基于自发参量下转换源二阶激发过程产生四光子超纠缠态

何英秋; 丁东; 彭涛; 闫凤利; 高亭

doi:10.7498/aps.67.20172230

摘要

目前，多光子纠缠态的制备大多通过线性光学器件演化自发参量下转换一阶激发过程产生的纠缠光子对得到.本文考虑由自发参量下转换源二阶激发产生四个不可区分的纠缠光子制备四光子超纠缠态的情况.通过几组分束器、半波片和偏振分束器等线性光学器件设计量子线路演化四光子系统，结合四模符合探测，可得到同时具有偏振纠缠和空间纠缠的四光子超纠缠态.

关键词:

Nowadays,the generation of multiphoton entangled states is almost realized by combining the coupled entangled photons emitted from spontaneous parametric down-conversion (SPDC) with the first-order term.In this case,one may focus mainly on the first-order term,and then avoid multipair emission events by restricting experimental parameters.On the other hand,for the higher-order terms in SPDC source,these emitted entangled photons have interesting features.For example,they are entangled maximally not only in photon number for the spatial modes,but also in polarization degree of freedom.In general,two photons,which are entangled in two or more degrees of freedom,are called hyperentangled pair of photons or hyperentangled state.We present a scheme to generate the four-photon hyperentangled state based on four indistinguishable photons emitted from SPDC source with the second-order term.Consider two SPDC sources with equal probability of emission of photons in respective spatial modes.With the passive linear optical devices,i.e., beam splitters,half wave plates,polarizing beam splitters,etc.,under the condition of registering a specified four-photon coincidence,we can obtain the four-photon hyperentangled state in which the photons are entangled in both polarization and spatial-mode degrees of freedom.Here,of course,for an arbitrary fourfold coincidence detection,one obtains a canonical four-photon Greenberger-Horne-Zeilinger (GHZ) state.Then we show the results of fourfold coincidence detections and the corresponding probabilities for the four-photon GHZ states,where the generation of the four-photon hyperentangled state is included as long as we are not to distinguish the two detectors located at the same locations. As a result,our scheme has two notable features.When we only consider the second-order emission,since it is not needed for us to distinguish between the two SPDC sources,the present scheme is simple and feasible.Also,based on the postselection with fourfold coincidence detection,our scheme is suitable for the normal first-order emission where we restrict the four photons emitted from the same source.In this sense,our scheme is efficient.In a word,we describe a method to generate the four-photon hyperentangled state with the second-order emission in SPDC source,which may contribute to the exploration of multipair entanglement with higher-order emissions from the SPDC source.

Keywords:

multiphoton entanglement /
spontaneous parametric down-conversion /
hyperentangled state

作者及机构信息

1.
承德医学院生物医学工程系, 承德 067000;

2.
华北科技学院理学院, 北京 101601;

3.
河北师范大学物理科学与信息工程学院, 石家庄 050024;

4.
河北师范大学数学与信息科学学院, 石家庄 050024

通信作者: 丁东, dingdong@ncist.edu.cn;flyan@hebtu.edu.cn ; 闫凤利, dingdong@ncist.edu.cn;flyan@hebtu.edu.cn ;

基金项目: 国家自然科学基金（批准号：11475054，11371005，11547169）、河北省自然科学基金（批准号：A2016205145，A2018205125）、承德医学院高层次人才科研启动基金（批准号：201701）、中央高校基本科研业务费专项资金（批准号：3142017069，3142015044）和河北省高等学校科学技术研究项目（批准号：Z2015188）资助的课题.

Authors and contacts

1.
Department of Biomedical Engineering, Chengde Medical University, Chengde 067000, China;

2.
College of Science, North China Institute of Science and Technology, Beijing 101601, China;

3.
College of Physics Science and Information Engineering, Hebei Normal University, Shijiazhuang 050024, China;

4.
College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, China

Corresponding author: Ding Dong, dingdong@ncist.edu.cn;flyan@hebtu.edu.cn ; Yan Feng-Li, dingdong@ncist.edu.cn;flyan@hebtu.edu.cn ;

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11475054, 11371005, 11547169), the Hebei Natural Science Foundation of China (Grant Nos. A2016205145, A2018205125), the Foundation for High-Level Talents of Chengde Medical University, China (Grant No. 201701), the Fundamental Research Funds for the Central Universities of Ministry of Education of China (Grant Nos. 3142017069, 3142015044), and the Research Project of Science and Technology in Higher Education of Hebei Province of China (Grant No. Z2015188).

