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Abstract
In this work, we present a new time-bin phase-encoding quantum key distribution (QKD), where the transmitter utilizes an inherently stable Sagnac-type interferometer, and has comparable electrical requirements to existing polarization or phase encoding schemes. This approach does not require intensity calibration and is insensitive to environmental disturbances, making it both flexible and high-performing. We conducted experiments with a compact QKD system to demonstrate the stability and secure key rate performance of the presented scheme. The results show a typical secure key rate of 6.2 kbps@20 dB and 0.4 kbps@30 dB with channel loss emulated by variable optical attenuators. A continuous test of 120-km fiber spool shows a stable quantum bit error rate of the time-bin basis within 0.4%∼0.6% over a consecutive 9-day period without any adjustment. This intrinsically stable and compatible scheme of time-bin phase encoding is extensively applicable in various QKD experiments, including BB84 and measurement-device-independent QKD.
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1. Introduction
Quantum key distribution (QKD) allows two distant parties, Alice and Bob, to communicate with information-theoretic security guaranteed by rules of quantum mechanics [1]. Unlike conventional public key cryptography schemes, which are vulnerable to decryption with increasing computational power, QKD is an ideal way to settle the problem regarding cipher text transfer. Legal communication entities, Alice and Bob, can evaluate the secrecy of the channel from the transmission bit error and can distill final secure keys from a set of received bits by post-processing. Since the introduction of the first QKD protocol by Bennett and Brassard in 1984 (BB84) [2], QKD has achieved considerable progress in both theoretical and experimental aspects [1,3–5]. Among various QKD protocols, the BB84 protocol is widely used in practical QKD systems due to its high performance, simple structure, and ease of maintenance. Several implementations of QKD systems exploiting various photonic degrees of freedom have been reported in recent years, including polarization encoding [6,7], phase encoding [8,9], and time-bin phase encoding [10,11]. Among these, the time-bin phase-encoding scheme exhibits advantages such as high key rates, long distances, and immunity to channel disturbance. Moreover, the time-bin basis is intrinsically stable in the receiver without requiring feedback, as opposed to the other two schemes that rely on polarization feedback [12–14] or phase feedback [8]. This inherent stability makes the system more reliable than the other two. Therefore, the time-bin phase-encoding scheme has gained increasing attention and is preferred for QKD systems.
The practical implementations of different encoding schemes can vary significantly. For a typical time-bin phase encoding implementation, two modulators are required: one phase modulator (PM) for phase encoding and one intensity modulation (IM) for time-bin encoding [10]. These modulators are often operated at twice the system frequency, and the time-bin states require an active feedback strategy to maintain stability due to the Mach–Zehnder interferometer structure. These requirements render the encoding setup either complex or unstable. A recent work [15] improves the stability with a Sagnac-type IM in a time-bin phase encoding structure. However, it still requires additional PM for X/Y phase basis. Another type of implementation employs the optical injection locking (OIL) technique [16,17] without the need of high-speed modulators. Instead, the driving pattern of master and slave lasers are modulated. However, the phase modulation is based on the amplitude modulate in the interval between the onsets of two slave pulses, which has higher requirement on the laser modulation. In comparison, polarization or phase encoding implementations typically generate a 4-level voltage loaded on single phase modulator [6–9] within a symmetric or asymmetric interferometer, allowing them to share the same electrical hardware.
In this work, we propose a novel scheme of time-bin phase encoding in QKD, based on Sagnac-type optical structure and compatible electrical hardware that reduces the requirement for high-operation-frequency modulation electronics for lasers or modulators. This new scheme allows for an intrinsically stable encoding of time-bin basis with the assistance of Sagnac-type optical structure. Furthermore, the modulation electronics only need to generate a 4-level voltage to obtain {0, π, π/2, 3π/2} phases, which also meets the requirement of polarization encoding or phase encoding. This means that the transmitter can be converted to the other two encoding schemes using the corresponding optical modules and configurable software. Based on this scheme, we have developed a compact QKD system and perform an experiment, demonstrating its secure key rate and attainable distance. We obtained a typical secure key rate of 6.2 kbps@20 dB and 0.4 kbps@30 dB. We have also demonstrated the system's stability by operating it for a long duration of 9 days, obtaining a stable quantum bit error rate (QBER) of approximately 0.4%∼0.6% in the time-bin basis, and a QBER of 2.0%–3.6% in the phase basis using an active phase-stabilization strategy. Our system can be applied to establish QKD networks between different cities, even different countries [18].
