## Abstract

Security certification of quantum key distribution (QKD) systems under practical conditions is necessary for social deployment. This article focused on the transmitter, and, in particular, investigated the intensity fluctuation of the optical pulses emitted by a gain-switched semiconductor laser used in QKD systems implementing decoy-BB84 protocol. A large intensity fluctuation was observed for low excitation, showing strong negative correlation between the adjacent pulses, which would affect the final key rate. The fluctuation decreased and the correlation disappeared as excitation increased. Simulation with rate equations successfully reproduced the experimental results and revealed that the large fluctuation originates from an intrinsic instability of gain-switched lasers driven periodically at a rate comparable to the inverse of carrier lifetime, as in GHz-clock QKD systems. Methods for further reduction of the intensity fluctuation were also discussed.

© 2017 Optical Society of America

## Corrections

31 January 2017: Corrections were made to Eqs. (1), (2), (4), and (10); and the body text.

## 1. Introduction

Quantum key distribution (QKD) was proposed to share cryptographic key between remote parties with unconditional security. It offers long term communication security against so-called “store now, read later” eavesdropping, where eavesdroppers store cipher texts as encrypted until they can decrypt with future technology. Currently, social deployment of secure communication network strengthened by QKD is becoming under serious consideration, because of advances in QKD technology. QKD systems based on BB84 protocol [1] with decoy method [2–4] are particularly well-developed to demonstrate stable key generation in practical optical fiber networks [5–10]. However, there still remain obstacles for deployment, such as lack of standards and security certification for practical devices, systems, and networks. Theoretically, the unconditional security of decoy-BB84 protocol has been established [11]. Nevertheless, the proofs rely on premises, some of which may not be fully satisfied in practice, because of the device imperfections. We need to guarantee the security of real systems by the following steps for security certification: (i) reduction of the assumptions in the security proofs into the device characteristics, (ii) characterization of the devices, and (iii) improvement of the devices or the protocol.

We here focus on the transmitter in decoy-BB84 QKD systems. The decoy method [2–4] has been proposed to guarantee security on QKD systems using attenuated laser pulses against photon number splitting attack [12,13]. The intensity, or mean photon number, plays an important role on security of decoy-BB84 protocol, which can be seen in the following simplest form of the decoy method. In the decoy method, a sender (Alice) transmits pulses to a receiver (Bob) with different mean photon numbers, *n̄*_{1}, *n̄*_{2}, . . . . The pulses are thus classified with the mean photon number. We here refer to the pulse with the mean photon number *n̄ _{k}* as

*k*-pulse. Though Eve is assumed to be able to measure the photon number

*n*contained in an optical pulse, she can not measure the mean photon number. Therefore, she has to apply the same eavesdropping strategy, described by transmission rate

*Y*and error rate

_{n}*E*for

_{n}*n*-photon states, to all the pulses sent by Alice. Then, for infinite samples, transmittance

*η*of

_{k}*k*-pulses can be related to

*Y*by

_{n}*p*

_{k|n}denotes the conditional probability that Alice sent a

*k*-pulse when Eve detects

*n*photons. Bayes’ theorem yields the conditional probability with photon number distribution

*p*

_{n|k}and the probability

*p*for Alice’s selection of

_{k}*k*-pulses as where the photon number distribution is a function of the mean photon number, such as Poisson distribution ${p}_{n|k}={\overline{n}}_{k}^{n}\text{exp}\left[-{\overline{n}}_{k}\right]/n!$ assumed for attenuated laser light. Similar equations apply to the error rates for

*k*-pulses and

*E*. From these equations, we can determine Eve’s strategy

_{n}*Y*and

_{n}*E*, which enables us to estimate the information leakage tightly, and to provide the final key rate proportional to the channel transmittance. This is a great advantage of the decoy method over the conventional one, where the final key rate decreases proportionally to the square of the channel transmittance. However, if the intensity of optical pulses deviates from the designed value, the probabilities

_{n}*p*

_{n|k}vary. Then Eqs. (1) and similar equations for error rate are no longer exact, so that we need to include the effect of deviation by introducing ranges to Eve’s strategy

*Y*and

_{n}*E*. This will loosen the estimation of Eve’s information, and thus reduce the final key rate. The accuracy of the optical pulse intensity has emerged as a serious issue in the decoy method. In contrast, it has little influence on the performances of conventional digital lightwave communication as long as enough signal-to-noise ratio is ensured.

