High-speed Gaussian-modulated continuous-variable quantum key distribution with a local local oscillator based on pilot-tone-assisted phase compensation

Heng Wang; Yaodi Pi; Wei Huang; Yang Li; Yun Shao; Jie Yang; Jinlu Liu; Chenlin Zhang; Yichen Zhang; Bingjie Xu

doi:10.1364/OE.404611

1. Introduction

Continuous-variable quantum key distribution (CVQKD) allows the sender (Alice) and the receiver (Bob) to distribute the key information securely through an insecure quantum channel controlled by the eavesdropper (Eve) [1–3]. Moreover, CVQKD has attracted significant interest in the practical implementation of quantum communications, because it has the advantages of high quantum efficiency based on coherent detection and good compatibility with commercial off-the-shelf components [4–6]. The Gaussian-modulated coherent-state (GMCS) CVQKD protocol can be easily implemented in practical experiment, and meanwhile it has also been proven secure against collective attacks [7,8] and coherent attacks [9,10]. Therefore, it is foreseeable that GMCS CVQKD becomes one of the main contenders of future QKD development [11–15].

There are two important CVQKD schemes, which have been proposed according to different arrangement of local oscillator (LO), named by the transmitting LO (TLO) and local LO (LLO) CVQKD [16–23], respectively. The TLO-CVQKD preserve good coherence between the quantum and LO signal that generated from the same laser source, and the homodyne detection can be performed efficiently at Bob’s site [24–29]. However, the TLO-CVQKD scheme suffers from the TLO intensity bottleneck and the potential security loopholes, where a countermeasure against the LO attacks consists of implementing a rigorous and robust real-time measurement of the shot noise [30]. Moreover, the TLO-CVQKD scheme based on homodyne detection requires extra random basis selection to realize the measurement of both quadrature X and P. By contrast, the LLO-CVQKD fundamentally removes the intensity bottleneck and the security loopholes of the TLO scheme by arranging the LO at Bob’s site [17,31], and it can flexibly operate at heterodyne detection instead of homodyne detection for the simultaneous measurement of both quadrature X and P in the absence of random basis selection [32]. However, in order to realize the coherent detection efficiently in LLO-CVQKD scheme, a reliable phase compensation is required for reducing the dominate phase noise resulted from the phase drift of two independent lasers and fiber channel disturbance in the practical implementation, so that an tolerable excess noise can be guaranteed for the secure key distribution.

In order to achieve reliable phase compensation in LLO-CVQKD scheme, the pilot-pulse-assisted methods, such as the pilot-sequential [34], pilot-delayline [18] and pilot-displacement [35] methods, have been proposed and experimentally demonstrated respectively, in which a bright pilot pulse is generated and propagated along with weak quantum signal by time multiplexing. However, the same balanced receiver has been used to measure the quantum signal and pilot pulse in the proposed methods [18,34,35], while the low saturation limit required for efficient detection of weak quantum signal restricts the detection of bright pilot pulse. Subsequently, the pilot-pulse-assisted methods are improved by separate transmission and detection of the quantum signal and pilot pulse by combining time and polarization multiplexing techniques [32,33]. Nevertheless, the high-speed LLO-CVQKD system based on time multiplexing requires the complicated electro-optical devices to generate optical pulse with high extinction ratio (ER) and achieve the precise time alignment between pilot signal and quantum signal for reducing the time crosstalk from the strong pilot pulse to the weak quantum signal in fiber channel, which limits its practice. So, a pilot-tone-assisted method based on frequency-multiplexing has been proposed to solve time-multiplexing problem [36–38]. Moreover, the polarization-multiplexing technique is also introduced for further avoiding the crosstalk from the remaining signal to the transmission and detection of the quantum signal in the same frequency band and guaranteeing different detection requirements of low-noise for quantum signal and high-saturation limitation for pilot tone [39]. However, most reported schemes are experimentally demonstrated based on discrete-modulated CVQKD protocol, which are not conducive to ensure maximum mutual information between Alice and Bob close to fiber channel capacity and sensitively monitor the eavesdropper Eve for the long distance CVQKD compared with GMCS CVQKD [6]. Therefore, the phase compensation methods that are capable of reducing the phase noise in LLO-CVQKD based on GMCS protocol, and at the same time avoiding the crosstalk between quantum and pilot signals are of great interest.

