A quantum network is an advanced networking infrastructure that utilizes quantum bits rather than classical bits to store and transmit information, different from the classical communication network [1]. In view of the potential impact that a quantum internet could have on the economy, society, and national security, research institutes are currently conducting extensive research and development on quantum networks worldwide. It is clear that quantum network research has made remarkable progress. Further, through quantum networks, previously difficult tasks have become possible. For example, quantum networks can improve the current global timing systems [2] to get a more accurate clock and more precise time synchronization [3], and also can facilitate new fundamental tests for a deeper understanding of the universe and multiple quantum computing nodes working together by utilizing the characteristics of quantum networks [4] to improve computing efficiency [5].

As one of the key applications in quantum networks, quantum key distribution (QKD) can ensure the security of its data due to its reliance on quantum mechanics to transmit data and its unconditional security [6,7]. With continuous variable quantum key distribution (CV-QKD) [8] as the branch of the QKD, point-to-point CV-QKD has made significant progress in both theory [9–19] and experiments [20–28] in recent years. Recently, a new digital signal processing analysis method was also proposed, which can better combine CV-QKD with a classical coherent optical communication system [29]. Due to these developments, the extension of point-to-point QKD to the networks [30–42] is getting more and more attention. As an important part of the networks, the access network can connect multiple end users to the backbone network and provide the so-called last-mile service. In a typical quantum access network, quantum network units (QNUs) are located at end-user sides, and the nodes that connect the network backbone are the quantum line terminals (QLTs). The intermediate side between the QNUs and the QLTs is the quantum distribution network (QDN). According to the direction of the signal, the quantum access network is divided into the upstream quantum access network and the downstream quantum access network. In the upstream quantum access network, multiple users need to be transmitted to the QDN through time-division multiplexing, and each user needs to have a corresponding time slot. However, as the number of users increases, the time slot corresponding to each user will narrow, and if there is no precise control, crosstalk between signals will easily occur. Such limitations have considerably less impact on the downstream quantum access network, since no multiplexing technique is demanded for it.

According to the location of the local oscillator (LO), CV-QKD is divided into the transmitting local oscillator (TLO) and the local local oscillator (LLO) CV-QKD, respectively. The TLO CV-QKD allows high-power LO and quantum signals to transmit through the channel together, while the LO is independently generated by the laser at the Bob’s side in the LLO CV-QKD. In contrast, the TLO CV-QKD has a certain disadvantage that the LO will attenuate through long-distance transmission and also has potential security loopholes resulting from the eavesdropper. The eavesdropper may utilize the potential loopholes to perform an LO fluctuation attack [43] or calibration attack [44]. Though the LLO CV-QKD can avoid the above problems, it also has to face the difficulty in eliminating the frequency offset between Alice’s and Bob’s individual lasers. Since the first two proposals in 2015 [22,45], numerous LLO CV-QKD experiments [20,25,28,46] have been realized by researchers, and various pilot schemes are designed to reduce the frequency offset and phase noise, such as pilot-sequential and pilot-multiplexing LLO CV-QKD. In recent years, there has been a lot of progress in the TLO CV-QKD access network continuously [37,41,42]. However, compared to TLO CV-QKD access networks, LLO CV-QKD access networks have all the advantages in point-to-point.

In this paper, we report the first experimental four-end-users LLO CV-QKD downstream access network. First, we propose a theoretical model of the quantum downstream access network structure and simulate its theoretical performance, and then give an experimental demonstration. In our proposed network model, we place a passive beam splitter in the QDN. This enables each user to have a higher sensitivity to excess noise as the number of users increases, compared to a point-to-point LLO CV-QKD. This also means that each user needs to have a higher precision compensation algorithm to get a lower level of excess noise. So, we classify the noise in the network subsequently and complete the analysis in turn. In particular, the phase noise caused by the frequency difference between the QLT’s and corresponding QNU’s laser is the important component of excess noise. In view of these, we first propose a new method based on particle filter (PF) [47] to estimate more accurate phase drift, so as to complete accurate phase compensation. Then, we propose an adaptive equalization algorithm—recursive least squares algorithm (RLS) [48]—to effectively reduce the remaining phase noise and improve the signal-to-noise ratio. Based on the theoretical model above, we experimentally achieved a secret key rate of 546 kbps, 535 kbps, 522.5 kbps, and 512.5 kbps for four QNUs under a transmission distance of 10 km with the block size of 1×108, respectively. This result provides significant experimental and theoretical validation of the practicality of the quantum downstream access network.

