◎ Guang Wu, Chunyuan Zhou, and Heping Zeng

　　Abstract：We introduce a stable, long-distance single-photon Sagnac interferometer, which has a balanced configuration to compensate efficiently phase drift caused by change of the fiber-optic path. By using time-division phase modulation, single-photon interference was realized at 1550 nm in a 5-km-long and 27-km-long Sagnac fiber loop, with a fringe visibility higher than 90% and long-term stability. The stable performance of the single-photon interference indicated that the time-division phase-modulated Sagnac interferometer might readily lead to practical applications in single-photon routing and quantum cryptography.
　　Key words：Sagnac single-photon interferometer； single-photon interference, single-photon routing, quantum cryptography

　
Ⅰ、Introduction
     Quantum cryptography is based on information processing with single quanta, typically single photons, as quantum bits (qubits). It is well-known that an unknown quantum state can never be duplicated and that any measurements will unavoidably demolish the quantum system. Therefore, eavesdroppers cannot take a measurement without being revealed [1]. As a result, quantum cryptography can guarantee a real absolute secrecy. Up to now, two methods have been demonstrated efficiently to encode qubits. One is to encode the quibts in the polarization of a single photon [2-4], which is mainly used in free-space quantum cryptography, while the other is to encode in the phase of a single photon, which is mostly used in optic-fiber quantum cryptography. And the latter is based on single-photon interference [5-8], which is critically dependent upon the which-way information the photon is encoded. A typical interferometer consists of two arms. At a certain port, the photon selects one arm, and interferes with itself at the output port due to influence from the other arm. Actually, the single-photon interference is determined by the path difference (phase difference) between the two arms. An optic-fiber Mach-Zehnder interferometer can be used to realize the single-photon interference, where the interference can be simply controlled with a phase modulator in one arm to produce a phase difference . However, the phase difference is detrimentally sensitive to the environmental influence due to either unavoidable thermal effects or mechanical stress on optic fiber. And polarization-mode dispersion (PMD) in long-distance fibers destabilizes the interference further. When the interferential arms are over several meters long, the single-photon interference becomes very unstable. In order to solve this problem, asymmetrical double Mach-Zehnder interferometers [10] were used in some schemes. Yamamoto and his coworkers proposed a scheme on the basis of differential-phase-shifted asymmetrical Mach-Zehnder interferometer [11]. Unfortunately, these schemes could not solve the problem very well. Recently, Faraday mirrors have been used in single-photon interferometers to compensate automatically the influence from the environmental fluctuation and PMD
    effects [12,13], which have already enabled successful realization of stable single-photon interference with fringe visibility up to 99.8% [12,13]. Nevertheless, its optic loop is relatively complicated. In this paper, we introduce an efficient and simple scheme by using a Sagnac optic-fiber loops instead of Faraday mirrors to compensate the slowly-varied phase drifts caused by the environmental fluctuations.
Ⅱ、Theory of the experiment
     In our experiment, we used a single-photon interferometer as shown in Fig.1. The Sagnac interferometer [14] was made up of a 50%:50% coupler, four polarization controllers, a phase modulator, a long fiber, and a delay fiber. The laser pulse was attenuated to a quasi single-photon level with an attenuator. The quasi-single-photon pulse entered the Sagnac loop through the coupler after a circulator, and interfered with itself at the coupler after passing through the Sagnac loop clockwise or counterclockwise, which was detected at both output ports after the coupler.
     As shown in Fig. 1, the single-photon a0 incident from port3 of the coupler, becomes the counter clockwise single-photon a1 or the clockwise single-photon a2, and returns to the coupler through the annular optic loop. We can then describe the process by the followings.
     ,
     ,
    where a0,a1,a3, and a4 denote the single-photon operators of the corresponding paths, respectively, r, t and r', t' are the reflectivity and transmission coefficients of port3 and port4, respectively. As the reflection/transmission ratio of the coupler is 50%:50%, the absolute values of r, t and r', t' equal to . Then, the photon numbers returning to port3 and port4 are
    
