i is the loaded Q factor of the signal (idler) resonance, Q_e,p is the external Q factor of pump resonance, Q_e,s=i is the external Q factor of the signal (idler) resonance, respectively. Given that the intrinsic Q of the resonator is Q_int, the relation between loaded Q, and external, intrinsic Q is 1/Q_loaded = 1/Q_int + 1/Q_e. The Q factors are obtained from the spectral analyses (please refer to Supplementary Note 3). The key parameters used here are: ω = 2πc/λ (where c is the light velocity in vacuum, λ = 1535 nm is the corresponding wavelength), v_g = c/2.1 ms^-1, γ = 1 W^-1m^-1, P = 14.12 mW, L = 2π × 228 μm.

Data availability
All the data used for plots generated in this study have been deposited in the database under accession code https://doi.org/10.6084/m9.figshare.30400042.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "In this paper, the authors demonstrated on-chip cavities with controllable Q-factors, enabling photon lifetime tuning over an order of magnitude. This control was achieved through thermal tuning of an auxiliary cavity’s resonance, effectively manipulating the coupling between cavities. The free spectral ranges of the two cavities were designed such that photon lifetime shortening occurs only for the signal and idler photons. The results are intriguing and the paper is well-written overall. I would recommend it for publication, provided the authors address the following concerns:\n1. The shortest photon lifetime reported is 4.1 ps, extracted from measurements with timing jitter around 70 ps. The authors should include an estimate of the uncertainty or error margin associated with this value.\n2. The authors claim that for high-Q resonances, the signal/idler lifetime depends solely on γ₁. However, even assuming perfect anti-resonance in the auxiliary ring, it is unclear why the coupling strength κ has no effect. In particular, the condition κ = γ_c/4 must be met to reach the exceptional point for the low-Q regime. In such a case, how is the effective critical coupling of the main ring maintained for the high-Q case?", "reviewer_2": "I reviewed the manuscript by N. Chen and co-workers entitled \"Quantum light sources with configurable lifetime leveraging parity-time symmetry\". The key result is that they apply a linewidth manipulation technique based on PT-symmetric coupling between two microring resonators side coupled to a bus waveguide to change the quality factor of the resonance enhancement experienced by signal and idler modes involved in a spontaneous four wave mixing process happening in the structure, without affecting the quality factor of the resonance employed for the pump mode. In particular, by tuning the auxiliary microring, which has half the length of the main one and thus twice the FSR, they can couple the two microrings' resonances and bring the coupled system to work at an exceptional point, where the quality factors of the signal and idler modes are significantly lower than the quality factor of the pump mode, not affected.\nOver the past 10 years, there has been a lot of work in the field of integrated nonlinear quantum photonic sources to engineer the properties of the generated quantum light to improve and tailor their capabilities to the needs of different quantum applications. Nice results have been reported but still work remains to be done. This work is the first report, to the best of my knowledge, of an integrated quantum light source in which the bandwidth-efficiency trade-off is addressed by means of a non-Hermitian structure, which is relatively simple to operate. And most notably, the measured tuning range is already pretty large but could be extended, which invites to apply this mechanism to other foreseable features of these quantum light sources that people typically work with and want to advance.\nOverall, the paper is well written, clear and pretty accessible for the reader. Theretical and experimental work carried out looks sound. My main concern is about the analysis of the experimental results, and therefore the interpretation of the exact behaviour of the source as well as conclusions drawn. I expect the authors to address the main points raised below before the manuscript can be recommended for publication in Nature Communications.\nThe first aspect to notice are the numbers of the quality factors and lifetimes, in comparison. I follow the experimental method the authors apply to assess the actual lifetime, however the results they get for photons lifetime do not seem to agree exactly with the quality factors plot in Figure 2. Unfortunately the quality factors are not given, but at maximum I can see a 18 or 20-fold reduction between 0 V and 6 V. I invite the authors to make this point more clear and sound, e.g. by including in Figure 4a and 4b an inset with the measured resonances, as reported for example in ref.20.\nThe second aspect to notice is the explanation of the changes in the PGR between 0 V and 6 V. As there is a reduction of signal and idler quality factors by more than a factor 10, one would expect a significant reduction in PGR, but surprisingly the PGR gets even larger, as well as the CAR does. The authors provide only qualitative reasons which are insufficent in my opinion. I invite the authors to apply the theory for SFWM in microrings involving modes of different quality factors, as it is presented e.g. in the paper by L. Zatti et al. PHYSICAL REVIEW A 107, 013514 (2023). The authors should find some agreement between their data and the theory, which has already been verified in many experiments, and therefore be able to clarify what is going on exactly.