## Abstract

We demonstrate fiber-optic quantum key distribution (QKD) at 1550 nm using single-photon detectors operating at 5 MHz. Such high-speed single-photon detectors are essential to the realization of efficient QKD. However, after-pulses increase bit errors. In the demonstration, we discard after-pulses by measuring time intervals of detection events. For a fiber length of 10.5 km, we have achieved a key rate of 17 kHz with an error of 2%.

© 2003 Optical Society of America

## 1. Introduction

Quantum key distribution (QKD) is a technique to share a private key of a random binary sequence between two remote parties, sender and receiver (called Alice and Bob, respectively) by exchanging qubits described by single photons or weak coherent pulses, in order to implement a secure one-time-pad encryption and decryption (for a good review, see [1]). Since any unknown qubit state cannot be perfectly copied, an eavesdropper (Eve) disturbs the transmitted qubits when extracting information. Its security relies on the laws of quantum mechanics. If a single-mode fiber is used as a quantum channel, the most desirable wavelength for low-loss transmission is 1550 nm. Recently, 80-km fiber-optic QKD experiments were reported at 1550 nm [2]. The performance of indium-gallium-arsenide (InGaAs) avalanche photodiodes (APDs) is widely investigated for single-photon detection at 1550 nm. A common method is referred to as the gated mode, in which the APD is pulse-biased above its breakdown [3–5]. During the gate-on time, a photon-induced avalanche can grow into a macroscopic pulse. However, a thermally excited carrier also triggers an avalanche, giving a dark count in detection. Furthermore, carriers are trapped every avalanche and if one emits during the next gate-on times, it can trigger a new avalanche. For 1550 nm, this avalanche (after-pulse) is frequently found in detection if the repetition frequency exceeds 1 MHz, resulting in a significant increase of bit errors in QKD [5–6]. However, high-speed single-photon detectors are essential to the realization of efficient QKD. Since only traps with an emission lifetime comparable to or longer than a reciprocal of the repetition frequency generate after-pulses, introducing a dead time in detection, during which, following an avalanche, no gates are applied to the APD, is an effective way to reduce bit errors in QKD [7].

In this paper, we demonstrate discarding of after-pulses by measuring time intervals of detection events to reduce bit errors in QKD. In the demonstration, single-photon detectors operating at a repetition frequency of 5 MHz are used. Furthermore, Alice is connected to Bob with two single-mode fibers. One is used as a quantum channel while the other for clock sharing. Such a system removes bit errors related to backscattered photons from the clock pulse. Here, we have achieved a key rate of 17 kHz with an error of 2% for a fiber length of 10.5 km.

## 2. Single-photon detectors

The evaluation method for gated-mode single-photon detectors described here makes it possible to measure the quantum efficiency and the after-pulse probabilities per gate from the same data [8]. Here, only important parts are summarized. To obtain the probability distribution, we measure time intervals of detection events. Let *p*
_{interval}(*Δt*) denote the probability of finding *Δt* among those measured. Also, let *p*
_{after-pulse}(*Δt*) denote the (conditional) probability that an after-pulse is observed after *Δt* following a previous avalanche. Then, one finds that for each interval *Δt _{n}*=

*n*/ν with

*n*=1,2,3…

Here, ν is a repetition frequency; *η* is a quantum efficiency and *µ* is an average of photons per incoming pulse. The probability of finding no after-pulses within an interval of *Δt _{n}* can be written as (

*n*=2,3,4…)

For long intervals such that *p*
_{after-pulse}(*Δt _{n}*) ~ 0,

*c*(

*Δt*) becomes

_{n}*n*-independent, enabling us to determine

*η*from the slope of ln

*p*

_{interval}(

*Δt*). Here, ln stands for natural logarithm. Furthermore, considering that

_{n}*c*(

*Δt*

_{1})=1,

*p*

_{after-pulse}(Δ

*t*) is calculated by substituting the estimated value of

_{n}*η*into Eq. (1). In the following, two single-photon detectors (D0 and D1) operating at ν=5 MHz are evaluated. Figure 1 shows ln

*p*

_{interval}(

*Δt*) of D0 measured at

_{n}*µ*=0.015. After-pulses are observed as a nonlinear decrease of the measured data for