In the classical downstream access network, the main structure mainly includes the optical line terminal, optical network unit, and optical distribution network [49–51]. In the downstream access network, the optical line terminal continuously sends signals to the optical distribution network. The passive beam splitter (BS) applied to the optical distribution network then passively separates the signal into N parts, and the separated signal is then transmitted to each optical network unit in the network through a separate fiber. The passive BS-based optical distribution network does not process the received signal, but simply distributes the signals sent by each optical network unit. For example, suppose that the optical line terminal prepares four packets for four optical network units. The optical line terminal sends the four packets together to the optical distribution network, which then divides the received packets into four pieces and distributes them to the four optical network units. Based on classical downstream access, the physical structure of our quantum downstream access network is shown in Fig. 1. There are the QLT, QDN, and QNU corresponding to the classical structure. Generally, we place the quantum transmitter in the QLT, a passive 1:N BS in the QDN, and the quantum receivers in the QNUs, respectively. In the model, we place the sender Alice and the receiver Bob of LLO CV-QKD in the QLT and QNU, respectively. In Gaussian modulation coherent state (GMCS) LLO CV-QKD, Alice first uses the laser source to prepare the original coherent states |a0⟩ with constant intensity and phase. Then, Alice draws two random numbers {XA,PA} from a set of Gaussian random numbers and loads it on |a0⟩ to generate the GMCS |XA+iPA⟩. Next, it needs to adjust the attenuation to get the optimal variance of the output coherent states. Then, they are divided into N pieces through the BS, and each piece will be sent to the corresponding Bob. After being transmitted through the quantum channel, a strong continuous wave generated locally by Bob’s laser source is used to interfere with the received coherent state. Next, the coherent state is detected by a balanced homodyne detector. In LLO CV-QKD, there exists a problem: the phase difference between two remote independent lasers is expected to fluctuate rapidly. Therefore, the output of the detector can be obtained through a series of digital signal processing methods to obtain the quadrature components XB and PB, respectively. The theoretical security of GMCS LLO CV-QKD is proved and the achievable secret key rate at a specific distance has also been estimated [22].

In the downstream access network, ensuring the security of data received by each user is crucial, especially since data is distributed randomly to every user. In the classical downstream access network, encrypting data packets can effectively protect data security. Specifically, when the optical line terminal prepares data packets, each data packet will be encrypted. For example, when packet 1 is sent to optical network unit 1, the optical line terminal will encrypt packet1 with the symmetric key between the optical line terminal and optical network unit1. The encrypted packet1 is then sent to the network. In this case, although packet1 is distributed to all optical network units, since only optical network unit1 has the key that can be used for decryption, only optical network unit1 can decode the encrypted packet1, which ensures the security of data transmission. While due to the nature of QDN broadcast in the downstream access network, it is not feasible to apply the quantum device such as QKD directly to the downstream access network. This is because the final key generated by QKD also needs to be kept secret between the QLT and the corresponding QNU, and other QNUs in the access network cannot get the key message about the QLT and the appointed QNU. However, the key information generated by the QLT is forwarded to each QNU when passing through the QDN, which means that the key information will be obtained by other QNUs in the network, which obviously destroys the security of QKD. Similarly, it is not advisable to encrypt the key information of the QKD downstream access network, because the final purpose is to obtain the symmetric key between the QLT and the specified QNU, and it is obviously not a good way to encrypt the key information again.