In this paper, we experimentally demonstrate a high-speed Gaussian-modulated LLO-CVQKD scheme by using pilot-tone-assisted phase compensation. The demonstrated LLO-CVQKD scheme not only properly guarantees the different detection requirements for quantum signal and pilot tone, but also eliminates the crosstalk between them with the frequency-multiplexing and polarization-multiplexing techniques. Moreover, the scheme realizes simultaneous measurement of both quadrature (X and P) by heterodyne detection, avoiding the use of extra random basis selection. Besides, the dominant phase noise of the LLO-CVQKD can be well compensated in the experiment by the proposed reliable pilot-tone-assisted phase compensation method, achieving a secure key rate of 7.04 Mbps in the asymptotic regime and a secure key rate of 1.85 Mbps under the finite-size block of N=10⁷ with an low level of excess noise at the transmission distance of 25 km.

2. Scheme setup

As is shown in Fig. 1, the Alice configuration consists of a heterodyne interferometer (HI). In the upper branch, a continuous optical carrier with central frequency f_A from Alice laser is modulated in AM1 to generate an optical pulse with repetition frequency f_rep, and then modulated in the cascaded AM2, PM1 and PM2 for achieving the Gaussian modulation. Note that the generated optical pulse is sent into two cascaded PMs instead of one PM in traditional method for halving the driven voltage requirement of the RF modulated signal. The optical signal in the upper branch is attenuated by a variable optical attenuator (VOA) to be quantum signal with GMCS $|{{x_A} + j{p_A}} \rangle = |{{A_{sig}}{e^{j{\phi_A}}}} \rangle$, also given by [31]

(1)$${E_{sig}} = {A_{sig}}\cos ({\textrm{2}\pi {f_A}t + {\phi_A} + {\varphi_A}} )$$

where the modulated amplitude A_sig and phase ϕ_A follow the Rayleigh distribution and the uniform distribution, respectively, and φ_A denotes the phase of Alice laser. In the lower branch, the optical carrier is injected into a Mach-Zehnder modulator (MZM) biased at minimum transmission to implement carrier suppression double sideband (CS-DSB) modulation for generating the desired pilot tone. Under small signal condition, the pilot tone can be expressed as:

(2)$${E_{ref}} = {A_{ref}}{J_1}(m )\cos ({2\pi {f_A}t \pm 2\pi {f_m}t + {\varphi_A}} )$$

where A_ref denotes the amplitude of pilot tone in the lower branch. f_m and m correspond to modulation frequency and modulation index of the CS-DSB modulation, respectively, and J₁(•) is the 1th-order Bessel function of the first kind. The quantum signal and the pilot tone are combined and transmitted into the SMF channel in different frequency band and orthogonal polarization state for avoiding the crosstalk from strong pilot tone to weak quantum signal.

Fig. 1. Schematic diagram of the proposed LLO-CVQKD scheme. BS: beam splitter, AM: amplitude modulator, PM: phase modulator, CS-DSB: carrier suppression double sideband, MZM: Mach-Zehnder modulator, VOA: variable optical attenuator, PBC: polarization beam combiner, SMF: single mode fiber, PC: polarization controller, PBS: polarization beam splitter, OC: optical coupler, BHD: balance homodyne detector, ADC: analog digital conversion, DSP: digital signal processing. The insets (a) optical spectrum of the Gaussian modulated quantum signal, (b) optical spectrum of the CS-DSB modulated pilot tone, (c) optical spectrum of the transmitting optical signals in SMF, (d) output quantum electrical spectrum, (e) output pilot electrical spectrum (f_m>Δf_AB in practical experiment).

