Space-division multiplexed phase compensation for quantum communication: concept and field demonstration

We conceptualize our phase compensation scheme in Fig. 1. The space-division multiplexed phase compensation prepares two neighboring fibers, one fiber for quantum communication (signal fiber) and the other for real-time phase-sensing (reference fiber). The phase drift in the reference fiber provides information on the phase drift of the signal fiber, assuming the correlation of the phase drifts in the two fibers (Fig. 1). We can implement the space-division multiplexed phase compensation scheme either with real-time feedback or in data post-processing. One can use the common actuator shown in Fig. 1 for real-time feedback implementation. Phase compensation in data post-processing is often valid for single-photon interference schemes Liu2021PRL ; Chen2021 ; Wang2022 ; Liu2023 ; Zhou2023 . A fundamental question of the space-division multiplexed compensation is the correlation of phase drift patterns between the signal and reference fibers in a field environment, especially in a metropolitan area. To address this point, we investigate the phase drift pattern of commercial fibers in the Osaka metropolitan area.

Figure 2 explains our experimental setup using data post-processing. We installed our measurement system at a building near Osaka Castle, a central part of the Osaka metropolitan area (Fig. 2a). Cable $1$ goes through a central area of Osaka and returns at the turnaround point to the starting point (Figs. 2a and b). Similarly, Cable $2$ starts from the same building and returns at another turnaround point (Figs. 2a and b). Cable $1$ and Cable $2$ are entirely buried fibers (see Table 1). We used four-fiber ribbon in both Cable $1$ and $2$. The signal fiber is the nearest neighbor of the reference fiber (Fig. $2$c). Hence, we have prepared the commercial fibers to investigate the correlation of the phase drift between the two neighboring fibers. A heterodyne interferometric measurement detects the phase drifts of the signal and reference fibers in a sub-wavelength precision (Fig. 2a and Methods). We obtain a differential phase by subtracting the phase drift of the reference fiber from that of the signal fiber in post-processing. We emphasize that our measurement of the phase drifts continues for $7$ days. Our experimental time window is sufficiently long to evaluate the long-term robustness of the space-division multiplexed phase compensation in a metropolitan fiber network.

Figure 3 analyzes the phase drift measured in the buried fibers. The phase drifts of the reference and signal fibers are overlayed in Fig. 3a. Fig. 3b compares the power spectral density (PSD) of the phase as a function of frequency ranging from $1$ Hz to $5\times 10^{5}$ Hz. The PSD of the reference fiber is correlated with that of the signal fiber. Due to the phase drift correlation, the baseline of the PSD of the differential phase is approximately $100$ times lower than that of the signal fiber. These results validate our space-division multiplexed phase compensation between the two neighboring fibers.

We evaluate the impact of our space-division multiplexed phase compensation on quantum communication. Fig. 3c shows the QBER computed from the phase measurements. As expected, our phase compensation improves the QBER of the signal fiber. Here, we evaluate the interval until the QBER reaches $3$% in the absence of phase compensation, $t_{0.03}$. Without and with the space-division multiplexed phase compensation, $t_{0.03}$ is $1.4\times 10^{-5}$ s and $2.5\times 10^{-3}$ s, respectively. Thus, our space-division multiplexed phase compensation improves the interval $t_{0.03}$ by a factor of $180$.

Moreover, we investigate the long-term robustness of our space-division multiplexed phase compensation performance in the buried fibers. (See Supplemental Figs. 1 and 2 for the full results of the phase data in spectrograms and estimated QBERs for $7$ days, respectively.) Figure 4 summarizes the time variation of the interval $t_{0.03}$. Both the measured and compensated data are almost constant. Our space-division multiplexed phase compensation robustly improves the interval $t_{0.03}$ for $7$ days. Considering the simple setup of our compensation scheme, this robustness and improvement should be notable.

A field fiber network usually combines buried fibers and aerial fibers. Thus, our space-division multiplexed phase compensation should be investigated using fibers, including aerial parts. We have prepared another fiber setup in the Osaka metropolitan fiber network as shown in Supplemental Fig. 3. The field fibers in the second setup consist mostly of aerial and partially buried fibers (for detail, see the caption of Table 1). Hereafter, these fibers are simply called aerial fibers unless it is confusing. We have measured the phase of the two neighboring fibers for $5$ days.

Figure 5a shows the phase drift measured within an interval of $10$ s. Although the phase of the aerial fiber obviously fluctuates at a few Hz, the phase drift pattern shows an apparent smilarity between the reference and signal fibers. In Fig. 5b, the PSD analysis confirms the correlation of the phase drift between the two fibers. Indeed, the space-division multiplexed phase compensation decreases the baseline of the phase PSD by a factor of approximately $100$ (see curves labeled as Reference and Differential in Fig. 5b). Figure 5c shows the QBER of the reference and compensated phase. Again, we set the QBER threshold at $3$% and compute the interval $t_{0.03}$ of the aerial fiber. Our phase compensation improves the interval $t_{0.03}$ from $2.5\times 10^{-4}$ s (without compensation) to $4.4\times 10^{-3}$ s (with compensation) in data post-processing. These results validate our space-division multiplexed phase compensation, even in aerial fibers.

We also check the long-term robustness of our space-division multiplexed phase compensation in aerial fibers. (See Supplemental Figs. 4 and 5 for the full results of the spectrograms and estimated QBERs of the aerial fibers for $5$ days.) In the aerial fibers, the phase drift fluctuates in a daily cycle. The interval of $t_{0.03}$ without compensation ranges from $2.5\times 10^{-5}$ s to $4.4\times 10^{-4}$ s, whose fluctuation amplitude is significantly larger than that of buried fibers shown in Fig. 4. Nonetheless, our space-division multiplexed phase compensation scheme has improved the interval $t_{0.03}$ of the aerial fibers by a factor of $18$ in our experiment.

