Open Access
Add to BibSonomy
Abstract
We report a directly modulated distributed feedback laser operating in gain-switching mode for preparing the coherent states required for the Gaussian-modulated coherent-state (GMCS) continuous-variable quantum key distribution (CV-QKD) protocol. The proposed single-component quantum transmitter design eliminates the need for external modulators, decreasing the complexity of GMCS CV-QKD systems. The experimental results demonstrate a potential asymptotic secret key rate value of 2.63 Mbps over an 11-km fiber link, making the directly modulated GMCS transmitter particularly suitable for metropolitan optical networks where compactness, robustness, and low cost are key desirable features.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Quantum key distribution (QKD) enables two distant users (generally referred to as Alice and Bob) to share a secret key by sending quantum states via an untrusted channel [1,2]. The fundamental principles of quantum physics are the basis of the information-theoretic security of QKD [3]. Continuous-variable QKD (CV-QKD) [1,2] that encodes the information on the quadratures of the light field is a strong alternative to discrete-variable QKD (DV-QKD) for various reasons. First, CV-QKD can be implemented using coherent optical fiber technology already available in the telecom industry, rendering it compatible with current communication infrastructures. Second, contrary to DV-QKD, CV-QKD does not require complex single-photon detectors [1,2], as it uses homodyne or heterodyne detection at room temperature. Third, CV-QKD can offer higher secret key rates over short (tens of km) metropolitan links [1,2]. In addition, CV-QKD is ideally positioned to be mass-produced and deployed using photonic integrated circuits (PICs) [4].
The Gaussian-modulated coherent-state (GMCS) protocol has been widely used for implementing CV-QKD because of its practicality [1,2,5]. In terms of security, the GMCS protocol has theoretically been shown to be robust against arbitrary collective [6] and coherent attacks [7] and, in some cases, under finite-size conditions [8–13]. In experiments, this protocol has been implemented in both laboratory and field tests [14–27]. For example, in the former, transmission distance of up to 202.81 km [23] or high speed of up to 66.8 Mbps [24] has been achieved. In the latter, the coexistence with a classical channel in a field-installed fiber has also been demonstrated [25,26]. In all these experiments, true random number generators are required for preparing the coherent states constituting the secret key.
Quantum random number generators (QRNGs) have emerged as certifiable sources of true random numbers [28]. QRNGs based on the detection of single photon events [29–34], amplified spontaneous emission (ASE) [35], vacuum fluctuations [36], and phase noise in continuous-wave [37–39] and pulsed semiconductor laser diodes [40–44] have been demonstrated. The latter approach has achieved among the highest random number generation rates to date using off-the-shelf optical fiber components. In particular, rates up to 68 Gbps have been demonstrated using a gain-switched (GS) distributed-feedback (DFB) laser and a coherent detector [41]. By continuously modulating the GS DFB laser from below to above its threshold, optical pulses with nearly identical amplitudes and completely randomized phases can be generated. The coherent detector converts the phase fluctuations into amplitude fluctuations, which, after proper digitization, are transformed into random numbers. The benefits of this scheme include its simplicity, robustness, low cost, and multi-clock frequency flexibility. PIC-based QRNG solutions using a GS DFB laser have also been demonstrated [45,46].
In the majority of GMCS CV-QKD implementations, the generation of random numbers and the coherent state preparation are performed by two different and independent devices, increasing the system complexity and cost. This also complicates the standardization of GMCS CV-QKD [5–26]. To address this problem, a passive-state-preparation (PSP) GMCS CV-QKD protocol has recently been proposed and experimentally validated in [47–50]. In this case, the amplitude and phase modulators required for the state preparation and QRNGs have been replaced by an ASE source. Although this PSP approach can in principle be suitable for high-speed and PIC-based CV-QKD solutions with lower complexity and cost, as no active modulation is required, the absence of modulators makes frame synchronization technically challenging. That is, PSP CV-QKD schemes require additional bulk optical delay lines [50]. An alternative cost-effective solution, which allows the preparation of GMCSs and robust synchronization mechanisms without the use of modulators while providing moderate complexity and high compactness, is a directly modulated DFB laser. On the one hand, coherent states with specific phase values can be generated for synchronization purposes by exploiting the inherent frequency chirp of the DFB laser at large biasing currents [51–53]. On the other hand, the generation and preparation of random coherent states can be performed by setting the DFB laser in the gain-switching mode in conjunction with a QRNG.
In this study, we introduce and demonstrate a simple transmitter for GMCS CV-QKD protocol using a pulsed GS DFB laser. The generated coherent states are then transmitted over an 11-km fiber link. To correct the phase errors caused by the laser drift and fiber channel length fluctuations, a phase recovery process based on a differential scheme is implemented, achieving a potential secret key rate value of 2.63 Mbps in the asymptotic regime. We believe that the newly proposed transmitter is particularly suitable for metropolitan optical networks. Moreover, compared to narrow-linewidth lasers, DFB lasers can easily be integrated in PICs, leading to ultra-compact and ultra-low-cost GMCS CV-QKD components and subsystems.
The paper is structured as follows. Section 2 describes the operation principle for the generation and preparation of GMCSs using a pulsed GS DFB laser as well as the interferometric structure for the detection. Section 3 demonstrates the GMCS CV-QKD protocol over an 11-km link using the GS DFB laser for preparing the coherent states. This section includes a detailed explanation of the experimental setup (Section 3.1), the implementation of digital signal processing with the proposed quantum state phase recovery process (Section 3.2), and the experimental results achieved using the proposed GMCS CV-QKD transmitter (Section 3.3). Finally, Section 4 presents the conclusion of the work.