_{n}The intensity deviation contains drift and fluctuation, the former refers to wandering mean value of the intensity, while the latter irregular variation around the mean value. We investigate the intensity fluctuation of the laser light source, since the property of the light source determines the ultimate characteristics of a system. The drift is usually slow and can be compensated with a slow optical power meter and an attenuator.

Most of QKD systems implementing the decoy-BB84 protocol use a gain-switched semiconductor laser (GSSL) as the light source, where the laser is excited from below the threshold by a strong current pulse for each pulse generation. The use of GSSL is favorable, because it simplifies the construction of the transmitter. Another important reason is that it emits phase randomized pulses [14], which is assumed in most security proofs. The phase correlation between pulses increases the distinguishability between non-orthogonal bases and between the signal and decoy pulses, and thus increases the required amount of the sacrifice bits [15–17]. The gain-switching operation, however, implies a large and fast variation of the carrier density in the laser. The nonlinear nature of the laser oscillation may amplify a small noise in excitation current to a large intensity fluctuation of optical pulses. It is thus necessary to characterize the intensity fluctuation in the GSSL, to investigate its mechanism, and to find methods for reduction.

In the following, we report the measurement and analysis on the intensity fluctuation in a GSSL. This article is constructed as follows: we measured the intensity distribution of optical pulses from a GSSL to clarify the dependence on the operating conditions in sec. 2. We analyzed the pulse oscillation by solving the rate equations of GSSL in sec. 3. We found instability in the periodically driven GSSL, which explained the intensity fluctuation observed in the experiment. We discussed mechanism of the unstable operation of GSSLs, and the operating condition to obtain optical pulses with stable intensity in sec. 4. We also consider the criterion of intensity fluctuation. Sec. 5 concludes the paper.

## 2. Intensity fluctuation of a periodically driven gain-switched semiconductor laser

We measured the intensity fluctuation of a GSSL used to generate phase randomized optical pulses. Figure 1 shows the experimental setup. We tested a distributed feedback (DFB) laser diode (NEL, NLK5C5EBKA) designed for 10-GHz direct modulation to emit optical pulses in a single longitudinal and transversal mode. The lasing wavelength was around 1560 nm at the threshold current (*J _{th}*) of 9.5 mA. The laser was driven by a 1.25-GHz pulse current (

*J*) of 100-ps duration and a DC bias current (

_{AC}*J*) combined with a bias-T. The electrical pulse from a pulse-pattern-generator (PPG) was amplified to the required value of

_{DC}*J*. We set the pulse current to one of the three peak-to-peak amplitudes, 2

_{AC}*J*, 3

_{th}*J*, and 4

_{th}*J*. We investigated the dependence of the laser intensity fluctuation on the excitation by changing the DC bias current. The optical pulse from the laser was detected by a high speed photodetector of 10-GHz bandwidth. Optical power was controlled by an attenuator to avoid saturation in the photodetector. The waveform was recorded by an oscilloscope of 6-GHz bandwidth. Each pulse area was measured and analyzed to investigate the statistical distribution of the pulse intensity. We defined the pulse intensity as the pulse area obtained for one period centered at the pulse peak, as shown by an arrow in Fig. 2(a), to circumvent the effects of the pulse jitter of the GSSL on the accuracy of the pulse intensity measurement. We collected 10000 samples for the statistical analysis.

_{th}Temporal behavior of the lase pulses is shown in Figs. 2(a)–2(c). The pulse intensity fluctuated strongly from pulse to pulse at low excitation *J _{DC}* = 0.65

*J*, but it became stable as the excitation increased. To investigate the intensity fluctuation quantitatively, we analyzed the statistical distribution of the optical pulse intensity, as shown in Figs. 3(a)–3(c), when the pulse current was set to

_{th}*J*= 3

_{AC}*J*with the DC bias current values

_{th}*J*= 0.65

_{DC}*J*, 0.7

_{th}*J*, and 0.9

_{th}*J*. The fluctuation of the laser intensity

_{th}*x*is presented by the normalized deviation from the mean value

*μ*=

*x*as (

*x*−

*μ*)/

*μ*. The laser intensity fluctuated strongly for a small drive current, 0.65

*J*, as seen in Fig. 3(a). The histogram of the laser intensity shows an asymmetric distribution broaden to the lower side. As increasing the DC bias current, the intensity fluctuation decreased, and the distribution became symmetric as shown in Figs. 3(b) and 3(c). We employed the triplication of the normalized standard deviation 3

_{th}*σ/μ*as a measure of the fluctuation for quantitative investigation, where the standard deviation was defined as usual by

*σ*

^{2}=

*x*

^{2}−

*μ*

^{2}. The fluctuation 3

*σ/μ*read 1.69, 1.09 and 0.11, for

*J*= 0.65

_{DC}*J*, 0.7

_{th}*J*, and 0.9

_{th}*J*, respectively.

