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Abstract
In fiber-based quantum information processing with energy-time entangled photon pairs, optimized dispersion compensation is vital to preserve the strong temporal correlation of the photon pairs. We propose and experimentally verify that, by simply tuning the wavelength of the entangled photon pairs, nonlocal dispersion cancellation (NDC) can provide a widely flexible and finely adjustable solution for optimizing the dispersion compensation, which cannot be reached with the traditional local dispersion cancellation (LDC) instead. By way of example, when a 50 km-long single-mode fiber (SMF) is dispersion compensated by a 6.2-km-long commercial dispersion compensating fiber (DCF) based on the LDC configuration, it will lead to an almost invariant over-compensation in the wavelength range of 1500-1600 nm which restricts the observed temporal coincidence width of the self-developed energy-time entangled photon-pairs source to a minimum of ∼110 ps. While in the NDC configuration, the dispersion compensation can be readily optimized by tuning the signal wavelength to 1565.7 nm and a minimum coincidence width of 86.1 ± 0.7 ps is observed, which is mainly limited by the jitter of the single-photon detection system. Furthermore, such optimized dispersion compensation can also be achieved as the fiber length varies from 48 km to 60 km demonstrating the wide flexibility of NDC. Thanks to these capabilities, elaborate dispersion compensation modules are no longer required, which makes NDC a more versatile tool in fiber-based quantum information and metrology applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Benefitting from their tight temporal correlation, energy-time entangled photon pairs generated by spontaneous parametric down conversion (SPDC) [1,2] have played an important role in the study of quantum physics [3–5] and quantum information processing technologies [6–15]. Due to the low transmission loss and widespread deployment, optical fibers have become a natural choice for distributing entangled photons in both intra- and intercity links [16,17]. Nevertheless, chromatic dispersion in the fiber link will impose significant broadening on the temporal correlation width of the distributed photon pairs [18], ultimately limiting the performance of fiber-based quantum information applications such as maximum reachable distance of quantum communication, achievable key rate of quantum key distribution (QKD) systems [11], and timing resolution of quantum clock synchronization (QCS) systems [13]. To cancel the dispersion effect, the conventional method is to employ a lumped dispersion compensation module (DCM) possessing opposite sign of dispersion in the fiber link, which is also called local dispersion cancellation (LDC). In practice, however, the almost constant dispersion parameter of the given DCM restricts its ability to optimize the dispersion compensation in both fine tunability and dynamic range.
As a unique quantum feature of energy-time entangled photon pairs, nonlocal dispersion cancellation (NDC) was proposed by Franson in 1992 [19]. In contrast to LDC, NDC gives a distributed way to compensate the dispersion experienced by the photons after propagating through a long-haul fiber link. Since it was first proposed, NDC has been intensively investigated [20–24] for its wide application in many fiber-based quantum information processing systems such as quantum nonlocality testing [14], quantum key distribution [25], and quantum time transfer [13,15]. Recently, Ruiming Chua et al. [24] achieved a near-ideal dispersion compensation of telecom O-band photon pairs via NDC. As the center wavelengths of the signal and idler photons were located on opposite sides of the zero-dispersion wavelength ($\sim$1316 nm) of the single-mode fiber (SMF), fine-grained NDC was realized by adjusting the individual propagation distance of the photon pairs. This achievement, however, was at the cost of troublesome fiber increments (>1km) [24] thus inconvenient for on-the-fly testing. Searching for a simpler way to optimize the dispersion compensation is crucial in field fiber-based quantum information processing applications.