2. Preparation and measurement of Gaussian-modulated coherent states (GMCSs)
Figure 1(a) shows the scheme for the preparation and measurement of the GMCSs used in this study. Aiming at the generation of short and phase-independent optical pulses, a DFB laser is set to work in the gain-switching mode. In this operation mode, the gain of the DFB laser is periodically switched from below to above its threshold level by applying modulating current pulses [40–44]. Below threshold, the cavity field of the DFB laser is strongly attenuated, and ASE dominates. This reduces any prior coherence to a negligible level while the ASE provides a masking field with a true random phase [40–44]. To boost the random phase field inside the cavity, the DFB laser is briefly driven above threshold, emitting a short coherent optical pulse with a random phase of quantum nature. The measurement of the coherent pulses containing the quantum states is performed by a phase-diversity homodyne detector in which the optical signal (S) generated by the DFB laser interferes with a continuous-wave (CW) local oscillator (LO) [54]. Here, the homodyne term refers to that used in the telecom industry. More specifically, the LO wavelength matches the S laser wavelength and both quadratures (${\rm X}$ and ${\rm P}$) are measured simultaneously. The detector consists of a 90° optical hybrid (OH) and a pair of balanced photodetectors (BPDs).
Fig. 1. (a) Schematic for preparation and measurement of Gaussian-modulated coherent states (GMCSs). (Acronyms) BPD: balanced photodetector, CW: continuous-wave, DFB: distributed-feedback, GS: gain-switched, LO: local oscillator, OH: optical hybrid. (b) Normalized histogram of the expected ${\rm X}$ and ${\rm P}$ projections at the output of the BPDs (Gaussian distribution) in arbitrary units (a.u.). (Inset) Squared Rayleigh distribution of the amplitude of the pulses at the input of the GS DFB laser.
In the absence of noise, the normalized photocurrents obtained at the receiver output (${\rm X}$ and ${\rm P}$) can be defined as ${\rm X}({\rm t} )= \sqrt {{\rm I}_{\rm S}({\rm t} )} {\rm cos}({{\mathrm{\Delta}}\mathrm{\phi} ({\rm t} )} )$ and ${\rm P}({\rm t} )= \sqrt {{\rm I}_{\rm S}({\rm t} )} {\rm sin}({{\mathrm{\Delta}}\mathrm{\phi} ({\rm t} )} )$ [54], where ${\rm I}_{\rm S}({\rm t} )$ is the pulsed signal used for the modulation of the DFB laser, and ${\mathrm{\Delta}}\mathrm{\phi} ({\rm t} )= \mathrm{\phi} _{\rm S}({\rm t} )-\mathrm{\phi} _{{\rm LO}}({\rm t} )$ is the random phase difference between the input signal S and LO. Under the gain-switching condition of the DFB laser, ${\mathrm{\Delta}}\mathrm{\phi} ({\rm t} )$ can be described as a random parameter following a Gaussian distribution with a large variance [39–44]. However, because of the modular nature of the cosine/sine functions, values exceeding -π and π are wrapped over the interval [-π, π). Therefore, a uniform distribution ${\rm U}$(-π, π) is a suitable approximation for ${\mathrm{\Delta}}\mathrm{\phi} ({\rm t} )$ [40–44]. Finally, the amplitude of the pulses of ${\rm X}$ and ${\rm P}$ signals can be controlled by appropriately selecting the amplitude of the electrical pulses defined by ${\rm I}_{\rm S}({\rm t} )$. For example, electrically driving the DFB laser with a pulsed signal in which the amplitudes follow a squared Rayleigh distribution will lead to optical pulses where the quadrature distribution projections of ${\rm X}$ and ${\rm P}$ obey a Gaussian probability distribution, as shown in Fig. 1(b).
Figure 2 shows the experimental characterization and validation results for the structure presented in Fig. 1 when a 2.5-GHz-bandwidth GS DFB laser emitting at 1550 nm is used for preparing the GMCSs. Figure 2(a) shows the output power response of the DFB laser as a function of the DC bias current (${\rm I}_{{\rm DC}-{\rm BIAS}}$); in the figure, a threshold value of 9.0 mA can be observed. Figure 2(b) shows the optical power spectrum measured at the laser output in the presence and absence of the ${\rm I}_{\rm S}({\rm t} )$ signal when ${\rm I}_{{\rm DC}-{\rm BIAS}}$ = 6.5 mA. The ${\rm I}_{\rm S}({\rm t} )$ amplitudes were Rayleigh-distributed with a mean of $\mathrm{\mu}$ = 0.5 mA and a standard deviation of $\mathrm{\sigma}$ = 5.6 mA. The pulse width (${\rm PW}$) and pulse rate (${\rm R}$) of ${\rm I}_{\rm S}({\rm t} )$ were 1 ns and 100 Mpulses/s, respectively. For this ${\rm I}_{\rm S}({\rm t} )$ signal configuration, Fig. 2(b) shows the cavity field of the DFB laser experiencing an attenuation of 40 dB when modulating from above to below the threshold level. Figure 2(c) shows the temporal ${\rm I}_{\rm S}({\rm t} )$ pulse shape used to drive the GS DFB laser for three different amplitudes. Figure 2(d) shows the optical output of the GS DFB laser for the three amplitudes considered in Fig. 2(c). As shown in Fig. 2(d), the optical pulses suffer from an initial overshooting that relaxes to a steady state in which the chirp is minimized. Also, it can be observed that the steady-state regime is reached after about 0.5 ns. The red line in Fig. 2(d) represents the temporal instant in which the optical pulses are sampled after beating them with the LO laser.
