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Analysis of Faraday Mirror in Auto-Compensating Quantum Key Distribution *

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Analysis of Faraday Mirror in Auto-Compensating Quantum Key Distribution * ISSN: 0256-307X 中国物理快报 Chinese Physics Letters Volume 32 Number 8 August 2015 A Series Journal of the Chinese Physical Society Distributed by IOP Publishing Online: http://iopscience.iop.org/0256-307X http://cpl.iphy.ac.cn C HINESE P HYSICAL S OCIET Y CHIN. PHYS. LETT. Vol. 32, No. 8 (2015) 080303 Analysis of Faraday Mirror in Auto-Compensating Quantum Key Distribution * WEI Ke-Jin(韦克7), MA Hai-Qiang(ê海r)**, LI Rui-Xue(oaÈ), ZHU Wu(Á武), LIU Hong-Wei(4宏伟), ZHANG Yong(张]), JIAO Rong-Zhen(焦Jû) School of Science and State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876 (Received 24 April 2015) The ‘plug & play’ quantum key distribution system is the most stable and the earliest commercial system in the quantum communication field. Jones matrix and Jones calculus are widely used in the analysis of this systemand the improved version, which is called the auto-compensating quantum key distribution system. Unfortunately, existing analysis has two drawbacks: only the auto-compensating process is analyzed and existing systems do not fully consider laser phase affected by a Faraday mirror (FM). In this work, we present a detailed analysis of the output of light pulse transmitting in a plug & play quantum key distribution system that contains only an FM, by Jones calculus. A similar analysis is made to a home-made auto-compensating system which contains two FMs to compensate for environmental effects. More importantly, we show that theoretical and experimental results are different in the plug & play interferometric setup due to the fact that a conventional Jones matrixof FM neglected an additional phase 휋 on alternative polarization direction. To resolve the above problem, we give a new Jones matrix of an FM according to the coordinate rotation. This new Jones matrix not only resolves the above contradiction in the plug & play interferometric setup, but also is suitable for the previous analyses about auto-compensating quantum key distribution. PACS: 03.67.Dd, 03.67.Hk DOI: 10.1088/0256-307X/32/8/080303 Quantum key distribution (QKD), known as the crucial component in AC QKD. Its Jones matrix is most promising field to real world applications among widely used in theoretical analyses of AC QKD sys- various quantum information technologies, allows two tems. For the analysis, the main stream is describing distant parties (commonly named Alice and Bob) to the fact, which is first presented by Martinelli[12] that generate a string of unconditional secret key guar- a light, after propagating through an optical fiber and anteed by the law of quantum physics.[1] Since Ben- being reflected by FM, returns orthogonally polarized nett and Brassard presented the first QKD protocol in with a total round-trip phase which is independent of 1984, researchers have implemented many QKD sys- the initial polarization state.[13] Unfortunately, exist- tems, with entangled source, single photon, and con- ing analyses have two drawbacks. First, most of them tinuous source, via free space and optical fiber.[2−8] just followed the main stream and analyzed the com- Generally, the polarization of optical pulses is used pensating process. However, none of them showed ex- for free space QKD, and the phase is known to be the plicitly the output after the light transmits in an AC most promising encoding resource in optical fibers, QKD system, especially, when the system contains the thanking the robustness during transmission. The odd number of FMs.[14−16] Secondly, most of them cite phase encoding QKD system is usually based on the a conventional Jones matrix of FM, as Eq. (6), with- double Mach–Zehnder interferometer, one side in Al- out tracing to its source and its affect to the output in ice and the other in Bob. To make the interferom- AC QKD. Thus existing analyses do not fully consider eter identical and stable, an auxiliary compensating light phase affected by FMs. Recently, some works system is required. However, it is extra costs and based on AC QKD systems have been reported.[17;18] complex.