For this problem, using protocols based on discrete variable quantum key distribution (DV-QKD) can avoid it. DV-QKD uses a single photon as a quantum source, based on the basic principle of quantum mechanics. When a single photon passes through the passive BS, it will be randomly distributed to a certain way. This also means that in DV-QKD, any single photon quantum state is distributed as key information, and finally only one QNU will be reached. Therefore, it perfectly avoids the problem that all QNUs will receive the key information mentioned above. However intuitively, using DV-QKD in the downstream access network may not be optimal. As mentioned in Ref. [34], the downstream implementation applied in DV-QKD has two major shortcomings. First, every user in the network requires a single-photon detector, which is often expensive and difficult to operate. Second, it is not possible to deterministically address a user. As a result, the complexity of the post-processing processes will be increased. If cost and deterministic key distribution are not taken into account, DV-QKD can also be utilized for downstream access networks. In contrast, coherent and squeezed states can be used to distribute key information to each QNU accurately in CV-QKD. Furthermore, the balanced homodyne detectors are much cheaper and also more robust. However, there is still the risk in the network that key information will be eavesdropped on by other QNUs when it is distributed to a specific QNU. To solve the mentioned problem, we conducted the security analysis of the CV-QKD downstream access network again.

In order to simplify the analysis, we only analyze the case of four QNUs here, and the number of QNUs can be arbitrarily selected in practice. While the practical implementations are constructed based on the prepare-and-measure (PM) model, its corresponding security analysis is usually conducted by using the entanglement-based (EB) model. In the EB model, EPR states are prepared at the QLT, and one of the modes performs homodyne measurement, projecting the other mode of the EPR state into a coherent state of displacement, which is equivalent to preparing a GMCS in the PM model. Subsequently, the QLT sends the quantum state into the channel. The transmitted quantum state first reaches the QDN. Then, the quantum state at the QDN is further divided into four paths and sent to various QNUs through respective channels. Assuming that the security analysis focuses on the key distribution process between the QLT and the QNU1, the formula for calculating the secret key rate can be generally expressed as K=βIAC1−χEC1,(1)where β is the reconciliation efficiency, IAC1 is the mutual information between the QLT and QNU1, and χEC1 represents the amount of information that Eve can obtain. Taking into account that each QNU has received quantum states from the QLT, there is an association between the QLT and each QNU in the network. To obtain the secret key between the QLT and QNU1, it is necessary to remove the associations with other QNUs in the network. Therefore, the formula for calculating the secret key rate mentioned above needs to be further modified as K′=K−χC1C2−χC1C3−χC1C4,(2)where χC1C2, χC1C3, and χC1C4 are the amount of information between other QNUs and QNU1. According to Eq. (1), it can be observed that although the final key is only generated between QLT and QNU1; the participation of other nodes is required to calculate the secret rate between them. Especially when there are more ONUs in the network, the calibration process becomes more complex. Clearly, this is impractical in real-world scenarios. Given the above considerations, the calculated secret key rate should be secure against other QNUs in the network. In the meantime, it should be decided in a more efficient way. So, we redefined the definition of Eve in the security assumptions. In the new definition, we assume that Eve has enhanced capabilities, allowing it to control all nodes in the network except for the QLT and the designated QNU that are involved in key generation. This approach not only enables security analysis to be independent of other nodes in the downstream access network but also simplifies the security process, thereby enhancing practicality. Hence, Eq. (1) can be rewritten as K=βIAC1−χE′C1,(3)where E′ represents the enhanced Eve in the model. Based on above, detailed secret key rate calculation can be found in Appendix C.

For a quantum network, the number of QNUs that can be accessed and the secret key rate that each QNU can achieve are important indicators. Hence, we simulated the relationship of these variables, and the results are shown in Fig. 2. Figure 2(a) shows the simulation results of the secret key rate that can be obtained under different number of QNUs and different transmission distances in the network, and the secret key rate results are represented by the false-color graph. It can be seen from the results that the secret key rate that can be obtained for the QLT and specified QNU in downstream access varies greatly under different conditions. In order to better observe the situation of the downstream access secret key rate, we provide the relationship between the number of QNUs and secret key rate in Fig. 2(b) and the relationship between individual transmission distance and secret key rate in Fig. 2(c). By comparison, when the number of QNUs in the network increases, the secret key rate changes more obviously, especially when the number of QNUs is from 0 to 10. In addition, it can still be seen that even in the case of the 64 QNUs and the transmission distance of 30 km, it is still observed that a secret key rate can be obtained.