\nFinally, related to the Franson experiment to prove the energy-time entanglement, some information must be added. Despite the fact that a proof less prone to loopholes issues would require two distinct unbalanced interferometers, one for signal and one for idler photons, in my opinion the experiment presented by the authors is still fine as there has been already plenty of other experimental confirmations. However, they should explain it clearly to the reader, add references for the type of experiment and better explain which type of entanglement they are measuring and how this is related to the original energy-time entanglement. At the moment, the unexperienced reader is left basically with no explanation and a lot of understatements. This can be fixed by adding a short paragraph with references.\nMinors:\nline 208, exponential not exceptional", "reviewer_3": "This article describes a method of generating quantum light with spontaneous four wave-mixing where the signal/idler modes have adjustable lifetimes. The system consists of two ring resonators with high Q contrast, and the coupling between the resonator creates supermodes with detuning-dependent linewidths. The authors claim a measured 38-fold photon lifetime tuning range between the decoupled modes and the coupled modes near the exceptional point. Other characterizations of correlation properties such as energy-time entanglement and g2 have also been measured.\nThe usage of EPs for controlling linewidth has frequently been applied to classical sensing applications but is a relatively new concept in quantum light generation. The experimental methods are overall solid, and necessary quantities have been measured. However, the theoretical descriptions of the method are inconsistent and confusing. There are also potential problems with data interpretation that are related to the central claim of the results. In view of the above, I could not recommend publication in its current state.\n- Using “PT-symmetric” to describe the system is questionable in at least two aspects. First, both modes of the system are lossy, although one may argue that the common loss can be subtracted from the effective Hamiltonian to resemble a gain-loss system. Second, since the two resonators differ in size, the P operation could not be properly defined in the real space (especially in the TMM framework).\n- The definition of “photon lifetime” becomes ambiguous if two or more optical modes are within the vicinity of each other. This is especially true at the EP, where the mode spectrum is no longer Lorentzian but super-Lorentzian [see e.g., Nat. Commun. 13, 3281 (2022)]. The authors should clarify this. In addition, what is the TMM result for the mode FWHM at the EP and does it agree with Eq. 4?\n- The measurement presented in Fig. 4b deviates significantly from exponentially decaying profiles and the fitting shows systematic error. For example, directly reading the 1/e width off the data would give a larger result of \\tau. This is potentially related to the previous point where the lineshape of the mode deviates from a Lorentzian. In addition, the authors should present error analysis on the obtained \\tau numbers since these are obtained by subtracting the jitter contributions and are extremely close to 0.\n- What controls the location of the signal and idler modes (C54 and C48)? If SFWM occurs in the main ring and there are uncoupled modes in the main ring with higher Q (e.g. at C53 and C49), why does the SFWM process not prefer these modes?\n- Line 208: I’m not sure what “double exceptional function” means, and I assume it should be a “double exponential function” or similar. Same for Line 204 in the Supplementary Material."}, "author_response": "Response to Reviewers Letter – manuscript NCOMMS-25-15128\nQuantum light sources with configurable lifetime leveraging parity-time symmetry\nIn this Letter of responses to reviewers, we have addressed all the comments from\nreviewers point by point. Our responses are highlighted in blue. The modifications\nmade in the revised manuscript and supplementary information, including main text,\nfigure captions, references etc., are all enclosed in boxes and highlighted in red.\nReviewer's comment:\nIn this paper, the authors demonstrated on-chip cavities with controllable Q-factors,\nenabling photon lifetime tuning over an order of magnitude. This control was achieved\nthrough thermal tuning of an auxiliary cavity’s resonance, effectively manipulating the\ncoupling between cavities. The free spectral ranges of the two cavities were designed\nsuch that photon lifetime shortening occurs only for the signal and idler photons. The\nresults are intriguing and the paper is well-written overall. I would recommend it for\npublication, provided the authors address the following concerns:\nAuthor's reply:\nWe sincerely thank the reviewer for summarizing our work and giving positive\nfeedback on the results. A point-by-point response addressing all the concerns is\nprovided below.\nReviewer's Q1:\n1. The shortest photon lifetime reported is 4.1 ps, extracted from measurements with\ntiming jitter around 70 ps. The authors should include an estimate of the uncertainty or\nerror margin associated with this value.\nAuthor's A1:\nAfter carefully reading all the comments from all the reviewers, we were advised that\nthe curve fittings regarding the lifetime of short-lived photons must be improved since\nthey deviate from data points, as shown in the left panel of the following figure (i.e.,\nFig. 4b in the original manuscript):\nFIG.4b Original fitting (exponential function) vs. revised Monte Carlo simulations\n(composite distribution of exponential and Gaussian functions)\nThis is because the exponential function fails to describe a large timing jitter of\nthe detection equipment, which is mainly affected by thermal and electrical noises. This\nsystematic issue caused an overestimated result of the short-lived photon lifetime, i.e.,\n4.1 ps, in the original manuscript.\nIn order to address this issue, a 1000-time Monte Carlo simulation is used,\napplying a composite distribution of exponential and Gaussian functions to fit the\n2\nshort-lived photon lifetime, following the methods in Sidorova et al., Phys. Rev. B 96,\n184504 (2017) and Wen et al., Phys. Rev. Applied 18, 024059 (2022):\n|𝑡−𝑡 | (𝑡−𝑡 )2\n0 0\nℎ(𝑡)=𝐴exp(− )∙exp(− )\n𝜏′ 2𝜎′2\nThe superimposed Monte Carlo curves are shown in the right panel of the above\nfigure (the revised Fig. 4b), which matches well with the measured data points.