Quantum key distribution has drawn considerable attention as a method of achieving a shared absolutely secure private key. If a sender and a receiver of information can share the key, secure communications of information composed of bit strings become possible. However, there are imperfections in the real world which can make it difficult to guarantee security and long distance (ultimately global) communications. For instance, loss by absorption in the quantum communication channel, imperfections in the light source, or inefficiencies of detectors in the detection system. Recent progresses in the field of quantum key distribution have made it possible to utilize a weak coherent source which has a finite multiphoton probability. However, it is still desirable to obtain single photon state for a higher secure key generation rate at longer distances. Here we consider a heralded photon source (HPS) which utilizes spontaneous parametric down conversion (SPDC). Photon pairs are generated by SPDC process in a χ(2) nonlinear crystal. One pump photon simultaneously splits into the photon pair (called signal and idler) where total energy and momentum are conserved. A HPS utilizes the property of the spontaneous generation which can be used in order to increase the communication distance. By getting coincidence between signal (heralded signal) and idler (heralding signal), we can substantially decrease the dark count probability and attain longer distance. Important features of the system are the availability of room-temperature operation, stability, and long-time operability guaranteed by nonlinear crystals which are nonbreakable against intense light. These are essentially required for the practical QKD. However, there is a higher multiphoton probability in the case of relatively large mean photon number close to unity. In this thesis an analysis of QKD utilizing a multiphoton reduced HPS and experimental demonstration of multiphoton reduction by a photon number resolving detector are shown.

　In chapter 3, quantitative estimations of a QKD system utilizing a HPS is given. If one can use a photon number resolving detector as a trigger detector of a HPS, the communication distance and secure key generation rate can be improved. Here time-multiplexed detector (TMD) consisting of commercially available single-photon detectors and optical fibers is considered. Because it can be operated at room temperature, HPS with a TMD can easily be implemented in a practical QKD system.

　The calculated secure key generation rate utilizing decoy state method is shown in Figure 1. The maximum distance of HPS (〓 171 km) becomes longer than that of weak coherent pulse (〓 140 km) even in the case of an imperfect trigger detector efficiency. Because cutoff distance is affected by a dark count probability, the decrease of dark count by coincidence detection leads to the longer distance. Though maximum secure key generation rate is lower than weak coherent pulse which is due to the imperfect detection efficiency and thermal property of the photon number distribution of a HPS, the secure key generation increases by utilizing a TMD compared with a detector which does not have an ability to resolve photon number.

　In chapter 4, a second-order correlation function was measured to prove removal of multiphoton. The multiphoton reduction by a TMD was real-time processed by fast logic gates. This measurement is different from usual measurement of correlation function on the point of the triggering (Fig. 2). The intensity ratio of the main peak to neighboring peaks is given by

while a threshold detector is given by

Here p(1) (p(2)) is the probability of emitting one (two) photon pair(s) in a pulse, P(lm) is TMD's detection probability of l counts for m incident photons. It is assumed that the detection efficiencies of two detectors after the 50/50 beam splitter are both equal to η.Since the 1st term in the denominators ≫ 2nd term (p(1) ≫ p(2) under the low μ condition in the present experiment), the denominators of the two ratios (I(TMD); I(th)) are nearly equal to each other. By using the following approximations, I(TMD) 〓 〓=〓，I(th)〓 〓=〓, the ratio of I(TMD) to I(th) is shown to give the degree of removal (I(reduction) = I(TMD)=I(th)〓P(1｜2)=(1 - (1 - ηA)2)). P(1｜2) is a probability that a TMD detects two photon as single photon and (1-(1-ηA)2) is a detection probability of a threshold detector. Because photon number is not resolved by a threshold detector, all signals including at least one photon are used as a key. Thus, I(reduction) is a probability that a TMD detects two photon as single photon normalized by that of a threshold detector. I(reduction) becomes zero in an ideal case. However, due to several imperfections such as inefficiencies, low coupling rates of detectors, and failure of mode separation in the TMD (0.25 for two photon in four mode TMD), the resulting peak is not very low (Fig. 3). Table 1 shows the ratios for four cases of setting (band width of interference filters = 1.5 nm and 10 nm, coupling fibers for the signal are singlemode and multimode fibers.) Photon number distribution changes thermal distribution to Poissonian in highly multimode cases. However, we can predict the value of ratio I(reduction) is constant irrespective of the photon number distribution, as the experimental results shown in Table 1 verify. It is shown that the multiphoton probability was reduced to 0.89 compared with a threshold detector was utilized. If it is assumed that detection efficiencies of the two trigger detectors are equal, the triggering efficiency is calculated as ηA 〓 0.28.

　The degree of improvement in the secure key generation rate by the adoption of the TMD is finally evaluated. Figure 4 shows the fractional improvement of secure key generation rate

R(decoy HPS-TMD)=R(decoyHPS-th) (R(decoyHPS-th) (R(decoyHPS-th)) is the secure key generation rate for the case of a TMD (threshold detector) is utilized as a trigger detector). In the case of the perfect trigger detection (ηA = 1), the degree of improvement is about 1.8 and rapidly increases from about 100km. This is due to the fact that in the case of a perfect trigger detection we can utilize just single photon signals in most of the case. However, as ηA decreases, the slope of key generation rate becomes steeper, and R(decoyHPS-TMD)/R(decoyHPS-th) is relatively reduced. About 10% improvement of the secure key generation rate by utilizing a TMD is attained in the case of ηA = 0:28 under the present experimental condition. The rate of improvement is nearly constant in the range up to 171km.