In the section, we conducted the security analysis of the CV-QKD downstream access network again and demonstrated numerous simulation results to evaluate the performance of CV-QKD in the downstream access network. Although when more optical network units access the network, the split ratio of the BS is inevitably increased, resulting in the decline of the secret key rate; the results provide a strong feasibility that up to 64 users can simultaneously access the network. Moreover, we successfully demonstrate the first four-end-users quantum downstream access network in continuous variable QKD with a local local oscillator as follows. Each user can achieve lower excess noise, and under our proposed network theoretical model and security analysis, each user can reach the secret key rate of 546 kbps, 535 kbps, 522.5 kbps, and 512.5 kbps under a transmission distance of 10 km, respectively, with the finite-size block of 1×108. Compared to the results in Ref. [37] that show the average key rates could reach about 0.25 kbps, 2 kbps, 6 kbps, and 10 kbps, respectively, our results can achieve higher secret key rates. In conclusion, the theoretical model has good performance and high practicability.

In the access network, in addition to the above theoretical security and performance, the practical security of the system is also important, in which the total noise of the system plays an important role in the practical security. Therefore, we will complete the noise analysis and classify the total noise in our downstream access network and explain the extent of their impact on the overall system in this section.

In our LLO CV-QKD downstream access network, the total noise of each QNU can be divided into trusted noise and untrusted noise. Trusted noise refers to noise in the system that has been fully quantified and modeled, with known characteristics and predictable effects. These noises are typically beyond Eve’s control and can be calibrated and controlled by the QLT. Under the modeling conditions of trusted detectors, we consider the receiver’s detector noise and the noise resulting from the analog-to-digital sampling accuracy to form electrical noise vel in the analysis of security. In contrast, untrusted noise refers to noise in the system that cannot be accurately quantified and modeled, with characteristics and effects that cannot be fully predicted or controlled. This type of noise may be introduced by malicious attacks, channel imperfections, or device non-idealities, leading to unpredictable effects on the key distribution process. Untrusted noise is typically within Eve’s control and contributes to excess noise. So, the total noise εtotal can be expressed as εtotal=εtrust+εuntrust=εDet+εADC+εRIN+εDAC+εMod+εPhase,(4)where εtrust is the trusted noise, εuntrust is the untrusted noise, εDet is the detection noise, εADC is the analog-to-digital conversion (ADC) quantization noise, εRIN is the intensity noise of two independent lasers, εDAC is the digital-to-analog conversion (DAC) quantization noise, εMod is the modulation noise, and εPhase is the phase noise.

Trusted noise mainly includes the detection noise and the ADC quantization noise; the two terms together are thought as electrical noise vel. We will introduce in detail respectively as follows.

The noise of detector mainly comes from shot noise and thermal noise, of which the thermal noise has the greatest impact on the detection ability of the detector, mainly from the irregular movement of free electrons or charge carriers inside the resistance. Therefore, the heterodyne detection noise at the QNU’s side εDet can be expressed as εDet=2NEP2BτℏfPLO,(5)where NEP is the noise-equivalent power, ℏ is the Planck’s constant, B is the actual effective bandwidth of the detector, τ is the pulse duration, and f and PLO are the frequency and the power of the LO, respectively. Here, in our experiment, NEP is 5.5pWHz−1/2, B is 100 MHz, τ is 10 ns, f is 193.5 THz, and PLO is 13.5 dBm. Based on the parameters, the quantization noise εADC can be theoretically calculated to be 0.0215.

After heterodyne detection, it is necessary to convert the analog signal output by the detector into a digital signal through the ADC before data processing can be carried out. However, due to the limitation of ADC sampling bits, the signal cannot be perfectly quantized and quantization error will exist between the quantized signal and the initial signal, so a certain noise will be introduced to the signal, which is called quantization noise εADC, given by εADC=2τ⏧fg2ρ2PLO112RU222n,(6)where RU and n are the full voltage range and quantization bits of ADC, respectively, g is the gain factor of the amplifier, and ρ is the responsivity of the PIN diodes. In our experiment, g is 16×103V/A, ρ is 0.9 A/W, and RU and n are set as 737 mV and 10 bits. Based on the parameters, the quantization noise εADC can be theoretically calculated to be 1.48×10−3.