    where is the additional phase difference after twice reflection in the coupler, is the phase difference between the clockwise and counterclockwise pathways in the Sagnac loop, is the mean input photon number at port3.
     It is obvious that the phase difference equals to zero at the static state, and the interference is independent of the modulating voltage. We can see , that means the single photon always reaches detector D1. If there is a time-division phase difference between the clockwise and counterclockwise photon pulses, the single photon reaches the detectors D1 and D2 with probabilities of and , respectively.
     In order to control the single-photon interference, we employed a scheme based on time-division phase modulation. We modulated the laser diode by a synchronizing signal to produce the laser pulse, and sent the synchronizing signal to the photon counter and delay signal generator. The time-division phase modulation is schematically shown in Fig.2, where t1 and t2 are the times a1 and a2 reaches phase modulator, respectively, t is the full-width at half-maximum (FWHM) of the single-photon laser pulse, is the width of the modulating pulse, is the delay period from the time the delay signal generator receives the synchronizing signal to the time it sends the modulating signal to the phase modulator, and is the periodic duration of the repetition cycle of the single-photon pulses. By adjusting the delay period and modulating pulse width , we can modulate a1 or a2 selectively. Only a1 is modulated when and , while only a2 is modulated when and . As the modulation voltage amplitude is changed during , different phase difference is obtained according to , where Vp is the half-wave voltage of the phase modulator.
    By time-division phase modulation, the interference is controlled effectively. Moreover, in a Sagnac single-photon interferometer, the single photons travel the same optic path with the same phase changes, the same chromatic dispersion and PMD. So the phase and polarization drifts caused by the environmental changes are well compensated, which will consequently eventuate quite high interference stability.
Ⅲ、Experimental setup and results
     We constructed a Sagnac interferometer as Fig.1. The laser diode was AVTECH AVO-9C-C-N-QTKA DFB at 1550 nm. And the single-photon detector was made of InGaAs/InP avalanche photon diode (EG&G C30644EJT-07). The avalanche photo diode (APD) was operated in so-called Geiger mode, at 210 K by Peltier cooling. The avalanche of the APD was quenched by a standard gate mode [15]. To reduce the dark count, we collected avalanche signals by using a coincidence-count technique, which switched on the coincidence gate only during the single-photon pulse reached the detector, thus the counter was only available during the coincident counting.
    To get an optimum condition for coincident counting, we compared the measurements with the various coincidence-gate widths from 5 to 20 ns in the experiments. The details of the experimental setup as schematically shown in Fig.1 are as follows. The lengths of the delay fiber and fiber-loop were 2 and 3 km, respectively. The average laser power was 7 nW, and the laser pulse had a pulse width less than 2 ns and a repetition rate of 10 kHz. The spectral width (FWHM) of output laser was about 6 nm. Note that the single-photon energy at 1550 nm is about . The average photon number in each pulse after a total attenuation of -75 dB was therefore . In our experiments, we selected the coincidence-gate width as 5, 10, and 20 ns, and scanned the voltage applied on the phase-modulator from -24~24V. We obtained a single-photon interference diagram as shown in Fig. 3, from which we can clearly see that the maximum count increased with the increase of the coincidence-gate width. Nevertheless, the dark count also increased with the increase of the coincidence-gate width. And consequently, there exists an optimum coincidence gate width to obtain a high fringe visibility for the single-photon interference. When the coincidence-gate width was 20 ns, the dark count increased observably. The interference fringe visibilities measured with the selected coincidence-gate widths of 5, 10, and 20 ns for the single-photon interferences at port3 and port4 were 95% & 97.8%, 97.7% & 97.6%, and 92% & 91.7%, respectively. The results show that the interference fringe visibilities obtained with 5-ns and 10-ns gates for the coincident counting were quite comparable, and better than that with a 20-ns gate. Note that the rising times of most avalanche signals were typically less than 5 ns under our experimental condition. While the dark count was in principle proportional to the coincidence-gate width, which implies that further increase of the coincidence-gate width as larger than 10 ns detrimentally increased the dark count rather than contributed to catch avalanche signals. According to the above experimental measurements, a coincidence-gate width within 5~10 ns was optimal to obtain a high Signal-to-Noise ratio for single-photon interference fringe visibility.
     Quasi-single-photon condition can be further improved by decreasing the laser power. To check the quasi-single-photon interference at different input conditions, we decreased the average output laser power down to 5 nW, which corresponded to an average photon number about 0.1 in each pulse. Under this circumstance, the probability that one laser pulse contains two photons was less than 0.5%. We selected a 10-ns gate for coincident counting. The obtained single-photon interference diagram is presented in Fig 4, where the fringe visibilities at port3 and port4 are 94% and 95%, respectively.