Fig. 2. Experimental characterization and validation of a GS DFB laser used for the preparation of Gaussian-modulated coherent states when the amplitudes of the modulation pulses are Rayleigh-distributed with $\mathrm{\mu}$ = 0.5 mA and $\mathrm{\sigma}$ = 5.6 mA. (a) Output power response of the DFB laser as a function of the bias current (${\rm I}_{{\rm BIAS}}$). (b) Measured optical power spectrum below the threshold level (9 mA) with and without the modulation signal when ${\rm I}_{{\rm DC}-{\rm BIAS}}$ = 6.5 mA. (c) Electrical pulse shape used to drive the GS DFB laser for three different amplitudes. (d) Optical output of the GS DFB laser when it is driven with the pulses shown in Fig. 2(c). (e) Phase-space density measured at the coherent receiver output in arbitrary units (a.u.). Normalized histograms for the amplitude (f) and phase (g) of the measured coherent states. (h) Absolute value of the autocorrelation function of a sequence of ${\rm N}$ = 105 coherent states up to a lag of ${\rm k}$ = 100.
Figure 2(e) shows the phase-space density of the measured coherent states at the receiver, whereas Figs. 2(f) and 2(g) show the normalized histograms of their amplitudes and phases, respectively. From Figs. 2(e)–2(g), it can be observed that the measured coherent states are suitably fitted to a Gaussian distribution in which the coherent state phases are uniformly distributed. Finally, Fig. 2(h) shows the absolute value of the normalized autocorrelation function $|{{\rm r}({\rm k} )} |= \left|{\mathop \sum \nolimits_{{\rm i} = 1}^{{\rm N}-{\rm k}} ({{\rm Z}_{\rm i}-\bar{{\rm Z}}} )({{\rm Z}_{{\rm i} + {\rm k}}-\bar{{\rm Z}}} )/\mathop \sum \nolimits_{{\rm i} = 1}^{{\rm N}-{\rm k}} {({{\rm Z}_{\rm i}-\bar{{\rm Z}}} )}^2} \right|$ of a sequence ${\rm Z}$ of ${\rm N}$ = 105 coherent states up to a lag of ${\rm k}$ = 100. Note that each coherent state in the sequence Z (${\rm Z}_{\rm i} = {\rm X}_{\rm i} + {\rm j}{\rm P}_{\rm i}$) corresponds to a sampled pulse as shown in Fig. 2(d). From Fig. 2(h), it can be observed that $|{{\rm r}({\rm k} )} |$ exhibits a delta-function-like behavior, proving the random nature of the coherent states prepared with the GS DFB laser. As shown in Fig. 2(h), the non-lagged autocorrelation (${\rm k}$ = 0) value is higher than that of the 30-dB level.
3. Gaussian-modulated coherent-state (GMCS) quantum key distribution (QKD)
3.1 Experimental setup
Figure 3 shows the block diagram of the experimental setup. At Alice’s site, the DFB laser characterized in Section 2 was used to generate and prepare the coherent states containing the symbols to be transmitted. To ensure true randomness in both quadratures, the DFB laser was configured to operate in the gain-switching mode by biasing it below its threshold level [40–44]. The threshold level of the DFB laser was 9 mA at 1550 nm (see Fig. 2(a)). To avoid back reflections into the cavity of the DFB laser, a 35-dB optical isolator (ISO) was added. The DFB laser was also temperature-stabilized by a proportional-integral-derivative controller.
Fig. 3. (a) Block diagram of the experimental setup. (Inset) Electrical signal used for the direct modulation of the GMCS transmitter in which the quantum and reference pulses are interleaved in time. (Acronyms) AWG: arbitrary waveform generator, BPDs: balanced photodetectors, BS: beam splitter, CW: continuous-wave, DFB: distributed-feedback, GS: gain-switched, I: DC current source for DFB laser biasing, ISO: optical isolator, LO: local oscillator, MZM: Mach–Zehnder modulator, OBPF: optical bandpass filter, PC: polarization controller, PoM: power meter, RTO: real-time oscilloscope, SMF: single-mode fiber, V: DC voltage source for the MZM biasing, VOA: variable optical attenuator, OH: optical hybrid. (b) Optical pulse measured at the output of the GS DFB laser with and without OBPF.
Figure 3 (Inset) depicts the electrical pulsed signal used for the direct modulation of the GS DFB laser. References and quantum symbols were interleaved in time to ensure high phase recovery accuracy, making it possible to estimate the phase difference between the LO and the GS DFB laser, as well as the phase fluctuations due to the channel fluctuations. Whereas reference pulses are constant in amplitude, the amplitudes of the quantum signal are set to follow a squared Rayleigh random distribution (see Fig. 1(b)), enabling the implementation of the GMCS protocol [5]. According to this protocol, both quadratures (${\rm X}$ and ${\rm P}$) must be independent and modulated following a zero-centered Gaussian random distribution. In particular, a sequence of 104 Rayleigh-distributed pseudo-random values with $\mathrm{\mu}$ = 0.5 mA and $\mathrm{\sigma }$ = 5.6 mA was used to generate and prepare the quantum states. It is worth noting that in a practical application, a separate QRNG would be required to generate those random values. The temporal features of the modulation pulses and the DC biasing condition of the DFB laser were set with the same values used in Section 2, that is, ${\rm PW}$ = 1 ns, ${\rm R}$ = 100 Mpulses/s, and ${\rm I}_{{\rm DC}-{\rm BIAS}}$ = 6.5 mA.
To reduce the spectrum spreading of the DFB laser when it is configured in the gain-switching mode [55], a tunable optical bandpass filter (OBPF) based on fiber Bragg grating with a 3-dB bandwidth of 2.5 GHz was used. This not only reduces the possibility of side-channel attacks [56] but also mitigates the driving current-dependent effects on pulse width, chirped tail, and pulse shape as shown in Fig. 3(b). We note that further work is required to fully assess the contribution of these effects to the security of the implemented protocol. After the OBPF, we can consider that at this point the phase-space distribution of the subset of quantum pulses approximates Fig. 1 (b).