*Δt*<10µs. However, for longer intervals, ln

_{n}*p*

_{interval}(

*Δt*) decreases linearly, yielding a quantum efficiency of

_{n}*η*=13%. Figure 2 shows ln

*p*

_{interval}(Δ

*t*) of D1 measured at the same value of

_{n}*µ*. Figure 3 shows the calculated

*p*

_{after-pulse}(

*Δt*), where solid and open circles correspond to those of D0 and D1, respectively. Although they have different probabilities for

_{n}*Δt*<4 µs, each after-pulse is mostly found within 10 µs following a previous avalanche. Table 1 summarizes operating conditions and evaluation results of D0 and D1. Here,

_{n}*d*

_{thermal}and

*d*

_{after-pulse}are dark-count probabilities per gate resulting from thermally excited carriers and after-pulses, respectively. The former is evaluated after excluding after-pulses by measuring time intervals of dark counts and discarding those with

*Δt*<10 µs (=

_{n}*Δt*

_{after-puse}). The remaining dark counts are found with a probability of

*d*

_{thermal}exp(-

*d*

_{thermal}ν

*Δt*

_{after-pulse}), which becomes nearly equal to

*d*

_{thermal}if

*d*

_{thermal}ν

*Δt*

_{after-pulse}≪1. Then, the difference between the dark-count probabilities per gate with and without discarding after-pulses coincides with

*d*

_{after-pulse}.

## 3. Quantum key distribution

#### 3.1 Quantum channel

Figure 4 shows a schematic diagram of the system employed for the QKD demonstration (Bennett’s two-coherent-protocol: B92 [9]). Alice is connected to Bob with two 10.5-km dispersion-shifted single-mode fibers (DSF1 and DSF2) with a loss of 0.21 dB/km. DSF1 is a quantum channel while DSF2 is used for clock sharing. Here, only the system related to the quantum channel is explained. A gain-switched laser diode (LD1) produces a sequence of light pulses at a repetition frequency of 5 MHz, each having a width of 50 ps with a bandwidth of 10 nm centered at 1550 nm. With a polarization controller (PC), the polarization state is aligned for transmission through a polarizing beam splitter (PBS1). With a half-wave plate (HWP) and PBS2, the pulse is divided into signal and reference pulses. After transmission through a quarter-wave plate (QWP) and reflection by a mirror (M), the reference pulse leaves Bob, going to Alice ahead of the signal pulse. With a fiber-optic delay line (DL) and a Faraday rotator mirror (FRM), the signal pulse is delayed for 100 ns compared with the reference pulse. Due to birefringence fluctuations in DSF1, both polarization states become unknown when arriving at Alice. With a polarization-independent phase modulator (PM) [10], Alice modulates the signal pulse at random but evenly between 0 and π while letting the reference pulse unchanged. Alice’s FRM guarantees that Bob receives the signal and reference pulses with polarization states linear but orthogonal to their initial states [7, 10]. A pseudo-random number generator (PRNG) outputs a voltage pulse with a width of 50 ns, which is applied to PM only when the bit value is 1. Alice returns the signal and reference pulses to Bob after attenuating those pulses with an attenuator (AT) such that the signal pulse has an average of 0.05 photons. At Bob, the reference pulse is delayed compared with the signal pulse, both re-arriving at PBS2 simultaneously with identical intensity. The recombined pulse is horizontally or vertically polarized, which only reflects Alice’s phase choice. The private key can be established by interpreting single-photon detectors D0 and D1 as “0” and “1”, respectively