_{th}The dependence of the intensity fluctuation on the drive current is depicted in Fig. 4, where the normalized fluctuation is plotted as a function of the DC bias current for the pulse currents 2 *J _{th}*, 3

*J*, and 4

_{th}*J*. The fluctuation 3

_{th}*σ/μ*fell to 0.08 at the highest excitation

*J*= 4

_{AC}*J*and

_{th}*J*= 0.9

_{DC}*J*in the present experiment. However, the observed intensity fluctuation was more than three times as large as that of the pulse current 3

_{th}*σ*/ 〈

_{AC}*J*〉 = 0.027, even for a large drive current set at

_{AC}*J*= 3

_{AC}*J*and

_{th}*J*= 0.9

_{DC}*J*. This suggests that the laser contains a mechanism to increase the fluctuation of the drive current when it is converted to the optical output.

_{th}We also observed intensity correlation between the adjacent pulses. The autocorrelation function of optical pulse intensity showed strong negative correlation between the adjacent pulses as shown in Fig. 5. The negative correlation decreased as the excitation increased. The correlation disappeared for high excitation at *J _{AC}* = 3

*J*and

_{th}*J*= 0.9

_{DC}*J*.

_{th}## 3. Numerical simulation with rate equations

Strong fluctuation was observed in a GSSL operation in a weak excitation region. The experimental results suggest existence of an intrinsic mechanism in the laser emission that amplifies the fluctuation of the drive current. In the following, we numerically analyzed the behavior of the GSSL oscillation with rate equations to explore the amplifying mechanism. It is known that the dynamics of semiconductor lasers can be described well with the rate equations on carrier density *N* and photon density *I*, where the light field phase plays little role on the laser dynamics. Because light intensity is proportional to photon density, we use both words without distinction.

We employed the following rate equations

*N*is transparent carrier density, Γ is confinement factor, and

_{g}*τ*

_{(p,N)}are photon and carrier lifetime, respectively. The spontaneous emission contribution to the lasing mode is represented by spontaneous emission factor

*β*. Pumping rate Λ is proportional to the sum of DC bias current and pulse current (Λ = Λ

_{sp}*+ Λ*

_{DC}*∝*

_{AC}*J*+

_{DC}*J*.) In the following, the pumping rate represents the normalized value by the threshold pumping rate,

_{AC}*i.e.*, Λ

*= 1. We used the simplest form of gain linear to the carrier density, and neglected the gain saturation.*

_{th}We simulated the behavior of the carrier density and photon density in a periodically excited GSSL with Eqs. (3) and (4). Since we aimed to obtain qualitative insight of the laser dynamics, the simulation used typical semiconductor laser parameters given in Table 1 without fitting. The pumping pulse was assumed to be rectangular shape of amplitude Λ* _{AC}* = 3, duration 100 ps, and frequency 1.25 GHz to mimic the experimental conditions.

Figure 6 shows the temporal behavior of the light intensity from a GSSL, where three values of the pumping rate were used: (a) Λ* _{DC}* = 0.58, (b) Λ

*= 0.6, and (c) Λ*

_{DC}*= 0.8. For low excitation, Λ*

_{DC}*= 0.58, we observed instability in laser oscillation, where a strong pulse was followed by a weak one. As seen in Figs. 6(b) and 6(c), such instability became small and disappeared, and stable laser operation was observed for high pumping rate. These alternating pulse intensities agree with the observation of the negative correlation between the adjacent pulses shown in Fig. 5.*

_{DC}To confirm the instability in periodic pulse operation, we analyzed the Poincaré map on the photon density and the carrier density. Period *T* is defined by the inverse of the pulse frequency (1.25 GHz), that is, *T* = 800 ps in the present simulation. Sampling the solutions of (3) and (4) with the period *T* yields a series of discrete solutions {(*I*_{0}, *N*_{0})*,* · · · , (*I _{n}*,

