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Abstract
Data acquisition in a continuous-variable quantum key distribution (CV-QKD) system is a necessary step to obtain secure secret keys. And the known data acquisition methods are commonly based on the assumption that the channel transmittance is constant. However, the channel transmittance in free-space CV-QKD fluctuates during the transmission of quantum signals, and the original methods are not applicable in this scenario. In this paper, we propose a data acquisition scheme based on the dual analog-to-digital converter (ADC). In this scheme, two ADCs with the same sampling frequency as the pulse repetition rate of the system and a dynamic delay module (DDM), which are used to construct a high-precision data acquisition system, eliminate the effect of transmittance fluctuation by a simple division operation of the data from the two ADCs. Simulation and proof-of-principle experimental results show that the scheme is effective for free-space channels and can achieve high-precision data acquisition under the condition of fluctuation of channel transmittance and very low signal-to-noise ratio (SNR). Furthermore, we introduce the direct application scenarios of the proposed scheme for free-space CV-QKD system and verify their feasibilities. This method is of great significance to promote the experimental realization and practical application of free-space CV-QKD.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantum key distribution (QKD) [1–3] enable two remote partys to share unconditional secure secret key. QKD can be divided into two directions: continuous-variable quantum key distribution (CV-QKD) and discrete-variable quantum key distribution (DV-QKD). CV-QKD has the advantages of high bit rate, low cost and easy compatibility with traditional optical fiber system, which have attracted great attentions recently [4–6]. So far, it has been experimentally demonstrated that the secure transmission distance of Gaussian-modulated coherent-state (GMCS) CV-QKD can achieve longer than 200 km through ultra low-loss optical fiber channel [7] and the secret key rate can reach Mbps orders of magnitude within metropolitan area [8,9].
In Ref. [10], the feasibility of free-space CV-QKD with coherent polarization states under real atmospheric conditions was demonstrated for the first time. Benefiting from the ability of coherent detection to resist background noise and high detection efficiency [10–13], all-day free-space quantum communication using coherent detection has be further demonstrated theoretically and experimentally feasible. Recently, in order to promote experimental realization of free-space CV-QKD, some techniques such as fading channel estimation [14], phase compensation [15], dynamic polarization control [16], excess noise suppression [17], frame synchronization [18] have been proposed, and the feasibility of CV-QKD through fog was further experimentally demonstrated [19].
In a CV-QKD system, a homodyne or heterodyne detector is usually used to measure the modulated key information encoded on the quadratures X and P of the coherent-states pulses with a electronic pulse output. And then a analog-to-digital converter (ADC) is applied to acquire the raw key data via pulse peak sampling. Typically, in the theoretical analysis of CV-QKD, it is generally considered that the sampling bandwidth of the ADC used for data acquisition is infinite. However, for practical fiber-based CV-QKD system, finite sampling bandwidth (FSB) effects [20] were found to affect its performance and practical security. The deviation of the sampling results will lead to errors in the parameter estimation, which will reduce the lower bound of the key rate and inhibit the linear promotion between the key rate and the system repetition rate.
To remove the FSB effects, an efficient scheme is proposed in Ref. [21] using dynamic delay module (DDM) and statistical power feedback control algorithm. In this way, the pulse peak values of the transmitted pulse can be accurately sampled, thus eliminating the FSB effects of fiber-based CV-QKD. Similarly, the FSB effect also exists in the free-space CV-QKD system. Unfortunately, the premise of the above scheme is that the channel transmittance is constant, while the atmospheric channel transmittance is fluctuating, so this scheme is inapplicable since the stability of the channel transmittance is the precondition of effective feedback control. And the algorithm of this scheme cannot support the system to work in the very low signal-to-noise ratio (SNR) environment. So far, there is no available high-precision data acquisition method for the scenario of quantum channels with fluctuating transmittance and very low SNR outputs, which quite restricts the field implementation of free-space CV-QKD.