#### 3.2 Clock sharing

Figure 4 also shows how to share the clock between Alice and Bob. A gain-switched laser diode (LD2) produces a sequence of light pulses at a repetition frequency of 1 MHz, each having a width of 50 ps with a bandwidth of 10 nm centered at 1550 nm. To synchronize LD1 with LD2, a two-channel synthesized function generator (SFG) is used, which also triggers a delay generator (DG) whose outputs become timing signals for gated-mode operation of D0 and D1. Alice detects clock pulses sent by Bob via DSF2 with a conventional avalanche photodiode (C-APD), whose output pulse is converted into a square wave with a frequency of 1 MHz, which becomes a reference signal of a frequency synthesizer (FS). A sinusoidal wave with a frequency of 10 MHz is generated from FS, and is applied as a time-base signal to an arbitrary wave function generator, which is used as PRNG. Since the signal/reference pulse and the clock pulse are transmitted through separate fibers, the difference in temperature between DSF1 and DSF2 will cause the relative temporal walk-off between those pulses. However, such a walk-off is estimated to be ~0.6 ns/K [2], and is not a significant problem because it is much smaller than the width of the voltage pulse applied to PM (=50 ns). Compared with the signal and reference pulses, the clock pulse is strong enough to produce a large number of backscattered photons. The presented system prevents those photons from entering single-photon detectors. Thus, only half of backscattered photons from the signal and reference pulses (much weaker than the clock pulse) become bit errors in QKD.

## 4. Results and discussion

Figure 5 shows the quantum bit-error rate (QBER) of D0 after discarding detection events with intervals *Δt _{n}*<

*Δt*

_{discard}. Solid circles are the measured results while open circles are corresponding key rates. For

*Δt*

_{discard}<5 µs, after-pulses are effectively discarded, leading to a significant decrease in QBER. However, if

*Δt*

_{discard}exceeds 5 µs, the QBER slowly decreases and then becomes

*Δt*

_{discard}-independent. Meanwhile, the key rate shows an exponential decrease such that

Here, *k*=*ηµ*exp[-(*αL*+*β*)/10]. Note that *η* is a quantum efficiency of Bob’s single-photon detector (D0) whereas *µ* is an average of photons of the signal pulse measured by Alice. *α* is a fiber loss in dB/km; *L* is a fiber length (km) and *β* is an internal loss (dB) of Bob’s system. In the demonstration, *η*=13%, *µ*=0.05, *α*=0.21, *L*=10.5 and *β*=3. A curve in Fig. 5 is obtained by substituting those parameters into Eq. (3). Figure 6 shows the measured results corresponding to D1. A curve in this figure is also obtained by substituting the same parameters as D0 except that *η*=11% into Eq. (3). Approximately, the QBER can be written as

In this equation, the first and second terms on the right-hand side express contributions to bit errors of thermally excited carriers and after-pulses, respectively. In the following calculation, we assume that *p*
_{after-pulse} ~ 0 for *Δt _{n}*>10 µs whereas others are presented as solid and open circles in Fig. 3. The third term on the right-hand side of this equation is the QBER induced by backscattered photons from the signal and reference pulses in the quantum channel, internal reflections at Bob’s system and other imperfections of optical and electrical components. In the demonstration,

*e*

_{others}~1% and is independent of

*Δt*

_{discard}. Open squares in Figs. 5 and 6 are those calculated with Eq. (4), agreeing with the results obtained in QKD experiments (solid circles). Since the key rate decreases with

*Δt*

_{discard}, we have to properly determine

*Δt*

_{discard}for D0 and D1. For example, if we choose

*Δt*

_{discard}=7.6 µs for D0 and 5 µs for D1, respectively, the total key rate becomes 17 kHz with an error of 2%.

It is well known that Benett-Brassard-84 protocol (BB84) [11] is the most popular protocol and is more secure than the demonstrated B92 protocol. However, since the systems of two protocols are so similar, the demonstrated discarding method seems to be effective for both protocols to reduce bit errors in QKD. We are planning to increase the repetition frequency of the single-photon detectors for realizing more efficient QKD although after-pulses are more often found in detection. In the demonstration, the sender is connected to the receiver with two single-mode fibers. One is a quantum channel and the other is used for carrying strong clock pulses. In this case, it is important to consider what new strategies could be applied by Eve and see how those strategies influence the security of the QKD. Studying such a security problem will be necessary in the future.

## 5. Summary

We have demonstrated fiber-optic quantum key distribution at 1550 nm using single-photon detectors operating at 5 MHz. After-pulses are discarded by measuring time intervals of detection events, leading to a significant reduction of the quantum bit-error rate. For a fiber length of 10.5 km, we have achieved a key rate of 17 kHz with an error of 2%.

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