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At Bob’s site, the quantum signal and the pilot tone are separated by the polarization beam splitter (PBS), and then coupled into two balanced homodyne detectors (BHDs) with LLO signal from Bob laser for heterodyne detection, respectively. The generated photocurrents can be expressed as, respectively

(3a)$$\begin{array}{l} {I_{sig}} = {R_1}{\eta _1}{|{{E_{sig}} + {E_{lo1}}} |^2} - {R_1}{\eta _1}{|{{E_{sig}} - {E_{lo1}}} |^2}\\ \quad \;{\kern 1pt} {\kern 1pt} = 2{R_1}{\eta _1}{A_{lo1}}{A_{sig}}\cos ({\textrm{2}\pi \Delta {f_{AB}}t + {\phi_A} + \Delta {\varphi_{fast}} + {\varphi_{sig}}} )\end{array}$$

(3b)$$\begin{array}{l} {I_{ref}} = {R_2}{\eta _2}{|{{E_{ref}} + {E_{lo2}}} |^2} - {R_2}{\eta _2}{|{{E_{ref}} - {E_{lo2}}} |^2}\\ \quad {\kern 1pt} \,\, = \textrm{2}{R_2}{\eta _2}{A_{lo2}}{A_{ref}}{J_1}{\kern 1pt} (m )\cos ({2\pi \Delta {f_{AB}}t \pm 2\pi {f_m}t + \Delta {\varphi_{fast}} + {\varphi_{ref}}} )\end{array}$$

with the frequency difference Δf_AB=f_A-f_B and the fast-drift laser phase Δφ_fast=φ_A-φ_B, where f_B and φ_B represent the central frequency and phase of Bob laser, respectively. A_lo₁ and A_lo₂ are the amplitudes of the LLO signals E_lo₁ and El_o₂ generated from the Bob laser, respectively. R_i and η_i denote the equivalent resistance and quantum efficiency of two BHDs (i=1, 2), respectively. φ_sig and φ_ref represent the slow-drift phase of quantum signal and pilot tone introduced by quantum channel disturbance, respectively. It is observed from Eq. (3a) that the state received by Bob is degraded by the fast-drift laser phase and slow-drift channel phase with respect to the GMCS prepared by Alice [14,19]. Therefore, a pilot-tone-assisted phase compensation method is proposed for eliminating the phase drift in the LLO-CVQKD scheme, which is based on the pilot assisted channel equalization recently applied in optical fiber LLO-CVQKD [31,35]. It is noteworthy from inset (e) in Fig. 1 that the positive frequency components are only shown in the real frequency domain and the modulation frequency f_m is set to be larger than the frequency difference Δf_AB in the proposed pilot-tone-assisted LLO-CVQKD scheme.

After the bandpass filtering, the heterodyning products of the quantum signal and the pilot tone are orthogonally down-converted according to Eqs. (3a) and (3b) respectively, which can be written as:

(4a)$${I_{sig}} \cdot \cos ({\textrm{2}\pi \Delta {f_{AB}}t} )= {R_1}{\eta _1}{A_{lo1}}{A_{sig}}\left[ \begin{array}{l} \cos ({{\phi_A} + \Delta {\varphi_{fast}} + {\varphi_{sig}}} )\\ + \cos ({\textrm{4}\pi \Delta {f_{AB}}t + {\phi_A} + \Delta {\varphi_{fast}} + {\varphi_{sig}}} )\end{array} \right]$$

(4b)$${I_{sig}} \cdot \sin ({\textrm{2}\pi \Delta {f_{AB}}t} )= {R_1}{\eta _1}{A_{lo1}}{A_{sig}}\left[ \begin{array}{l} \sin ({\textrm{4}\pi \Delta {f_{AB}}t + {\phi_A} + \Delta {\varphi_{fast}} + {\varphi_{sig}}} )\\ - \sin ({{\phi_A} + \Delta {\varphi_{fast}} + {\varphi_{sig}}} )\end{array} \right]$$

(4c)$${I_{ref}} \cdot \cos ({2\pi {f_m}t - 2\pi \Delta {f_{AB}}t} )= {R_2}{\eta _2}{A_{lo2}}{A_{ref}}{J_1}{\kern 1pt} (m )\left[ \begin{array}{l} \cos ({\Delta {\varphi_{fast}} + {\varphi_{ref}}} )\\ + \cos ({4\pi {f_m}t - 4\pi \Delta {f_{AB}}t + \Delta {\varphi_{fast}} + {\varphi_{ref}}} )\end{array} \right]$$