The filter output was split by a 50:50 fiber beam splitter (BS1) into two paths: one to locally measure the symbols sent by Alice using a homodyne detector as described in Section 2, and the other to reduce the signal power to the quantum level and send the generated symbols to the channel. In this path, a Mach–Zehnder modulator (MZM) was exclusively used to control the ratio ($\mathrm{\rho }$) of the intensity of the reference pulses with respect to that of the quantum pulses by adjusting the related driving electrical signal ratio. The MZM was then fed with a two-amplitude pulsed signal in which the pulse rate and the pulse width were 100 Mpulses/s and 2 ns, respectively. The pulse width of 2 ns was chosen not to modify the shape of the pulses coming from the DFB laser. To guarantee suitable accuracy of the phase recovery process, $\rho $ was set to 115 and the MZM was biased at its null transmission point [57]. Note that the use of the MZM can be avoided by either increasing the number of reference pulses or using a phase recovery process based on the maximization of the correlation between the symbols disclosed by Alice and Bob [50]. The former approach reduces the effective transmission rate, whereas the latter increases the complexity of the digital signal processing.
After the MZM, Alice’s modulation variance (${\rm V}_{{\rm mod}}$) was adjusted using a variable optical attenuator (VOA). ${\rm V}_{{\rm mod}}$ can be estimated as double of the mean photon number $\langle {\rm n}\rangle $ measured at Alice’s output. To measure $\langle {\rm n}\rangle $ and calculate the ${\rm V}_{{\rm mod}}$ in real-time, a 90:10 fiber beam splitter (BS2) was employed to send 90% of the light to an optical power meter (PoM). The remaining 10% was launched into an 11-km single-mode fiber (SMF) with a loss coefficient of 0.22 dB/km. The GS DFB laser and the MZM were both driven by an arbitrary waveform generator (AWG) with a 3-dB 1-GHz electrical bandwidth and set as 2.5 GSa/s. The electrical signal of each AWG output was amplified using a 20-dB gain driver amplifier.
At Bob’s site, the LO was a tunable CW external cavity laser with a linewidth of 10 kHz. It was biased to emit 48 mW at 1550 nm. Moreover, 10% of the LO output power was sent to Alice to perform detection using a 90:10 fiber beam splitter (BS3). The other output of BS3 was connected to Bob’s homodyne detector. In practical applications, instead of transmitting the LO from Bob to Alice, it would be more appropriate to generate Alice’s LO locally [49]. The electrical bandwidth of the BPDs was 350 MHz. A 10-GSa/s real-time oscilloscope (RTO) was employed to digitize the outputs of the BPDs. The electrical bandwidth of each RTO channel was fixed at 250 MHz. In this configuration, an average clearance value of 14.9 dB was obtained. At Bob’s site, the clearance, in decibel units, is calculated as the difference between the shot noise power and the electronic noise power. For Alice, since the measurement of the coherent states was performed at a classical level, the effects associated with shot and electronic noise were negligible. The signal polarization was manually adjusted to that of the LO using a polarization controller (PC) at Alice’s and Bob’s input. Finally, to prevent Trojan horse attacks, a 35-dB ISO was placed at Alice’s and Bob’s output [58].
3.2 Digital signal processing
Digital signal processing (DSP) was performed offline, comprising mainly four processes. The first process was the downsampling of Alice’s and Bob’s signals (${\rm Z}^{\rm Y} = {\rm X}^{\rm Y} + {\rm j}{\rm P}^{\rm Y}$, with ${\rm Y} = \{{{\rm Alice},\; {\rm Bob}} \}$). The samples per symbol (state) were subsequently reduced to one by periodically choosing those samples that maximize the energy of the resulting signal. The second process was the phase recovery of quantum states. Since the phases of the optical pulses leaving the GS DFB laser are random, reference pulses with a specific phase cannot be established. In this case, a phase recovery process based on a differential approach is a better solution. Accordingly, redefining ${\rm Z}^{\rm Y}$ as the set of reference and quantum states obtained for Alice or Bob after the downsampling process, we obtained ${\rm Z}^{\rm Y} = \{{{\rm R}_{\rm i}^{\rm Y} ,{\rm Q}_{\rm i}^{\rm Y} ,\; {\rm R}_{{\rm i} + 1}^{\rm Y} ,{\rm Q}_{{\rm i} + 1}^{\rm Y} ,{\; } \ldots } \}$, where ${\rm R}_{\rm i}^{\rm Y} $ and ${\rm Q}_{\rm i}^{\rm Y} $, complex numbers, are the i-th reference and quantum states, respectively. The corrected i-th quantum state ($\hat{{\rm Q}}_{\rm i}^{\rm Y} )$ can be calculated using the phase information of the preceding reference state, as indicated in Eq. (1). After this process, the references can be discarded. The third process was frame synchronization. The time offset was removed by performing a cross-correlation between the sequence of coherent states measured by Alice and Bob.
The last process involved in the DSP was the estimation of excess noise, channel transmittance, and the secret key rate. This parameter estimation stage required the calibration of the shot noise variance (${\rm N}_{\rm o}$). ${\rm N}_{\rm o}$ can be estimated using Eq. (2) [59], where $\hat{{\rm N}}_{\rm o}$ and $\hat{{\rm V}}_{{\rm elec}}$ are the variances of the total detection noise and electronic noise, respectively, in V2. $\hat{{\rm N}}_{\rm o}$ can be measured as the average of the output-voltage variance of Bob’s detectors when only the LO is switched on. Similarly, $\hat{{\rm V}}_{{\rm elec}}$ can be measured when the signal and LO are both switched off.
