1. Introduction

Quantum communication is one of the most rapidly growing and exciting fields of physics in recent years. Its most mature application is quantum key distribution (QKD), which was invented by Bennett and Brassard in 1984 and which ensures the distribution of a secret key between two parties [1]. Following the first experimental demonstration of QKD over a distance of 40 cm in free space [2], development of a long-distance fiber-optic QKD system has been the main issue in the way of putting QKD into practice.

The first invention to realize long-distance fiber-optic QKD is a time-multiplexed interferometer based on Faraday mirrors, which can achieve a stable interference by Faraday ortho-conjugation over multikilometer distances of optical fiber links [3, 4, 5]. The second invention is a single-photon detector at 1550 nm using gated InGaAs/InP avalanche photodiodes (APDs), which can achieve a dark-count probability of ~ 10^-5 per gate in the temperature range of thermoelectric cooling [6, 7, 8, 9]. The QKD systems, composed of a time-multiplexed interferometer and single-photon detectors at 1550 nm, enabled us to exchange keys in the laboratory over a 100-km single-mode fiber spool [10] and in the field over a 67 km installed single-mode fiber link [11].

The long-distance fiber-optic QKD system can be applied for short-distance fiber links as well, when the fiber links are composed of single-mode fibers. However, short-distance fiber links, i.e., local area networks (LANs), are often composed of multimode fibers for the reduction in cost. Since a multimode fiber can support many spatial modes, modal interference and dispersion may cause instabilities or errors in QKD. This situation is similar to the case in which a wavelength in the first optical fiber communication window (~ 830 nm) is employed for QKD over a standard single-mode telecommunication fiber link. Although the standard single-mode fiber can support more than one spatial mode at the wavelength, the stable single-mode behavior was readily attained over a 1.1 km single-mode fiber spool [12].

In this paper, we show that the long-distance QKD system designed for operation over standard single-mode fibers can also be used to distribute keys over multimode fibers. QKD over a multimode fiber is very important in practice, because we can use any installed LANs for a quantum channel without any additional costs. First, we consider a possibility of a graded-index (GI) multimode fiber for a quantum channel, and we show that transmission of light in an underfilled mode distribution is available for a quantum channel. Then, we show experimental results of a QKD using an installed GI multimode fiber link. The time-multiplexed interferometer that we adopted as a QKD system is similar to that presented in Ref. [5], except that the losses at the receiver’s station are minimized by use of polarization-maintaining optical fibers and LiNbO₃ waveguide optical modulators [13]. The losses at the receiver’s station must be as low as possible in QKD over a multimode fiber, because the coupling losses between the single-mode fiber at the receiver’s station and the multimode fiber are fairly large.

2. Measurement of group velocity dispersion

If a multimode fiber link is adopted as a quantum channel, an interference visibility of the QKD system deterioriates as the result of a large group velocity dispersion (GVD) in the multimode fiber. Therefore, it seems that a quantum communication cannot be realized using the multi-mode fiber link. However, if only a few propagation modes are excited in a multimode fiber without transition to the equivalent mode distribution (EMD), namely an underfilled mode distribution (Fig. 1), the GVD in the multimode fiber can be suppressed. In particular, a performance of the multimode fiber is similar to a single-mode fiber when only one mode is excited in the multimode fiber. As a consequence, transmission of light in the extremely underfilled mode distribution is capable of applying the multimode fiber to a quantum channel.

 

Fig. 1. Illustration of the mode distribution in a graded index multimode fiber: (a)Equivalent mode distribution (b) Underfilled mode distribution