In this paper, we begin with a realistic example of compensating the dispersion in a 50 km-long SMF with a 6.2 km commercial dispersion compensating fiber (DCF). According to theoretical analysis, the LDC configuration leads to an almost invariant over-compensation over the wide wavelength tuning range of the signal photons from 1500 nm to 1600 nm. On the other hand, when NDC is applied, over-compensation can be readily switched to under-compensation by merely tuning the wavelengths of the signal photon across 1565 nm. For experimental verification, the self-developed energy-time entangled photon pair source [26] is utilized. By changing the working temperature of the periodically poled lithium niobate (PPLN) waveguide, the signal photon wavelength can be varied from 1537 nm to 1568 nm while the photon pair generation efficiency remains almost constant. Over this wavelength tuning range, the observed temporal coincidence width in the LDC configuration showed a trivial variation around $110$ ps, whilst in the NDC configuration, a minimum temporal coincidence width of $86.1\pm 0.7$ ps was achieved when the signal photon wavelength was tuned to 1565.7 nm. Furthermore, with the same DCM, the dispersion compensation optimization in SMFs with other lengths varying from 48 km to 60 km was also investigated by tuning the signal wavelength. It was shown that the minimum temporal coincidence width can be always less than 96 ps. The wide flexibility and fine adjustability of NDC would further enhance its versatility in various fiber-based quantum information processing applications.
2. Theoretical simulation
The typical NDC configuration is shown in Fig. 1(a). The signal and idler photons, whose joint spectral density function is depicted in Fig. 1(b), respectively propagate through media of positive dispersion (PD) and negative dispersion (ND). The temporal coincidence width is observed with the help of the single-photon detectors (D1, D2) and a coincidence counter (C.C.). Without loss of generality, the dispersion slope of the PD can be assumed to be exactly opposite to that of the ND (i.e. $D_{PD}(\lambda )=-D_{ND}(\lambda )$), which is depicted by the orange and green lines in Fig. 1(c). Due to the spectral anti-correlation between the energy-time entangled photon pairs, the wavelengths of the signal and idler photons are related by $\lambda _{i}=\frac {\lambda _{p}\lambda _{s}}{\lambda _{s}-\lambda _{p}}$, where $\lambda _{p}$ denotes the pump wavelength. Thus, as the signal photon wavelength increases, the corresponding idler photon wavelength will decrease. In this case, the positive dispersion experienced by the signal photons with longer wavelengths ($\lambda _{{s+}}$) will be greater than the negative dispersion experienced by its paired idler photons with shorter wavelengths ($\lambda _{{i-}}$) and vice versa. As a result, it is easy to show in Fig. 1(d) that the net dispersion of the NDC system can be varied from negative to positive by tuning the wavelength of the signal photons.
Fig. 1. Scheme for optimizing NDC with wavelength tuning of entangled photons. (a) Typical NDC configuration; the signal (s) and idler (i) photons propagate through media of positive dispersion (PD) and negative dispersion (ND), respectively. The temporal correlation measurement system contains two single photon detectors (D1, D2) and a coincidence counter (C.C.); (b) Joint spectral intensity plot of frequency anti-correlated photons; (c) Dispersion curves of positive and negative dispersive media; (d) Net dispersion of the NDC versus the photon wavelength. The under-compensation and over-compensation regions are colored in orange and green, respectively.