*N*)

_{n}*,*· · · }, where

*I*and

_{n}*N*are defined by

_{n}*I*=

_{n}*I*(

*nT*) and

*N*=

_{n}*N*(

*nT*), respectively. Perturbations Δ

*I*and Δ

_{n}*N*on the solutions (

_{n}*I*,

_{n}*N*) will alter the solution after one period as

_{n}*λ*

_{1}and

*λ*

_{2}. The periodic solution is unstable when one of the absolute value of the eigen values |

*λ*

_{1}| and |

*λ*

_{2}| exceeds one. We found the unstable region of the periodic solutions, which supports the instability of the periodically driven GSSL, which was suggested in temporal analysis of the laser pulses shown in Fig. 6(a).

So far, the simulation using Eqs. (3) and (4) assumed fixed pumping rate, so that it neglected the fluctuations of carriers and photons induced by the pump current noise. Thus, the calculated optical pulses showed regular alternation, as seen in Fig. 6(a). However, since the instability of the laser is expected to amplify the electrical pumping noise effect on the intensity of optical pulses, we introduced a noise term *F*(*t*) to the pumping rate to simulate the intensity fluctuation observed in the experiment as

*F*(

*t*) is assumed to represent white Gaussian noise with zero mean. The standard deviation

*σ*was determined from the observed pulse current fluctuation 3

_{AC}*σ*/ 〈

_{AC}*J*〉 = 0.027. The DC pumping rates were chosen to provide similar probability distribution to the experimental results: (d) Λ

_{AC}*= 0.6, (e) Λ*

_{DC}*= 0.63, and (f) Λ*

_{DC}*= 0.8. We calculated the optical output of the GSSL, as shown in Figs. 2(d)–2(f). The calculated temporal behavior reproduced the measurement result quite well.*

_{DC}We also derived the probability distribution of the intensity as shown in Figs. 3(d)–3(f). The simulated intensity distribution showed a broad and asymmetric in low excitation as shown in Fig. 3(d), but became narrow and symmetric in high excitation. The shapes of the distribution agreed qualitatively with the experimental results seen in Figs. 3(a)–3(c). The effect of the electrical noise was clearly demonstrated by comparing the results with those calculated without the electrical noise. Figures 3(g)–3(i) shows that the intensity distribution concentrated in a single column; no large intensity fluctuation occurs without the electrical noise. The calculation shows that unstable lasing appears slightly below Λ* _{DC}* = 0.6, as seen in Fig. 6(a). A large intensity
fluctuation at Λ

*= 0.6 would result from the electrical noise, which decreases the excitation from stable lasing regiion to unstable region.*

_{DC}The measure of the intensity fluctuation 3*σ/μ* was calculated as a function of DC pumping from the simulated data with fixed pulse pumping Λ* _{AC}* = 3, as shown in Fig. 7 (a), as well as the autocorrelation function of laser intensity between adjacent pulses in Fig. 7 (b). The intensity fluctuation reached as high as 2 for low excitation Λ

*= 0.6, when the autocorrelation function shows the largest negative correlation. As the excitation increased, the values of the fluctuation reduced to about 0.5 for medium excitation Λ*

_{DC}*= 0.63, and finally to less than 0.1 for high excitation Λ*

_{DC}*= 0.8. Similar behavior was observed in the autocorrelation function, which approached to zero for high excitation. The calculated intensity fluctuation shown in Figs. 3 and 7 reproduced the experimental observations quite well, if we consider the inaccuracies in the laser parameters. This agreement between the simulation and experiment indicates the validity of the model used in the calculation.*

_{DC}## 4. Discussion

Intensity fluctuation can be also measured by second order correlation function defined by

which can be obtained with photon number resolving photon detectors [18]. If the photon state is given as a statistical mixture of Poisson distributions with different values of mean photon number*n̄*

*s*

^{2}= 〈

*n̄*

^{2}〉 − 〈

*n̄*〉

^{2}of the mean photon number as where 〈

*n̄*〉 estimated with the distribution function

*f*(

*n̄*) refers to the expectation value of

*n̄*.