(4d)$${I_{ref}} \cdot \sin ({2\pi {f_m}t - 2\pi \Delta {f_{AB}}t} )= {R_2}{\eta _2}{A_{lo2}}{A_{ref}}{J_1}{\kern 1pt} (m )\left[ \begin{array}{l} \sin ({4\pi {f_m}t - 4\pi \Delta {f_{AB}}t + \Delta {\varphi_{fast}} + {\varphi_{ref}}} )\\ - \sin ({\Delta {\varphi_{fast}} + {\varphi_{ref}}} )\end{array} \right]$$

After the low-pass filtering, the desired baseband components can be obtained as:

(5a)$${X_{sig}} = {R_1}{\eta _1}{A_{lo1}}{A_{sig}}\cos ({{\phi_A} + \Delta {\varphi_{fast}}} )$$

(5b)$${P_{sig}} ={-} {R_1}{\eta _1}{A_{lo1}}{A_{sig}}\sin ({{\phi_A} + \Delta {\varphi_{fast}}} )$$

(5c)$${X_{ref}} = {A_{ref}}{R_2}{\eta _2}{A_{lo2}}{J_1}(m )\cos ({\Delta {\varphi_{fast}} + \Delta {\varphi_{slow}}} )$$

(5d)$${P_{ref}} ={-} {A_{ref}}{R_2}{\eta _2}{A_{lo2}}{J_1}(m )\sin ({\Delta {\varphi_{fast}} + \Delta {\varphi_{slow}}} )$$

where Δφ_slow=φ_ref -φ_sig means the slow-drift relative channel phase between quantum signal and pilot tone. Next, the fast-drift laser phase Δφ_fast is compensated based on Eq. (5) and both quadrature values of the quantum signal can be simultaneously obtained as:

(6a)$${X^{\prime}} = {A_{sig}}\cos ({{\phi_A} - \Delta {\varphi_{slow}}} )= \frac{{{X_{sig}}{X_{ref}} + {P_{sig}}{P_{ref}}}}{{\sqrt {X_{ref}^2 + P_{ref}^2} }} \cdot \frac{1}{{\sqrt {{N_0}} }}$$

(6b)$${P^{\prime}} = {A_{sig}}\sin ({{\phi_A} - \Delta {\varphi_{slow}}} )= \frac{{{X_{sig}}{P_{ref}} - {P_{sig}}{X_{ref}}}}{{\sqrt {X_{ref}^2 + P_{ref}^2} }} \cdot \frac{1}{{\sqrt {{N_0}} }}$$

where N₀ corresponds the shot noise variance of the LLO signals E_lo₁. It is seen from Eqs. (6a) and (6b) that both quadratures of the quantum signal are normalized to shot noise unit (SNU). Finally, the slow-drift channel phase Δφ_slow can be estimated by a bit of training sequence disclosed in quantum signal, and the received quadratures X_B and P_B at Bob’s site can be more precisely compensated as

(7a)$${X_B} = {X^{\prime}}\cos ({\Delta {\varphi_{slow}}} )- {P^{\prime}}\sin ({\Delta {\varphi_{slow}}} )$$

(7b)$${P_B} = {X^{\prime}}\sin ({\Delta {\varphi_{slow}}} )+ {P^{\prime}}\cos ({\Delta {\varphi_{slow}}} )$$

It is worth noting that the proposed LLO-CVQKD scheme avoids the crosstalk from strong pilot tone to weak quantum signal by properly separating the weak quantum signal and the strong pilot signal in different frequency sideband and orthogonal polarization state. Meanwhile, the scheme has the advantage of guaranteeing the sufficient optical power to achieve the low-noise detection for quantum and high-saturation limitation detection for pilot by providing a “locally” local laser, which gets rid of the intensity bottleneck and security hole in the TLO-CVQKD [35]. Moreover, one can see from Eqs. (7a) and (7b) that both quadratures (X and P) are simultaneously obtained by heterodyne detection, avoiding extra random basis selection in the conventional CVQKD based on homodyne detection. Besides, compared with time-multiplexing LLO-CVQKD scheme with coarse time resolution in practical experiment, it is conductive for the proposed frequency-multiplexing LLO-CVQKD scheme with ultrahigh frequency separation between the quantum signal and pilot tone to promote the repetition frequency of the LLO-CVQKD.