\nUncertainty estimations are obtained by the Monte Carlo method during the extraction\nof all the photon lifetimes and timing jitters of detection systems. Now, a 7.1 ± 1.5 ps\nshort-lived photon lifetime is redetermined. The value of 7.1 ps also agrees well with\nthe theoretical estimations from the Q factors of the signal and idler resonances (6.5\nand 9.8 ps, Table. S2 in supplementary information S2) in spectral analyses. This\nrelatively large uncertainty (~20 %) is because the contribution of the short photon\nlifetime to the total variance of the coincidence count histogram is minimal.\nThe revised part is shown in the following:\n(Line 210 in revised manuscript)\nThe coincidence counts (CCs) of generated photon pairs under the bias voltages\nof 0 V and 6.6 V are recorded in Fig. 4a and 4b, respectively. A 1000-time Monte\nCarlo simulation is used to fit the coincidence count histograms. A composite\nfunction of exponential and Gaussian distribution is applied [60,61], since the timing\njitters of the detection systems are included in the coincidence peaks. The total\nstandard deviations of the two counting histograms are 𝑤 = 153.7±1.3 ps and\n1\n𝑤 = 83.0±1.0 ps, respectively. Accordingly, the fittings giving the lifetimes \\tau\n2\nof long and short-lived photons are 129.6 ± 1.9 ps and 7.1 ± 1.5 ps, respectively, and\nthe timing jitter of the detection system 𝜎 is 82.8 ± 0.9 ps. The uncertainties are\nextracted from the Monte Carlo simulations. A near 20-fold photon lifetime adjusting\ncontrast is achieved, matching the Q-factor variations in spectral tests given earlier.\nReviewer's Q2:\n2. The authors claim that for high-Q resonances, the signal/idler lifetime depends solely\non γ₁. However, even assuming perfect anti-resonance in the auxiliary ring, it is unclear\nwhy the coupling strength κ has no effect. In particular, the condition κ = γ_c/4 must be\nmet to reach the exceptional point for the low-Q regime. In such a case, how is the\neffective critical coupling of the main ring maintained for the high-Q case?\nAuthor's A2:\nThis is indeed a counterintuitive phenomenon. Due to FSR difference of the two rings,\nhalf of the high-Q resonances that exist in the main ring satisfy anti-resonance condition\nin the auxiliary ring (an example is shown by the resonances given in the blue dashed\nbox in the following figure). In other words, no effective modes are coupled from the\nauxiliary to the main ring. These modes composed the effective pump mode and we use\nthe transfer matrix method (TMM) to describe the performance of the pump mode\nwithout approximation (see Supplementary information S1). That is to say, the CMT is\n3\nused in this work to describe the mode coupling effect in the signal and idler modes (as\nshown by the red dashed box).\nBefore going into mathematical details, the quasi-critical coupling of the main ring\nfor the high-Q resonances can be understood in a simple way. At the anti-resonant\nfrequency, the auxiliary ring can be regarded as a section of waveguide, and the pump\nwave is coupled into the main cavity by firstly being coupled from the bus waveguide\nto the auxiliary ring and then coupled from the auxiliary ring to the main ring, resulting\nin an effective coupling coefficient that is roughly the product of the two coupling\ncoefficients. Since each coupling coefficient is smaller than 1, the product of the two\nbecomes even smaller (equals about 0.23, derived in Supplementary information S2).\nThe relatively small coupling coefficient leads to a quasi-critical-coupling condition,\nresulting in the high-Q resonances appearing.\nFor a more rigorous analysis, we theoretically investigate this effect by applying\nthe method, e.g., Hafezi et al., Nature Photonics 7,12 (2013) (we now add this paper to\nthe manuscript as Ref. [56]), i.e., TMM, to analyze system responses without\napproximation. The results are shown in Supplementary information S2. We then\ndeveloped an equivalent single ring to model the high-Q resonances at anti-resonance\nconditions. The auxiliary ring and bus-waveguide are treated as an equivalent\nwaveguide directly coupling to the main ring, with an effective coupling coefficient\n𝑢′ = 0.142 (Eq. (S13), close to critical coupling condition, note that the critical-\nc\ncoupling condition in this case is 𝑢 = 0.17 ), and a hopping phase is\ncritical−couping\ndetermined by 𝜙′ = 𝜙 /2 , where 𝜙 is the round-trip phase accumulation of the\n2 2\nauxiliary ring.\nTherefore, by designing the device parameters (𝛾 ,𝛾 and 𝜅 in CMT), we can\n1 𝑐\nmanipulate the high-Q resonances to operate near the quasi-critical-coupling condition,\nat which the linewidth of the resonances is about 2𝛾 . More details can also be found\n1\nin Kim et al., elight 4,6 (2024).\nWe now add a notice to readers referring to Supplementary information S2 in the\ncontext of describing the linewidth of the high-Q resonances:\n(Line 146 in revised manuscript)\n… The signal and idler resonances for long-lived photons and high-Q pump\nresonances can be regarded as resonances from a single ring under quasi-critical\ncoupling with a linewidth of ∆𝜔 = 2𝛾 when detuning is large, e.g., δ𝜔 ≈ 𝐹𝑆𝑅 /2\n1 1\n(the high-Q pump resonance functions in the same way. Detailed analyses refer to\nAlso, Supplementary information S2 is revised to discuss all the high-Q\nresonances, unlike in the original version, where only the pump resonance is considered.