     To increase the length of the single-photon Saganc interferometer, we replaced the 3-km fiber by a 25-km one. The whole distance of the Sagnac fiber loop correspondingly became 27 km. The system to send and receive synchronizing signals was the same as used in the above experiments. To avoid for the sake of simplicity in analyzing the experimental results, we intentionally decreased the repetition rate to 5 kHz. We emphasize that there in principle exist no difficulties to increase the repetition rate up to 10 KHz or larger. In this experiment, the average output laser power was 15 nW, which was attenuated with a total attenuation of -76-dB to reach the average photon number per laser pulse. The coincidence gate width was selected as 5 ns. We obtained the single-photon interference diagram as given in Fig. 5, where the interference fringe visibilities at port3 and port4 were 94% and 95%, respectively.
    Ⅳ、Discussion and Conclusion
     It is of practical significance that long-distance single-photon interference can be realized with very high long-term stability. Our experimental results have clearly demonstrated that interference fringe visibility up to 98% could be obtained at the average photon number down to 0.1 and 0.2 in a 5-km-long Sagnac interferometer, and that single-photon interference fringe visibility could reach 95% in a 27-km-long Sagnac interferometer. Furthermore, the single-photon interferometer can work very stably and continuously in a long duration. Our measurements of the 5-km-long interference have run for 2.5 hours without any adjustments of the optic loop or polarization controllers inside the Sagnac loop. It indicated that the system could be maintained with a very high long-term stability. All these owe to the annular geometric configuration of the Sagnac loop.
    First, the clockwise and the counter-clockwise photons traveled the same optic loop, and the phase drifts of the two pathways caused by non-linear effects are only associated with the energy difference of laser pulses passing through the two pathways. Note that the laser energy was less than and the beam split ratio of the coupler was 50%:50%. Therefore, the energy difference was negligible, and accordingly, the phase drifts from any non-linear effects could be neglected. The variations of the environmental temperature and mechanical stress caused the lengths of the optic fibers to fluctuate, and thus the phase difference between two arms drifted. Since the clockwise and counter-clockwise photon pulses passed through a certain point of the optic loop at times differing at most the duration that a photon traveled around the whole loop (Tloop), during which the phase drifts varied negligibly, the clockwise and counter-clockwise single-photon pulses traveled the same optical path and therefore suffer the same slowly-varied phase drifts due to the annular geometric configuration of the Sagnac loop. Under standard (non-critical) circumstances, the fluctuation of fiber length is negligible in a time period less than Tloop, so the annular configuration of the Sagnac interferometer can enable efficient automatic compensation of the slowly varied phase drifts. On the other hand, PMD in the Sagnac fiber loop made the photon polarization fluctuate. In our experiments, we used Lucent single-mode fiber, , the PMD of 5-km fiber was less than 0.3 ps, while the spectral FWHM of the laser wavelength was 6 nm, corresponding to a coherence time , which was much longer than PMD. Furthermore, clockwise and counter-clockwise single-photons encountered the same PMD, which was also automatically compensated at the exit port of the Sagnac interferometer. We also used four Lefevre polarization controllers to adjust the polarization states of photons in the Sagnac loop in order to improve the stability of the single-photon interference.
    In summary, we have achieved single-photon interference with interference fringe visibilities up to 98% and very high long-term stability in a long-distance time-division phase-modulated Sagnac interferometer, which has clearly indicated that our Sagnac single-photon interferometer is of promising prospect in applications for quantum cryptography and single-photon routing. As an example of practical applications, we illustrated in Fig. 6 a quantum cryptographic system on the basis of a single-photon Sagnac interferometer. By just adding a phase modulator in the Sagnac loop, we can form the key sender (Alice) and receiver (Bob) for the quantum key distribution, with either the protocol BB84 [17] or B92 [9]. Up to now, we have already completed the cryptographic system by using a time-division phase-modulated single-photon Sagnac interferometer, and have realized key distribution primarily. Experiments to achieve long-distance quantum cryptography demo system with very high long-term stability are still in progress.
    
    Acknowledgements : This work was supported by Shanghai Priority Academic Discipline and the National Fundamental Research Program (No. 2001CB309301).
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    Fig.1. Schematic of Sagnac single-photon interferometer
    Fig.2. Signal and modulation timing diagram
     Fig.3. Comparison of single-photon interference detected with the coincidence-count gate width of 5, 10, and 20 ns
    Fig.4. Single-photon interference with a 5-km fiber loop at an average photon number =0.1.
    Figure 5. Single-photon interference with a 27-km fiber loop at an average photon number =0.6.
    Fig.6. Schematic of quantum cryptography with a time-division phase-modulated single-photon

　　作者简介：吴光，男，华东师范大学物理系光学专业2002级硕士研究生，研究方向为量子信息学；
周春源，男，华东师范大学物理系光学专业2001级博士生，研究方向为量子信息学。

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