3. Experimental demonstration

We experimentally performed the proposed LLO-CVQKD scheme based on the setup shown in Fig. 1. At Alice’s site, a continuous optical carrier comes from Alice laser (NKT Photonic Basik) with narrow linewidth and low relative intensity noise (RIN), which is split into two branches of the HI by a BS with intensity ratio of 99:1. The weak optical carrier in the upper branch is sent into an amplitude modulator (AM1, EOSPACE) and modulated to be an optical pulse signal with repetition frequency of 100 MHz. The generated optical pulse is injected into another amplitude modulator (AM2, EOSPACE) for the required Rayleigh distribution modulation and then sent into two cascaded phase modulators (PM1 and PM2, EOSPACE) for the required uniform distribution modulation. So, the Rayleigh distribution and uniform distribution modulation are cascaded to provide a centered Gaussian distribution modulation. Note that the quantum signals need to be pulsed signals instead of continuous signals for more precisely extracting the heterodyne-detected products with better signal to noise ratio (SNR) in practical experiment. A VOA is used to adjust the GMCS with an optimized variance of V_A. The pulse modulation electrical signal and the Gaussian modulation electrical signal are generated from two arbitrary waveform generators (Keysight M8195A and Ceyear 1652A). The strong optical carrier in the lower branch is modulated in a Mach-Zehnder modulator (MZM, EOSPACE) according to a sinusoidal microwave signal generated from microwave source (MS, Rohde & Schwarz SMB) for forming CS-DSB modulated pilot tone. At Bob’s site, the quantum signal and the pilot tone are separately detected using two commercial BHDs (THORLABS PDB480C), and the corresponding optical powers of LLO signals supplied by Bob laser (NKT Photonic Basik) are 10.04 dBm and 3.08 dBm, respectively, for the accurate detection of quantum signal and pilot signal. A 25 km fiber spool (with loss coefficient 0.2 dB/km) is used as the transmission channel in our experiment. The polarization controller (PC, General Photonics PSY PSY-201) is used to eliminate the deterioration of the polarization due to fiber channel disturbance in favor of the clear separation between the quantum signal and the pilot tone at Bob’s site. Two BHDs’ electrical outputs are monitored and collected with a real-time oscilloscope (Keysight DSAZ254A) and an electrical spectrum analyzer (Rohde & Schwarz FSW43) for the subsequent offline digital signal processing (DSP) with Matlab-code on a high-performance computer. A pilot-tone-assisted phase compensation method in the offline DSP is designed to extract the phase sharing of the pilot tone for compensating the raw data of the quantum signal, from which the final key is distilled. In our experiment, the transmitter and receiver are triggered by the same synchronous source to achieve clock synchronization at the transmitter and receiver for simplicity. Moreover, a marking signal is embedded in the pilot signal for further clock calibration. All the optical components are connected with angle polished finishes to reduce the residual reflection as much as possible.

Figure 2 shows two BHDs’ output time waveforms and frequency spectra of the quantum signal and the pilot tone, respectively, in the case of the repetition frequency f_rep=100 MHz and the CS-DSB modulation frequency f_m=2 GHz. It is worth noting that the duration of the quantum signal pulse is 100 ms in our experiment due to the repetition frequency of 100 MHz and the total block size of 1 × 10⁷, in spite of time scale of 500 us only shown in Fig. 2(a). Moreover, the fluctuating response in Fig. 2(a) is mainly originated from the Gaussian modulated quantum random signal with the large voltage amplitude fluctuating. Besides, compared with the inset (e) in Fig. 1, the detected electrical component f_m+f_AB (2.7 GHz) cannot be experimentally shown in Fig. 2(d) because it is out of the response range of BHD. One can see from Fig. 2(b) that the peak at every repetition frequency of 100 MHz is presented in frequency spectrum of the quantum signal due to a bit of sinusoidal training sequences disclosed in the raw data of the quantum signal. It is noteworthy that the frequency difference alignment between Alice laser and Bob laser is performed by precise laser wavelength control in order to ensure a much smaller optical-frequency deviation.

Fig. 2. Heterodyne-detected products of the quantum and pilot optical signals, where (a) output quantum time waveform, (b) output quantum electrical spectrum, (c) output pilot time waveform, (d) output pilot electrical spectrum.