2. Scheme and setup
The time-bin phase-encoding scheme is based on the structure of a Sagnac interferometer and an asymmetric M-Z interferometer (AMZI). The schematic structure is shown in Fig. 1.
Fig. 1. Experimental setup of the transmitter based on the proposed time-bin phase-encoding scheme and the receiver based on the Michelson interferometer. LD: laser Diode; PM: phase modulator; CIR: circulator; BS: beam splitter; VOA: variable optical attenuator; ISO: isolator; FBG: fiber Bragg grating; DWDM: dense wavelength division multiplexer; PS: phase shifter; FM: faraday mirror; FDL: fiber delay line; APD: InGaAs/InP avalanche photodiode.
We adopted a Sagnac-based intensity modulation unit (S-IM) [19–22], including a beam splitter (BS) and a phase modulator (PM) in the Sagnac loop. The BS splits the signal pulse into two coherent parts, traveling in the Sagnac loop in opposite directions, i.e., the clockwise and anti-clockwise directions. The two parts can be modulated with different phases $\Delta \emptyset $ when traveling through the PM at different time slots, such as PM modulates the phase of the unit-clockwise pulse to introduce this phase difference, as shown in Fig. 2(a). The time difference is set to be optimal 1.6 ns, and its accuracy relies on the precision of the fiber length in the Sagnac loop and can be controlled at a level of <5 ps. Thus, they interfere at the BS with this phase difference and could generate two interference pulses outputted by ports 0 and 1 of the BS, with the complex amplitudes ${\alpha _0},{\alpha _1}$ as follows:
Fig. 2. (a) Relationship between optical pulses and electrical signals in S-IM2 and AMZI. (b) Intensity correlation and phase correlation between the two pulses.
Moreover, the two pulses the S-IM1 outputs exhibit both intensity and phase correlations, as depicted in Fig. 2(b). In terms of intensity correlation, the two pulses demonstrate complementary intensity correlation, particularly when the phase difference $\Delta \emptyset $ is 0 or π, resulting in a single output pulse. When $\Delta \emptyset $ is π/2 or 3π/2, both pulses output half the intensity. In terms of phase correlation, comparing the two scenarios of Δ∅︀ within the ranges of 0∼π and π∼2π, a stable π difference (noted as geometric phase [23,24]) exists between the relative phases of two pulses. Notably, we consider the situations of $\Delta \emptyset $ set to π/2 and 3π/2, although both pulses output the same half intensity, there is a stable π disparity between their respective relative phases.
With these correlations, we ultilize S-IM1 and cascaded passive AMZI to prepare the time-bin phase states for BB84 protocol, as shown in Fig. 2(a). We apply four phase settings: {0, π/2, π, or 3π/2} of $\Delta \emptyset $ on the PM within S-IM1. Its two output pulses are then transmitted to the following AMZI through two different optical paths with a time difference ${t_1} - {t_0}$, and are then coupled by the BS in AMZI to obtain the four BB84 states:
Table 1. Relationship between phase $\Delta \emptyset $ and the corresponding quantum state
Based on this time-bin phase-encoding scheme, we develop a QKD transmitter clocked at a system repetition rate of 312.5-MHz. This transmitter includes a signal laser source that generates phase-randomized light pulses. S-IM2 is used to modulate the pulse intensity for the decoy-state method [25–28], which ultilizes the same modulation mechanism of S-IM1. Then, the time-bin phase-encoding apparatus is used to prepare four BB84 states. We consider an asymmetric BB84 protocol [29] with the “signal + vacuum + weak” decoy-state scheme [28]. The pulses are then attenuated by two variable optical attenuators (VOA) to the single-photon level, with the photon numbers of the signal, decoy, and vacuum states set to $\mu $ = 0.4, $v$ = 0.1, $\rho \ll 0.001$. A true random number generator (Model: QCTWNG from QuantumCTek) [30] sets the probabilities of sending the signal, decoy, and vacuum states to ${p_\mu }$= 0.5, ${p_\nu }$= 3/8, and ${p_\rho }$= 1/8, respectively. The basis probabilities for the signal state (decoy or vacuum states) are set to ${p_{z|\mu }}$= 7/8, ${p_{x|\mu }}$= 1/8 (${p_{z|v}}$ = 2/8, ${p_{x|v}}$= 6/8, ${p_{z|\rho }} = {p_{x|\rho }} = 1/2$). A 99:1 BS and a monitor are used to measure the signal intensity for stabilization. Before the transmitter outputs the quantum signal, an isolator, a Fiber Bragg Grating filter with a circulator, and another monitor are used for better isolation and monitoring of any injected attacking light.