参考文献

[1]	Kok P, Munro W J, Nemoto K, Ralph T C, Dowling J P, Milburn G J 2007 Rev. Mod. Phys. 79 135
[2]	Pan J W, Chen Z B, Lu C Y, Weinfurter H, Zeilinger A, Żkowski M 2012 Rev. Mod. Phys. 84 777
[3]	Burnham D C, Weinberg D L 1970 Phys. Rev. Lett. 25 84
[4]	Kiess T E, Shih Y H, Sergienko A V, Alley C O 1993 Phys. Rev. Lett. 71 3893
[5]	Kwiat P G, Mattle K, Weinfurter H, Zeilinger A 1995 Phys. Rev. Lett. 75 4337
[6]	Kok P, Braunstein S L 2000 Phys. Rev. A 61 042304
[7]	Simon C, Weihs G, Zeilinger A 2000 Phys. Rev. Lett. 84 2993
[8]	Wieczorek W, Schmid C, Kiesel N, Pohlner R, Ghne O, Weinfurter H 2008 Phys. Rev. Lett. 101 010503
[9]	Yao X C, Wang T X, Xu P, Lu H, Pan G S, Bao X H, Peng C Z, Lu C Y, Chen Y A, Pan J W 2012 Nat. Photon. 6 225
[10]	Wang X L, Chen L K, Li W, Huang H L, Liu C, Chen C, Luo Y H, Su Z E, Wu D, Li Z D, Lu H, Hu Y, Jiang X, Peng C Z, Li L, Liu N L, Chen Y A, Lu C Y, Pan J W 2016 Phys. Rev. Lett. 117 210502
[11]	Ou Z Y, Rhee J K, Wang L J 1999 Phys. Rev. Lett. 83 959
[12]	Lamas-Linares A, Howell J, Bouwmeester D 2001 Nature 412 887
[13]	Simon C, Bouwmeester D 2003 Phys. Rev. Lett. 91 053601
[14]	de Riedmatten H, Scarani V, Marcikic I, Acín A, Tittel W, Zbinden H, Gisin N 2004 J. Mod. Opt. 51 1637
[15]	Nagata T, Okamoto R, O'Brien J L, Sasaki K, Takeuchi S 2007 Science 316 726
[16]	Ding D, He Y Q, Yan F L, Gao T 2017 arXiv:1705.00392[quant-ph]
[17]	Kwiat P G 1997 J. Mod. Opt. 44 2173
[18]	Barreiro J T, Langford N K, Peters N A, Kwiat P G 2005 Phys. Rev. Lett. 95 260501
[19]	Vallone G, Ceccarelli R, de Martini F, Mataloni P 2009 Phys. Rev. A 79 030301
[20]	Du K, Qiao C F 2012 J. Mod. Opt. 59 611
[21]	Ding D, He Y Q, Yan F L, Gao T 2015 Acta Phys. Sin. 64 160301 (in Chinese) [丁东, 何英秋, 闫凤利, 高亭 2015 64 160301]
[22]	Deng F G, Ren B C, Li X H 2017 Sci. Bull. 62 46
[23]	Walborn S P, Pádua S, Monken C H 2003 Phys. Rev. A 68 042313
[24]	Wei T C, Barreiro J T, Kwiat P G 2007 Phys. Rev. A 75 060305
[25]	Sheng Y B, Deng F G, Long G L 2010 Phys. Rev. A 82 032318
[26]	Ren B C, Wei H R, Hua M, Li T, Deng F G 2012 Opt. Express 20 24664
[27]	Li X H, Ghose S 2016 Phys. Rev. A 93 022302
[28]	Xia Y, Chen Q Q, Song J, Song H S 2012 J. Opt. Soc. Am. B 29 1029
[29]	Sheng Y B, Deng F G 2010 Phys. Rev. A 81 032307
[30]	Ren B C, Long G L 2014 Opt. Express 22 6547
[31]	He Y Q, Ding D, Yan F L, Gao T 2015 J. Phys. B: At. Mol. Opt. Phys. 48 055501
[32]	Ren B C, Du F F, Deng F G 2014 Phys. Rev. A 90 052309
[33]	Du F F, Li T, Long G L 2016 Ann. Phys. 375 105
[34]	Liu H J, Xia Y, Song J 2016 Quantum Inf. Process. 15 2033
[35]	Ren B C, Wang H, Alzahrani F, Hobiny A, Deng F G 2017 Ann. Phys. 385 86
[36]	Ren B C, Wang G Y, Deng F G 2015 Phys. Rev. A 91 032328
[37]	Li T, Long G L 2016 Phys. Rev. A 94 022343
[38]	Wei H R, Deng F G, Long G L 2016 Opt. Express 24 18619
[39]	Wang T J, Song S Y, Long G L 2012 Phys. Rev. A 85 062311
[40]	Sheng Y B, Zhou L 2015 Sci. Rep. 5 7815
[41]	Jiang Y X, Guo P L, Gao C Y, Wang H B, Alzahrani F, Hobiny A, Deng F G 2017 Sci. China: Phys. Mech. Astron. 60 120312
[42]	Wu F Z, Yang G J, Wang H B, Xiong J, Alzahrani F, Hobiny A, Deng F G 2017 Sci. China: Phys. Mech. Astron. 60 120313
[43]	Nemoto K, Munro W J 2004 Phys. Rev. Lett. 93 250502
[44]	Munro W J, Nemoto K, Beausoleil R G, Spiller T P 2005 Phys. Rev. A 71 033819
[45]	Barrett S D, Kok P, Nemoto K, Beausoleil R G, Munro W J, Spiller T P 2005 Phys. Rev. A 71 060302
[46]	Lin Q, He B, Bergou J A, Ren Y H 2009 Phys. Rev. A 80 042311
[47]	Ding D, Yan F L 2013 Phys. Lett. A 377 1088
[48]	Ding D, Yan F L, Gao T 2014 Sci. China: Phys. Mech. Astron. 57 2098
[49]	He Y Q, Ding D, Yan F L, Gao T 2015 Opt. Express 23 21671
[50]	Zhou L, Sheng Y B 2015 Phys. Rev. A 92 042314
[51]	Sheng Y B, Pan J, Guo R, Zhou L, Wang L 2015 Sci. China: Phys. Mech. Astron. 58 060301
[52]	He Y Q, Ding D, Yan F L, Gao T 2017 Sci. Rep. 7 15356