Figure 1: (a) Secure key generation rate vs distance. (b) optimal mean photon number vs distance. Detection efficiency of single photon detectors of TMD is ηA = 0.6.

Figure 2: Experimental setup. Ti:Sa: pulsed-Ti:sa laser, doubler: frequency doubler, BBO: type II beta-barium borate crystal, PBS: polarizing beam splitter, 50/50: half beam splitter, IF: interference filter, TIA: time interval analyzer, SPCM: single photon count-ing module (threshold detector), coinc.: coincidence detection system.

Figure 3: Triggered measurements of correlation function (left) TMD, (right) threshold detector. (band width of interference filters =1.5nm; a single mode fiber for the coupling of the signal (zoom around central peak). Noisy structure of TMD measurement is due to noise in the output voltage pulse from the logic circuit. Detailed results are in TABLE 1

Table 1: I(reduction), I(TMD), and I(th) for four parameter settings. All fibers on the side of trigger detectors are multi mode fibers.

Figure 4: Dependence of Improvement in the key generation rate between HPS-TMD scheme and HPSthreshold detector scheme R(decoyHPS-TMD)=R(decoyHPS-th) on trigger detection efficiency. ηA = 1, 0:28; and 0.1 from above. The photon number distribution is assumed to be thermal. Right edge of the figure is 171 km which is the same maximum distance realized by using an ideal single photon source.

　本論文は5章からなる。第1章は、イントロダクションであり、量子鍵配布の原理、実験上の問題点、および本研究の意義が述べられている。第2章は、量子鍵配布の基礎事項に関するものであり、量子鍵配布の具体的手順および安全鍵生成率の定式化、伝令光子源および時間多重検出器の概要などが述べられている。第3章では、伝令光子源の安全鍵生成率が定量的に解析されている。第4章では、時間多重検出器による伝令光子源の多光子低減の実験について述べられている。第5章では、本研究のまとめと今後の展望が述べられている。

　量子鍵配布の本質は、1ビットの情報を1光子によって伝送する点にある。しかしながら、理想的な単一光子源は未だに開発されていない。これまでの量子鍵配布実験には、1パルスあたりの平均光子数が1以下であるようなレーザー光(コヒーレント光)のパルス(weak coherent pulse)が光源として主に用いられてきた。しかし、コヒーレント光パルスの光子数はポアソン分布しており、2つ以上の光子を含む光パルスが一定の割合で存在する。そのような複数光子パルスは、第三者による盗聴を可能とするため、結果的に安全鍵生成率の劣化を招く。本研究の目的は、パラメトリック下方変換によって生成した光子対および光子数識別器を用いることにより、コヒーレント光パルスの欠点を回避した量子鍵配布の実用的光源を開発することにある。

　パラメトリック下方変換によって生成した光子対の一方を受信側のトリガーとして検出し、他方を信号(伝令)光子として受信側に送ることにより、受信側の光子検出器のダークカウントを実効的に減少させることができる。第3章では、盗聴を見破るための強力な手法であるデコイ法と呼ばれる手法を前提として、パラメトリック下方変換を用いた伝令光子源の安全鍵生成率を初めて定量的に解析した。この解析により、伝令光子源の最長伝送距離がコヒーレント光パルスの140kmから、理想的な単一光子源の最長伝送距離である170kmにまで伸びることが示された。

　トリガーの検出器として光子数を識別できる検出器を用い、この検出器が単一光子を検出した場合にのみ受信側にトリガー信号を送れば、複数光子が低減された(実効的に単一光子の)伝令光子源が実現できる。このアイデアに基づき、現存する単一光子検出器(threshold detector)を複数個組み合わせて時間多重検出器を構成し、光子数識別を実現した場合の安全鍵生成率も第3章で解析されている。その結果、時間多重検出器を用いることによって、全ての伝送可能距離に対して約2倍の安全鍵生成率の向上が見られることが確認された。

　本研究では、パラメトリック下方変換を用いた伝令光子源に対し、実際に時間多重検出器を構築し、複数光子低減を始めて実現した。その詳細は第4章に述べられている。本研究では、複数光子低減の確認のために、伝令光子源が2光子を発生したときのみ、光源の強度相関を測定するというオリジナルな手法を用いた。その結果、時間多重検出器を用いることによって、約10%の複数光子低減率が実現されていることが確認された。この低減率は時間多重検出器の量子効率(約28%)によって制限される値である。本研究は、単に伝令光子源の複数光子低減を実現しただけでなく、高速論理回路を用いてリアルタイムに信号を処理して受信側にトリガー信号を送るスキームを構築した点にも意義があり、現実的な量子鍵配布への利用に繋がるものである。

　なお、本論文の第3章は小林孝嘉氏、王海波氏、佐々木秀貴氏、第4章は小林孝嘉氏、竹野唯志氏、藪下篤史氏との共同研究であるが、論文提出者が主体的に研究を遂行したものであり、論文提出者の寄与が十分であると判断する。

　したがって、博士(理学)の学位を授与できると認める。