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Abstract
Quantum key distribution (QKD) promises unconditional security for communication. However, the random choices of the measurement basis in QKD usually result in low key creation efficiency. This drawback is overcome in the differential-phase-shift QKD, provided that each photon can be prepared in a large number of time slots with a proper waveform. In this work we develop a miniature room-temperature 1550-nm single-photon source to generate narrowband single photon in 50 time slots with a nearly optimal waveform for achieving unity key creation efficiency. By utilizing these single photons in the field test, we demonstrate the differential-phase-shift QKD with a key creation efficiency of 97%. Our work shows that the practical QKD can benefit from the narrowband single photons with controllable waveforms.
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1. Introduction
Quantum key distribution (QKD) promises unconditionally secure communication granted by the laws of physics [1]. The first demonstration of QKD [2], proposed by Bennett and Brassard [3], was based on the polarization encoding. Since then, different protocols [4–10] have been demonstrated to avoid the polarization distortion in optical fibers and extend the transmission distance [11–29]. While the weak coherent pulses (WCP) were widely used in these demonstrations due to their simple realization, the quantum light sources offer additional benefits. For example, the use of single photons can tolerate higher loss [30,31] and higher quantum bit error rate (QBER) [32]. The use of entangled photons can extend the distance of QKD with the help of quantum repeaters [33].
Recently, single photons with long coherence time and narrow bandwidth have been exploited for enhancing the key creation efficiency (KCE) of QKD [34]. By using the 795-nm narrowband single photons generated from the cold atoms, Liu et. al increased the KCE of differential-phase-shift (DPS) QKD [5] from 22.2% to 66.6% with unconditional security. However, the wavelength and complexity of the used single-photon source are not suitable for practical QKD. To realize the QKD in a fiber network with multiple users, compact single-photon sources in the telecommunication band are necessary for high scalability and low transmission loss.
In this paper we demonstrate the field test of QKD with highest reported KCE thus far to the best of our knowledge. This is accomplished by implementing a miniature 1550-nm narrowband single-photon source at the room temperature. The long coherence time (198.8 ns) and narrow bandwidth (2.3 MHz) of the single photons allow us to increase the KCE and reduce the QBER [35], respectively. Moreover, the double-exponential waveform of the single photons is nearly optimal for achieving the unity KCE. By shaping each photon into 50 equally spaced time slots, with the width and spacing $T_s = 1.65$ ns, we achieve a KCE of 97% and a QBER of 3.2% below the threshold level of unconditional security [36]. The KCE is 94% and 11.8% higher than that of the BB84 and T12 protocol [37], respectively. Our work shows that the single photons with long coherence time and narrow bandwidth are feasible for practical QKD.
2. High key creation efficiency
In the DPS QKD [5], Alice (the sender) prepares each photon in three time slots with a flat-top envelope. The bits (0 or 1) are then randomly encoded in the phase difference (0 or $\pi$) between the adjacent time slots. After receiving the photons, Bob (the receiver) measures the bits with an one-bit delay Mach-Zehnder interferometer. If the photon is not detected in the first and last time slots, the bits are extracted by correlating the exit port of the photon to the phase difference. The KCE (namely, the probability of successfully creating one bit per photon) is thus equal to the probability of finding the photon at the correct exit port and in the time slots other than the first and last slots after the interference. In Fig. 1(a) we show the calculated KCE for DPS QKD by considering each photon prepared in $N \geq 3$ time slots [colored areas in the inset of Fig. 1(b)] with an envelope of various waveforms [dashed line in the inset of Fig. 1(b)]. The width and spacing of the time slots are both $T_s = 1.65$ ns and the non-flat waveforms are assumed to have a $1/e$ decay time constant of 52 ns. While the high number $N$ of time slots is important to obtain high KCE (which approaches 100% at very large $N$), the waveform of the envelope has a notable impact on the KCE. In general, the symmetrically decaying waveforms such as the triangle, double exponential, and Gaussian waveforms increase the KCE by lowering the probability of finding the single photons in the first and last time slots. However, due to the mismatch in the amplitudes between the adjacent time slots, the interference visibility of all non-flat-top waveforms deviate from 100%. Nevertheless, the QBERs induced by the amplitude mismatch are negligible compared to the threshold level 4.12% of the unconditionally secure DPS QKD [36]. This can be seen in Fig. 1(b), where we calculate the QBER by computing the probability of finding the photon, which is not in the first and last time slots after the interference, at the wrong exit port.