In this paper, in order to achieve accurate pulse peak sampling for free-space CV-QKD, we present a high-precision data acquisition, which is independent on fluctuations of channel transmittance. In particular, by using two ADCs with the same sampling frequency, a DDM and the corresponding algorithm can thoroughly eliminate the influence of the fading effect of free-space quantum channel and work in the low SNR environment. The simulation and proof-of-principle experiment verify the feasibility and efficiency of the proposed scheme. In practical application, since the fluctuation of channel transmittance will also deactivate the feedback of the traditional dynamic polarization control. Here, the proposed high-precision scheme can just be used for free-space polarization compensation. Moreover, intuitively, the scheme can be directly applied for high-precision data acquisition of electronic output of homodyne detection under very low SNR condition. It is of great significance to promote the experimental realization and practical application of free-space CV-QKD.
The data acquisition process in the fiber-based CV-QKD system is reviewed and the free-space high-precision data acquisition method is proposed in section 2.. In the section 3., we carry out the simulation and a proof-of-principle experiment, and analyze the performance of the scheme. Finally, the application scenarios of the scheme are introduced and analyzed in section 4..
2. Free-space CV-QKD data acquisition scheme
In the CV-QKD system, the function of the data acquisition module is to extract the key information encoded on the quadratures X and P, and data acquisition is usually performed by means of oversampling. However, the sampling bandwidth of the ADC is finite, which will directly affect the accuracy of the sampled data. The inaccurate sampling results will affect the performance and practical security of the CV-QKD system [20]. In this section, we present the previous data acquisition scheme in the first subsection, however, it cannot be applied to free-space channels and low SNR conditions. In the second subsection, we describe in detail the proposed scheme, which is able to be applied to free-space channels in low SNR conditions compared to the previous scheme.
2.1 Limitation of the existing scheme
In order to improve the sampling accuracy and remove the FSB effects, a scheme based on co-frequency sampling was proposed [21], which can greatly reduce the excess noise induced by inaccurate pulse peak sampling. As shown in the Fig. 1, the function of DDM is to dynamically adjust the clock signal to the specified position under the control of the feedback signal. By using DDM to adjust the clock delay, the pulse signal at $M$ ($M=t_r/t_s$) different positions is sampled in one pulse period. Here, $t_s$ is the minimum step size of DDM, $t_r=1/f_r$ and $f_r$ is the system repetition. After sampling each location for a period of time $T_c$, the power can be calculated in the following way
where $i\in [0,M-1]$, $L=T_c/t_r$, $T$ represents the transmittance of the quantum channel between Alice and Bob and $A_{i(j)}$ is the amplitude of the $j$th sampled value at the relative position $i$. By comparing different $P_i$ we can find the largest one. And then the corresponding position for the largest value is the optimal sampling position.
Fig. 1. The structure in previous CV-QKD system. ADC: analog-digital converter; DDM: dynamic delay module.
As shown in simulation results of Ref. [21], this scheme can reduce the sampling error to a great extent. To be more specific, when the duration of the light pulse is much smaller than the response time of the homodyne detector, the output waveform of the detection signals are generally Gaussian [22], which can be expressed as
where $A_{peak}$ is the pulse peak value of the Gaussian curve, $\mu$ is the mean value and $\sigma$ is the variance. Here, without loss of generality, we choose that $\mu =t_r /2$ and $\sigma ^2=(t_r /8)^2$. The amplitude of sampled data can then be expressed as where $\delta \in [-t_{s}/t_{r}, t_{s}/t_{r}]$. It can be seen from Eq. (3) that the accuracy of the sampled data is only determined by the sampling time precision $t_s$. Therefore, the maximum normalized error of the sampled data is given byHowever, when the scheme is used in the free-space CV-QKD i.e., with fluctuating channel transmittance, the Eq. (1) will become
where $i$ denotes the sampling position and $T_i$ represents the transmittance at this time. Since the transmittance $T_i$ is fluctuating, $P_i$ cannot be directly compared to find the maximum value of $A_i$. Therefore this scheme is not suitable for free-space CV-QKD system.2.2 Data acquisition scheme based on dual sampling
In order to achieve high-precision data acquisition for free-space CV-QKD, a scheme for free-space CV-QKD based on dual sampling is proposed. As shown in Fig. 2, the key part of the scheme is a pulse peak sampling module (PPSM). Firstly, the electronic signal to be processed is split into two identical signals by the power divider. Then a $\Delta t$ (it is set to $t_r/10$ in this paper, and the reason for this is described in subsection 3.1) delay line is used to create a delay between the two signals. After that the two signals are sampled and transmitted to the data processing module. Finally, the feedback signal is obtained by the algorithm to control the DDM to adjust the clock signal delay, such that the rising edge of the clock signal is aligned with the peak position of the pulse.