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The single-mode fiber enables light to propagate in a single mode with a small field diameter. Therefore, when a butt junction is used through which light from the single-mode fiber is coupled into a multimode fiber, only one mode can be excited in the center of the core of the multimode fiber. The remarkable point is whether the input state in the extremely underfilled mode distribution is destroyed or not in the multimode fiber. In order to demonstrate it, we measured GVDs in a multimode fiber. Figure 2 shows the experimental setup for GVD measurements based on photon counting [14]. A 1550 nm distributed feedback (DFB) diode laser (Hama-matsu PLP-01) sends ≃ 50 ps pulses to Fiber 1. They are coupled into Fiber 2 through the butt junction and finally detected by a single-photon detector with a 1 m single-mode fiber pigtail. Here, the standard FC/PC couplers (fiber-to-fiber coupler) were used to connect these fibers. The detector is InGaAs/InP avalanche photodiode (EPM239BA Epitaxx) operated by a gated mode [9]. Here, the gate width is set to 20 ns, which is much longer than the light pulse width. (More details of the single-photon detector are in Section 3.) The multichannel scaler (MCS) records an arrival time of photons in the pulse. We can obtain a profile of light pulses from the tested fiber by constructing a histogram of the time of each photon’s arrival. We tested three combinations of fibers (see Table 1). The GI multimode fiber we used has NA of 0.275 and core/cladding of 62.5/125 (the size is in microns), and the single-mode fiber has NA of 0.11 and core/cladding of 9/125. In Test 1, we measured the pulse width after the ideal single-mode transmission in the 50 m single-mode fiber. In Tests 2 and 3, we measured the pulse width after transmission in the 550 m GI multimode fiber (an installed LAN). Here, notice that the input states of Tests 2 and 3 are the EMD for the GI multimode fiber and the single-mode (the extremely underfilledmode distribution), respectively. Measured pulse profiles of these tests are shown in Fig. 3. Although the laser pulse width is ~50 ps, measured pulse width is ~300 ps (Test 1). The 50 m long (dispersion shifted) single-mode fiber has the dispersion slope of only ~ 1ps/nm/km ² (at the zero dispersion wavelength of 1550 nm). Therefore, the pulse width was not widened via the dispersion of the single-mode fiber, and it is caused by the timing resolution of the single-photon detector [8]. However, the timing resolution of the measurement system is enough to confirm that the pulse width is widened via the large GVD in the GI multimode fiber (Test 2). The remarkable result is in Test 3. Although the light pulse passes through the GI multimode fiber, the GVD is very small, and the profile of the light pulse is almost identical with that of Test 1. This result suggests that almost all the energy of the light pulse is in the single mode and a small fraction of the energy is shifted to higher-order propagation modes, although the light pulse propagates in the GI multimode fiber.

 

Fig. 2. Measurement of group velocity dispersion based on a photon counting.

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Fig. 3. Profiles of light pulses after transmission through (a) only the single-mode fiber (Test 1), (b) only the multimode fiber (Test 2), (c) the multimode fiber after the single-mode fiber (Test 3).

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Table 1. Tested combination of fibers

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To obtain another aspect of the property, we also measured the NA of an output from the GI multimode fiber through a conventional knife-edge experiment. When the combination of fibers matched that of Test 2, the measured NA was 0.27. The result is in good agreement with the NA of GI multimode fiber. On the other hand, when the combination of fibers matched that of Test 3, the measured NA was 0.12. This value is close to the NA of the single-mode fiber rather than the multimode fiber. The butt junction through which light from the GI multimode fiber is coupled into the single-mode fiber usually has large optical losses (~30 dB), resulting from a mismatch of the core diameter and the NA. However, since the NA and the mode field diameter in the extremely underfilled distribution match the single-mode fiber, optical losses of the junction were reduced to 3~5 dB. Moreover, the energy rejected at the junction is in higher-order modes excited in the multimode fiber. In other words, through the filtering effect, only the energy having the single-mode behavior in the GI multimode fiber is efficiently coupled into the single-mode fiber. As described above, the extremely underfilled transmission improves performances of the GI multimode fiber link.

However, when the light propagates through the much longer GI multimode fiber, the more energy propagating in GI multimode fiber is in higher-order modes. This indicates that the underfilled behavior deteriorates and the propagation-mode distribution would eventually reach the EMD. In the EMD, the optical losses in the connection between the GI multimode fiber and the single-mode fiber will be huge. Therefore, a necessary length for the transition to the EMD will set a limit of the communication distance in the GI multimode fiber channel. Assuming that the fiber is a high-quality silica fiber, the necessary length is generally more than 10 km. However, the length may be shortened by an environment of a fiber channel, for example, a stress on the fiber advances the transition to the EMD.