\n(Line 98 in revised Supplementary information)\nIn the case of all the high-Q resonances, including all the resonances under 0 V bias\nand pump resonances under 6.6 V bias, we use an equivalent single ring to model the\ncoupling state. High-Q resonances feature a high extinction ratio (ER) and a narrow\nlinewidth, which is close to the critical coupling conditions. This is …\nReviewer's comment:\nI reviewed the manuscript by N. Chen and co-workers entitled \"Quantum light sources\nwith configurable lifetime leveraging parity-time symmetry\". The key result is that they\napply a linewidth manipulation technique based on PT-symmetric coupling between\ntwo microring resonators side coupled to a bus waveguide to change the quality factor\nof the resonance enhancement experienced by signal and idler modes involved in a\nspontaneous four wave mixing process happening in the structure, without affecting the\nquality factor of the resonance employed for the pump mode. In particular, by tuning\nthe auxiliary microring, which has half the length of the main one and thus twice the\nFSR, they can couple the two microrings' resonances and bring the coupled system to\nwork at an exceptional point, where the quality factors of the signal and idler modes are\nsignificantly lower than the quality factor of the pump mode, not affected.\nOver the past 10 years, there has been a lot of work in the field of integrated\nnonlinear quantum photonic sources to engineer the properties of the generated\nquantum light to improve and tailor their capabilities to the needs of different quantum\napplications. Nice results have been reported but still work remains to be done. This\nwork is the first report, to the best of my knowledge, of an integrated quantum light\nsource in which the bandwidth-efficiency trade-off is addressed by means of a non-\nHermitian structure, which is relatively simple to operate. And most notably, the\nmeasured tuning range is already pretty large but could be extended, which invites to\napply this mechanism to other foreseeable features of these quantum light sources that\npeople typically work with and want to advance.\nOverall, the paper is well written, clear and pretty accessible for the reader.\nTheoretical and experimental work carried out looks sound. My main concern is about\nthe analysis of the experimental results, and therefore the interpretation of the exact\nbehaviour of the source as well as conclusions drawn. I expect the authors to address\nthe main points raised below before the manuscript can be recommended for\npublication in Nature Communications.\nAuthor's reply:\nWe sincerely thank the reviewer for the summary of our work and the comments on the\nnovelty and impact. We have revised our manuscript carefully according to all the\nconstructive advice.\nReviewer's Q1:\nThe first aspect to notice are the numbers of the quality factors and lifetimes, in\ncomparison. I follow the experimental method the authors apply to assess the actual\nlifetime, however the results they get for photons lifetime do not seem to agree exactly\nwith the quality factors plot in Figure 2. Unfortunately the quality factors are not given,\nbut at maximum I can see a 18 or 20-fold reduction between 0 V and 6 V. I invite the\nauthors to make this point more clear and sound, e.g. by including in Figure 4a and 4b\nan inset with the measured resonances, as reported for example in ref. 20.\nThe reviewer pointed out a very important issue of the relation between the Q factors\nof the merged signal(idler) resonance and the actual photon lifetime that was measured\nin the experiment.\nFirstly, we have supplemented the Q factors data in Fig. 2 by adding a concise\nsummary of Qs of pump, signal, and idler resonances in the figure caption.\n(Caption of Fig. 2 in revised manuscript)\nFig. 2. Spectra characterizations. a System transmission spectrum under the 0 V\nbias (left axis). A series of high-Q resonances comes from the main ring, and the\nhighlighted yellow lines are the resonances of the auxiliary ring. b System\ntransmission spectrum under the 6.6 V bias (left axis). The green circles in a and b\nare the marks of the high-Q factors of the main ring’s resonances (right axis). The\norange circles in b are the low-Q factors of merged resonances. The dashed blue,\ngreen, and red regions represent the locations of signal (C48), pump (C51), and idler\n(C54) resonances. Under 0 V bias, the Q factors of three resonances are: 1.53×105,\n1.53×105, and 1.43×105, respectively; Under 6.6 V bias, Q factors are 8.03×103,\n2.05×105, and 1.12×104, respectively. The pump resonance remains a high Q, while\nthe Q factors of signal and idler resonances are significantly decreased.\nMeanwhile, a table documenting all Q factors and expected photon lifetimes of seven\nresonances ranging from 1534 nm to 1540 nm in Fig. 2 is added in Supplementary\ninformation S2:\n(Line 156 in revised Supplementary information)\nFrom the spectra, the Q factors and expected photon lifetimes of seven involved\nresonances ranging from 1534 nm to 1540 nm presented in the main text are listed in\nTable. S2. We choose the resonance in C51 channel as the pump resonance, while the\nresonances in C48 & C54 are chosen to be the signal and idler resonances,\nrespectively.\nTable. S2. Q factors and expected photon lifetimes of resonances.\nSecondly, according to another reviewer's comments, after carefully evaluating\nour experimental data and data processing methods, we found that there was a\nsystematic issue with our analyses of coincidence count histograms. We noticed that in\nthe case of the short-lived photon tests and the measurements of timing jitters of\nthe experimental equipment, the exponential function fittings have deviated from\nthe data points, see e.g., the following result:\n(composite distribution of exponential and Gaussian functions)\nThis deviation caused disagreements in the results of lifetimes, especially the short-\nlived photons, and the results from the unsuitable data fittings (which give a 4.1 ps short\nphoton lifetime and a 38-fold lifetime tunable range in the original manuscript) were\noverestimated.\nWe notice that the exponential decays of photon wave packets and the Gaussian\nnoises of detection equipment are included together in the coincidence count peak.