Quantum key distribution (QKD) serves a method for securely exchanging information between remote parties, providing unconditional security, even in the presence of an eavesdropper^[1]. As QKD technology evolves, secure communication networks have been employed in practice. Currently, QKD systems based on the BB84 protocol with the decoy method exhibit stable key generation in optical fiber networks^{[2, 3]}. Although unconditional security has been established, there are still numerous obstacles, such as lack of industry standards, security certification and imperfections in practical devices, systems, and networks^[4−6]. The decoy method exploits attenuated laser pulses with typically three different intensities to fend off photon number splitting attacks^{[6, 7]}. When the intensity of the optical pulses deviate from their intended values, the system may underestimate the eavesdropper’s information and thus inaccurately determine the final secure key rate (SKR)^[8−10]. Consequently, the accuracy of the output pulses intensities is crucial for the security of QKD. The intensity deviation comprises drift and fluctuation, which respectively correspond to the variation of mean value and the noise of each individual pulse. The drift can easily be compensated for with optical powermeter and an attenuator, but the intensity fluctuation is related to the correlation between adjacent pulses, potentially opening a security loophole^[11−14].

QKD systems are typically equipped with a gain-switched semiconductor lasers (GSSLs) as the light source. This simplifies the construction of the transmitter and naturally provides phase randomization between encoded optical pulses required by the BB84 protocol^[15]. Any inter-pulse phase correlation increases the distinguishability of non-orthogonal bases, and thus more bits need to be sacrificed in the privacy amplification stage to maintain security^[16]. To ensure phase randomization, the GSSL is excited from below the threshold and driven by a strong current pulse for each pulse generation. However, this approach lacks scientific rigor for security quantification. For each QKD setup, one needs to experimentally determine whether the working state of the GSSL meets the requirements. With the rapid development of high-speed QKD systems, it is necessary for the lasers to be pulsed at GHz-repetition rates, which implies a large and fast variation of carrier density. During the process of stimulated radiation, laser oscillation exhibits nonlinear characteristics, and small excitation currents noise may amplify, leading to intensity fluctuations of the output light pulse. Therefore, it is urgent to investigate the intensity fluctuation in the GSSL at GHz repetition. In previous work^[13], large intensity fluctuations and strong negative correlation between adjacent pulses at 1.25 GHz were observed. However, only intensity correlation between adjacent pulses was considered. Thus, the high-order intensity correlation at higher repetition rates needs to be explored.

In this work, we conducted a detailed measurement and analysis of a GSSL at a repetition rate of 2.5 GHz. We measured the dependence of intensity fluctuation and correlation distribution on driving currents. Through rate equation simulation of GSSL, we analyzed the mechanism of the evolution in intensity correlation distribution. Finally, we considered the influence of intensity fluctuation on secure key rate.

Fig. 1(a) presents the experimental setup. A distributed feedback (DFB) laserdiode (Gooch & Housego, AA0701), boasting a 10 GHz modulation bandwidth, is utilized. The auto-correlation function of optical pulses under 1.25 GHz modulation is shown in Fig. S1(a) which is consistent with previous report^[13]. Fig. S1(b) presents the LIV curve of the DFB laser. The threshold current is 12.5 mA and the working voltage is approximately 0.93 V. Figs. S1(c) and S1(d) show the output spectra of the laser below and above the threshold, respectively. As the driving current exceeds the threshold, the laser transitions from the spontaneous emission process to the stimulated emission process. For the purpose of investigating the higher-order intensity correlation effects, the DFB laserdiode is driven by a 2.5 GHz pulse voltage (V_pp = 1.6 V) with a duration of 100 ps and a DC current (J_DC). An optical attenuator (ATT) is employed to control the optical power, thus preventing saturation of the photodetector (PD) with a bandwidth of 10 GHz. The waveform is recorded by a 16-GHz bandwidth oscilloscope (OSC). We explore the statistical distribution of pulse intensity through the integral area of optical pulse. The intensity fluctuation is measured by the standard deviation (σ) of intensity distribution normalized by the mean value (μ). Here, four adjacent pulses are marked as "T", "X", "Y", and "Z", respectively. It's just for the convenience of analyzing the intensity correlation among them. Essentially, there is no difference among them. To circumvent the effect of pulse jitters, 10 000 samples are collected for statistical analysis.

The temporal behavior of the laser pulses is depicted in Fig. 1(b). The intensity fluctuated strongly at a lower driving current (12 mA). As the driving currents increased, optical pulses gradually stabilized. To quantitatively investigate the intensity fluctuation, we analyzed the statistical distribution of pulse intensity as shown in Fig. 1(c). Here, we employed the normalized standard deviation (σ/μ) as the intensity fluctuation. When the driving currents are greater than 17 mA, the fluctuation is less than 0.04. However, the fluctuation of the electronic pulse σAC/〈JAC〉 from PG is about 0.009. This indicates that the electrical signal noise is indirectly amplified during the conversion to optical pulse output.

