## Abstract

In this paper, we propose a polarization-insensitive phase modulation scheme based on frequency modulation of light waves using either one or a pair of acousto-optic modulators. A stable Sagnac quantum key distribution (QKD) system employing this technique is also proposed. The interference visibility for a 40km and a 10km fiber loop is 96% and 99% respectively, at single-photon level. We ran standard BB84 QKD protocol in a simplified Sagnac setup (40km fiber loop) continuously for one hour and the measured quantum bit error rate stayed within 2%–5% range.

© 2006 Optical Society of America

## 1. Introduction

One important practical application of quantum information is quantum key distribution (QKD), whose unconditional security is based on the fundamental law of quantum mechanics [1, 2]. In phase-coding QKD systems, the legitimate users (Alice and Bob) achieve phase encoding/decoding with phase modulators (PM) and Mach Zehnder interferometers (MZI) [3, 4]. However, there are a few practical difficulties if QKD is to be implemented over long distance through fiber: namely, phase and polarization instabilities. Promising progresses have been achieved by using active feedback control to stabilize the interferometer [4], but better stability was demonstrated with the “plug & play” auto-compensating QKD structure employing a Faraday mirror [5]. Like the “plug & play” system, the Sagnac loop also offers phase stability and polarization stability, as the two interfering signals travel through the same path, but its structure is in principle much simpler than the “plug & play” scheme [6–9]. However, all reported Sagnac QKD systems employed polarization-sensitive phase modulators, requiring complicated polarization controls, which makes this scheme unattractive. For example, four polarization controllers were employed in [8], and the interference visibility for a 5 km fiber loop was only 87%.

We note that the design of QKD system will be greatly simplified if a high-speed polarization insensitive phase modulator is available. In this paper, we present a design for such a polarization-insensitive phase modulation scheme together with a Sagnac QKD system, and demonstrate stable QKD operation over one hour without feedback control.

## 2. Phase modulation with frequency shifters

Figure 1(a) shows the basic structure of the phase modulation scheme based on frequency shifter, which consists of an acousto-optic modulator (AOM), followed by a fiber with length *L*.

For the first order diffracted light, the AOM will introduce a frequency shift equal to its driving frequency *f* (due to the Doppler effect). The phase of the diffracted light is also shifted by an amount of-*ϕ*(*t*), which is the phase of acoustic wave at the time of diffraction [10]. Assuming two light pulses, *S _{1}* and

*S*, are in phase and are sent to the phase modulator at the same time from opposite directions as shown in Fig. 1(a). They will reach the AOM at different times with a time difference

_{2}Here *n* is effective index of fiber and *c* is the speed of light in vacuum. The phase difference between *S _{1}* and

*S*after they go through the phase modulator will be

_{2}By modulating AOM’s driving frequency *f*, the relative phase between *S _{1}* and

*S*can be modulated. This is the basic mechanism of our AOM-based phase modulator. We remark that the frequency of light will be up-shifted by this phase modulator by an amount

_{2}*f*. To remove this “side-effect”, we can add another frequency down shifter at the other end of the fiber, as shown in Fig. 1(b). In Fig. 1(b), the two AOMs, which are driven by the same driver, will shift the frequency of light by the same amount but with different signs. So the net frequency shift will be zero. Since a down-shift AOM will shift the phase of the diffracted light by an amount of -

*ϕ*(

*t*), the resulting phase difference between

*S*and

_{2}*S*after they go through the phase modulator can be derived as

_{1}Compared with the *LiNbO*
_{3} waveguide-based phase modulator, this phase modulator is insensitive to the polarization state of the input light, and the phase delay can be controlled precisely by the acoustic frequency *f*.

The ultimate operating rate of this phase modulation scheme is mainly determined by two factors: the fiber length L in Fig. 1 and the response time of the AOM. We tested the phase modulation rates of both type 1 [Fig. 1(a)] and type 2 [Fig. 1(b)] phase modulators by putting them in a fiber Sagnac loop (similar to the design in Fig. 3). The interference patterns were recorded while the driving frequency of the AOM hopped between two values (corresponds to 0 or π phase delay). Figure 2 shows the testing results of the two phase modulators at 1MHz phase modulation rate (L ~15m). The sharp edge indicates that a phase modulation rate of 10 MHz could be achieved. Note in Fig. 2(b), there is a 240ns step in the middle of each edge. This is induced by the asymmetry between the two AOMs (for different AOMs, the optical beam intercepts the acoustic wave at a slightly different lateral location). In practice, we could eliminate this problem by choosing two matched AOMs or by introducing a fixed delay in one of the driving signals to the AOM.