Equation (3) defines the excess noise at Bob ($\mathrm{\varepsilon}$) in shot-noise units (SNU) [21,60]. There, $\mathrm{\nu}_{{\rm elec}} = \hat{{\rm V}}_{{\rm elec}}/{\rm N}_{\rm o}$, and ${\rm V}_{{\rm Bob}|{\rm Alice}}$ is the conditional variance and can be calculated as indicated in Eq. (4) [21,60], in which ${\mathrm{\eta}}$ is the detection efficiency; ${\rm q}_{{\rm Alice}} = {\; }\{{{\rm{\Re}}\{{{\rm Q}_{\rm i}^{{\rm Alice}} } \},{\; \; }{\rm {\frak J}}\{{{\rm Q}_{\rm i}^{{\rm Alice}} } \}} \}$ and ${\rm q}_{{\rm Bob}} = {\; }\{{{\rm{\Re}}\{{{\rm Q}_{\rm i}^{{\rm Bob}} } \},{\; \; }{\rm {\frak J}}\{{{\rm Q}_{\rm i}^{{\rm Bob}} } \}} \}$ are Alice’s and Bob’s measured coherent states, respectively; and ${\rm T}$ is the channel transmittance that can be calculated using Eq. (5), where $q_{Alice}q_{Bob}$ is the inner product of Alice’s and Bob’s measured states [21,60]. Eq. (5) corresponds to the case in which an eavesdropper is in charge of controlling the channel loss.
Finally, the secret key rate (${\rm SKR}$) was estimated considering the asymptotic limit and reverse reconciliation. Eq. (6) defines the ${\rm SKR}$ in bits per second (bps), where $\mathrm{\beta}$ is the reconciliation efficiency [15], ${\rm I}_{{\rm AB}}$ is the mutual information between Alice and Bob posed in Eq. (7) [21,60], ${\mathrm{\chi}}_{{\rm BE}}$ is the Holevo bound, and ${\rm R}_{{\rm eff}}$ is the effective quantum pulse rate. In our case, ${\rm R}_{{\rm eff}} = \; {\rm R}/2$.
3.3 Experimental results
Phase correction is required to compensate for the phase errors between Alice and Bob. In a practical fiber link, the phase errors are caused by the fluctuation of the transmission optical path length that causes the optical wavefront to reach the detector of Alice and Bob changing with time, resulting in a phase mismatch that hinders the extraction of the secret key. Considering that the optical path changes slowly during transmission (< 1 kHz) and the transmission rate is relatively high (> 1 MHz), the relative phase drift between two consecutive symbols can be assumed constant. Therefore, subtracting the phase of a reference to its precedent quantum state, as indicated in Eq. (1), compensates for any phase difference between Alice and Bob owing to the optical path. Moreover, since the phase recovery process indicated in Eq. (1) follows a differential approach, the phase errors caused by the frequency mismatch between the signal laser and LO can also be compensated. Figure 4 shows the comparison of the phase difference of 100 states measured at Alice’s and Bob’s site without (green line) and with (blue line) the proposed phase recovery process for 11-km SMF and ${\rm V}_{{\rm mod}}$ = 30 SNU. From Fig. 4, it can be seen that the phase difference between the states measured at Bob’s site and Alice’s site is close to zero only when the phase recovery process is applied. Fig. 5 shows a measurement of the relationship of the states measured at Alice and Bob’s site for the ${\rm X}$ quadrature (105 states) after the phase recovery for three different ${\rm V}_{{\rm mod}}$ values considering 11-km SMF. In particular, Fig. 5(a) corresponds to ${\rm V}_{{\rm mod}}$ = 30 SNU, Fig. 5(b) to ${\rm V}_{{\rm mod}}$ = 3.35 SNU, and Fig. 5(c) to ${\rm V}_{{\rm mod}}$ = 1.50 SNU. In Fig. 5(c), almost no correlation can be observed between Alice’s and Bob’s data.
Fig. 4. Comparison of the phase difference ($\Delta \emptyset $) between Alice and Bob without and with the proposed phase recovery process for 100 measured states (symbols). These symbols are measured using the 11-km SMF and = 30 SNU.
Fig. 5. Obtained scatter plot of 105 ${\rm X}$-quadrature states measured at Alice’s and Bob’s sites for different ${\rm V}_{{\rm mod}}$ values: (a) 30 SNU, (b) 3.35 SNU, and (c) 1.50 SNU. Signals are measured after propagation trough 11-km SMF and processed using the proposed phase recovery process.
The parameters used for QKD transmission are summarized in Table 1. Fig. 6 shows the experimental results for the GMCS CV-QKD system shown in Fig. 3 in which the coherent states are prepared by a GS DFB laser, and the transmission is over an 11-km SMF. In particular, Fig. 6(a) and (b) present 10 consecutive measurements of $\mathrm{\varepsilon}$ and $ {\rm T}$ with ${\rm SKR}$, respectively, obtained over 20 min. Each measurement featured a block size of 105 coherent states. Moreover, the values of $\mathrm{\varepsilon}$, ${\rm T}$, and ${\rm SKR}$ were independently estimated for each block of coherent states. The calibration of ${\rm N}_{\rm o}$ was manually performed before each measurement. The mean values of $\mathrm{\varepsilon}$ (Eq. (3)) and ${\rm T}$ (Eq. (5)) were 36 mSNU and 0.581, respectively. From Fig. 6(b), it can be observed that the ${\rm T}$ parameter was stable with a relative standard deviation of 0.4%. The asymptotic ${\rm SKR}$ was estimated using Eq. (6), accounting for the values obtained for $\mathrm{\varepsilon}$ and ${\rm T}$. A mean ${\rm SKR}$ value of 2.63 Mbps was achieved, as observed in Fig. 6(b). Finally, Fig. 6(c) shows a simulation of ${\rm SKR}$ versus the transmission distance, using the experimental parameters estimated above. From Fig. 6(c), a positive ${\rm SKR}$ could be obtained up to a distance of 20.5 km. This simulation has considered that $\mathrm{\varepsilon}$ remains constant at longer distances.