An example of the system sampling process is shown in Fig. 3, with the front and back panels corresponding to the two sampling processes before and after the sampling process. To simplify the illustration, we temporarily ignore the influence of noise and assume that the sampling position at the beginning is aligned with the position of $t=0$ in Eq. (2). After n delays, the sampling values obtained by ADC1 and ADC2 can be expressed as
It can be clearly seen that the effect of the fluctuations of transmittance on the signal amplitude is eliminated by the division operation. When setting $G(1)=1, G(n+1)=G(n)k_{n}$ to recover the approximation of the original waveform $G$, we can use Gaussian Fitting method to perform Gaussian fitting on this group, and get $\mu$ through fitting calculation. Here, $\mu$ can then be used to obtain the best pulse peak position. The delay steps of DDM can be calculated as
This indicates that accurate pulse peak sampling can be achieved by controlling the DDM to delay m steps from the initial time point. And if higher precision data is required, $t_s$ can be further reduced.
3. Simulation and proof-of-principle experiment
In this section, simulation as well as proof-of-principle experiment are performed to verify the feasibility of the scheme proposed in this paper. In the first subsection, it is verified by simulation that the scheme can be operated well with low SNR and fluctuating channel transmittance. In the second subsection, the practical feasibility of the scheme is verified by proof-of-principle experiments.
3.1 Simulation verification
For simplicity, the analysis in subsection 2.1 is performed without considering the influence of noise. However, in practice, the signals are inevitably affected by noise to a certain extent. In order to suppress the influence of noise, the method of averaging process is used. We specify a period of time $T_c$ for ADC sampling, and perform averaging operation on these collected data as
where $L=T_c/t_r$. Then, Eq. (7) will be changed into and the effect of noise on $k$ will be greatly suppressed.However, when the SNR is low or the required sampling time precision $t_s$ is particularly high, the optimal sampling point calculated by the scheme may remain some deviations. In order to test the performance and effectiveness of the scheme under different noise conditions, we have carried out simulations under different SNRs.In the simulation, the system repetition rate is set to 10MHz and the duty cycle of pulse is set to 30%. The signal lengths used for simulation are $N=2\times 10^5$ and $N=10^6$, respectively. First, a Gaussian signal is generated and its pulse peak position is recorded, then noise and fluctuating transmittance are added to the signal, and finally the algorithm is performed on the signal. Where the channel transmittance is randomly generated and its fluctuations are shown in Fig. 4. The simulated free-space signal can be obtained by multiplying the transmittance with the Gaussian pulse. If the pulse peak position obtained by the algorithm is equal to the actual pulse position, then the algorithm is judged to be successful. It is of attention that $\Delta t$ is critical to the success rate of finding the best sampling point, so we need first to find out the length of $\Delta t$ that best fits our scheme. As shown in Fig. 5, when $t_s=t_r/200$ and $SNR=0, -5, -10, -15$dB, the scheme always performs best when $\Delta t/tr=8\%-10\%$, so we can choose $\Delta t=0.1t_r$ for the subsequent simulation.
Fig. 5. The success rate as a function of $\Delta t/t_r$. The four curves are obtained with $SNR=0,-5,-10,-15$dB, respectively. The duty cycle of pulse is set to 30% and $N=10^6$.
As shown in Fig. 6, when setting $t_s\ge 0.25$ns and $N=10^6$, we can find the success rate of the algorithm can keep in a high level (larger than 80%) when SNR ranges from 5dB to -15dB. It can be seen that as the SNR gradually decreases, especially below -10dB, the success rate of the algorithm also gradually decreases. In addition, comparing the success rates of $N=2\times 10^5$ and $N=10^6$, it can be seen that under the same SNR, the longer the data length, the higher the success rate. In order to keep the success rate of high-accuracy sampling above 80%, the SNR should be maintained not less than -15dB and $N\ge 10^6$. If we need to adapt to a lower SNR scenario or a smaller sampling time precision $t_s$, we can improve its performance by increasing the signal length, i.e., the amount of data used in the algorithm. Similarly, in Fig. 7, when the SNR is fixed to 0dB and -10dB, the success rate gradually decreases with the sampling time precision $t_s$. Obviously, when $t_s\ge 0.25$ns, the success rate of the algorithm can reach 90% under both signal-to-noise ratio conditions.