## 3. QKD based on fiber Sagnac interferometer

A fiber agnac QKD system employing this novel phase modulator is shown in Fig. 3. To realize the BB84 protocol, Alice randomly encodes the relative phase between the clockwise and counterclockwise light pulses with the AOM-based phase modulator PM1, while Bob randomly chooses his measurement basis with phase modulator PM2. We remark the transmittance of AOM can also be modulated by modulating the amplitude of its driving signal, so, the same device can function as an amplitude modulator as well as a phase modulator.

To demonstrate the feasibility of our design, we have performed experiments with a simplified system as shown in Fig. 4.

Here, only Alice holds a phase modulator, while Bob always chooses the same measurement basis. This simplified system allows us to evaluate the essential properties of the proposed QKD system (Fig. 3), namely stability and quantum bit error rate (QBER), with only one AOM. Here, the cw output from a *1550nm* laser (L) is modulated by an amplitude modulator (AM) to generate 500ps laser pulses. Each laser pulse is split into *S _{1}* and

*S*at a symmetric fiber coupler, which go through a long fiber loop (

_{2}*L*+

_{1}*L*~40km) in the clockwise and counterclockwise directions, respectively. The interference patterns at Ch1 and Ch2 are measured by two InGaAs single photon detectors (SPD, Id Quantique, id200), which work in gated mode. For a 5ns gating window, the overall detection efficiency is ~10% and the dark count probability is 5×10

_{2}^{-5}per gating window. A fiber-pigtailed AOM (Brimrose inc.) is placed inside the fiber loop asymmetrically (

*L*–

_{1}*L*is about 700m, which is the length difference of two 20km fiber loop. There is no specific requirement for the length difference). Due to this asymmetry, phase modulation between

_{2}*S*and

_{1}*S*can be achieved by modulating AOM’s driving frequency, similar to the phase modulator shown in Fig. 1(a). Because of the birefringence in the fiber loop, the polarization states of S1 and S2 could be different after they go through the fiber loop [12]. This is compensated by a polarization controller (PC). The synchronization is achieved as follows: A pulse generator (PG), which is triggered by a function generator (FG1), drives the amplitude modulator (AM) to produce 500ps laser pulses. FG1 also triggers a delay generator (DG, Stanford research system, DG535), which in turn produces two gating signals for SPD1 and SPD2, and one trigger signal for a data acquisition card (NI, PCI-6115). The AOM is driven by another function generator FG 2, whose frequency is controlled by the data acquisition card.

_{2}## 4. Experimental results

We measured the interference visibility by scanning the frequency of FG2 while recoding the outputs from the two SPDs. The average photon number per pulse (out from Alice’s side) was set to be 0.8, which matched with signal photon level in a decoy state QKD system [13]. The measured visibility for a 40km fiber loop was about 96%, as shown in Fig. 5 (For a 10km fiber loop, it was about 99%). Note the optical loss in Ch2 is higher than that in Ch1 due to additional loss from the fiber circulator. This could be compensated by adjusting the efficiencies of the two SPDs.

To run the BB84 protocol, a random number file (1Kbits) is preloaded to the buffer of the data acquisition card. This random file contains a sequence of four discrete values corresponding to the four phase values in the BB84 protocol {0, π/2, π, 3π/2}. Once triggered, the data acquisition card reads out a value from the random file and sends it to FG2 to encode Alice’s phase information. The data acquisition card also samples the outputs from SPDs (Bob’s measurement results) into its input buffer. In this preliminary setup, Bob always uses the same basis for his measurement. After transmitting 100K bits, Alice and Bob can estimate the QBER by comparing the data contained in Alice’s random file and Bob’s measurement results. We ran the system continuously for about one hour without any adjustment, the QBER drifted slowly from 2% to 5%, as shown in Fig. 6. In practice, to improve the long-term stability, simple recalibration process can be employed, as in other QKD systems. During this experiment, the pulse repetition rate was set to 1 KHz for easy synchronization.