Fig. 1. (a) The calculated key creation efficiencies and (b) the QBERs for different numbers of time slots (colored areas) and waveforms or envelopes of the single-photon wavepackets (dash lines). The inset shows an example of single-photon wavepacket with a double-exponential waveform.
3. Miniature telecom single-photon source with controllable waveforms
To increase the KCE, temporally long and symmetrically decaying single-photon wavepackets are advantageous. Figure 2(a) illustrates the setup of our single-photon source. Degenerate photon pairs at 1550 nm are generated from a periodically poled and type-II phase-matched KTiOPO$_4$ (PPKTP) crystal inside a monolithic resonator, which is implemented by spherically polishing the end faces of the crystal and depositing a 1550-nm high-reflection coating. The pump is a 775-nm cw external cavity diode laser (ECDL) frequency-stabilized by a high-precision wavemeter (1-MHz resolution and 30-MHz absolute accuracy). By depositing an additional high-reflection coating at the pump wavelength on the rear end, we realize the double-pass pumping. Together with the monolithic resonator, this enables the photon pairs to be generated in a single longitudinal mode without the need of external filters [38–40]. After the photon pair are separated by a polarizing beamsplitter (PBS), a single (signal) photon is heralded by detecting an idler photon with a single-photon counting module (SPCM, ID Quantique ID230). Accounted for the fiber-coupling efficiency (75%), transmission loss (5%), and detector’s quantum efficiency (15%), the generation rate is $4.2 \times 10^5$ s$^{-1}$ per mW of the pump power with a fluctuation within 15% about the average rate [Fig. 2(b)]. Figure 2(c) shows the measured second-order quantum coherence function at zero time delay $g^{(2)}(0)$ [41] of the heralded single photons. For the pump power below 600 $\mu$W, the antibunching ($g^{(2)}(0) < 0.5$) is clearly evident.
Fig. 2. (a) Experimental setup of the miniature room-temperature 1550-nm single-photon source based on the monolithic doubly resonant parametric down-conversion. The abbreviations stand for the data acquisition (DAQ), beamsplitter (BS), polarizing beamsplitter (PBS), and electro-optic modulator (EOM). (b) The generation rate versus the time. (c) The pump power is kept low to ensure good single-photon quality ($g^{(2)}(0) < 0.5$). Due to the finite dead time ($\sim 2\ \mu$s) and the lack of the photon-number resolvability of the detectors, the measured $g^{(2)}(0)$ saturates when the pump power is above 600 $\mu$W. (d) The temporal wavepacket (blue area) of the heralded single photons. The red curve is the fit of the double-exponential function. (e) The electro-optic modulation of the single-photon wavepacket into a pulse train. The integration times are 180 s and 120 s in (d) and (e), respectively.
As shown in Fig. 2(d) (blue area), the wavepacket of the single photons has a long temporal length and a double-exponential waveform. The wavepacket is described by the Glauber correlation function [38],
4. Field test of high-KCE DPS QKD
The field test of the DPS QKD is carried out in the daytime using a pair of the inter-university fibers between the campuses of National Tsing Hua University (NTHU) and National Yang Ming Chiao Tung University (NYCU) in the Hsinchu City, Taiwan. The fiber pair, both extending from the General II Building in NTHU to the IT Service Center in NYCU [Fig. 3(a)], are connected at the NYCU site to form a fiber loop with a total length of 3.4 km. The attenuation of the fiber pair is 2 dB mainly due to the existing fiber-to-fiber connectors in the network. Figure 3(b) illustrates Alice’s setup at the NTHU site. The single photons, after being shaped into multiple time slots with a pulse width of 1.65 ns by an electro-optic intensity modulator (EOIM) with an insertion loss of 1.7 dB, are sent into a fiber-coupled electro-optic phase modulator (EOPM) with an insertion loss of 1.6 dB. Alice uses the phase modulator to code the bits by applying random phase differences 0 and $\pi$ on adjacent time slots. To account for the broadening of the time slots by the modulator’s finite bandwidth, the pulse width of the phase modulation is chosen to be wider than the pulse width of the intensity modulation.