Fig. 6. The success rate as a function of SNR. The solid curves indicate $N=2\times 10^5$ and the dashed curves indicate $N=10^6$. From bottom to top, the three curves are obtained with $t_s=0.125$ns, $0.25$ns, and $0.5$ns, respectively. The system repetition rate is set to 10MHz and the duty cycle of pulse is set to 30%.
Fig. 7. The success rate as a function of sampling time precision $t_s$. From left to right, the two bars are obtained with $SNR=0$dB and $-10$dB, respectively. The system repetition rate is set to 10MHz, the duty cycle of pulse is set to 30% and $N=10^6$.
3.2 Proof-of-principle experimental verification
Here we construct a practical short-distance free-space channel to verify the feasibility of the proposed dual sampling method in practical applications.
As shown in Fig. 8, Alice sends a pulsed laser signal with 10MHz repetition rate and 30% duty cycle, which is received by Bob after passing through a 1 meter long free-space channel. At Bob’s side, the received optical signal is converted into an electronic signal, and then it is divided into two parts by a power divider. Finally, two electronic signals are collected by a oscilloscope for algorithm verification. It is important to mention that the signal length N in experiments is the same as in the simulation, which is $10^6$.
Fig. 8. Proof-of-principle experimental setup. AM: amplitude modulator; VOA: variable optical attenuator; PD: photodetector; Osc: oscilloscope.
For the sake of simplicity, we preprocess the data after over-sampling by oscilloscope without violating the principle of the algorithm. For each pulse, only one point is saved, and the dynamic delay module is simulated in this way. Finally, the processed data is used to verify the feasibility of the algorithm. In order to simulate the influence of turbulence on transmittance in the real channel, heated airflow simulations were employed to simulate transmittance fluctuations caused by atmospheric turbulence. The spectrum of the channel transmittance is shown in Fig. 9. It can be found that the fluctuating frequency of channel transmittance is basically within 1kHz, which is well consistent with the one of the real atmospheric channel as shown in Refs. [11,15].
During the experiment, we found that the shape of the output signal of the practical detector is not a perfect Gaussian waveform, so there is a certain gap between the pulse peak position pointed by the result of Gaussian fitting and the practical pulse peak position. To solve this problem, we propose a preprocessing method. In particular, when the detector is not saturated, inputting homologous optical signals of different powers to the detector will result the same waveform of the output electronic signal. Therefore, we can train this deviation before pulse peak search experiment. In the training process, the optical signal with a large SNR is used to directly observe the practical optimal sampling position, and then use the Gaussian fitting method to calculate the theoretical optimal sampling position. Finally, the compensation value is obtained by subtracting the two sampling positions.
After setting the compensation value, we can get the correct pulse peak position by adding the theoretical pulse peak position to the compensation value in the experimental process of pulse peak search. As shown in Fig. 10, firstly, the pulse peak value of the strong light under oversampling is found by the power comparison method for verification the effectiveness of the proposed scheme. Then the dual sampling method is used to find the pulse peak of the weak light signal. The results show that even when the waveform is a non-perfect Gaussian, the optimal sampling time under different attenuation conditions can still be found by the algorithm.
Fig. 10. When $t_s=0.2$ ns, the optimal sampling position of the algorithm under different channel fading conditions is found in experiments.
4. Application scenarios
In this section, we present two possible application scenarios of the present scheme and analyze their performance. In the first subsection, the scenario where the scheme is used to implement pulse peak sampling in dynamic polarization control of free-space CV-QKD is presented. In the second subsection, we analyze the FSB effect of free-space CV-QKD. In the raw key data acquisition session of free-space CV-QKD, the excess noise increases with the system frequency for the conventional oversampling schemes due to the FSB effect. While using this scheme, the effect of the FSB effect can be fundamentally reduced.
4.1 Dynamic polarization control
In the free-space CV-QKD, if the polarization multiplexing of signal and local oscillation (LO) is carried out, dynamic polarization control is usually required at the Bob’s side to reduce the influence of LO leakage.