Moreover, we find that the field environment (buried or aerial) affects the low-frequency phase drift. The aerial and buried fibers have unique peaks at $\sim 2$ Hz (Figs. 5a and b) and $\sim 20$ Hz (Fig. 3b), respectively. These peaks indicate (i) wind galloping on the aerial fiber Wuttke2003 ; Ding2017 and (ii) mechanical vibration from transportation to the buried fiber HAO2001 . Although the low-frequency phase drifts have little impact on our results of $t_{0.03}\sim 10^{-3}$ s, understanding the phase drift in a full frequency range may be helpful for the further improvement of QBER in a field fiber network.

Our field study has demonstrated that space-division multiplexed phase compensation is applicable in both buried and aerial fiber conditions. Now, we address the technical limitations of our setups for the future improvement of space-division multiplexed phase compensation. In the first setup (buried fibers), the compensation has improved the interval $t_{0.03}$ to $2.5\times 10^{-3}$ s (Fig. 3c). This means that the high-frequency noise above $\sim 10^{3}$ Hz limits the further improvement of QBER. In the PSD, the reference and signal fiber phase noises periodically dip at $83$ kHz and its higher harmonics (Fig. 3b). These dips are explained as the laser frequency noise cancellation because their frequency matches the theoretical prediction (Methods and Table 1). It will be the case when the lasers used for the signal and reference fibers are the same (like our demonstration setup) or as long as they are synchronized. Therefore, the reduction of the laser frequency noise for the reference measurement should improve QBER.

In the second setup (aerial fibers), the compensated QBER shows a significant fluctuation in a daily cycle. We question if the “precision” of our phase compensation also fluctuates in the aerial fiber condition. Supplemental Fig. 6 shows the long-term change in the interference visibility of our system. The visibility of the second setup significantly fluctuates in comparison with that of the buried fiber. Thus, the visibility fluctuation in the second setup comes from the aerial part of the fiber. The fluctuation of polarization in the aerial fibers is a sensible explanation for the visibility fluctuation in the second setup. This explanation is supported by previous field investigations on the polarization drift of a field fiber bersin2023 , where the polarization drift couples with weather conditions (e.g., wind speed and temperature). Hinted from the previous reports, we compare the visibility fluctuation with the weather data of the Osaka metropolitan area. The visibility of the second setup tends to become unstable when the wind speed is relatively high. This tendency indicates that the weather conditions might change the interference visibility of the second setup. Taken together, the additional compensation for polarization drift may improve the long-term robustness of our space-division multiplexed phase compensation, especially when the fiber is in an aerial environment.

Figure 2(c) shows the schematic of the phase measurement. We use an external cavity diode laser (RIO ORION) with a wavelength of $1550\pm 2$ nm and a Lorentzian linewidth of $<1$ kHz. A beam splitter just after the laser source splits the laser beam for two cables. An acousto-optic modulator device shifts the frequency of the laser light incident to one cable by $40$ MHz for the heterodyne interferometry. The laser light is split to two paths (for signal and reference) before the cable input. A data logger with a sampling rate of $1$ MSa/s collected a phase dataset for $10$ s and paused the data collection for $50$ s as shown in Supplemental Fig. 7. We repeated the data collection cycles in the first setup for $7$ days and in the second setup for $5$ days. The total data of $1.4$ TB is analyzed.

Table 1 summarizes the one-way lengths of buried and aerial fibers of each of the cables. $\Delta{L}$ is calculated as $|L_{1}-L_{2}|\times 2$, where $L_{1}$ and $L_{2}$ are the one-way length of a cable and its complementary cable in a setup, respectively. To explain the dip of the high-frequency noise in Fig. 3b, the frequency interval ($\Delta{f}$) is calculated as $\Delta{f}=c/n\Delta{L}$, where $c$ and $n$ are the speed of light in vacuum and the refractive index of the optical fiber. Note that we set the coefficients as $c=3.0\times 10^{8}$ m/s and $n=1.5$. In our phase measurement, laser frequency noise is canceled at every $\Delta{f}$ along the frequency axis. In the first setup, the calculation of $\Delta{f}$ shows quantitative agreement with the experimental result of Fig. 3b. In the second setup, the frequency interval of $\Delta{f}$ is out of our measurement range (Fig. 5b).

Here, we consider the single photon interference for the quantum communication with an interferometer having two output ports. When the interference condition is set to be constructive for one output port, the photon is always detected in that port. The probability of the photon detection in the other port increases when the interference condition is shifted. This interference condition shift leads to the communication error. QBER is an error metric from the point of view of QKD Clivati2022 . Here, we ignore the other factors increasing QBER, such as the non-unity quantum efficiency of the detector. QBER in a given measurement time window is formulated as

where $\phi_{i}$ $(i=1,2,3,...,n)$ is the round-trip phase difference of the two arms of the interference, and $n$ is the number of phase data points. Here, $\phi_{i}=0$ is defined as the constructive phase condition explained above.

In our study, one phase dataset has a duration of $10$ s with a sampling rate of $1$ MSa/s. The dataset is split into subsets so that one subset corresponds to a duration of $t_{\rm g}$ (see Supplemental Fig. 7). We follow the equation (1) to compute QBER from a phase data subset, where $t_{\rm g}=10^{-5+k/4}$ s and $k=0,1,2,...,24$. Then, QBER is averaged over the subsets in a phase dataset. In Figs. 4 and 6, QBER is additionally averaged over phase datasets (Supplemental Fig. 7). In our analysis, $t_{0.03}$ corresponds to the time when ${\rm QBER}(t_{\rm g})$ exceeds $3$%.