\nTherefore, the experimental data are now analyzed by a 1000-time Monte Carlo\nsimulation, applying a composite distribution of exponential and Gaussian\nfunctions, following the methods in, e.g., Sidorova et al., Phys. Rev. B 96, 184504 (2017),\nWen et al., Phys. Rev. Applied 18, 024059 (2022):\n|𝑡−𝑡 | (𝑡−𝑡 )2\n0 0\nℎ(𝑡)=𝐴exp(− )∙exp(− )\n𝜏′ 2𝜎′2\nThe lifetime of short-lived photons is re-determined as 7.1 ± 1.5 ps. The lifetime\nuncertainty range is estimated by Monte Carlo method according to other referees'\nadvice. Detailed analyses are given in Supplementary information S4.\nThis part is revised as below:\n(Line 210 in revised manuscript)\nThe coincidence counts (CCs) of generated photon pairs under the bias voltages\nof 0 V and 6.6 V are recorded in Fig. 4a and 4b, respectively. A 1000-time Monte\nCarlo simulation is used to fit the coincidence count histograms. A composite\nfunction of exponential and Gaussian distribution is applied [60,61], since the timing\njitters of the detection systems are included in the coincidence peaks. The total\nstandard deviations of the two counting histograms are 𝑤 = 153.7±1.3 ps and\n1\n𝑤 = 83.0±1.0 ps , respectively. Accordingly, the fittings giving the lifetimes 𝜏\n2\nof long and short-lived photons are 129.6 ± 1.9 ps and 7.1 ± 1.5 ps, respectively, and\nthe timing jitter of the detection system 𝜎 is 82.8 ± 0.9 ps. The uncertainties are\nextracted from the Monte Carlo simulations. A near 20-fold photon lifetime adjusting\ncontrast is achieved, matching the Q-factor variations in spectral tests given earlier.\nMeanwhile, Fig. 4 is also revised.\nOur results show an effective photon lifetime tuning method by harnessing a PT-\n8\nsymmetric microring system. However, as a proof-of-concept device, the tuning range\nis limited by the losses of the waveguide that composes the ring (intrinsic Q of the ring).\nA significantly larger tuning range is expected, upper to 80-fold if the waveguide loss\ncould be optimized to 0.1 dB/cm, as discussed in Supplementary information S6.\nFinally, due to the timing jitters of detection systems, the discrepancy of the\nreciprocal of the resonance linewidths 1⁄Δ𝜔 and the 1/e width of coincidence count\npeaks √𝜏2+𝜎2 is quite large, therefore, we did not place the figures of the resonances\nand the coincidence count peaks together to avoid any misunderstanding (for other\npapers, this is acceptable since 𝜏 ≫ 𝜎 ,). Nonetheless, we add a comparison of the\nFourier transform of the intracavity fields and a standard Laplace distribution (double\nexponential functions) to verify the effectiveness of the composite distribution fitting\nthe coincidence counts in the Supplementary information S4. Very close results of\nphoton lifetimes are obtained, showing the correctness of our method:\n(Line 223 in revised Supplementary Information)\nFig. S10. Fourier transforms (FTs) of the intracavity resonances of the main ring and the\ncomparison of a standard Laplace distribution (double exponential function), as well as the\nanalytical short-lived photon wave packet.\nThe second aspect to notice is the explanation of the changes in the PGR between 0 V\nand 6 V. As there is a reduction of signal and idler quality factors by more than a factor\n10, one would expect a significant reduction in PGR, but surprisingly the PGR gets\neven larger, as well as the CAR does. The authors provide only qualitative reasons\nwhich are insufficent in my opinion. I invite the authors to apply the theory for SFWM\nin microrings involving modes of different quality factors, as it is presented e.g. in the\npaper by L. Zatti et al. PHYSICAL REVIEW A 107, 013514 (2023). The authors should\nfind some agreement between their data and the theory, which has already been verified\nin many experiments, and therefore be able to clarify what is going on exactly.\nAuthor's A2:\nWe thank the reviewer for providing a relevant reference about the PGR calculation.\nIndeed, someone might be puzzled by the increase in PRG in our Q-factor-tunable\ndevices, as intuitively, a decrease in Q factors of involved resonances would be\nexpected to lead to a significant decrease in PGR. A theoretical calculation of the PGRs\nis now supplemented. Since our work focuses on lifetime-tunable quantum light sources\nand the PGRs may not be the core conclusion, we provide a brief quantitative\nclarification in the main text to explain the trend in PGRs, and include the\ncomplete theoretical calculation in the Supplementary information S4. The main\nresult is that a ~1.65 times larger PGR in the linewidth-manipulated case is expected\nand agrees well with the experimental measurements (Figs. 4c and 4d in the main text),\nafter excluding the losses and detection efficiency of SNSPDs.\n(Line 233 in revised manuscript)\nIt is noteworthy that the CCs increase surprisingly, even though the Q factors\ndecrease when comparing the states before and after the rings are aligned. This can\nbe attributed to the over-coupling of signal and idler resonances with an enhanced\nphoton escape probability, which can be effectively modelled by taking into account\nthe different Q factors of evolved three resonances in the SFWM process [35,36, 62–\n64]. Theoretical calculation reveals an on-chip PGR of high-Q case (0 V bias) of 8.5\n×104 Hz (under 14.12 mW on-chip pump); and a higher PGR of 1.4×105 Hz is\nexpected after the linewidth manipulation (6.6 V bias), matching well with the\nexperimental results in Figs. 4c and 4d (a 1.65 times larger PGR in the linewidth-\nmanipulated case), excluded the system losses and detection efficiency of SNSPDs\n(see details in Supplementary information S5).\nRevisions in Supplementary information:\n(Line 276 in revised Supplementary information)\nIn light of Figs. 4c and 4d in the main text, the photon-pairs generation rate\n(PGR) becomes larger after the thermal tuning. Here, a theoretical calculation is\ngiven. In general, the PGR of a microresonator system, in which different Q factors\nof three resonances are involved, can be calculated by [5–8]\n8𝑣4 2𝛾𝑃 𝑄4𝑄3\nPGR = 𝑔 ( )2 𝑝 𝑠/𝑖\n𝜔3 𝐿 𝑄2 𝑄2\n𝑒,𝑝 𝑒,𝑠/𝑖\n10\n(S16)\nwhere 𝜔 is the resonance frequency, 𝑣 is the group velocity, 𝛾 is the nonlinear\n𝑔\ncoefficient, 𝑃 is the pump power, 𝐿 = 2𝜋𝑅 is the circumference of the main ring,\n1\nrespectively. 