Fig. 6. Experimental results of (a) excess noise ($\mathrm{\varepsilon}$), and (b) channel transmittance (${\rm T}$) with secret key rate (${\rm SKR}$) for 10 measurements acquired over 20 min through an 11-km SMF. Each measurement corresponds to a block size of 105 coherent states. (c) Simulation of the ${\rm SKR}$ as a function of link distance in the asymptotic regime.
Table 1. Transmission parameter summary
4. Conclusions
In this study, we demonstrated the use of a directly modulated DFB laser for the preparation of coherent states required for the GMCS CV-QKD protocol. To ensure true randomness in both quadratures and provide high rates, the DFB laser was configured to operate in gain-switching mode. An asymptotic secret key rate value of 2.63 Mbps over an 11-km transmission distance was demonstrated experimentally. A phase recovery scheme based on a differential approach has also been proposed and implemented. The proposed scheme allows the generation of optical pulses with random phases without the use of additional phase modulators. For its simplicity, compactness, low cost, and high performance over short fiber links, the proposed transmitter scheme is particularly suitable for metropolitan CV-QKD networks.
Funding
Horizon 2020 Framework Programme (820466, 820405); H2020 Marie Skłodowska-Curie Actions (713729); MICIIN with funding from European Union NextGenerationEU (PRTR-C17.I1); Generalitat de Catalunya; CEX2019-000910-S (MCIN/AEI/10.13039/501100011033); Fundació Cellex; Fundació Mir-Puig; Generalitat de Catalunya through CERCA.
Disclosures
V. Pruneri, S. Etcheverry, and R. Valivarthi are co-inventors of a patent application US2022085895 (A1), related to the content of this paper.
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.
References
1. S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. Shamsul Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, “Advances in quantum cryptography,” Adv. Opt. Photonics 12(4), 1012–1236 (2020). [CrossRef]
2. F. Xu, X. Ma, Q. Zhang, H. K. Lo, and J. W. Pan, “Secure quantum key distribution with realistic devices,” Rev. Mod. Phys. 92(2), 025002 (2020). [CrossRef]
3. H. K. Lo and H. F. Chau, “Unconditional security of quantum key distribution over arbitrarily long distances,” Science 283(5410), 2050–2056 (1999). [CrossRef]
4. G. Zhang, J. Y. Haw, H. Cai, F. Xu, S. M. Assad, J. F. Fitzsimons, X. Zhou, Y. Zhang, S. Yu, J. Wu, W. Ser, L. C. Kwek, and A. Q. Liu, “An integrated silicon photonic chip platform for continuous-variable quantum key distribution,” Nat. Photonics 13(12), 839–842 (2019). [CrossRef]
5. F. Grosshans and P. Grangier, “Continuous Variable Quantum Cryptography Using Coherent States,” Phys. Rev. Lett. 88(5), 057902 (2002). [CrossRef]
6. M. Navascues, F. Grosshans, and A. Acin, “Optimality of Gaussian Attacks in Continuous-Variable Quantum Cryptography,” Phys. Rev. Lett. 97(19), 190502 (2006). [CrossRef]
7. R. Renner and J. I. Cirac, “de Finetti Representation Theorem for Infinite-Dimensional Quantum Systems and Applications to Quantum Cryptography,” Phys. Rev. Lett. 102(11), 110504 (2009). [CrossRef]
8. A. Leverrier, F. Grosshans, and P. Grangier, “Finite-size analysis of a continuous-variable quantum key distribution,” Phys. Rev. A 81(6), 062343 (2010). [CrossRef]
9. A. Leverrier, “Composable Security Proof for Continuous-Variable Quantum Key Distribution with Coherent States,” Phys. Rev. Lett. 114(7), 070501 (2015). [CrossRef]
10. A. Leverrier, “Security of continuous-variable quantum key distribution via a Gaussian de Finetti reduction,” Phys. Rev. Lett. 118(20), 200501 (2017). [CrossRef]
11. F. Furrer, T. Franz, M. Berta, A. Leverrier, V. B. Scholz, M. Tomamichel, and R. F. Werner, “Continuous variable quantum key distribution: finite-key analysis of composable security against coherent attacks,” Phys. Rev. Lett. 109(10), 100502 (2012). [CrossRef]
12. A. Leverrier, R. Garcia-Patron, R. Renner, and N. J. Cerf, “Security of continuous-variable quantum key distribution against general attacks,” Phys. Rev. Lett. 110(3), 030502 (2013). [CrossRef]
13. C. Lupo, C. Ottaviani, P. Papanastasiou, and S. Pirandola, “Continuous-variable measurement-device-independent quantum key distribution: composable security against coherent attacks,” Phys. Rev. A 97(5), 052327 (2018). [CrossRef]
14. J. Lodewyck, M. Bloch, R. García-Patrón, S. Fossier, E. Karpov, E. Diamanti, T. Debuisschert, N. J. Cerf, R. Tualle-Brouri, S. W. McLaughlin, and P. Grangier, “Quantum key distribution over 25 km with an all-fiber continuous-variable system,” Phys. Rev. A 76(4), 042305 (2007). [CrossRef]