Fig. 3. (a) The field test exploits the inter-university optical fiber network (orange line) between the General II Building (yellow dot) at NTHU (bounded by the yellow dash line) and the IT Service Center (red dot) at NYCU (bounded by the red dash line) in Hsinchu City, Taiwan. Alice’s and Bob’s setup are illustrated in (b) and (c), respectively. The abbreviations stand for half-wave plate (HWP), beamsplitter (BS), polarizing beamsplitter (PBS), piezoelectric actuator (PZT), photodiode (PD), and superconducting nanowire single-photon detector (SNSPD). (d) The interference visibility of the Mach-Zehnder interferometer with (black curve) and without (red curve) the active stabilization. (e) The Poincaré sphere shows the fluctuation of a single-frequency laser’s polarization state after passing through the fiber pair. The polarization state starts at the green arrow (the horizontal) and ends up in the orange arrow after 24 hours. (f) The time-resolved interference of single photons prepared in 20, 30, and 50 time slots (from the left to right). The integration time and coincidence time bin are 600 s and 0.2 ns, respectively.
The bits are later decoded by Bob with a one-bit delay Mach-Zehnder interferometer [Fig. 3(c)] at the NTHU site. To keep the interferometer stable during the field test, the long arm (total length of 1.1 m) is folded to reduce the overall size. In addition, the interference of a counter-propagating single-frequency laser, offseted in position to avoid the light leaking into the detectors, is continuously monitored. The interference signal is then fed to a piezoelectric actuator on a mirror to actively stabilize the optical path difference. As shown in Fig. 3(d), the interference visibility with the active stabilization (black curve) is nearly unchanged compared to that without the stabilization (red curve). In practice, the fluctuation of polarization in the fibers can also degrade the interference visibility. Such fluctuation is shown in Fig. 3(e), where a single-frequency laser’s polarization state (initially prepared in the horizontal) after transmitting through the fiber loop is recorded over 24 hours by a polarimeter. To achieve high visibility and low QBER, a polarizing beamsplitter is placed in front of the interferometer as a polarization filter, resulting in a reduction of the count rate and key rate by 2%.
After a single photon enters the first beamsplitter of the interferometer, the wavefunctions taking the short and long paths interfere at the second beamsplitter. Depending on the phase difference between the adjacent time slots, constructive or destructive interference occurs between the time slots of the two wavefunctions, resulting in a click at one detector in the corresponding time slot. The time tags of these detection events are then sent back to Alice through a public channel for generating the bit string. In the demonstration of the field test, the single-photon counting modules are replaced by the superconducting nanowire single-photon detectors (Single Quantum Eos), which have a quantum efficiency higher than $82$% at 1550 nm. Examples of the single-photon interference are shown in Fig. 3(f) for single photons prepared in 20, 30, and 50 time slots. All phase differences between the adjacent time slots are set to 0 in these examples. By analyzing these time-resolved interferences, useful parameters can be obtained. Figure 4(a) (dots) shows the KCE, which is given by the probability of the successful interference, for different numbers of time slots. When the single photons are prepared in 50 time slots, the KCE reaches 97% and is in good agreement with the theory (solid curve). The probability of the failed interference (in the time slots except the first and last ones), on the other hand, can cause the photons to show up at the wrong exit ports, thus resulting in bit error and QBER as shown in Fig. 4(b) (circular dots). Finally, as the photons taking part in the interference contribute to the key, the count rate of the single photons gives an estimate of the achievable sifted key rates, provided that the modulation bandwidths are not limited. Such sifted key rates are also plotted in Fig. 4(b) (triangles). At higher numbers of time slots, because a larger portion of the single-photo wavepacket is used and the KCEs are higher, the sifted key rates are also higher.