𝑄 is the loaded Q factor of the pump resonance, 𝑄 is the loaded\n𝑝 𝑠/𝑖\nQ factor of signal (idler) resonance, 𝑄 is the external Q factor of pump resonance,\n𝑒,𝑝\n𝑄 is the external Q factor of signal (idler) resonance, respectively. Given that the\n𝑒,𝑠/𝑖\nintrinsic Q of the resonator is 𝑄 , the relation between loaded Q, and external,\n𝑖𝑛𝑡\nintrinsic Q is 1/𝑄 = 1/𝑄 +1/𝑄 . According to Eq. (S16), we can\n𝑙𝑜𝑎𝑑𝑒𝑑 𝑖𝑛𝑡 𝑒\ntheoretically obtain that PGR = 85 kHz in the high-Q case (0 V bias), and PGR = 140\nkHz in the linewidth-manipulated case (6.6 V bias), respectively. A 1.65 times larger\nPGR in the linewidth-manipulated case is estimated and agrees well with the\nexperimental results shown in Figs. 4c and 4d in the main text. The intrinsic Q factor\nof the main ring is extracted from the spectrum fittings of 𝑄 = 3.5×105.\n𝑖𝑛𝑡\nThe key parameters used in simulations are: 𝜔 = 2𝜋𝑐/1535×10−9 𝐻𝑧 (𝑐 is the light\nspeed in vacuum), 𝑣 = 𝑐/2.1 m/s , 𝛾 = 1 𝑊−1 m−1 , 𝑃 = 14.12 m𝑊 , 𝐿 =\n𝑔\n2𝜋×228 μm.\nReviewer's Q3:\nFinally, related to the Franson experiment to prove the energy-time entanglement, some\ninformation must be added. Despite the fact that a proof less prone to loopholes issues\nwould require two distinct unbalanced interferometers, one for signal and one for idler\nphotons, in my opinion the experiment presented by the authors is still fine as there has\nbeen already plenty of other experimental confirmations. However, they should explain\nit clearly to the reader, add references for the type of experiment and better explain\nwhich type of entanglement they are measuring and how this is related to the original\nenergy-time entanglement. At the moment, the unexperienced reader is left basically\nwith no explanation and a lot of understatements. This can be fixed by adding a short\nparagraph with references.\nAuthor's A3:\nWe agree with the reviewer at this point. An explanation about the test of energy-time\nentanglement is added within the context of the Franson interference experiment:\n(Line 247 in revised manuscript)\n… interferometer (UMI) is shown in Fig. 3c. The entanglement can be characterized\nthrough a Franson-type two-photon interference experiment, by violating Bell’s\ninequality [65,66]. As shown in Fig. 3c, we measured two-photon interference fringes\nusing a folded Franson interferometer configuration implemented via an unbalanced\nMichelson interferometer (UMI) [28,32,67,68]. This arrangement can be regarded as\na folded version of the standard Franson interferometer, in which the two unbalanced\ninterferometers for the signal and idler photons are effectively combined into a single\ninterferometer, employing the symmetry of the original configuration. Figures 5a and\n5b give the CCs of …\nMinors:\nline 208, exponential not exceptional\nAuthor's A4:\nThanks for catching this spelling error. We have corrected the word.\nReviewer's comment:\nThis article describes a method of generating quantum light with spontaneous four\nwave-mixing where the signal/idler modes have adjustable lifetimes. The system\nconsists of two ring resonators with high Q contrast, and the coupling between the\nresonator creates supermodes with detuning-dependent linewidths. The authors claim a\nmeasured 38-fold photon lifetime tuning range between the decoupled modes and the\ncoupled modes near the exceptional point. Other characterizations of correlation\nproperties such as energy-time entanglement and g2 have also been measured.\nThe usage of EPs for controlling linewidth has frequently been applied to classical\nsensing applications but is a relatively new concept in quantum light generation. The\nexperimental methods are overall solid, and necessary quantities have been measured.\nHowever, the theoretical descriptions of the method are inconsistent and confusing.\nThere are also potential problems with data interpretation that are related to the central\nclaim of the results. In view of the above, I could not recommend publication in its\ncurrent state.\nAuthor's reply:\nWe thank the reviewer for giving a concise review of our work and for pointing out the\nshortcomings in our theoretical analysis. We have revised the manuscript according to\nthe reviewer’s constructive comments.\nReviewer's Q1:\n- Using “PT-symmetric” to describe the system is questionable in at least two aspects.\nFirst, both modes of the system are lossy, although one may argue that the common loss\ncan be subtracted from the effective Hamiltonian to resemble a gain-loss system.\nSecond, since the two resonators differ in size, the P operation could not be properly\ndefined in the real space (especially in the TMM framework).\nAuthor's A1:\nFirstly, we agree with the referee that a common loss should be subtracted from the\nHamiltonian to construct a lossy PT-symmetric system resembling an actual gain-loss\nsystem. We add this part after the definition of Hamiltonian in the main text, referencing,\ne.g., Guo et al., Phys. Rev. Lett. 103, 093902 (2009), Wang et al., Science 171(6560)\n1261-1265 (2021):\n(Line 126 in revised manuscript)\n… the detuning between the two rings. It is worth noting that the Hamiltonian is\ndefined in a lossy PT-symmetric configuration by factoring out an average loss (𝛾 +\n1\n𝛾 +𝛾 )/4 , in which the unbalanced losses compose an equivalent gain-loss\n2 𝑐\nmechanism in this gauged PT-symmetric system [47,55]. Solving the complex\neigenfrequencies of the Hamiltonian, …\nNext, within the context discussed in this work, the microresonators are modeled\nby the coupled mode theory (CMT), and the development of the system Hamiltonian is\ndirectly defined in the frequency domain. Differing from optical waveguide systems\n13\nthat are usually described by the wave vector 𝛽⃗ , the Hamiltonian of the coupled\nmicroresonator can be written as,\n𝜔 −𝑖\n𝛾1\n𝜅\n1\n𝐻̂ = [ 2 ],\n𝜅 𝜔 −𝑖\n𝛾2+𝛾c\n2\n2\nwhich remains symmetric under the P operation (swap the diagonal