The second order correlation function method and the present method are complementary ways to characterize the fluctuation in the following aspects. Though the second order correlation function method provides the fluctuation of intensity or mean photon number indirectly through the photon number distribution, it can measure the fluctuation of photon number, which directly affects the decoy method. The second order function method works for attenuated pulses down to single photon level. It yields statistical information on the fluctuation. In contrast, the present method directly measures the intensity fluctuation, however, the photon number fluctuation is estimated indirectly with the photon number distribution. Since it measures the pulse intensity before attenuation, we needs to assume that ratio of mean intensity to fluctuation is conserved in attenuation, however it would be plausible. The present method yields the information on intensity of each pulses, which is useful to investigate the mechanism of the fluctuation as given in this article. The present method can be performed with standard photonics laboratory equipment, such as a high-speed photodetecor and a oscilloscope. Therefore, it would be useful to test QKD transmitters at development and manufacture sites.

In the following, we examine mechanism of the intensity fluctuation in GSSLs. We observed a negative correlation between the adjacent laser pulses in the experiment, as shown in Fig. 5. The simulation revealed instability in the pulse generation for low excitation, which results in the negative correlation. This instability also causes the large intensity fluctuation.

Let us briefly recall the principle of the gain-switching operation to investigate the origin of the instability. A GSSL produces strong and short pulses through the following mechanism. In the initial state, the carrier density is determined by *J _{DC}* and

*τ*), which is set below the threshold

_{N}*N*=

_{th}*N*+ 1/(Γ

_{g}*gτ*). Then, a current pulse increases the carrier density. When it reaches the threshold, the GSSL begins to lase. However, since the photon density is small in the initial stage of lasing, the carrier density still increases. Then, the optical gain is higher than the threshold, which results in strong amplification of light. The strong light induces large stimulated emission to consume carriers. The carrier density may fall below the threshold by the strong light, even if the current pulse remains. If the current pulse disappears before the carrier density recovers from this undershoot, the GSSL ceases lasing, which yields the optical pulse shorter than the current pulse. If the pulse interval is longer enough than the carrier lifetime

_{p}*τ*, the carrier density returns to the initial value. Then, the gain-switching operation repeats under the same initial condition. However, if the pulse interval is comparable to or shorter than the carrier lifetime, the carrier density may not return to the initial value. Then, the lasing starts under a different initial condition, which will alter the lasing behavior. It should be noted that the stimulated emission reduces the effective carrier lifetime while the light remains in the laser active layer. The stronger light results in the shorter effective lifetime, as described by the nonlinear stimulated emission term

_{N}*g*(

*N*−

*N*)

_{g}*I*in Eq. (4).

The instability in the periodically driven GSSLs can be explained by the overshooting of carrier density and the light-intensity dependent effective carrier lifetime, as depicted in Fig. 8. When DC bias and pulse current are small but just enough to excite carriers to the threshold carrier density, the optical gain is equal or slightly larger than the cavity loss, as seen in 0–0.1 ns of Fig. 8. Since the laser light is amplified only a little, the GSSL emits a weak optical pulse as seen in 0.1–0.3 ns. The excited carriers decay mostly by the spontaneous emission lifetime, because of the small effect of the stimulated emission by weak light. Since the time interval of the current pulses is comparable with the carrier lifetime (1.2 ns, in the present simulation), the next current pulse is applied before the carrier density reaches the initial value, as seen at 0.8 ns. The carrier density is the sum of the remaining carriers excited by the first pulse and the ones by the second pulse. Therefore, it exceeds the threshold carrier density greatly to yield a strong optical pulse as seen at around 1 ns. Since the intense optical pulse reduces the effective carrier lifetime by the strong stimulated emission, the carrier density can decrease to the initial value when the third pulse current is applied, as seen at 1.6 ns. Therefore, the intensity of the third optical pulse is again weak. This cycle results in the observed negative correlation between the adjacent optical pulses. When DC bias current and pulse current are large to excite the carriers sufficiently higher than the threshold carrier density, strong optical pulses always accelerate the carrier decay to the enough rate to return the initial value. Thus, the instability disappears. The instability takes place in the GSSLs operated in high repetition rate, where the pulse period is comparable to the carrier lifetime. A transmitter for the decoy-BB84 QKD system with GHz-clock frequency is one that satisfies this condition.