3. Experimental setup

Next, we performed the QKD experiment to demonstrate the performance of the GI multimode fiber link for a quantum channel. A schematic diagram of our QKD system is shown in Fig. 4. The system is composed of a polarization-splitting time-multiplexed interferometer and In-GaAs/InP single-photon avalanche photodiodes operated in gated mode. A sender (Alice) and a receiver (Bob) are separated by a 10.5 km spool of dispersion-shifted single-mode optical fiber (Corning LEAF fiber) or a 550 m installed LAN of GI multimode optical fiber. At Bob’s station, a 1550 nm (≃ 50 ps) pulses is injected into the polarization-maintaining fiber (PMF) connected to the input port of a circulator. The polarization of the optical pulses is oriented along the slow axis of the PMF. The slow axis of the PMF from the output port of the circulator is rotated by 45 degrees and connected to the input port of the first polarizing beam splitter (PBS₁). Then the pulses are again horizontally polarized as they pass through PBS₁. The slow axis of the PMF from the output port of PBS₁ is rotated by 45 degrees and connected to the input port of the second polarizing beam splitter (PBS₂). Then PBS₂ evenly divides the pulse into orthogonal components P₁ and P₂. P₁ passes straight through PBS₂ and onto the 10.5 km single-mode fiber spool or the 550 m installed multimode fiber LAN, while P₂ is sent through a polarization-maintaining fiber loop, which delays it by ≃ 70 ns before it returns to PBS₂ and is sent onto the spool or the LAN. A LiNbO₃ waveguide phase modulator PMb (Sumitomo Osaka Cement PM4-140-11; insertion loss = 1.8 dB) in the delay loop imparts no phase shift to outgoing pulses.

Each orthogonally polarized pulse pair arrives at Alice’s station in unspecified elliptical polarization states. They pass through a 50/50 fiber-optic coupler, an attenuator, and Alice’s phase modulator PM_A (PM4-140-15). The Faraday mirror reflects the pulses and rotates the polarization states by 90 degrees, which interchanges the elliptical polarization states of P₁ and P₂. Due to this polarization interchange, P₁ and P₂ will accumulate equal overall phase shifts in the course of completing a round trip back to PBS₂, provided that the properties of the optical path change slowly relative to the time required for a round trip. If Alice deliberately violates this condition by rapidly stepping the voltage V_A applied to PM_A between the times that P₁ and P₂ pass through it, she can encode information as a differential phase shift ϕ_A = ϕ _P2 - ϕ _P1 between the P₁ and P₂ pulses. She implements Bennett’s four-state protocol for encoding key data by randomly choosing between voltages V_A = -1.45, 0, 1.45, and 2.9 V that give the four phase shifts ϕ_A = -π/2,0, π/2, and π.

 

Fig. 4. Schematic diagram of quantum key distribution system: C Circulator; PBS₁, PBS₂ Polarizing beam splitters; SMF Single-mode fiber; PMF Polarization-maintaining fiber; MMF Multimode fiber; FC 50/50 fiber-optic coupler; AT Attenuator; PM_A, PM_B Phase modulators; FM Faraday mirror; D₁, D₂ Single-photon detectors; PG₁, PG₂ Pulse generators; AWG_A, AWG_B; Arbitrary waveform generators.

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The returning pulse pair is attenuated to the single-photon level as it leaves Alice’s station. The Faraday orthoconjugation guarantees that the returning pulse components arrive at PBS₂ linearly polarized and orthogonal to their outgoing polarization. Thus the P₁ component arrives vertically polarized and is sent through the delay loop, while the initially delayed P₂ component arrives horizontally polarized and passes straight through PBS₂. For each returning pulse pair, Bob can use the phase modulator PM_B to give a phase shift ϕ_B to P₁ as it passes through the delay loop. By randomly choosing ϕ_B = 0 or -π/2, Bob can switch between two nonorthogonal bases for analyzing the returning photons, as required by the four-state protocol.