elements) and T\noperation (time reversal symmetry) after subtracting a common loss term, i.e., 𝑖(𝛾 +\n1\n𝛾 +𝛾 )/4. For passive PT-symmetry discussion, we here consider only the resonant\n2 𝑐\nmodes that exist both in the main ring (R1) and the auxiliary ring (R2), as illustrated in\nthe following:\nwhile the other high-Q modes that only exist in the main ring are calculated using the\ntransfer matrix method (TMM). Similar treatment has been used in a number of works\nstudying coupled frequency modes related to the synthetic frequency dimension, e.g.,\nYu et al. PRL 130, 143801 (2023), Cheng et al. Light Sci Appl 12, 158 (2023). Actually,\nPT-symmetry in the microresonators that in other applications may not have a general\nreal-space P symmetry, e.g., Hou et al. PRL 132, 256902 (2024) (the same spatial mode\nis distributed within a single microring.), Lin et al. Frontiers in Optics, FTu5E.3 (2018)\n(Having the cw and ccw at different frequency modes within the same microring).\nTherefore, PT operations of Hamiltonian are generally accepted and used in the\nfrequency domain, which is also the case of our system configuration.\nReviewer's Q2:\n- The definition of “photon lifetime” becomes ambiguous if two or more optical modes\nare within the vicinity of each other. This is especially true at the EP, where the mode\nspectrum is no longer Lorentzian but super-Lorentzian [see e.g., Nat. Commun. 13,\n3281 (2022)]. The authors should clarify this. In addition, what is the TMM result for\nthe mode FWHM at the EP and does it agree with Eq. 4?\nAuthor's A2:\nThanks for this comment. In our work, the photon lifetime is defined as the inverse of\nthe resonance linewidth, 𝜏 = 1/∆𝜔, same as many other works, e.g., O.-Ricardo et al.,\nPhoton. Res. 9, 2237 (2022), and Wen et al., Phys. Rev. Applied 18, 024059 (2022).\nThe TMM yields similar results with Eq. 4 as long as near EP operation, which is\n14\napproximately the case of our work. For the parameters that are further away from EP\nand give super-Lorentzian lineshape, a synthetic linewidth as discussed in Kim et al.,\nelight 4, 6 (2024) can be used. Therefore, to keep discussions simple and show clear\nphysics, Eq. 4 is an approximation and is used to catch the essence of photon-lifetime\nat EP.\nWe are very grateful for this comment, which pointed out an understanding gap in\nthe original manuscript. We provide a clarification in the main text:\n(Line 158 in revised manuscript)\nAt this point, the photon lifetime is defined as the inverse of the mode linewidth by\n1\n𝜏 = .\nlow−Q\n𝛾\nc\n(4)\nMeanwhile, this issue can be numerically verified by accessing the Fourier\ntransform of the intracavity resonances. We examined this by comparing the photon\nwave packet and the standard Laplace distribution (double exponential function), as\nwell as the analytical expression of the Fourier transform of super-Lorentzian lineshape\nin the time domain:\n(Line 215 in revised Supplementary information)\n1. For photon lifetimes, Fig. S10 shows the normalized time-domain photon\nwave packets obtained through the Fourier transforms of the intensity enhancement\n(Eq. (S8)) at the signal resonance (1534.6 nm, C48 channel) within the main ring.\nTogether, for comparison, at 0 V bias, a standard Laplace distribution (i.e., double\nexponential function, the FT of a standard Lorentzian lineshape) 𝐿(𝑡) =\n(1/𝜏)exp(−|𝑡|/𝜏) is drawn (𝜏 = 135 ps), as shown in Fig. S10a. The photon wave\npacket slightly deviates from the Laplace distribution because the high-Q resonance\nis slightly influenced by the resonance of the auxiliary ring. However, its 1/e lifetime\n𝜏 is very close to that of Laplace distribution.\n𝑙𝑜𝑛𝑔\ncomparison of a standard Laplace distribution (double exponential function), as well as the\nanalytical short-lived photon wave packet.\nAt 6.6 V bias, since the resonance within the ring is near the EP, it is no longer\na standard Lorentzian lineshape but rather a super-Lorentzian shape [4]. Consider the\nIE under the framework of TMCT (Eq. (S6)) with a merged two-Lorentzian\nexpression. The Fourier transform of this lineshape can be derived as:\n𝐿 (𝑡) = 4𝐴𝑖𝜅√𝛾 (𝑒−𝑥1|𝑡| +𝑒−𝑥2|𝑡|)\n𝑙𝑜𝑤−𝑄 𝑐\n(S15)\nwhere 𝐴 is the normalization coefficient, 𝑥 , 𝑥 are the complex roots of the\n1 2\nquadratic equation in the denominator of the super-Lorentzian function, 𝑥 =\n1,2\n(𝛾 +𝛾 +𝛾 )±√4𝜅2+𝛾 (𝛾 +𝛾 )−(𝛾 +𝛾 +𝛾 )2 . For this merged exponential\n1 2 𝑐 1 2 𝑐 1 2 𝑐\nfunction at EP, its 1/e width is determined by the real part of the complex roots,\n𝜏 = 1/(𝛾 +𝛾 +𝛾 ) . This result is the same as the definition of the photon\n𝑠ℎ𝑜𝑟𝑡 1 2 𝑐\nlifetime in the main text, when 𝛾 ≫ 𝛾 ,𝛾 , we defined that 𝜏 = 1/𝛾 (Eq. (5)\n𝑐 1 2 short c\nin the main text). When the system deviates from the EP, the discrepancy of decay\nrate of 𝑥 (non-zero value under the square root) will cause oscillatory decay in\n1,2\nthe time domain (when mode splitting occurs in PT-symmetric region). In fact, our\nsystem is operating in the PT-symmetric region but very close to EP (analyses are\ngiven in Supplementary information S5). Figure S10b gives the comparison of the\n16\nnumerical calculated photon wave packet and the analytical expression derived from\nthe FT of TCMT Eq. (S15). A slight oscillatory decay can be found. For simplicity,\nwe still use the EP condition to demonstrate the effectiveness of photon lifetime\nadjustment. We also compare the short photon wave packet with a standard Laplace\ndistribution and found that the overall 1/e width are almost consistent (remaining\naround 7 ns), proving that our model and experimental results are valid.\nIt is well known that a Lorentzian lineshape in the frequency domain transforms into a\nLaplace distribution in the time domain after the Fourier transform. At 6.6 V bias, a\nsuper-Lorentzian shape is formed. Its Fourier transform (Fig. S10b) has a shorter tail\ncompared to the standard Laplace distribution; however, the overall 1/e width of the\nwave packet remains close to the width of the Laplace distribution (~7 ns, also close to\nour experimental results), proving that the definition of photon lifetime is valid.