As mentioned in introduction, the intensity fluctuation affects the security analysis on decoy-BB84 protocol. However, it is not straightforward to obtain quantitative criteria for the intensity fluctuation. The effects of the intensity fluctuation on the decoy method has been studied by Wang, *et al.* [19,20]. Recently, Hayashi and Nakayama [21] and Mizutani, *et al.* [22] analyzed its impact on the final key rate in detail including the effect of finite data. These studies concluded different requirements for the intensity fluctuation, mainly because of the different assumptions on the probability distribution of the intensity and security analysis methods. The one that assumes Gaussian distributions [21] showed no significant decrease of the final key rate up
to the standard deviation *σ* = 0.1*n̄*, where *n̄* stands for the designed value of the mean photon number. On the other hand, the estimation considering more general distribution concluded that the intensity fluctuation should be less than 10 % [22]. Effects of the fluctuation also depend on the parameters used to implement the decoy protocol. Therefore, it is hard to establish a general criterion. In other words, the exact criteria should be addressed with the methods of security analysis and the values parameters used in the system design. Nevertheless, to proceed experimental studies, we would propose a rough target value as 3*σ* = 0.1*μ*, which lies in between the analyses [21–23]. The probability that deviation from the expectation values exceeds 10 %
is $\text{erfc}(0.1\mu /\left(\sqrt{2}\sigma \right)=2.7\times {10}^{-3}$ for the Gaussian distribution of *σ* = 0.033*μ*.

The instability of GSSLs may affect the validity of the security analysis. The negative correlation between the adjacent pulses would break the assumption of independent and identically distributed probabilistic variables, on which a number of security proofs rely. Some theories assume only martingale process, but still it is not obvious whether the assumption is satisfied. Though further study is required to quantify the effect of the intensity correlation on the security, it is safe to suppress any correlations. As we observed, the correlation disappears for high excitation. It is also necessary to operate a GSSL in such an operating condition in terms of the intensity correlation.

Before closing the section, we consider methods to obtain pulses with low intensity fluctuation and correlation. The experiment and simulation suggest that the intensity fluctuation decreases by increasing DC bias current and pulse current. However, the DC bias current cannot exceed the threshold to obtain the gain switching. Moreover, the inter-pulse phase correlation may appear when the DC bias current is too close to the threshold. The DC pumping is limited by the condition that the pulse interval should be much longer than the effective photon lifetime defined by [14]

For example, we estimated the upper-limit of DC pumping from Eq. (10) when the effective photon lifetime reaches 80 ps, the tenth of the pulse duration 800 ps in a 1.25-GHz clock QKD system. The estimated upper-limit is ${\mathrm{\Lambda}}_{\mathit{DC}}^{\mathit{max}}={J}_{\mathit{DC}}^{\mathit{max}}/\mathit{Jth}=0.9$ with the cavity lifetime of 8 ps. On the other hand, there is no limit for the pulse current in principle as long as the GSSL admits. However, in practice, it is limited by the output power of the laser driver. In the present experiment, the minimum intensity fluctuation was limited to 3*σ/μ*= 0.08 by the pulse current of 4

*J*from the driver. A commercially available laser driver with higher output power will reduce the intensity fluctuation to a few percents. These values fall below our target and would be enough for most QKD systems. If further decrease of the fluctuation would be required, the GSSLs might not satisfy the requirement. A possible method would be the combination of direct modulation on a semiconductor laser and electro-absorption (EA) modulation by an integrated modulator. The direct modulation produces long optical pulses to reduce the inter-pulse phase correlation by setting the drive current in OFF-state smaller than the threshold. Then, EA modulator cuts the pulse to short duration required for the QKD system. This combination provides an advantage that it will reduce the chirping, the frequency shift during the pulse, because the change of the carrier density can be slower than that in the gain switching. Another method is to give up the pulse operation of the laser. Pulses are generated from the continuous wave (CW) laser light with an intensity modulator, which can be done by an EA-modulator integrated with a laser To apply this method, the phase correlation should be suppressed with an external phase modulator. This method solves the issues on the lasers, but may make the system complicated and increase the production cost.