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In practical experiment, the orthogonal polarization states of the quantum signal and the pilot tone should be carefully maintained by the PC for guaranteeing the separate detection of the pilot tone and the quantum signal and avoiding the influence of the remaining optical carrier resulted from incomplete CS-DSB modulation on extraction of the quantum signal. Moreover, it is beneficial for the clear separation between the quantum signal and the pilot tone to meet the low-noise heterodyne detection required by quantum signal and high-saturation limitation heterodyne detection required by pilot tone. It is shown in Figs. 2(b) and 2(d) that the desired quantum signal and the desired pilot tone component 1.31 GHz (f_m-Δf_AB) are efficiently separated by using the frequency-multiplexing and polarization-multiplexing technique. Meanwhile, comparing the results in Figs. 2(b) and 2(d), it is obvious that the detected remaining carrier sideband 0.69 GHz (Δf_AB) is practically isolated from the quantum signal due to the modulation ER of more than 20 dB and the polarization isolation of more than 40 dB in the experiment, which verifies that the remaining pilot tone has no influence on the extraction of the quantum signal. It is noteworthy that the proposed pilot-tone-assisted LLO-CVQKD might get rid of the use of the multiplexing technique by guaranteeing no remaining frequency in the desired quantum frequency band and the BHD with large dynamic range applicable for different detection requirement of the quantum signal and the pilot tone.

The quantum and pilot heterodyne-detected signals are recorded and digitized by the real-time oscilloscope with sample rate of 10 GSa/s for the offline DSP illustrated in Fig. 3. In the off-line DSP, every pulse is sampled and averaged by 40 points when the duty cycle of our system is set to be 2/5 in our experiment. Firstly, the frequency difference Δf_AB between Alice laser and Bob laser is estimated to be 0.69 GHz based on the designed frequency offset estimation algorithm from the recorded pilot data after fast Fourier transform (FFT), and the frequency f_m-Δf_AB of the desired pilot tone can be determined to be 1.31 GHz. Secondly, the desired quantum signal is obtained by band-pass filtering the quantum frequency spectrum with the center frequency of 0.69 GHz, and the desired pilot tone is also obtained from the pilot frequency spectrum by using a band-pass filter with the center frequency of 1.31 GHz. Note that the filter bandwidths for quantum and pilot are chosen based on the bandwidth of Gaussian modulated quantum signal and the expected optical-frequency deviation in the experiment. Thirdly, the band-pass filtered quantum signal is orthogonally down-converted with frequency of 0.69 GHz (Δf_AB) to the baseband for extracting the in-phase component X_sig and the quadrature component P_sig. Similarly, the in-phase component X_ref and quadrature component P_ref of the pilot tone can also be obtained by the orthogonal down-conversion with frequency 1.31 GHz (f_m-Δf_AB) and baseband filtering. Finally, after the clock recovery, both quadratures of the quantum signal can be simultaneously obtained and normalized to SNU according to Eqs. (6a) and (6b) after the compensation for the fast-drift laser phase Δφ_fast from the phase sharing of the pilot tone. Subsequently, both quadratures of the quantum signal can be more precisely compensated based on Eqs. (7a) and (7b) by estimating slow-drift channel phase Δφ_slow from a bit of training sequences disclosed in the raw data of the quantum signal. Note that the fast-drift laser phase Δφ_fast and slow-drift channel phase Δφ_slow are divided according to time scales of 100 ms [14,19]. In our experiment, the time scale of a measurement block including quantum key sequence and the disclosed quantum training sequence should be controlled within 1 ms as much as possible and the amount of training sequence can be further increased without the influence on the required amount of the final quantum key, so that the slow phase drift of quantum key sequence can be more precisely compensated by sharing phase from the disclosed quantum training sequence in the same measurement block. As is shown in Fig. 4, both quadratures (X and P) between Alice and Bob are compared before and after phase compensation, which verifies the proposed pilot-tone-assisted phase compensation method. It is worth noting that both quadratures (X and P) can be simultaneously obtained in our experiment because the proposed pilot-tone-assisted LLO-CVQKD based on heterodyne detection requires more than double bandwidth of Gaussian modulated quantum signal compared with the former pilot-pulse-assisted LLO-CVQKD based on homodyne detection, which can be proved by comparing the heterodyne detection noise χ_het=[(2-η) + 2υ_el]/η [32] and the homodyne detection noise χ_hom=[(1-η)+υ_el]/η [33].