Due to channel fading, it is quite difficult to make the dynamic polarization control required in CV-QKD work properly. A dynamic polarization control scheme for free-space CV-QKD was proposed in Ref. [16]. As show in Fig. 11, after the signal and the LO are collected by the telescope, their polarization state is changed by a DPC so that the signal and the LO can be separated by a PBS. Part of the incident light is split using two BSs, and subsequently, the division operation of the two PDs output generates the feedback signal of the DPC, which completely satisfied for secure key distribution.
Fig. 11. Dynamic polarization control applications. DPC: dynamic polarization controler; BS: beam splitter; PBS: polarizing beam splitter.
However, because this scheme does not have a pulse peak acquisition module, it is almost impossible to collect information with a co-frequency ADC when using a pulsed laser, so it can only be verified by a CW laser. If a pulsed signals are used instead, it can be combined with our scheme for polarization control. As shown in Fig. 12, we add a PPSM after the PD and then the pulse peaks of the two signals can be collected by the PPSM. In this way, the pulse peak value of the feedback signal can be collected thus to complete the free-space CV-QKD dynamic polarization control process. Combined with the cascade control algorithm proposed in Ref. [16], the polarization isolation can reach 40dB. The excess noise introduced due to the leakage of LO can be expressed as
where $n_A$ is the mean photon number of the LO sent by Alice, $R_{AM}$ is the extinction ratio of the amplitude modulator at Alice’s side, and $R_PI$ is the polarization isolation. Without loss of generality, let $n_A$ be $10^8$ and $R_{AM}$ be 60dB, then when $R_{PI}$ is 40dB, the excess noise is 0.01, which completely satisfied for secure key distribution.Fig. 12. Schematic diagram of the application of dual sampling scheme in dynamic polarization control. DPC: dynamic polarization controler; BS: beam splitter; PBS: polarizing beam splitter; MCU: microcontroller unit. PPSM: pulse peak search module; DDM: dynamic delay module; ADC: analog-to-digital converter.
4.2 Raw key data acquisition
The significant application scenario of the proposed data acquisition scheme is to collect the raw key data output from Bob’s homodyne detection in the free-space CV-QKD system. As shown in Fig. 13, the polarization beam splitter (PBS) separates the signal from the LO for homodyne detection operation, and then the output data is collected by PPSM for data post-processing.
Fig. 13. The structure of Bob’s apparatus in the proposed free-space CV-QKD system; BS: beam splitter; PBS: polarizing beam splitter; PM: microcontroller unit. PPSM: pulse peak search module; DDM: dynamic delay module.
Under the premise of ensuring the effectiveness of the proposed data acquisition scheme, we analyze its pulse peak sampling accuracy and its influence to secret key rate. The fluctuation frequency of atmospheric channel is commonly about 1kHz, while, the system repetition rate is typically of the order of several tens of MHz. In other words, the channel transmittance is relatively stable within 1ms in most cases. Therefore, in parameter estimation, we can divide the channel into several stable sub-channels. Each sub-channel is regarded as a normal linear model, then the related variables of Alice and Bob can be expressed as
where $i$ represents the ith subchannel, $t_{i}=\sqrt {\eta T_{i}}$, $\eta$ is the detection efficiency, $T_{i}$ is the transmittance, $z_{i}$ is the Gaussian noise with variance of $\sigma _{i}^2=(N_{0i}+\eta T_{i}\xi _{i} + V_{el})$ and mean value zero. $N_{0i}$ is granular noise, $\xi _{i}$ is excess noise, $V_{el}$ is electronic noise. We can getUsing the previous estimator, we can get $T_i$ and $\xi _i$
However, there are deviations between the practical sampling value and the ideal sampling value. In a realistic CV-QKD model [22,23], Eve cannot manipulate Bob’s detection device. Therefore, the detection efficiency $\eta$ as well as the electronic noise $V_{el}$ level will not change. Then we can get
It indicates that if Alice and Bob are not notice of the deviations between the data collected in the practical system and the theoretical data, Eq. (13) will be used to calculate the final key rate of relevant parameters. Accordingly, the following relation will be derived as
The estimated excess noise as a function of system repetition rate is depicted in Fig. 14. Due to the limitation of the sampling bandwidth, the excess noise introduced by the traditional over-sampling method increases with the system pulse rate. However, as introduced in subsection 2.2, our scheme uses a Gaussian fitting scheme to find pulse peak value. When the value of $t_r/t_s$ stays the same, the success rate of the scheme is only related to the duty cycle and SNR of the signal. Therefore the excess noise will not increase with the system pulse rate and our scheme can accommodate higher system pulse rate. As shown in Fig. 15, the key rate will increase linearly with the system pulse rate. And the key rate bound as a function of distance is shown in Fig. 16, where, the relation between the horizontal path distance and $\langle T\rangle$ is as follows [24]
where the total extinction coefficient $\alpha (\lambda )$ comprises the aerosol scattering, aerosol absorption, molecular scattering, and molecular absorption terms:As shown in Fig. 16, the secure transmission distance increases with the reduction of $t_s$, which directly demonstrates the feasibility and efficiency of the proposed data acquisition scheme.