In a practical situation, the standard SMF (ITU-T G.652) can be considered as a typical PD medium. According to Corning’s technical note [27], the group delay dispersion (GDD) versus wavelength ($\lambda$) of standard SMF in the region of 1200 nm-1600 nm can be approximated by the following equation,
where the parameters $B$ and $C$ denote the linear and nonlinear dependence of the GDD on the photon wavelength respectively. DCF is normally employed as the ND medium to compensate for the dispersion introduced by the SMF. Without loss of generality, its GDD versus wavelength can be assumed to follow the perfect dispersion slope matching except for a weighting factor $\eta$, that is [28]In the LDC and NDC configurations, the net dispersion ($D_{net}$) can be respectively written as
According to the anti-correlated relation between the signal and idler photon wavelengths, the net dispersion in the case of NDC ($D_{net,n}$) can be rewritten as a function of the signal wavelength ($\lambda _{s}$), i.e.,
Subsequently, the measured temporal coincidence widths of the photon pairs in both cases of LDC and NDC can be given by [22],
In our experiment, a SMF of 50 km is under test, for which the parameters $B= 0.016$, $C= 6.56\times 10^{10}$ were obtained by fitting the experimentally measured dispersion data to Eqn. (1). When a factor of $\eta =1.05$ is taken for the GDD expression of the DCM, the dependencies of the net dispersion for both cases of LDC and NDC on the signal wavelength within the range of 1500-1600 nm can be plotted in Fig. 2(a). It is clear to see that in the case of LDC, the net dispersion remains negative and shows rather slight change with respect to the signal wavelength. In the case of NDC, the net dispersion varies significantly from negative to positive with the increasing wavelength and reaches zero at the signal photon wavelength of 1565 nm. Substituting the experimental parameters presented in Section 3 (Experiment) into the theoretical equations, we then investigate the temporal coincidence width as a function of the signal photon wavelength for different SMF lengths ranging from 48 km to 60 km, with the results shown in Fig. 2(b). It can be seen that, by optimizing the signal photon wavelength, the temporal widths for different SMF lengths of 48-60 km can be minimized to an almost constant level of 88 ps that is barely greater than the detection system jitter (78.4 ps). Benefiting from this wide flexibility and fine tunability of the dispersion compensation with NDC, the temporal coincidence width broadening due to incomplete dispersion compensation arising from constant DCM elements can be mitigated effectively. We further consider the case that a broadband SPDC source is utilized, simulations show that temporal coincidence widths for different SMF lengths can be also minimized close to the detection system jitter and a more sensitive tunability with wavelength tuning is observed. Therefore, the presented method can be applied to broad photon spectra such as those generally produced with Type-0 SPDC.
Fig. 2. Theoretical simulation of (a) net dispersion versus the signal wavelength in the range 1500-1600 nm under conditions of LDC and NDC; (b) temporal coincidence widths at the full-width-at-half-maximum (FWHM, $2 \sqrt {2 \ln 2} \Delta _n$) in NDC configuration for different SMF lengths versus the signal wavelength range of 1550-1570 nm. The experimental parameters in Section 3 are utilized for the simulation: $\Delta _{jit}=33.3$ ps, $\Delta _{0, \tau }=0.5$ ps, $\sigma _{s}=0.8$ nm and $\sigma _s^c=0.02$ nm.
3. Experiment
3.1 Experimental setup
To verify the above theoretical prediction, we employed a self-developed all-fiber photon pair source [26] for experimental examinations. As shown in Fig. 3(a), a pigtailed distributed Bragg reflector single-frequency laser (DBR, DBR780PN, Thorlabs) at 780.6 nm was used to pump a 15 mm-long type-II PPLN waveguide (HC Photonics) with a poled period of $\sim 8 \mu$m. Polarization orthogonal and time-energy entangled photon pairs were then generated. After filtering out the residual pump beam using the combination of a customized fiber wavelength division multiplexer (WDM) and a band-pass filter (BP), the remaining signal and idler photons were separated by the subsequent fiber optic polarization beam splitter (FPBS). As depicted in Fig. 3(b)-(d), the signal and idler photons were detected by two superconducting nanowire single-photon detectors (SNSPD1$\&$SNSPD2, Photon Technology Co., Ltd.). Because the jitter in each detector depends on its detected count rate [12,24], we adopted the common approach of fixing the count rates at $\sim$100 kHz during the measurements. The coincidence measurement was accomplished by a time tagging unit (TTU, Time Tagger Ultra, Swabian Instruments), with a nominal RMS jitter less than 9 ps for each channel. Based on the dispersive wavelength-to-time mapping technique [29,30], the signal photon wavelength variation as a function of the PPLN waveguide temperature was investigated with the measurement setup in Fig. 3(b). With a DCM based on the fiber Bragg grating (DCM-FBG, Proximion DCM-CB) in the signal photon path while the idler photon being fed to the SNSPD directly, the measured coincidence count distribution would experience a shift in time as the temperature of the PPLN waveguide is varied. According to the pre-calibrated relationship between the signal photon wavelength and the measured coincidence peak shift [31], we are then able to determine the signal photon wavelength as a function of the PPLN waveguide temperature. The setups for studying and comparing the dispersion compensation performance under configurations of NDC and LDC are shown in Fig. 3(c) and (d), respectively. For both cases, the PD propagation was implement by placing the spooled SMF into the signal photon path. When measuring the NDC performance, the 6.2-km dispersion compensating fiber (DCM-50, YOFC Ltd.) was placed in the idler path as shown in Fig. 3(c). When measuring the LDC performance, the same DCF was placed in the signal path directly after the SMF, as shown in Fig. 3(d).