The P₁ and P₂ components then arrive simultaneously at PBS₂ and recombine into a single pulse P₀ with a polarization state that depends only on the difference of the applied phase shift, Δϕ = ϕ_A - ϕ_B. The polarization of P₀ is oriented along the fast axis of the PMF connected to PBS₁ when the phase difference Δϕ = 0. Then the P₀ is reflected by PBS₁ and detected by the single-photon detector D₂. On the other hand, the polarization of P₀ is oriented along the slow axis of the PMF when the phase difference Δϕ = π. Then the P₀ passes through PBS₁ and is detected by the single-photon detector D₁. For mismatched basis choices, where Δϕ = -π/2, π/2, or 3π/2, the recombined pulses will be circularly polarized, and PBS₁ will randomly divide the photons between the two detectors with equal probability.

Our single-photon detectors are Epitaxx EPM239BA APDs operated in gated mode at -55 degree Celsius [9]. The DFB laser triggers first a delay generator (Stanford Research Systems DG535), which provides the timing of our QKD system. Then the delay generator triggers pulse generators (PG₁ and PG₂) that generate gate pulses. Each gate pulse has an amplitude of 4.5 V and a full width at a half-maximum of approximately 1 ns. The repetition frequencies of the gate pulses, i.e., laser pulses, are 9 kHz for the 10.5 km spool and 100 kHz for the 550 m LAN, which are limited by the longest delay in the system. The delays are set such that the photons impinge upon the APD’s sensitive area when it is gated on. A gated passive quenching circuit [15] with a 47 kΩ series resistance providing the dc bias and the APDs are placed in a cryostat. The gate pulse was superimposed on the dc bias voltage and the signal was measured over a 50 Ω resistance. We controlled the excess bias voltage above breakdown by adjusting the dc bias voltage. In the QKD experiments, the excess bias voltage was set to 3.5 V. The avalanche signal was finally discriminated and registered by a photon counter (Stanford Research Systems SR 400). The delay generator also triggers arbitrary waveform generators (AWG_A and AWG_B), which control the timing when the phase modulators are biased.

 

Fig. 5. Count rates of single-photon detectors D₁ and D₂ versus Alice’s modulator voltage. The average photon number per pulse was ≃ 2, and Bob’s phase shift Δϕ_B was set to 0 or -π/2. (a) 10.5 km single-mode fiber spool, (b) 550 m multimode fiber local area network.

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4. Experimental results

The interference fringes for the 10.5 km spool and the 550 m LAN are shown in Fig. 5. The single-photon counting was performed using D₁ and D₂ while changing Alice’s modulator voltage from -5 to 5 V when Δϕ_B = 0 or -π/2. Here the average photon number per pulse was set to ~2 by the monitoring of another output of the 50/50 coupler using one of the single-photon detectors. The count rate of D₁ results from attenuation by the 10.5 km spool (L ₁ ~2.4 dB) or the 550 m LAN (L ₂ ~5 dB including the coupling losses between the multimode and single-mode fibers), the optical losses leading to D₁ in Bob’s station (L _D1 ~4.0 dB), and the quantum efficiency of D₁ (η ₁ ~ 23 %). On the other hand, the count rate of D₂ results from attenuation by L ₁ or L ₂, the optical losses leading to D₂ in Bob’s station (L _D2 ~3.5 dB), and the quantum efficiency of D₂ (η ₂ ~19%). Here the optical losses L _D1 and L _D2 were measured by connecting the 50/50 coupler directly to PBS₂. The optical losses in Bob’s station are much smaller than those in Ref. [5] (~ 10dB), which are realized by rotating PMFs instead of inserting polarization rotators and using a low-loss LiNbO₃ waveguide optical modulator.