_{th}## 5. Conclusion

We measured the intensity fluctuation of a gain-switched semiconductor laser, which commonly used in transmitters for decoy-BB84 QKD systems. Our concern on the intensity fluctuation originates from security certification in practical QKD systems, but this issue also provides a novel insight on semiconductor laser dynamics. We observed large intensity fluctuation and negative correlation between the adjacent pulses for low excitation, which may affect the security certification of QKD systems. The gain-switching operation was simulated with simple rate equations to investigate the mechanism of the large fluctuation. The simulation successfully reproduced the experimental observations. It also showed that instability of the pulsed laser operation arises in the low excitation regime, which resulted in the large fluctuation and the negative correlation. The instability was explained as an intrinsic property of the gain switched lasers driven at high reputation rate, where the pulse period is comparable to the carrier life time, as used in GHz-clock decoy-BB84 QKD systems. The instability was suppressed with high excitation, so that pulses with low intensity fluctuation and correlation were obtained. Gain-switched semiconductor lasers can be used for decoy-BB84 systems by setting DC bias current close to the threshold with large pulse current. This study was achieved by a combination of device physics and QKD theory, which would provide a foundation of the practical security certification for wide deployment of the QKD systems.

## Funding

National Institute of Information and Communications Technology (NICT) contract research project (157-B); ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Government of Japan) “Advanced Information Society Infrastructure Linking Quantum Artificial Brains in Quantum Network.”

## References and links

**1. **C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems, and Signal Processing (1984), pp. 175–179.

**2. **W.-Y. Hwang, “Quantum Key Distribution with High Loss: Toward Global Secure Communication,” Phys. Rev. Lett. **91**, 057901 (2003). [CrossRef] [PubMed]

**3. **X.-B. Wang, “Beating the photon-number-splitting attack in practical quantum cryptography,” Phys. Rev. Lett. **94**, 230503 (2005). [CrossRef] [PubMed]

**4. **H.-K. Lo, X. Ma, and K. Chen, “Decoy state quantum key distribution,” Phys. Rev. Lett. **94**, 230504 (2005). [CrossRef] [PubMed]

**5. **M. Peev, C. Pacher, R. Alléaume, C. Barreiro, J. Bouda, W. Boxleitner, T. Debuisschert, E. Diamanti, M. Dianati, J. F. Dynes, S. Fasel, S. Fossier, M. Fürst, J.-D. Gautier, O. Gay, N. Gisin, P. Grangier, A. Happe, Y. Hasani, M. Hentschel, H. Hübel, G. Humer, T. Länger, M. Legré, R. Lieger, J. Lodewyck, T. Lorünser, N. Lütkenhaus, A. Marhold, T. Matyus, O. Maurhart, L. Monat, S. Nauerth, J.-B. Page, A. Poppe, E. Querasser, G. Ribordy, S. Robyr, L. Salvail, A. W. Sharpe, A. J. Shields, D. Stucki, M. Suda, C. Tamas, T. Themel, R. T. Thew, Y. Thoma, A. Treiber, P. Trinkler, R. Tualle-Brouri, F. Vannel, N. Walenta, H. Weier, H. Weinfurter, I. Wimberger, Z. L. Yuan, H. Zbinden, and A. Zeilinger, “The SECOQC quantum key distribution network in Vienna,” New. J. Phys. **11**, 075001 (2009). [CrossRef]

**6. **T. E. Chapuran, P. Toliver, N. A. Peters, J. Jackel, M. S. Goodman, R. J. Runser, S. R. McNown, N. Dallmann, R. J. Hughes, K. P. McCabe, J. E. Nordholt, C. G. Peterson, K. T. Tyagi, L. Mercer, and H. Dardy, “Optical networking for quantum key distribution and quantum communications,” New J. Phys. **11**, 105001 (2009). [CrossRef]

**7. **W. Chen, Z.-F. Han, T. Zhang, H. Wen, Z.-Q. Yin, F.-X. Xu, Q.-L. Wu, Y. Liu, Y. Zhang, X.-F Mo, Y.-Z. Gui, G. Wei, and G.-C. Guo, “Field experiment on a “star type” metropolitan quantum key distribution network,” IEEE Photonics Technol. Lett. **21**, 575–577 (2009). [CrossRef]

**8. **T.-Y. Chen, J. Wang, H. Liang, W.-Y. Liu, Y. Liu, X. Jiang, Y. Wang, X. Wan, W.-Q. Cai, L. Ju, L.-K. Chen, L.-J. Wang, Y. Gao, K. Chen, C.-Z. Peng, Z.-B. Chen, and J.-W. Pan, “Metropolitan all-pass and inter-city quantum communication network,” Opt. Express **18**, 27217–27225 (2010). [CrossRef]