The correlation distribution of intensity between adjacent pulses under different driving currents is shown in Fig. 2. Here, the X-axis corresponds to the intensity of pulse "T", while the Y-axis corresponds to the intensity of pulses “X”, “Y”, and “Z”, respectively. Fig. 2(a) shows the intensity correlation distribution of adjacent pulse "TX". When the driving current is around 9.5 mA, intensity correlation distribution is manifested as a slightly right oblique elliptical shape, indicating a weak positive correlation between pulse "T" and pulse "X". As the driving current increases to 10.7 mA, the axis of symmetry of the ellipse is symmetrical to the coordinate axis, suggesting that the linear correlation coefficient of "TX" is zero. Continuing to increase the driving current to 12.9 mA, the intensity correlation exhibits a significant left oblique elliptical shape, indicating that there is a significant negative correlation between adjacent pulses. Fig. 2(b) shows the intensity correlation distribution between pulse "T" and pulse "Y". Similar to adjacent pulse "TX", intensity correlation has a significant dependence on the driving currents. When the driving current is below 11 mA, the intensity correlation of "TY" is consistent with "TX". However, it exhibits a clear positive correlation when the driving current is 12.9 mA. Fig. 2(c) shows the intensity correlation between pulse "T" and pulse "Z". Almost identical to the trend of adjacent pulse "TX" in Fig. 2(a), the correlation characteristic is slightly weaker when the driving current is 12.9 mA. When the driving currents increases to 16.5 mA, the intensity correlation distribution all behave in a circular shape, which means intensity correlation disappears.

In Fig. 2(d), we used autocorrelation functions to characterize the intensity correlation.

1R(τ)=E[(Xt−μ)(Xt+τ−μ)]σ2.

Here, X represents the sequence of pulse intensity. To evaluate the correlation between adjacent pulses, τ is set to 1, 2, and 3 to correspond to "T-X", "T-Y", and "T-Z", respectively. E(X) and σ represent the mean and standard deviation of X, respectively. Fig. 2(d) presents the dependence of auto-correlation function on driving currents. The correlation coefficient of "TX" and "TZ" changes from positive to negative and ultimately disappears. The correlation coefficient of "TY" is opposite. It starts with weak positive correlation, decreases to zero, reaches maximum positive correlation, and finally disappears. The experimental results demonstrate that the output characteristics of the laser strongly depend on the driving currents. The QKD experiment requires the laser to operate in the optimal state where the intensity fluctuation of the output light pulse should be as weak as possible, and there should be no correlation between adjacent pulses.

To understand the physical mechanism, we qualitatively analyze the working state of a GSSL using rate equations,

2dIdt=Γg(N−Ng)−Iτp+ΓβspNτN,

3dNdt=Λ−g(N−Ng)I−NτN.

Here g is differential gain coefficient (g = 2 × 10⁻⁶ cm³·s⁻¹ ), N_g transparent carrier density (N_g = 1.5 × 10¹⁸ cm⁻³), Γ confinement factor (Γ = 0.2), and parameters τ(p,N) are photon and carrier lifetimes (τp = 5 ps, τN = 1.2 ns), respectively. βsp is spontaneous emission factor (βsp = 1.0 × 10⁻⁵), which represents the spontaneous emission contribution to the lasing mode. Pumping rate Λ (cm⁻³·s⁻¹) is proportional to the sum of DC bias and AC pulse currents (Λ=ΛDC+ΛAC∝JDC+JAC).

We simulated the behavior of carrier density and photon density in GSSL excited at 2.5 GHz repetition rate. The electrical pulse was assumed to be rectangular shape of amplitude J_AC = 0.12, ranging from −0.06 to 0.06, which is three times the threshold current, and a duration of 100 ps to imitate the experimental conditions. Fig. 3 shows the temporal behavior of carrier density and photon density from a GSSL. The number of solutions of the rate equation determines the output state of the laser pulse. For low excitation (J_DC = 0.065) as shown in Fig. 3(a), the equations have only one steady-state solution and the carrier density is below the threshold. This indicates that the laser operates in the spontaneous emission. When the driving current increases to 0.072 as shown in Fig. 3(b), the equations have two steady-state solutions and the carrier density exceeds the threshold. The laser begins to enter the stimulated radiation, but cannot maintain a stable state. There is an overshooting in adjacent carrier density, resulting in different heights of front and rear pulses exceeding the threshold, ultimately generating alternating light pulses. As the driving current increases to 0.074, the equations have four steady-state solutions, which reveals that intensity correlation does not only occur between adjacent pulse ("TX").