Fig. 3. The offline DSP of the proposed LLO-CVQKD scheme.

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Fig. 4. Measured both quadratures (X and P) (200 000 points) in shot noise units between Alice and Bob before and after phase compensation, (a) in-phase X and (b) quadrature P before phase compensation, (c) in-phase X and (d) quadrature P after the fast-drift laser phase compensation, (e) in-phase X and (f) quadrature P after the slow-drift channel phase compensation.

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4. Excess noise and secure key rate

The excess noise estimation is a crucial step in CVQKD, which has direct influence on the secure key rate of the designed system. In order to estimate the excess noise and evaluate the secure key rate, the LLO-CVQKD system is considered to be a normal linear model $y = \sqrt {\eta T} x + z$ for Alice and Bob’s correlated variables (x_i, y_i) i=1,2,…,n, where η is the quantum efficiency and T is the transmittance efficiency, and z follows a centered normal distribution with a variance of σ²=ηTε+1+υ_el. Note that the excess noise ε and the electronic noise υ_el are normalized to SNU. For the proposed LLO-CVQKD scheme, one of the most critical contributions to the excess noise is the phase noise including laser phase noise and channel phase noise. The laser phase noise ε_fast=2πV_A(Δν_A+Δν_B)/f_rep is resulted from the fast-drift laser phase between Alice laser and Bob laser, where Δν_A and Δν_B are the linewidths of Alice laser and Bob laser respectively [35]. In our scheme, the laser phase noise can be eliminated by sharing fast-drift laser phase Δφ_fast of the pilot tone. The channel phase noise ε_slow is introduced from different fiber channel disturbance between quantum signal and pilot tone, which can be compensated by sharing slow-drift channel phase Δφ_slow from a bit of training sequence disclosed in the raw data of the quantum signal. In addition to the phase noise, the other noise sources, such as modulation noise due to finite dynamics and ADC quantization noise, should be reduced as much as possible for a tolerable excess noise ε in the experiment.

With the repetition frequency f_p=100 MHz, the quantum efficiency η=0.56, the ratio of electronic noise and shot noise υ_el/N₀=0.042, the modulation variance V_A=3.246 SNU, the secure transmission distance L=25 km (with 5 dB loss) and the reverse reconciliation efficiency β=95% [40,41], the experimentally measured excess noise ε with block of size 5 × 10⁶ and the worst excess noise ε_max with considering of the finite-size effect can be calculated by the following equations [32,33], respectively

(8)$$\varepsilon = \frac{{({{\sigma^2} - 1 - {\upsilon_{el}}} )}}{{\eta T}}$$

(9)$${\varepsilon _{\max }} = \frac{{({{\sigma^2} + \Delta \sigma_0^2 - 1 - {\upsilon_{el}}} )}}{{\eta T}}$$

with the total noise variance σ² and the channel transmittance T, and the offset $\Delta \sigma _\textrm{0}^\textrm{2}$ is related with the block of size 5 × 10⁶ used for excess noise estimation. As is shown in Fig. 5, the measured excess noise and the worst excess noise under finite-size effect are estimated and normalized to SNU over 30 minutes. Note that the rest excess noise is probably originated from the polarization drift due to inaccurate polarization control, ADC quantization noise, the modulation noise because of finite dynamic and the residual phase noise resulted by imperfect phase compensation scheme [32]. Moreover, the tolerable maximal value of excess noise at the secure transmission distance of 25 km is calculated and defined in the yellow solid line in Fig. 5. Besides, one can see from Fig. 5 that the mean of measured excess noise is around 0.022 and the mean of the worst excess noise under the finite-size effect is round 0.048, from which the final key rate under the infinite-size and the finite-size effect can be calculated by [32,33], respectively