After analyzing and calculating the trusted noise, we will introduce the rest of the noise as follows. The rest of the noise constitutes excess noise and is critical to each QNU’s final secret key rate. In particular, each user has a higher sensitivity to excess noise compared to a point-to-point LLO CV-QKD as the number of users increases in the network. Therefore, we analyze it in detail, and will propose a higher precision algorithm to reduce this part of the noise in the following content.

The first term εRIN is the intensity noise of the two independent lasers. Laser refers to the amplification of light by stimulated radiation, and the reversal of particle number caused by stimulated radiation is a necessary condition for generating laser. With the occurrence of stimulated radiation, due to the inherent nature of atoms, spontaneous radiation also inevitably occurs. The light generated by spontaneous radiation is different from stimulated radiation, because spontaneous radiation is random and irregular, and a part of the incoherent field generated by spontaneous radiation will be superimposed into the highly coherent light generated by stimulated radiation. So, the intensity of the laser generated by the stimulated radiation produces a certain jitter, namely intensity noise. The intensity noise is difficult to accurately describe in terms of both randomness and complexity using simple models and predict. So, the noise is considered as untrusted noise and can be expressed as εRIN=VARINsigΔvQLT+14TRINLOΔvQNUVRIN(q^),(7)where RINsig and RINLO are the relative intensity noise of QLT’s and QNU’s laser, respectively, and ΔvQLT and ΔvQNU correspond to their laser linewidths. Moreover, VRIN(q^)=TVA represents the quantum variance without taking count of the LO’s RIN.

The second term εDAC is the DAC quantization noise. Since when the digital signal generated by the FPGA passes through the DAC, due to the limitation of the quantization bits, there is a certain error between the analog modulated signal and the ideal signal. It is also difficult to accurately describe using simple models and is also subject to non-linear effects such as non-linear distortion in practical quantization, making it even more challenging to predict. So, the noise is considered as untrusted noise, given by εDAC=VA[πδVDACVDAC+π22(δVDACVDAC)2]2,(8)where VDAC is the output voltage of a self-developed DAC board mounted on an FPGA, and δVDAC represents the voltage error due to the limited quantization bits.

The third term εMod is modulation noise. In the pilot-sequential LLO CV-QKD, the strong pilot tone and the weak quantum signal are generated by the same modulator. However, in practice, the limited dynamics of the modulator will result in an additional noise to the signal. Specifically, when we generate pilot tone and quantum signal in the QNU, due to the finite dynamics of the modulator, if there is not enough extinction ratio, it will introduce leakage to the weak quantum signal. These factors make it more complex and unpredictable. Thus, the noise is considered as untrusted noise, expressed as [53] εMod=|as|210−ddB/10,(9)where as is the amplitude of the pilot tone and where ddB represents the extinction ratio of the in-phase/quadrature (IQ) modulator, and can be calculated as ddB=10log10(Emax2Emin2),(10)where Emax and Emin represent the maximal and minimal amplitudes that the IQ modulator can output in practice.

The last term is phase noise εPhase, which is also the largest proportion of excess noise. The noise is caused by random fluctuations in the phase of a signal through the transmission. These fluctuations are often random and unpredictable. So, the noise is also considered as untrusted noise. In the LLO CV-QKD scenario, the phase noise can be divided into two parts, given by εPhase=εfast+εslow,(11)where εfast is the fast-shift phase noise caused by the frequency difference between the QLT’s and QNU’s lasers, and εslow is the slow-shift phase noise caused by the signal transmission in the fiber. In general, εfast=2πVA(ΔvQLT+ΔvQNU)/Rsym, where Rsym is the repetition of the signal. Taking into account the non-linear effects of εfast, traditional methods often struggle to effectively track changes. Fortunately, particle filter [47] exhibits significant advantages in handling non-linear and non-Gaussian systems with substantial state changes. In particular, using the phase of the pilot signal as observation, based on the Monte Carlo method, a particle set is utilized to represent the posterior probability. The core idea is to randomly sample state particles from the posterior probability to represent its distribution and to estimate the state by progressively updating these particles. The state equation and measurement equation can be expressed as X(k)=H(k)X(k−1)+W(k−1),(12)Z(k)=g(X(k))+V(k).(13)