\nReviewer's Q3:\n- The measurement presented in Fig. 4b deviates significantly from exponentially\ndecaying profiles and the fitting shows systematic error. For example, directly reading\nthe 1/e width off the data would give a larger result of \\tau. This is potentially related\nto the previous point where the lineshape of the mode deviates from a Lorentzian. In\naddition, the authors should present error analysis on the obtained \\tau numbers since\nthese are obtained by subtracting the jitter contributions and are extremely close to 0.\nAuthor's A3:\nWe thank the reviewer for pointing out this important issue. As also been commented\nby other reviewers, after carefully evaluating our experimental data and data processing\nmethods, we now carry out the photon lifetime by using a 1000-time Monte Carlo\nsimulation, applying a composite distribution of exponential and Gaussian\nfunctions, since the thermal and electrical noises are affecting the coincidence count\nand follow different noise statistics.\nFIG.4b Original fitting (exponential function) vs. revised Monte Carlo simulations\n(composite distribution of exponential and Gaussian functions)\nThe lifetimes and timing jitters here are extracted, following methods in, e.g., Sidorova\net al., Phys. Rev. B 96, 184504 (2017), Wen et al., Phys. Rev. Applied 18, 024059\n17\n(2022):\n|𝑡−𝑡 | (𝑡−𝑡 )2\n0 0\nℎ(𝑡)=𝐴exp(− )∙exp(− )\n𝜏′ 2𝜎′2\nand thus, the photon lifetimes are redetermined as 7.1 ± 1.5 ps. A lifetime uncertainty\nrange is estimated according to other referees' advice, by employing the Monte Carlo\nmethod. This part is revised:\n(Line 209 in revised manuscript)\nThe coincidence counts (CCs) of generated photon pairs under the bias voltages\nof 0 V and 6.6 V are recorded in Fig. 4a and 4b, respectively. A 1000-time Monte\nCarlo simulation is used to fit the coincidence count histograms. A composite\nfunction of exponential and Gaussian distribution is applied [60,61], since the timing\njitters of the detection systems are included in the coincidence peaks. The total\nstandard deviations of two counting histograms are 𝑤 = 153.7±1.3 ps and 𝑤 = 83.0±\n1 2\n1.0 ps, respectively. Accordingly, the fittings giving the lifetimes \\tau of long and\nshort-lived photons are 129.6 ± 1.9 ps and 7.1 ± 1.5 ps, respectively, and the timing\njitter of the detection system 𝜎 is 82.8 ± 0.9 ps. The uncertainties are extracted from\nthe Monte Carlo simulations. A near 20-fold photon lifetime adjusting contrast is\nachieved, matching with the Q factors variations in spectral tests given earlier (see\nmore detailed analyses in Supplementary information S4).\nA detailed analysis is provided in Supplementary information S4. Accordingly, Figs. 4a\nand 4b in the main text are revised; please see the manuscript.\nAnother concern regarding the lineshape of the Gaussian-like coincidence count is\nmainly contributed to the relatively large system timing jitters swamping the photon\nwave packets. This has been analyzed in Q2.\nReviewer's Q4:\n- What controls the location of the signal and idler modes (C54 and C48)? If SFWM\noccurs in the main ring and there are uncoupled modes in the main ring with higher Q\n(e.g. at C53 and C49), why does the SFWM process not prefer these modes?\nAuthor's A4:\nAll the signal and idler resonance-pairs that satisfy phase-matching and frequency-\nmatching conditions can generate photon-pairs (C52 & C50, C53 & C49, C55 &\nC47, …), not only in the C54 & C48 channel-pair. The figure below illustrates this:\nfrequency combs (e.g., Ref. [3] Reimer et al. Science 351, 1176-1180 (2016), Ze et al.\nLight: Science & Applications 14, 164 (2025)), and is used for high-dimensional\nentanglement generation (Ref. [2] Kues et al. Nature 546, 622-626 (2017)), frequency\nmultiplexing single-photon sources (Ref. [28] Zeng et al. PRL 132(13), 133603 (2024)),\netc.\nIn our device, higher Q mode-pairs can generate photon-pairs, but we did not\ncharacterize them. As a proof-of-concept work, we focus on the resonances that are\nmanipulated (C50 & C52, C48 & C54, …) and their photon lifetime.\nOn the experimental side, C50 & C52 channels are too close to the pump resonance,\nwhich may cause the noise increase due to the imperfect pump-rejection filtering. C46\n& C56, along with other more distant channels, might be more significantly affected by\ndispersion, leading to a decrease in photon count rates, which is the reason why we did\nnot measure them.\nWe did not explain the reasons for channel selection in the manuscript. The\nreviewer helped point out this important point, so we have added the reason for channel\nselection to the main text.\n(Line 183 in revised manuscript)\n… Resonances that are three 𝐹𝑆𝑅 away from the pump resonance on both sides are\n1\nselected as the signal and idler resonances to avoid imperfect pump rejection filtering,\nwhich might cause large noise photon counts.\nReviewer's Q5:\n- Line 208: I’m not sure what “double exceptional function” means, and I assume it\nshould be a “double exponential function” or similar. Same for Line 204 in the\nSupplementary Material.\nAuthor's A5:\nThanks for catching this spelling error. We have corrected the wrong word.\n19"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The authors have addressed my concerns and I recommend the paper to be accepted.", "reviewer_2": "I reviewed the revised version of the manuscript by Nuo Chen and co-workers, after reading their replies to my concerns, as well as those of the other 2 reviewers. In my opinion the work is now fully clear and sound, therefore it can be published in Nature Communications.", "reviewer_3": "The authors have addressed my previous concerns regarding data processing and theoretical analysis. I believe the current results can be published in Nature Communications. I do not have further comments."}, "author_response": ""}]

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