Fig. 14. The estimated excess noise as a function of system repetition rate. Parameters are set as: $T_i=0.1$, $\eta =0.6$, $V_A=4$, $\beta =0.96$, $\xi =0.01$ and $V_{ele}=0.3$. From bottom to top, the three dashed curves are obtained with $t_s=t_r/800$, $t_r/400$ and $t_r/200$, while the three solid curves are obtained with $f_{samp}$ = 5 GHz, 2 GHz and 1 GHz, respectively.
Fig. 15. The key rate as a function of system repetition rate. Parameters are set as: $Var(\sqrt {T})=0.02$, $\langle T\rangle =0.6$, $\eta =0.6$, $V_A=4$, $\beta =0.96$, $\xi =0.01$ and $V_{ele}=0.1$. From top to bottom, the three dashed curves are obtained with $t_s=t_r/800$, $t_r/400$ and $t_r/200$, respectively.
Fig. 16. The key rate as a function of transmittance. Parameters are set as: $Var(\sqrt {T})=0.02$, $\eta =0.6$, $V_A=4$, $\beta =0.96$, $\xi =0.01$ and $V_{ele}=0.3$. The components of the total extinction coefficient are $\alpha ^{aer}_{sca}(\lambda )=1.64 \times 10^{-4}$, $\alpha ^{aer}_{abs}(\lambda )=3.35 \times 10^{-3}$, $\alpha ^{mol}_{sca}(\lambda )=2.52 \times 10^{-2}$, $\alpha ^{mol}_{abs}(\lambda )=5.49 \times 10^{-3}(km^{-1})$, which are the typical parameters for summer. From top to bottom, the three dashed curves are obtained with $t_s=t_r/800$, $t_r/400$ and $t_r/200$, respectively.
5. Conclusion
A high-precision data acquisition scheme for free-space CV-QKD system with fluctuating transmittance is proposed. The scheme can be used to achieve high-precision data acquisition independent on the variation of channel transmittance and work in very low SNR environment. Specifically, two ADCs are used for sampling at the same time to calculate the division of the two sampled values so as to remove the fluctuating effect of channel transmittance, and the sampling positions are simultaneously and continuously delayed to output a string of these division values. Then a Gaussian fitting is used to find the best sampling position so as to accurately obtain the encoded secret key information. After the simulation verification of the scheme, proof-of-principle experiments are carried out to verify the feasibility of the scheme. The results show that the scheme can be well applied for accurate data acquisition in free-space CV-QKD scheme. Finally, application scenarios of this scheme in free-space CV-QKD are introduced, and the performance of these scenarios is analyzed and verified. The application of this scheme in these two scenarios can greatly reduce the excess noise and improve the transmission distance and key rate of the free-space CV-QKD system. In particular, the oversampling method is affected by the FSB effect and requires extremely high sampling bandwidth, which is unrealistic in practical applications. While the scheme can achieve same-frequency high-precision sampling in free-space without the limitation of the system main frequency, which can well facilitate the implementation of free-space CV-QKD experiments and applications.
Funding
National Natural Science Foundation of China (61971276, 62101320); Shanghai Municipal Science and Technology Major Project (2019SHZDZX01); Key R&D Program of Guangdong province (2020B030304002).
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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