**9. **M. Sasaki, M. Fujiwara, H. Ishizuka, W. Klaus, K. Wakui, M. Takeoka, S. Miki, T. Yamashita, Z. Wang, A. Tanaka, K. Yoshino, Y. Nambu, S. Takahashi, A. Tajima, A. Tomita, T. Domeki, T. Hasegawa, Y. Sakai, H. Kobayashi, T. Asai, K. Shimizu, T. Tokura, T. Tsurumaru, M. Matsui, T. Honjo, K. Tamaki, H. Takesue, Y. Tokura, J. F. Dynes, A. R. Dixon, A. W. Sharpe, Z.-L. Yuan, A. J. Shields, S. Uchikoga, M. Legré, S Robyr, P. Trinkler, L. Monat, J.-B. Page, G. Ribordy, A. Poppe, A. Allacher, O. Maurhart, T. Länger, M. Peev, and A. Zeilinger, “Field test of quantum key distribution in the Tokyo QKD Network,” Opt. Express **19**, 10387–10409 (2011). [CrossRef] [PubMed]

**10. **B. Frohlich, J. F. Dynes, M. Lucamarini, A. W. Sharpe, Z. Yuan, and A. J. Shields, “A quantum access network,” Nature **501**, 69–72 (2013). [CrossRef] [PubMed]

**11. **V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, and M. P. Norbert Lütkenhaus, “The Security of Practical Quantum Key Distribution,” Rev. Mod. Phys. **81**, 1301–1353 (2009). [CrossRef]

**12. **G. Brassard, N. Lütkenhaus, T. Mor, and B. C. Sanders, “Limitations on practical quantum cryptography,” Phys. Rev. Lett. **85**, 1330 (2000). [CrossRef] [PubMed]

**13. **N. Lütkenhaus and M. Jahma, “Quantum key distribution with realistic states: photon-number statistics in the photon-number splitting attack,” New J. Phys. **4**, 44 (2002). [CrossRef]

**14. **T. Kobayashi, A. Tomita, and A. Okamoto, “Evaluation of the phase randomness of a light source in quantum-key-distribution systems with an attenuated laser,” Phys. Rev. A **90**, 032320 (2014). [CrossRef]

**15. **D. Gottesman, H.-K. Lo, N. Lütkenhaus, and J. Preskill, “Security of quantum key distribution with imperfect devices,” Quantum Info. Comput. **5**, 325 (2004).

**16. **H.-K. Lo and J. Preskill, “Security of quantum key distribution using weak coherent states with nonrandom phases,” Quantum Info. Comput. **7**, 431–458 (2007).

**17. **S.-H. Sun, M. Gao, M.-S. Jiang, C.-Y. Li, and L.-M. Liang, “Partially random phase attack to the practical two-way quantum-key-distribution system,” Phys. Rev. A **85**, 032304 (2012). [CrossRef]

**18. **J. F. Dynes, Z. L. Yuan, A. W. Sharpe, O. Thomas, and A. J. Shields, “Probing higher order correlations of the photon field with photon number resolving avalanche photodiodes,” Opt. Express **19**, (14), 13268–13276 (2011). [CrossRef] [PubMed]

**19. **X.-B. Wang, C.-Z. Peng, J. Zhang, L. Yang, and J.-W. Pan, “General theory of decoy-state quantum cryptography with source errors,” Phys. Rev. A **77**, 042311 (2008). [CrossRef]

**20. **X.-B. Wang, L. Yang, C.-Z. Peng, and J.-W. Pan, “Decoy-state quantum key distribution with both source errors and statistical fluctuations,” New. J. Phys. **11**, 075006 (2009). [CrossRef]

**21. **M. Hayashi and R. Nakayama, “Security analysis of the decoy method with the bennett–brassard 1984 protocol for finite key lengths,” New J. Phys. **16**, 063009 (2014). [CrossRef]

**22. **A. Mizutani, M. Curty, C. C. W. Lim, N. Imoto, and K. Tamaki, “Finite-key security analysis of quantum key distribution with imperfect light sources,” New J. Phys. **17**, 093011 (2015). [CrossRef]

**23. **Y. Nagamatsu, A. Mizutani, R. Ikuta, T. Yamamoto, N. Imoto, and K. Tamaki, “Security of quantum key distribution with light sources that are not independently and identically distributed,” Phys. Rev. A **93**, 042325 (2016). [CrossRef]