The receiver employs an asymmetric Michelson interferometer design. A single beam splitter (BS) determines the measurement basis, with the Z:X basis split at a ratio of 70%:30%. In the Z basis, the BS output port directly links to an InGaAs/InP avalanche photodiode (APD0). Detection of the former or latter time position in one period maps to bit 0 or 1 in the Z basis. For the X basis, the other BS output port connects to a Michelson interferometer, followed by APD1 and APD2. Detection at APD1 or APD2 translates to bit 0 or 1 in the X basis. The three detectors operate at a 1.25-GHz gate frequency, with a detection efficiency of approximately 15% and a dark count rate below 800 Hz. A 600-ps time window within a system period of 3.2 ns reduces the effective dark count rate to below 200 Hz.
With the scheme of the high-speed QKD transmitter and receiver, we develop a compact QKD system based on the electrical structure of a field-programmable gate array (FPGA) and CPU for real-time operation. As shown in Fig. 1, the transmitter/receiver is housed in a 4U 19-inch rack unit containing the optical module and the custom-made electrical modules. The electrical modules include the pulse modulation electrical boards for time-bin phase encoding, the APD module for time-bin phase decoding, and the main control electrical board for system control altogether with data sifting, error correction and privacy amplification. Standard 1 G Ethernet interfaces are used for classical communication between the transmitter and receiver. It is worth noting that, with the same electrical modules, we could only modify the optical module and the corresponding configuration parameters to implement the other QKD systems, such as for phase or polarization encoding.
3. Experiment results
To demonstrate the feasibility and stability of this compact QKD system, we firstly adopt VOA with different loss conditions to emulate the channel losses and measure the specifications in a lab environment, with the results shown in Fig. 3. The maximal loss can be extended to 30 dB, and a typical secure key rate of 6.2 kbps@20 dB and 0.4 kbps@30 dB are obtained. In the experiment, a block size of approximately 0.5Mbits is used for link loss of 0-25 dB, while 1 Mbit for 30 dB. Thus, simulated secure key rate for block sizes of 0.5 Mbits and 1 Mbits are both presented in Fig. 2. The QBER in the time-bin basis can be kept very stable and low within 1%. Meanwhile, the QBERs in the phase basis can be maintained within (1.0%–3%) below 25 dB, while increased up to 6.0% in 30-dB case. Here we adopt a phase-feedback strategy [31] to minimize the QBERs in the phase basis, using a feedback frame time-multiplexed with the quantum frame by 20% time consumption.
Fig. 3. The measured secure key rate and the QBER (Z/X basis) under different attenuation conditions using VOA.
In the case of a 30-dB loss, all the relevant parameters are listed in Table 2 for a typical running duration of $t = 256\;\textrm{ s}$ in our experiment to accumulate approximately 1-M bits sifted key. A final key rate $R = L/t$ of approximately 0.4 kbps. Details can be found in Appendix.
Table 2. Measured parameters and calculated results in the 30-dB case
To demonstrate the stabilization performance of this system, we implement a continuous test for 9 days with 120-km-fiber of 24 dB. As shown in Fig. 4, the QBER of the signal state of the Z basis remains very stable within (0.4%–0.6%), and the QBER of X basis can be maintained within (2.0%–3.6%) based on the aforementioned phase-feedback strategy [31].
4. Conclusion and discussion
In this work, we present a new time-bin phase-encoding transmitter for QKD based on a Sagnac-type optical structure and compatible electrical hardware. Using this transmitter, we implemented the QKD experiment in laboratory, obtaining a typical secure key rate of 6.2 kbps @20 dB and 0.4 kbps @30 dB. Furthermore, a 9-day continuous test demonstrated its stabilization performance, achieving a stable QBER of approximately 0.4%∼0.6% in the time-bin basis and a QBER of 2.0%–3.6% in the phase basis by an active phase-stabilization strategy.