A detailed explanation of Eqs. (12) and (13) and a more detailed process can be found in Appendix A. Through it, we can obtain a more accurate fast-drift phase for each QNU. After completing phase compensation, residual phase noise still exists in the system. This noise can affect the phase stability and accuracy of the signal, thereby reducing system performance. A common approach to mitigate this residual phase noise is to use the recursive least squares algorithm [48] for estimation and compensation. The algorithm has excellent adaptive capabilities and can enhance signal recovery and phase stability, thereby minimizing the impact of residual phase noise on the system. In particular, based on the minimum mean square error criterion, the weights of the filter w(n) are continuously updated according to the error between the input signal and the output signal, using the amplitudes of the X and P signals as input signals x(n), to make the output signal as close as possible to the desired output. The desired output y(n) can be given by y(n)=w(n)x(n).(14)

The detailed process of the algorithm is provided in Appendix B. After recursive least squares, the remaining phase noise εremain can be expressed as εremain=εerror−εRLS-com,(15)where εerror is the error between the ideal phase drift and the calculated phase according to the pilot tone and εRLS-com is the phase compensated by recursive least squares.

These are basically the components of untrusted noise (excess noise). In the experiment, the excess noise of each QNU is around 0.003–0.006, which can also be seen in the following experimental results.

A four-end-users downstream access network of the GMCS LLO CV-QKD experimental scheme is depicted in Fig. 3. In our work, we adopt a pilot-sequential structure in which pilot tone and quantum signal are transmitted together. The sender of the CV-QKD (Alice) is deployed at QLT’s side; the receivers of the CV-QKD (Bob1, Bob2, Bob3, and Bob4) are employed at QNUs’ side. At QLT’s side, the continuous-wave coherent light is generated by a frequency stabilized laser at 1550.12 nm with a narrow linewidth. Then, continuous light enters an IQ modulator to generate the GMCS signal with a repetition frequency of 100 MHz, where the modulated signals of the IQ modulator are generated by the FPGA equipped with a self-developed DAC board. It is worth noting that due to the influence of the actual environment and other factors on the operating point of the modulator, we use a bias controller to dynamically control the bias of the IQ modulator, so as to obtain a stable output of the modulated signal. The modulated signal is then attenuated by a variable optical attenuator (VOA) to obtain an optimal modulation variance VA around 4 in this paper. The quantum signal with GMCS and the pilot tone can be given by Esig=Asigcos(2πfAt+φQLT+φsig),(16)Eref=Arefcos(2πfAt+φQLT),(17)where the amplitude Asig and phase φsig follow the Rayleigh distribution and the uniform distribution, respectively, fA denotes the central frequency of the QLT’s laser, φQLT denotes the initial phase of the QLT’s laser, and Aref represents the amplitude of the pilot tone.

The signal was then sent through a quantum channel (10 km standard single-mode fiber whose attenuation is 0.2 dB/km). In order to maintain the optimal polarization state of the signal to improve the quality as much as possible, we use a manual polarization controller (MPC) to align the state of polarization (SOP) of the quantum signal. A passive beam splitter of a ratio of 1:4 is used as the passive optical network device in the QDN. The signal is evenly divided into four QNUs in the QDN. At the QNUs’ side, the structure is basically the same for four users (QNU1, QNU2, QNU3, and QNU4). Therefore, take one user QNU1 as an example to explain in detail. At the QNU1’s side, the LO is produced by another frequency-stable laser which is the same as the QLT’s side and can be expressed by ELO=ALOcos(2πfBt+φQNU1),(18)where ALO denotes the amplitude of the LO, fB denotes the central frequency of the QNU1’s laser, and φQNU1 denotes the initial phase of the QNU1’s laser.