15. P. Jouguet, S. Kunz-Jacques, A. Leverrier, P. Grangier, and E. Diamanti, “Experimental demonstration of long-distance continuous-variable quantum key distribution,” Nat. Photonics 7(5), 378–381 (2013). [CrossRef]
16. D. Huang, P. Huang, D. Lin, and G. Zeng, “Long-distance continuous-variable quantum key distribution by controlling excess noise,” Sci. Rep. 6(1), 19201 (2016). [CrossRef]
17. P. Jouguet, S. Kunz-Jacques, T. Debuisschert, S. Fossier, E. Diamanti, R. Alléaume, R. Tualle-Brouri, P. Grangier, A. Leverrier, P. Pache, and P. Painchault, “Field test of classical symmetric encryption with continuous variables quantum key distribution,” Opt. Express 20(13), 14030 (2012). [CrossRef]
18. D. Pan, S. Xin Ng, D. Ruan, L. Yin, G. Long, and L. Hanzo, “Simultaneous two-way classical communication and measurement-device-independent quantum key distribution with coherent states,” Phys. Rev. A 101, 012343 (2020). [CrossRef]
19. F. Karinou, H. H. Brunner, C.-H. Fred Fung, L. C. Comandar, S. Bettelli, D. Hillerkuss, M. Kuschnerov, S. Mikroulis, D. Wang, C. Xie, M. Peev, and A. Poppe, “Toward the integration of CV quantum key distribution in deployed optical networks,” IEEE Photonics Technol. Lett. 30(7), 650–653 (2018). [CrossRef]
20. Y. Zhang, Z. Li, Z. Chen, et al., “Continuous-variable QKD over 50 km commercial fiber,” Quantum Sci. Technol. 4(3), 035006 (2019). [CrossRef]
21. R. Valivarthi, S. Etcheverry, J. Aldama, F. Zwiehoff, and V. Pruneri, “Plug-and-play continuous-variable quantum key distribution for metropolitan networks,” Opt. Express 28(10), 14547–14559 (2020). [CrossRef]
22. I. H. Lopez Grande, S. Etcheverry, J. Aldama, S. Ghasemi, D. Nolan, and V. Pruneri, “Adaptable transmitter for discrete and continuous variable quantum key distribution,” Opt. Express 29(10), 14815–14827 (2021). [CrossRef]
23. Y. Zhang, Z. Chen, S. Pirandola, X. Wang, C. Zhou, B. Chu, Y. Zhao, B. Xu, S. Yu, and H. Guo, “Long-Distance Continuous-Variable Quantum Key Distribution over 202.81 km of Fiber,” Phys. Rev. Lett. 125(1), 010502 (2020). [CrossRef]
24. F. Roumestan, A. Ghazisaeidi, J. Renaudier, L. T. Vidarte, E. Diamanti, and P. Grangier, “High-Rate Continuous Variable Quantum Key Distribution Based on Probabilistically Shaped 64 and 256-QAM,” 2021 European Conference on Optical Communication (ECOC). [CrossRef]
25. D. Huang, P. Huang, H. Li, T. Wang, Y. Zhou, and G. Zeng, “Field demonstration of a continuous-variable quantum key distribution network,” Opt. Lett. 41(15), 3511 (2016).
26. A. Aguado, V. Lopez, D. Lopez, M. Peev, A. Poppe, A. Pastor, J. Folgueira, and V. Martin, “The Engineering of Software-Defined Quantum Key Distribution Networks,” IEEE Commun. Mag. 57(7), 20–26 (2019). [CrossRef]
27. S. Sarmiento, S. Etcheverry, J. Aldama, I. H. Lopez, L. T. Vidarte, G. B. Xavier, D. A. Nolan, J. S. Stone, M. J. Li, D. Loeber, and V. Pruneri, “Continuous-variable quantum key distribution over a 15 km multi-core fiber,” New J. Phys. 24(6), 063011 (2022). [CrossRef]
28. V. Mannalath, S. Mishra, and A. Pathak, “A Comprehensive Review of Quantum Random Number Generators: Concepts, Classification and the Origin of Randomness,” arXiv, arXiv.2203.00261 (2022). [CrossRef]
29. T. Jennewein, U. Achleitner, G. Weihs, H. Weinfurter, and A. Zeilinger, “A fast and compact quantum random number generator,” Rev. Sci. Instrum. 71(4), 1675–1680 (2000). [CrossRef]
30. Q. Yan, B. Zhao, Q. Liao, and N. Zhou, “Multi-bit quantum random number generation by measuring positions of arrival photons,” Rev. Sci. Instrum. 85(10), 103116 (2014). [CrossRef]
31. M. Stipčević and B. M. Rogina, “Quantum random number generator based on photonic emission in semiconductors,” Rev. Sci. Instrum. 78(4), 045104 (2007). [CrossRef]
32. H. Fürst, H. Weier, S. Nauerth, D. G. Marangon, C. Kurtsiefer, and H. Weinfurter, “High speed optical quantum random number generation,” Opt. Express 18(12), 13029–13037 (2010). [CrossRef]
33. W. Wei and H. Guo, “Bias-free true random-number generator,” Opt. Lett. 34(12), 1876–1878 (2009). [CrossRef]
34. T. Durt, C. Belmonte, L.-P. Lamoureux, K. Panajotov, F. Van den Berghe, and H. Thienpont, “Fast quantum-optical random-number generators,” Phys. Rev. A 87(2), 022339 (2013). [CrossRef]
35. A. Argyris, E. Pikasis, S. Deligiannidis, and D. Syvridis, “Sub-tb/s physical random bit generators based on direct detection of amplified spontaneous emission signals,” J. Lightwave Technol. 30(9), 1329–1334 (2012). [CrossRef]
36. Y. Shen, L. Tian, and H. Zou, “Practical quantum random number generator based on measuring the shot noise of vacuum states,” Phys. Rev. A 81(6), 063814 (2010). [CrossRef]
37. B. Qi, Y.-M. Chi, H.-K. Lo, and L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. Lett. 35(3), 312–314 (2010). [CrossRef]