The fringe visibility V₁ at the output port leading to D₁ is 0.9990, and V₂ at the output port leading to D₂ is 0.9975 for the 10.5 km single-mode fiber spool. On the other hand, the visibilities V₁ and V₂ for the 550 m LAN are 0.9973 and 0.9967, respectively. Here the optical losses of the 550 m LAN are approximately 2 dB. Although the optical losses of GI multimode fiber are only 0.3 dB/km, attenuation by several connectors in the LAN results in such large optical losses. In addition, the optical losses from the LAN to Bob’s station are approximately 3 dB, which is caused by NA and core diameter mismatches between the multimode and the single-mode fiber. As described in Section 2, the relatively small optical losses indicate that only a few modes are excited in the 550 m GI multimode fiber LAN. The single-mode fiber in Bob’s station plays the role of a filter to reject higher-order modes excited in the LAN. Therefore only the energy having the single-mode behavior in the LAN was efficiently coupled into the single-mode fiber in Bob’s station. As a result, the high fringe visibilities were obtained, although the LAN was composed of the multimode fiber.

Table 2. Summary of quantum-bit-error rates and bit rates

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In the case that the fiber channel is the GI multimode fiber, it seems that the channel security is lower than the single-mode fiber channel, resulting from a facility for extracting photons in higher-order modes. However, this eavesdropping is the same as the conventional beam-splitting attack, because photons in higher-order modes are excited at random with a transition probability and are detected by the eavesdropper. Moreover, the secret key sharing cannot be carried out using photons in higher-order modes, because the photons in higher-order modes are rejected by the single-mode fiber in Bob’s station. Therefore, the security of the GI multimode fiber channel is the same as the single-mode fiber channel, except that the additional optical losses caused by the coupling into single-mode fiber in Bob’s station increases bit errors (details of the bit errors are described below).

The performance of our QKD system should be evaluated by use of a quantum-bit-error rate (QBER) given by [11]

QBER_opt is simply the probability for a photon to hit the wrong detector. This is a measure of the quality of the optical alignment of the polarization maintaining components and the stability of the fiber link. QBER_dark and QBER_after, the errors due to dark counts and after-pulses, depend on the characteristics of the single-photon detectors. QBER_stray is the errors induced by stray light, essentially Rayleigh backscattered light [4]. In our experiments, QBER_after can be negligible because the repetition frequencies of the laser pulses (9 kHz or 100 kHz) are much lower than the frequency above which the afterpulsing increases the errors [8, 9]. Moreover, QBER_stray can also be negligible because an optical pulse is not emitted before the previous one returns in the repetition frequencies. Therefore we will discuss QBER_opt and QBER_dark below.

QBER_opt for the compatible phase setting is given by [16]

where C_right and C_wrong are the correct and false counts subtracting dark counts. Using the data in Fig. 5, we can calculate the mean value over all four possible compatible phase settings. The averaged QBER_opt for the 10.5 km spool is 1.23×10^-3 and that for the 550 m LAN is 1.46×10^-3. On the other hand, QBER_dark is given by

where P_dark,i is the probability of getting a dark count in detector D_i, and P_phot is the probability of detecting a photon given by

In our QKD experiments performed by randomly changing ϕ_A and ϕ_B, the average photon number per pulse μ was set to 0.1. The dark-count probabilities P _dark,1 and P _dark,2 are 2.2×10^-5 and 3.0 × 10^-5, respectively. The measured QBERs and bit rates with the values estimated using Eqs. (1)-(4) are summarized in Table 2. The measured QBERs and bit rates are in good agreement with the estimated ones.

5. Conclusions

In conclusion, we have developed a fiber-optic QKD system at 1550 nm based on a polarization-splitting time-multiplexed interferometer incorporating InGaAs/InP single-photon avalanche photodiodes operated in gated mode. Using the system, we have demonstrated QKD over a 550 m installed multimode fiber LAN with a QBER of 1.09 %. Coupling losses between the multimode fiber LAN and the single-mode fiber was reduced by the underfilled mode distribution. Moreover, high fringe visibilities are obtained because of not only the underfilled transmission but also a rejection of higher-order mode by the single-mode fiber in Bob’s station. However, the optical losses at the junction would be large if a number of modes are excited (transition from the underfilled mode distribution to the equivalent mode distributions) in the multimode fiber by extending the LAN. The coupling losses will set a limit to the communication distance in local area QKD using multimode fiber.