(10)$${R_1} = {f_{rep}}[{\beta {I_{AB}} - S_{BE}^{{ \in_{PE}}}} ]$$

(11)$${R_2} = {f_{rep}}\frac{n}{N}[{\beta {I_{AB}} - S_{BE}^{{ \in_{PE}}} - \Delta (n )} ]$$

with the total block size N, the number n for key distribution and the reverse reconciliation efficiency β. I_AB denotes the Shannon mutual information between Alice and Bob in the case of the heterodyne detection. $S_{BE}^{{ \in _{PE}}}$ represents the Holevo bound and Δ(n) is related to the security of the privacy amplification. Therefore, the experimental and simulated results of secure key rate corresponding to different secure transmission distance are plotted in Fig. 6. The red solid line denotes the secure key rate as a function of secure transmission distance under finite-size block of N=10⁷ and the black dash line defines the calculated secure key rate as a function of secure transmission distance in infinite-size scenarios. As is shown in Fig. 6, the experimental secure key rate with infinite size and finite-size block of N=10⁷ are calculated to be 7.04 Mbps and 1.85 Mbps, respectively. In addition, the previous experiment results are also shown in Fig. 6 for comparison, highlighting the proposed higher rate GMCS LLO-CVQKD.

Fig. 5. Measured excess noise in SNU over 30 minutes. The red circles represent the measured excess noise on the block of size 5 × 10⁶ and the blue squares represent the worst excess noise under the finite-size effect, while the red dash dot and the blue dash line define the mean of them, respectively. The yellow solid line represents the excess noise threshold of null key rate at the secure transmission distance of 25 km.

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Fig. 6. Secure key rate versus secure transmission distance curves. The red solid line represents secure key rate at different secure transmisstion distance under finite-size block of N=10⁷ and the black dash line represents secure key rate at different secure transmission distance in infinite-size scenarios, while the blue circle and the yellow square represent the experimental secure key rate with block size N=10⁷ and infinite-size, respectively. The reverse reconciliation efficiency β is 95%, the quantum efficiency η is 0.56, the ratio of electronic noise and shot noise υ_el/N₀ is 0.042 and the modulation variance V_A is 3.246 SNU in the experiment.

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5. Conclusion

We have experimentally demonstrated a high-speed pilot-tone-assisted GMCS LLO-CVQKD system. In order to compensate the dominate phase noise of the proposed LLO-CVQKD scheme, a pilot-tone-assisted method is proposed based on the frequency-multiplexing and polarization-multiplexing techniques. Superior to the former pilot-pulse-assisted method, the pilot tone is separately transmitted and detected with quantum signal in different frequency band and orthogonal polarization state, guaranteeing no crosstalk from the strong pilot tone to the weak quantum signal and different detection requirements of low-noise for quantum signal and high-saturation limitation for pilot tone. Moreover, compared with the conventional CVQKD based on homodyne detection, the proposed scheme based on heterodyne detection not only avoids the need of the random base selection, but also realizes the simultaneous measurement of both quadrature (X and P) in demand of more than double bandwidth. Besides, the phase noise caused by the fast-drift laser phase of two independent lasers and the slow-drift channel phase from fiber channel disturbance can be compensated in real time by sharing phase of the pilot tone and the disclosed quantum training sequence, so that a low level of excess noise at the secure transmission distance of 25 km can be obtained for the GMCS LLO-CVQKD with an asymptotic secure key rate of 7.04 Mbps and an secure key rate of 1.85 Mbps under the finite-size block of N=10⁷. It is worth noting that the final secure key rate might be further improved by robustly increasing the repetition rate of the proposed LLO-CVQKD, benefiting from the pilot-tone-assisted method based on the high-resolution frequency-multiplexing technique.

Funding

Applied Basic Research Program of Sichuan Province (2020YJ0482); National Natural Science Foundation of China (61702469, 61771439, 61901425, U19A2076); Sichuan Youth Science and Technology Foundation (2019JDJ0060); National Cryptography Development Fund (MMJJ20170120); CETC Fund (6141B08231115); Innovation Special Zone Funds (18-163-00-TS-004-040-01); Sichuan Province Science and Technology Support Program (2020YFG0289).

Disclosures

The authors declare that there are no conflicts of interest related to the article.

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High-speed Gaussian-modulated continuous-variable quantum key distribution with a local local oscillator based on pilot-tone-assisted phase compensation

Abstract

1. Introduction

2. Scheme setup

3. Experimental demonstration

4. Excess noise and secure key rate

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (6)

Equations (20)

Optics Express