For a typical AOM, its frequency modulation rate is in the range of 1–10MHz, which is compatible with the operation rate of today’s QKD system. We remark that in our simplified setup, the frequency of the laser pulse output from Alice depends on the encoded phase information (due to the frequency shift induced by the AOM). The difference between the shifted frequencies is in the order of 10^{5} Hz (for 700m fiber length difference), which is much smaller than the spectral width of a nanosecond pulse (~10^{9} Hz). It’s impossible for an eavesdropper (Eve) to decode the phase information by measuring the spectrum of a single photon pulse. On the other hand, Eve could explore this imperfection and launch a “phase remapping attack” [14]. We remark that this security loophole can be closed by employing the phase modulator based on a pair of AOM [as shown in Fig. 1(b)], which introduces no frequency shift. Another possible solution is that instead of modulating the frequency of the RF signal [as shown in Fig. 1(a)], we could modulate the phase of the RF signal directly. In this case, the frequency shift will be independent of Alice’s phase information.

## 5. Conclusion

We proposed a novel polarization-insensitive optical phase modulation scheme based on frequency shifts. We have demonstrated the operation of such a phase modulator at MHz rate, as well as a stable Sagnac QKD system employing this phase modulation technique. Compared with previous Sagnac QKD schemes based on polarization-sensitive phase modulators, our system achieved lower QBER over longer fiber. Experimental results showed an interference visibility of 96% and 99% for a 40-km and a 10-km fiber loop, respectively, at single photon level. With this novel polarization-insensitive phase modulation scheme, we expect the performance of many practical QKD systems can be greatly improved.

## Acknowledgments

Financial supports from NSERC, CRC Program, CFI, OIT, PREA, and CIPI are gratefully acknowledged.

## 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* (Institute of Electrical and Electronics Engineers, New York, 1984), pp.175–179.

**2. **N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. **74**, 145–195 (2002). [CrossRef]

**3. **R. J. Hughes, G. L. Morgan, and C. G. Peterson, “Quantum key distribution over a 48 km optical fiber network,” J. of Mod. Opt. **47**, 533–547 (2000).

**4. **Z. Yuan and A. Shields, “Continuous operation of a one-way quantum key distribution system over installed telecom fibre,” Opt. Express **13**, 660–665 (2005). [CrossRef]

**5. **A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, “Plug and play” systems for quantum cryptography,” Appl. Phys. Lett. **70**, 793–795 (1997). [CrossRef]

**6. **T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, “Circular type” quantum key distribution,” IEEE Photon. Technol. Lett. **14**, 576–578 (2002). [CrossRef]

**7. **C. Y. Zhou and H. P. Zeng, “Time-division single-photon Sagnac interferometer for quantum key distribution,” Appl. Phys. Lett. **82**, 832–834 (2003). [CrossRef]

**8. **C. Y. Zhou, G. Wu, L. E. Ding, and H. P. Zeng, “Single-photon routing by time-division phase modulation in a Sagnac interferometer,” Appl. Phys. Lett. **83**, 15–17 (2003). [CrossRef]

**9. **P. D. Kumavor, A. C. Beal, S. Yelin, E. Donkor, and B. C. Wang, “Comparison of four multi-user quantum key distribution schemes over passive optical networks,” J. Lightwave Technol. **23**, 268–276 (2005). [CrossRef]

**10. **A. Stefanov, H. Zbinden, N. Gisin, and A. Suarez, “Quantum entanglement with acousto-optic modulators: Two-photon beats and Bell experiments with moving beam splitters,” Phys. Rev. A **67**, 042115 (2003). [CrossRef]

**11. **E.-B. Li, J.-Q. Yao, D.-Y. Yu, J.-T. Xi, and J. Chicharo, “Optical phase shifting with acousto-optic devices,” Opt. Lett. **30**, 189–191 (2005). [CrossRef]

**12. **D. B. Mortimore, “Fiber loop reflectors,” J. Lightwave Technol. **6**, 1217–1224 (1988). [CrossRef]

**13. **Y. Zhao, B. Qi, X.-F. Ma, H.-K. Lo, and L. Qian, “Experimental quantum key distribution with decoy states,” Phys. Rev. Lett. **96**, 070502 (2006). [CrossRef]

**14. **C.-H. F. Fung, B. Qi, K. Tamaki, and H.-K. Lo, “Phase-remapping attack in practical quantum key distribution systems,” arXiv:quant-ph/0601115 (2006) http://xxx.lanl.gov/abs/quant-ph/0601115