The power of LO in our experiment is large enough to meet the limit of shot-noise-limited detection of the quantum signals. Then, LO and the signal will complete the stable interference in a BS with a fast axis cutoff. The two optical signals output by the BS will pass through the VOA of the same length and enter a balanced homodyne detector (BHD) with a bandwidth of 1.6 GHz. After heterodyne detection, the output electronic signal of the BHD is collected in a real-time oscilloscope working at 1 GSa/s and the generated photocurrents can be expressed as, respectively, $$\begin{gathered} Isig=Rη(|Esig+ELO|2−|Esig−ELO|2) \\ =2RηAsigALOcos(2πΔfABt+φsig+Δφfast+Δφslow), \end{gathered}$$(19)$$\begin{gathered} Iref=Rη(|Eref+ELO|2−|Eref−ELO|2) \\ =2RηArefALOcos(2πΔfABt+Δφfast), \end{gathered}$$(20)with the frequency difference of two lasers ΔfAB=fA−fB, the fast-drift laser phase Δφfast=φQLT−φQNU1, and the slow-drift channel phase Δφslow=φsigchannel−φrefchannel, in which φsigchannel and φrefchannel denote the phase drift of the quantum signal and pilot tone distributed by the quantum channel, respectively, and R and η denote the responsiveness and quantum efficiency of the BHD.

In the LLO CV-QKD downstream access network, since the actual center frequency of each QNU’s and QLT’s laser is not exactly the same, each QNU’s signal will have frequency offset and the resulting phase drift after receiving the signal and heterodyne detection. In order that each QNU can better recover and demodulate the original signal to obtain less excess noise and higher secret key rates, we designed the following signal processing, as shown in Fig. 4.

First, the frequency difference ΔfAB is estimated to be 248 MHz according to a fast Fourier transform performed on the sampling signals first in our experiment. Second, we utilize the frequency ΔfAB to obtain the X and P orthogonal components of the quantum signal and the pilot tone combining with low-pass filtering, given by $$\begin{gathered} Xsig=LPF(real(Isig·e−2πΔfABt)) \\ =RηAsigALOcos(φsig+Δφfast+Δφslow), \end{gathered}$$(21)$$\begin{gathered} Psig=LPF(imag(Isig·e−2πΔfABt)) \\ =−RηAsigALOsin(φsig+Δφfast+Δφslow), \end{gathered}$$(22)$$\begin{gathered} Xref=LPF(real(Iref·e−2πΔfABt)) \\ =RηArefALOcos(Δφfast), \end{gathered}$$(23)$$\begin{gathered} Pref=LPF(imag(Iref·e−2πΔfABt)) \\ =−RηArefALOsin(Δφfast), \end{gathered}$$(24)where LPF(·) represents the function of low-pass filtering; real(·) denotes the function to obtain the real part X of orthogonal components while imag(·) denotes the function to obtain the imaginary part P of orthogonal components.

Third, after data synchronization, we need to complete the phase compensation including fast-drift and slow-drift. We propose a phase estimation method based on PF (see further details in Appendix A) to obtain a more accurate fast-drift phase, so as to complete the fast-drift compensation better. Therefore, both quadrature values of the quantum signal after compensating fast-drift phase can be simultaneously obtained as [45] $$\begin{gathered} Xsig′=Xsigcosθfast−Psigsinθfast \\ =RηAsigALOcos(φsig+Δφslow), \end{gathered}$$(25)$$\begin{gathered} Psig′=−Xsigsinθfast−Psigcosθfast \\ =−RηAsigALOsin(φsig+Δφslow). \end{gathered}$$(26)

For slow-drift phase, we use a method called the phase searching algorithm to find θslow by traversing all angle values from 0 to 2π and then estimating it according to the cross-correlation values. The received quadratures XB and PB at the QNU’s side can be more precisely compensated as $$\begin{gathered} XB=Xsig′cosθslow−Psig′sinθslow \\ =RηAsigALOcos(φsig), \end{gathered}$$(27)$$\begin{gathered} PB=−Xsig′sinθslow−Psig′cosθslow \\ =−RηAsigALOsin(φsig). \end{gathered}$$(28)

Finally, there will still exist noise in the signal due to the imbalance of X and P components and the remaining phase noise caused by the imperfection of phase compensation. Aiming at these noises, we propose an adaptive equalization algorithm: RLS (see further details in Appendix B), which can effectively reduce noise and improve signal-to-noise ratio by amplitude equalization.