The effective lifetime of the carrier (τNeff∝−g(N−Ng)I) depends on the output optical intensity in Eq. (2). As the driving current increases, the effective carrier lifetime further decreases. When the driving current increases to 0.085, the rate equation returns to two steady-state solutions. As the driving currents further increase to 0.12, it can be observed that the rate equations have only one steady-state solution and the pulse width narrows. A high driving currents will significantly reduce the carrier lifetime, resulting in a narrower pulse width. Here, the simulation assumed there is no noise in the J_DC driving currents. In order to approach the experimental condition, we introduced a noise term N(t) to simulate the fluctuation of electrical pulses in the experiment as Γ(t)=ΓDC+ΓAC(t)+N(t) where represents white Gaussian noise with zero mean value. The standard deviation σAC was set to be 0.05J_AC. We calculated the intensity fluctuations and correlation of output optical pulses under different DC driving currents. When there are two steady-state solution, the result is similar to Nakata’s work^[13]. The unstable lasing appears slight below Γ_DC = 0.07. The intensity fluctuation reached as high as 1 for low excitation currents, when the autocorrelation function shows the largest negative correlation. However, when there are four steady-state solution, there is a significant difference in pulse intensity, so we cannot accurately define the intensity fluctuation. Thus, we only considered intensity correlation distribution here. The calculated intensity correlation reproduced the experimental observations quite well in Fig. 3(f). When the DC driving current is below 0.07, the GSSL is mainly dominated by spontaneous emission. Here, pulse "T" is taken as the reference target pulse. If its carrier concentration is below the threshold, and then due to the carrier concentration accumulation effect, the carrier concentration of the adjacent pulses "X" became higher, exhibiting a positive correlation characteristic. As the currents increases, the laser gradually was dominated by stimulated radiation. The higher the carrier concentration of pulse "T", the lower the initial carrier concentration of pulse "X" due to overshooting, showing a negative correlation. Similarly, this results in a higher carrier concentration for pulse "Y", showing a positive correlation. The higher the value is above the threshold, the faster the depletion rate. Therefore, as the carrier concentration further increases, the correlation distribution between pulse intensity gradually disappears. Thus, the DC driving currents determines the initial charge density and greatly changes the lasing behavior.

The instability of GSSLs may affect the validity of the security analysis. Based on the proposed model^[8], we evaluated the impact of intensity fluctuations of the GSSL laser under test on SKRs under different driving conditions, as shown in Fig. 4. Here, we use a system clockfrequency of 2.5 GHz. The intensity of signal and decoy state is 0.48 and 0.12, respectively. The encoding probability of signal, decoy and vacuum states is 2 : 1 : 1. The detection efficiency is 0.075, which includes both the detector efficiency and Bob’s apparatus transmission loss. The dark count rate is 8.0 × 10⁻⁶ per pulse and the error correction efficiency is 1.05. We also considered finite size effects with a post-processing block size of 100 Mb. As the driving currents increase, the fluctuation of light intensity decreases, resulting in a significant improvement in SKR and transmission distance. Minor changes in driving currents can have a significant impact between 14 and 17 mA. Therefore, we need to strictly control the working state of GSSL in the transmitter. The experiment and simulation suggest that the intensity fluctuation decreases by increasing DC bias currents and pulse currents. However, the DC bias currents cannot exceed the threshold to obtain the gain switching. When the DC driving currents is too close to the threshold, the phase correlation will appear^{[13, 16−19]}. Generally, the pulse interval should be much longer than the effective photon lifetime which is determined by DC driving currents (τeff=11−ΛDCτp). Thus, the upper-limit of the effective photon lifetime is 40 ps, a tenth of the pulse duration 400 ps in a 2.5 GHz clock QKD system^[13]. For the pulse currents in principle has no limits, but it is limited by the output power of the laser driver. The negative correlation between the adjacent pulses would break the assumption of independent and identically distributed probabilistic variables, upon which a number of security proofs rely. Though further study is required to quantify the effect of the intensity correlation on the security, it is a good practice to suppress any correlations. It is necessary to operate a GSSL in such an operating condition in terms of the intensity correlation.

We measured the intensity fluctuation of a GSSL operating at a repetition frequency at 2.5 GHz, which is the de facto light source in decoy-BB84 QKD systems. We observed large intensity fluctuation and high-order correlation distribution between adjacent pulses. As the driving currents increase, optical pulses gradually stabilize and correlation distribution disappears. Using rate equations, we confirmed the fluctuation arising from the instability of GSSLs at high repetition frequency where the electrical pulse interval is comparable to the carrier lifetime. We gain valuable insights into the transmitter in QKD, particularly regarding intensity fluctuations and correlation distribution between adjacent pulses. Our work contributes to the assessment and optimization of QKD transmitters at high repetition rate of GHz.