Using this transmitter, we can easily extend the applications for polarization or phase encoding in diversified situations, based on a compatible modulation electronics, flexible optical modules, and configurable software. It is because the modulation electronics of this transmitter generates a 4-level voltage, which also meets the polarization or phase encoding requirement. Moreover, the Sagnac-type optical structure of the transmitter ensures intrinsic stability of time-bin phase encoding and a very low encoding error, resulting in high stability and a high secure key rate. Last but not least, the traditional scheme using PM for phase encoding is not convenient to monitor the phase fluctuation of phase encoding in the transmitter. In this scheme, the geometric phase characteristics of S-IM structure are utilized to achieve stable phase encoding, and the phase fluctuation on the phase modulator in S-IM is transformed to intensity variation. Therefore, it is easier to design an intensity monitoring scheme rather than a direct phase monitoring one to stabilize the phase modulator status. Consequently, this scheme can better serve the needs of flexibility in encoding schemes while maintaining high performance. It offers a preferred approach contributed to the development of quantum communications toward diverse and complex environment.
Appendix
Based on the theory of asymmetric decoy-state QKD, we developed data postprocessing algorithms using an FPGA implementation to finish both error correction and privacy amplification. We consider the statistical fluctuation with a finite key length of 1 Mbits of ${M_{\mu zz}} + {M_{\mu xx}}$, the measured detection counts of Z and X bases after basis sift in signal state. We obtain the following formula to calculate the secure key length:
$M_{1zz}^L$ and $\textrm{e}_{1\textrm{z}z}^{pU}$ (similar for $M_{1xx}^L$ and $\textrm{e}_{1xx}^{pU}$) are calculated using the following formulas:
Here
${f^U}(x )= x + \beta + \sqrt {2\beta x + {\beta ^2}} ,{f^L}(x )= x - \frac{\beta }{2} - \sqrt {2\beta x + \frac{{{\beta ^2}}}{4}} $ with $\beta ={-} ln\frac{{{\varepsilon _{sec}}}}{{23}}$ are the lower and upper bounds of expected value for a given observed value x, respectively, by considering the statistical fluctuations using the variant of Chernoff bound with a failure probability $\varepsilon = {\varepsilon _{sec}}/23$. ${p_{z/x}}$ is the probability of choosing the Z/X basis in the transmitter. ${M_{\mu (\nu, \rho )zz}}$ is the measured detection count corresponding to choosing $\mu ({\nu ,\rho } )$ intensity in the transmitter and Z basis after basis sift. $M{E_{\mu (\nu )xx}}$ is the measured error detection count corresponding to choosing $\mu (\nu )\;\textrm{the }$ intensity in the transmitter and X basis after basis sift.
${\theta _x}$ is calculated according to Ref. [34,35] via a random-sampling theory without replacement:
The factor $\frac{1}{{23}}$ is due to ${\varepsilon _{\textrm{sec}}} = n\varepsilon + 2({2{\alpha_1} + {\alpha_2} + {\alpha_3}} )+ \bar{\nu } = 23\varepsilon $ [35], when ${\alpha _1} = {\alpha _2} = {\alpha _3} = \bar{\nu } \buildrel \Delta \over = \varepsilon $ and $n = 14$ ($n$ means the times to use the function g or f for parameter estimation). Thus, the secure parameter for each parameter estimation is ${\varepsilon _{\textrm{sec}}}/23$.
The error correction is realized using the Winnow algorithm, with CRC-64 for verification. The initial fragment length is set to 15, resulting in an error correction efficiency of approximately 1.8 for 1% error rate. The fragment length can be optimized to improve the error correction efficiency. Toeplitz matrix is employed for privacy amplification. The failure probability is set to ${\varepsilon _{\textrm{sec}}}\; = \; {10^{ - 9}}$.
Funding
Major Scientific and Technological Special Project of Anhui Province (202103a13010004); Major Scientific and Technological Special Project of Hefei City (2021DX007); Key R&D Plan of Shandong province (2020CXGC010105); China Postdoctoral Science Foundation (2021M700315).
Acknowledgments
We thank Enago for its linguistic assistance during the preparation of this paper.
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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