The experimental results of the LLO CV-QKD downstream access network are presented in the following. The time waveform of signals acquired by the oscilloscope working at 1 GSa/s and the frequency domain performance are shown in Figs. 5 and 6, respectively. As can be seen from Fig. 6, the peak value between the adjacent signals is 50 MHz, which means that our actual repetition frequency of quantum signal is only half of the frequency of the actual modulated signal because we use a pilot-sequential structure in which the pilot tone and the quantum signal are inserted at intervals.

In our downstream access network structure, excess noise mainly comes from phase noise based on the former noise analysis. Among them, the largest proportion is the fast-drift phase noise caused by the frequency difference between the QLT’s and QNUs’ lasers. The phase shift curves before and after PF are shown in Fig. 7. It can be clearly seen that the phase estimation results after PF are more convergent, which means that a more accurate phase can be compensated to the quantum signal. We describe the changes in the secret key rate and excess noise of QNU1 within 8 h, for example, and the results are shown in Fig. 8. Lower marks represent excess noise, exhibiting the initial excess noise, the excess noise after PF, and the excess noise after RLS from top to bottom. Upper marks indicate secret key rate, exhibiting the secret key rate after RLS, the secret key rate after PF, and the initial secret key rate from top to bottom. It can be clearly seen that after PF and RLS, the fluctuation and value of excess noise are both reduced. It means that we can get a more stable and higher secret key rate with our technology, which also can be seen in Fig. 8. Furthermore, we give the relationship between the secret key rate of four QNUs and the transmission distance and mark the experimental results in Fig. 9. The black solid line represents the PLOB [54] bound of the GMCS CV-QKD protocol with heterodyne detection. The solid line in different colors represents the secret key rate of each QNU respectively in infinite-size scenarios while the dashed line in different colors represents the secret key rate of each QNU respectively under a finite-size block of 1×108. The pentagram in different colors represents the experimental secret key rate in infinite-size scenarios, respectively. The diamond in different colors represents the experimental secret key rate with a block size of 1×108, respectively. The secret key rate of QNU1 is 1.01 Mbps, the secret key rate of QNU2 is 995.5 kbps, the secret key rate of QNU3 is 979.5 kbps, and the secret key rate of QNU4 is 967 kbps in infinite-size scenarios, respectively. The secret key rate of QNU1 is 546 kbps, the secret key rate of QNU2 is 535 kbps, the secret key rate of QNU3 is 522.5 kbps, and the secret key rate of QNU4 is 512.5 kbps under a finite-size block, which proves the feasibility of the physical structure and provides the foundation for the future multi-users quantum downstream access network.

In conclusion, we demonstrate the first experimental system of a quantum downstream access network with four end users in LLO CV-QKD via a standard optical fiber of 10 km. In theory, for the success and feasibility of the experiment, we made the following preparations. First of all, we introduce the theoretical model of our quantum downstream access network and simulate its performance. Among them, the more important network capacity and the secret key rates that each user can get are discussed. Second, we analyze the noise in the whole network structure. The noise is mainly divided into trusted noise and untrusted noise, in which untrusted noise constitutes excess noise and is at a lower level. In practice, we also made some innovations. In order to simplify the system, meet the practical application, and better combine CV-QKD and quantum downstream access network, we have made the following improvements. We use an IQ modulator instead of the “AM + PM” structure to complete GMCS more easily and adopt a simpler pilot-sequential LLO CV-QKD structure. To improve the performance and achieve the higher secret key rates, we have developed new technologies. First, we propose a new method based on PF to estimate more accurate fast-drift phase, so as to complete accurate phase compensation. Second, RLS is used to reduce the amplitude distortion of the signal, reduce the remaining phase noise, and improve the signal-to-noise ratio. Thus, the lower excess noise and higher secret key rate can be obtained than before. Finally, we realized a secret key rate of 546 kbps, 535 kbps, 522.5 kbps, and 512.5 kbps for each QNU under a transmission distance of 10 km with a block size of 1×108, respectively. Moreover, by increasing the repetition frequency, adding new automatic control technology, and proposing a new algorithm to reduce the phase noise or other noise in the future, the secret key rate will be further improved. More importantly, the successful demonstration of our quantum downstream access network in LLO CV-QKD also paves the way for secure broadband metropolitan and quantum access networks.