Fig. 3. Experimental setup. (a) The SPDC source is generated from a piece of type-II PPLN wave-guide pumped by a DBR laser at 780.6 nm [26]. (b) The signal photon wavelength measurement is implemented with the dispersive wavelength-to-time mapping technique. (c) Measurement configuration for nonlocal dispersion cancellation. (d) Measurement configuration for local dispersion cancellation. DBR laser: distributed Bragg reflector laser; PPLN: PPLN waveguide; TC: temperature controller; WDM: wavelength division multiplexer; BP: band-pass filter; FPBS: fiber optic polarizing beam splitter; DCM-FBG: dispersion compensation module based on the fiber Bragg grating; DCM-DCF: dispersion compensation module based on the dispersion compensating fiber; SMF: single mode fiber; SNSPD: superconducting nanowire single-photon detector; TTU: time tagging unit.
3.2 Results and discussions
With the help of Hong-Ou-Mandel quantum interference measurement [32], the initial temporal correlation width ($\Delta _{0, \tau }$) of the photon pairs was measured to be $\sim 0.5$ ps. Furthermore, the temperature of the PPLN waveguide for wavelength-degenerate phase matching was determined to be 23.3 $^{\circ }$C. In this case, the signal and idler photons should have the same wavelength of 1561.2 nm, which is exactly twice that of the CW pump laser. The joint spectral intensity of the photon pairs was measured with the hybrid frequency-time spectrograph method [31] and plotted in Fig. 4(a), indicating an evident frequency anti-correlation. The timing jitter of the detection setup ($\Delta _{jit}$) was measured by injecting both the signal and idler photons directly into the SNSPDs (not shown in Fig. 3). Under the setting of fixing the count rates at $\sim$100 kHz, the total timing jitter of the detection setup was evaluated as $78.4 \pm 0.7$ ps in FWHM. Based on the measurement setup shown in Fig. 3(b), the measured coincidence count distribution was observed to shift in time as the temperature of the PPLN waveguide was varied. According to the wavelength-to-time mapping relationship, the dependence of the signal photon wavelength on the PPLN waveguide temperature was identified and is shown in Fig. 4(b) with black squares. A linear fit to the measured data indicates a temperature tuning coefficient of 0.57 nm/$^{\circ }$C over the wavelength range of 1537-1568 nm. In addition, the count-rate of the generated photons is inserted in Fig. 4(b), showing slight fluctuations when the PPLN temperature was tuned from 12 $^{\circ }$C to 65 $^{\circ }$C.
Fig. 4. (a) Measured joint spectral intensity of the wavelength-degenerated photon pairs; (b) Signal wavelength and count-rate (insertion) versus PPLN temperature; (c) Measured coincidence width as a function of signal wavelength.