38. F. Xu, B. Qi, X. Ma, H. Xu, H. Zheng, and H.-K. Lo, “Ultrafast quantum random number generation based on quantum phase fluctuations,” Opt. Express 20(11), 12366–12377 (2012). [CrossRef]
39. J. R. Alvarez, S. Sarmiento, J. A. Lazaro, J. M. Gene, and J. P. Torres, “Random number generation by coherent detection of quantum phase noise,” Opt. Express 28(4), 5538–5547 (2020). [CrossRef]
40. M. Jofre, M. Curty, F. Steinlechner, G. Anzolin, J. Torres, M. Mitchell, and V. Pruneri, “True random numbers from amplified quantum vacuum,” Opt. Express 19(21), 20665–20672 (2011). [CrossRef]
41. Y. Q. Nie, L. Huang, Y. Liu, F. Payne, and J. Zhang, “The generation of 68 Gbps quantum random number by measuring laser phase fluctuations,” Rev. Sci. Instrum. 86(6), 063105 (2015). [CrossRef]
42. C. Abellán, W. Amaya, M. Jofre, M. Curty, A. Acín, J. Capmany, V. Pruneri, and M. Mitchell, “Ultra-fast quantum randomness generation by accelerated phase diffusion in a pulsed laser diode,” Opt. Express 22(2), 1645–1654 (2014). [CrossRef]
43. Z. Yuan, M. Lucamarini, J. Dynes, B. Fröhlich, A. Plews, and A. Shields, “Robust random number generation using steady-state emission of gain-switched laser diodes,” Appl. Phys. Lett. 104(26), 261112 (2014). [CrossRef]
44. R. Shakhovoy, V. Sharoglazova, A. Udaltsov, A. Duplinskiy, V. Kurochkin, and Y. Kurochkin, “Influence of chirp, jitter, and relaxation oscillations on probabilistic properties of laser pulse interference,” IEEE J. Quantum Electron. 57(2), 1–7 (2021). [CrossRef]
45. C. Abellan, W. Amaya, D. Domenech, P. Muñoz, J. Capmany, S. Longhi, M. W. Mitchell, and V. Pruneri, “Quantum entropy source on an InP photonic integrated circuit for random number generation,” Optica 3(9), 989–994 (2016). [CrossRef]
46. T. Roger, T. Paraiso, I. De Marco, D. G. Marangon, Z. Yuan, and A. J. Shields, “Real-time interferometric quantum random number generation on chip,” J. Opt. Soc. Am. B 36(3), B137–B142 (2019). [CrossRef]
47. B. Qi, P. G. Evans, and W. P. Grice, “Passive state preparation in the Gaussian-modulated coherent-states quantum key distribution,” Phys. Rev. A 97(1), 012317 (2018). [CrossRef]
48. B. Qi, H. Gunther, P. G. Evans, B. P. Williams, R. M. Camacho, and N. A. Peters, “Experimental Passive-State Preparation for Continuous-Variable Quantum Communications,” Phys. Rev. Appl. 13(5), 054065 (2020). [CrossRef]
49. X. Wu, Y. Wang, Y. Guo, H. Zhong, and D. Huang, “Passive continuous-variable quantum key distribution using a locally generated local oscillator,” Phys. Rev. A 103(3), 032604 (2021). [CrossRef]
50. P. Huang, T. Wang, R. Chen, P. Wang, Y. Zhou, and G. Zeng, “Experimental continuous-variable quantum key distribution using a thermal source,” New J. Phys. 23(11), 113028 (2021). [CrossRef]
51. I. N. Cano, J. C. Velásquez, and J. Prat, “7.5 Gb/s direct DFB phase modulation with 8-DPSK for 6.25 GHz spaced coherent UDWDM-PONs,” Optical Fiber Comm. Conf., 2016, pp. 1–3.
52. J. C. Velásquez, I. N. Cano, V. Polo, and J. Prat, “Direct beat phase modulated DFB for flexible 1.25–5 Gb/s coherent UDWDM-PONs,” Optical Fiber Comm. Conf., 2017, pp. 1–3.
53. I. N. Cano, A. Lerín, and J. Prat, “DQPSK Directly Phase Modulated DFB for Flexible Coherent UDWDM-PONs,” IEEE Photonics Technol. Lett. 28(1), 35–38 (2016). [CrossRef]
54. K. Kikuchi, “Fundamentals of Coherent Optical Fiber Communications,” J. Lightwave Technol. 34(1), 157–179 (2016). [CrossRef]
55. H. T. G. Bookey and A. K. Kar, “Characterisation and optimisation of a dual-channel picosecond gain-switched DFB laser system for use as a pump–probe source,” Opt. Commun. 248(1-3), 229–239 (2005). [CrossRef]
56. S. L. Braunstein and S. Pirandola, “Side-Channel-Free Quantum Key Distribution,” Phys. Rev. Lett. 108(13), 130502 (2012). [CrossRef]
57. D. B. S. Soh, C. Brif, P. J. Coles, N. Lütkenhaus, R. M. Camacho, J. Urayama, and M. Sarovar, “Self-Referenced Continuous-Variable Quantum Key Distribution Protocol,” Phys. Rev. X 5(4), 041010 (2015). [CrossRef]
58. N. Gisin, S. Fasel, B. Kraus, H. Zbinden, and G. Ribordy, “Trojan-horse attacks on quantum-key-distribution systems,” Phys. Rev. A 73(2), 022320 (2006). [CrossRef]
59. M. Zou, Y. Mao, and T. Y. Chen, “Rigorous calibration of homodyne detection efficiency for continuous-variable quantum key distribution,” Opt. Express 30(13), 22788–22797 (2022). [CrossRef]
60. F. Laudenbach, C. Pacher, C. H. Fred Fung, A. Poppe, M. Peev, B. Schrenk, M. Hentschel, P. Walther, and H. Hübel, “Continuous-Variable Quantum Key Distribution with Gaussian Modulation—The Theory of Practical Implementations,” Adv. Quantum Technol. 1(1), 1800011 (2018). [CrossRef]