Fig. 4. (a) The key creation efficiency (dots) versus the number of time slots. The solid and dash curves correspond to the theory considering realistic and 100% interference visibility, respectively. (b) The sifted key rates and the QBERs versus the number of time slots. The red dashed line at 4.12% represents the threshold level for the unconditional security [36]. (c) The QBER recorded over a two-hour of operation. (d) The sifted key rates (black dots) and secure key rates (blue dots) against the general attack of individual photons recorded over a two-hour of operation. The lengths of the outdoor fibers used in (a) [or (b)] and (c) [or (d)] are 3.4 km and 4.8 km, respectively.
With 50 time slots and 97% KCE, we obtain an average sifted key rate of $R_s = 3048$ bps and an average QBER of $e_b = 3.2$%, which is below the threshold level of the unconditionally secure DPS QKD [36]. Assuming an error correction coding efficiency approaching the Shannon limit and an ideal single-photon source, the secure key rates are $R_s\{-\log _2[1-e_b^2-(1-6e_b)^2/2]-H(e_b)\}=1117$ bps against the general attack of individual photons [36,43] and $R_s[1-H(e_b)-H((3+\sqrt {5})e_b)]=433$ bps against the coherent attack [36], where $H(x)=-x\log _2x-(1-x)\log _2(1-x)$ is the binary Shannon entropy. Figures 4(c) and (d) show the QBER, sifted key rate (black dots), and secure key rate (blue dots) against the general attack of individual photons in the QKD operation over two hours. For these measurements, the fiber pair are further extended from the General II Building in NTHU to the IT Service Center in NYCU to the Science Building III in NCYU, with a total length of 4.8 km and a total attenuation of 5 dB.
5. Conclusion
We have shown that both the single-photon waveform and the number of time slots in DPS QKD play important roles in enhancing the KCE. To generate the single photons with the waveform for achieving the perfect KCE, we implement a miniature room-temperature 1550-nm single-photon source based on the monolithic doubly resonant parametric down-conversion. The single-photon wavepacket has a nearly ideal waveform (double exponential) with a $1/e^2$ width of 198.8 ns and a generation rate of $4.2 \times 10^5$ s$^{-1}$mW$^{-1}$. With each photon shaped into 50 time slots, we demonstrate the field test of DPS QKD with 97% KCE and a QBER below the threshold level of unconditional security. To estimate the influence of the multiphoton emission, we approximate the photon statistics of our heralded single-photon source by a Poisson distribution (weak coherent state) with an average photon number of 0.37 [43]. We find that the secure key rate would decrease by 2%. By increasing the number of time slots and interference visibility, which require faster modulation and shorter optical path difference of the interferometer, the KCE can be further increased [see the dash line in Fig. 4(b)] and the QBER can be reduced. The approach demonstrated here can also be used for the narrowband single-photon sources based on the thermal atoms [44–47], integrated photonics [48,49], or cold atoms [50–55], provided that the operating wavelengths of the thermal- or cold- atom-based sources can be converted to the telecom band. Recently, a QKD network with a trusted relay have been set up across the campuses of NTHU and NYCU in Taiwan. The QKD system demonstrated here will be incorporated in this network to work with other QKD systems based on the weak coherent states. Our work shows that the practical QKD can benefit from the single photons with long coherence time, narrow bandwidth, and controlled waveforms.
Funding
National Science and Technology Council (110-2112-M-007-021-MY3, 111-2119-M-007-007).
Acknowledgments
The authors would like to thank the Computer and Communication Center of NTHU and the Information Technology Service Center of NYCU for their kind assistance with the inter-university optical fiber network. We also thank D.-S. Chuu, C.-Y. Mou, H.-S. Goan, I. A. Yu, C.-M. Li, A. Yabushita, Y.-C. Liang, Y.-N. Chen, R.-K. Lee, and C.-M. Wu for the fruitful discussion and experimental assistance.
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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