[9;10] A simple, while novel, plug & play QKD In this Letter, we present a detailed analysis of the prototype, proposed by Muller et al.,[11] provides an output after a light pulse transmits through the setup inexpensive and efficient way to solve this problem. in Fig. 1, which is the earliest commercial and most Figure1 shows the setup of a plug & play QKD famous scheme in the AC QKD that contains only an system, pulse transmits through a single mode fiber FM. A similar analysis is carried out to a home-made (SMF) which serves as a quantum channel from Bob to AC scheme which contains two FMs to compensate Alice. Then it is back from Alice to Bob after reflected for the environmental effects. These works can be re- by a Faraday mirror (FM). Since the pulse passes garded as a general Jones matrix analysis for the AC through the same channel twice in reverse order, en- QKD system. More importantly, we show that the vironmental effects during the transmissions, such as conventional Jones matrix of the FM neglects an ad- birefringence in fiber, are auto-compensated. Subse- ditional phase 휋 on alternative polarization direction. quently, many developed QKD systems have used this As a result, theoretical and experimental results are compensating way to deal with environmental effects different in the plug & play QKD setup (or Fig. 1). during transmission. We usually call this type of sys- To solve the gap between experiment and theory, we tems the auto-compensating (AC) QKD. An FM is a present a new Jones matrix of the FM. Thus it resolves *Supported by the National Natural Science Foundation of China under Grant No 61178010, the Fund of State Key Laboratory of Information Photonics and Optical Communications of Beijing University of Posts and Telecommunications under Grant No 201318, and the Fundamental Research Funds for the Central Universities of China under Grant No 2014TS01. **Corresponding author. Email: [email protected] © 2015 Chinese Physical Society and IOP Publishing Ltd 080303-1 CHIN. PHYS. LETT. Vol. 32, No. 8 (2015) 080303 the problem that theoretical and experimental results a delay line in the lower arm. This does not have any out are different in the mentioned one-FM system orthe effect on the results. Thus the output ofPBS E5 is two-FM system. Furthermore, it is also suitable for (︂ out )︂ the previous analyses. out E2 E5 = out : (4) E3 PBS Cir The entire system’s Jones matrix (for the forward and Ein Eout Eout 1 C 2 T 1 5 FM backward propagations through SMF) is thus given by Eout QC 3 Ein Ein LD 2 R 5 PMA Eout Eout Ttotal = expf푖휃gFold; (5) 4 3 D1 Ein DL PMB 3 where 휃 is the propagation phase, and the Jones ma- D2 Alice Bob trix of the FM Fold is Fig. 1. The plug & play interferometric setup. LD: laser (︂ 0 −1 )︂ F = : (6) diode, Cir: circulator, C: a 50/50 coupler, C: PBS: polar- old −1 0 ization beam splitter, FM: Faraday mirror, PMA, PMB: phase modulators, D1, D2: single photon detectors, QC: quantum channel. Furthermore, Fold is a conventional formula which is widely cited in the previous works about the AC Figure1 shows a plug & play interferometric setup, QKD.[14;15;19] widely used in QKD over SMF. In detail, a laser pulse After the light is reflected by the FM, the input of is sent by a laser diode (LD) and passes through a in the PBS E1 is circulator (Cir) to a 50/50 coupler (C) that splits it in out into two pulses propagating through an upper arm E5 = Ttotal · E5 : (7) and a lower arm (that includes a delay line) with the According to the principle of PBS, the coupler’s inputs same polarization, respectively. The pulses arrive at in in a polarization beam splitter (PBS) and leave the Bob E2 and E3 can be described by station by the same polarization beam splitters PBS (︂ 1 0 )︂ port 1: the lower arm pulse is vertically polarized and Ein = Ein; (8) 2 0 0 5 the upper one is horizontally polarized. Then, the up- per arm and the lower arm pulses are transmitted over (︂ 0 1 )︂ Ein = Ein: (9) SMF and reflected at the FM, which rotates their po- 3 0 0 5 larizations by 90∘, then transmits back. Both pulses out continue their backward propagation over the SMF Finally, the output vectors E1 corresponding to the out and are directed by the PBS to the opposite arms (in measurement result of D1 and E4 corresponding to regards to their forward propagation), thus they ar- the measurement result of D2 are found by the electric rive at the same time at the coupler C, where they field amplitude transmission equation for a symmetri- interfere. Depending on the phase difference ' (that cal coupler could be controlled by Alice’s phase modulator PMA p and Bob’s PM ) between the pulses, they are detected out 2 in in B E1 = (E2 + iE3 ); (10) to either detector D1 or D2 if the phase difference p2 ' = 0 ± 2n휋, or ' = 휋 ± 2푛휋, where n is an inte- 2 Eout = (iEin + Ein): (11) ger; or in both detectors with different probabilities 4 2 2 3 for other ' values. Thus the plug & play system is auto-compensated for propagation time difference be- After substituting all formula (already calculated in this section) one can finally obtain the following ex- tween the upper arm and lower arm pulses since they out out travel the same path length.
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