For the NDC configuration, the measured coincidence widths as a function of the signal photon wavelength are shown by blue dots in Fig. 4(c). When the signal photon wavelength was set to 1561.2 nm, which corresponds to the wavelength-degenerate phase matching condition, the measured coincidence width was about $122.8\pm 0.8$ ps as indicated by the blue star. This coincidence width evidently indicates a residual dispersion that contributes to the temporal broadening. However, by tuning the signal photon wavelength to 1565.7 nm, a minimum coincidence width of $86.1\pm 0.7$ ps was obtained. Substituting the experimental parameters into Eqn. (7), we generated a fitted curve shown by the blue solid line, which shows a good agreement with measured data. For comparison, the measured coincidence widths as a function of the signal photon wavelength via LDC and the corresponding simulation are also shown as red symbols and line respectively. We see that using LDC, the weak dispersion dependency on the wavelength would lead to a relatively constant coincidence width of about 110 ps, which is greater than the optimized value obtained using NDC.
To further explore the effectiveness of NDC, we continued to use the same DCF for compensating the dispersion of SMF spools with different lengths varying from 48 km to 60 km. The minimized coincidence widths, which were achieved by tuning the temperature of the PPLN, with respect to different fiber lengths are shown by blue dots in Fig. 5. As expected, by maintaining the signal photon wavelength at 1561.2 nm, the net dispersion would evolve from under-compensation to over-compensation when the SMF length was shortened to 48 km. However, when the signal wavelength was tuned at 1567.7 nm by reducing the PPLN temperature to 12 $^{\circ }$C, zero net dispersion could be approached and thus a minimum coincidence width of 95.8$\pm$1.4 ps was obtained. Vice versa, when the SMF was extended beyond 50 km, the increased dispersion under-compensation can also be corrected by raising the PPLN temperature for shifting the signal photon to a shorter wavelength. According to the measurements, the PPLN temperature was set to 52 $^{\circ }$C for optimizing the dispersion compensation of the SMF length of 60 km. The corresponding minimum coincidence width was measured as 88.2$\pm$0.6 ps. Therefore, the configuration of NDC displays wide flexibility in optimizing the dispersion compensation even for a fairly large fiber length variation from 48 km to 60 km.
For comparison, similar measurements were implemented with the configuration of LDC, which shows a very rapid broadening as long as the SMF length was other than 53 km, though at which length a minimum temporal width close to the system jitter was also achieved. Utilizing the theoretical model given by Eqn. (7) and (6), simulation for the results of NDC and LDC were also presented in Fig. 5 with solid lines, which show excellent agreements with the respective experimental results. Analogous to the concept of dispersion length [33], which describes the propagation distance of a non-chirped pulse when it is dispersively broadened by $\sqrt {2}$, a quantitative measure of the effective dispersion compensation range can be defined as the region in which the measured coincidence width remains less than $\sqrt {2}$ times the detection system jitter. It is easy to find in Fig. 5 that, for the fixed DCF of 6.2 km, the tight temporal correlation could be maintained only for the SMF length range of 49-54 km (area filled with light red) via LDC. In contrast, thanks to the strong dependence of dispersion on wavelength in NDC, the SMF length range can be extended to 46-73 km (area filled with green) which indicates a five-fold improvement and thus gives a solid proof of the wide flexibility of the NDC.
4. Conclusion
Based on the wavelength-dependent dispersion characteristics of standard fiber components, a simple method for flexible and precise dispersion compensation with the NDC configuration is presented in this paper. In our method, the photon wavelength is tuned by changing the temperature of the PPLN waveguide employed to generate the entangled photon pairs. Using a fixed length of dispersion compensating fiber, optimized NDC condition can be actively maintained over the so-called effective dispersion compensation range for entangled photon transmission in SMF varying from 46 to 73 km, which is five times greater than the distance variation range achievable using traditional local dispersion compensation. As our method avoids the need to use an elaborate dispersion compensation device for accurate dispersion pre-compensation, it provides a much easier and more flexible way to optimize dispersion mitigation, which is expected to play an important role in fiber-based quantum information and metrology applications.
Funding
National Natural Science Foundation of China (12033007, 12103058, 61801458, 61875205, 91836301); Chinese Academy of Sciences (QYZDB-SSW-SLH007, XAB2019B15, XAB2019B17, XDC07020200); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2021408, 2022413).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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