施引文献

[1]	Kok P, Munro W J, Nemoto K, Ralph T C, Dowling J P, Milburn G J 2007 Rev. Mod. Phys. 79 135
[2]	Pan J W, Chen Z B, Lu C Y, Weinfurter H, Zeilinger A, Żkowski M 2012 Rev. Mod. Phys. 84 777
[3]	Burnham D C, Weinberg D L 1970 Phys. Rev. Lett. 25 84
[4]	Kiess T E, Shih Y H, Sergienko A V, Alley C O 1993 Phys. Rev. Lett. 71 3893
[5]	Kwiat P G, Mattle K, Weinfurter H, Zeilinger A 1995 Phys. Rev. Lett. 75 4337
[6]	Kok P, Braunstein S L 2000 Phys. Rev. A 61 042304
[7]	Simon C, Weihs G, Zeilinger A 2000 Phys. Rev. Lett. 84 2993
[8]	Wieczorek W, Schmid C, Kiesel N, Pohlner R, Ghne O, Weinfurter H 2008 Phys. Rev. Lett. 101 010503
[9]	Yao X C, Wang T X, Xu P, Lu H, Pan G S, Bao X H, Peng C Z, Lu C Y, Chen Y A, Pan J W 2012 Nat. Photon. 6 225
[10]	Wang X L, Chen L K, Li W, Huang H L, Liu C, Chen C, Luo Y H, Su Z E, Wu D, Li Z D, Lu H, Hu Y, Jiang X, Peng C Z, Li L, Liu N L, Chen Y A, Lu C Y, Pan J W 2016 Phys. Rev. Lett. 117 210502
[11]	Ou Z Y, Rhee J K, Wang L J 1999 Phys. Rev. Lett. 83 959
[12]	Lamas-Linares A, Howell J, Bouwmeester D 2001 Nature 412 887
[13]	Simon C, Bouwmeester D 2003 Phys. Rev. Lett. 91 053601
[14]	de Riedmatten H, Scarani V, Marcikic I, Acín A, Tittel W, Zbinden H, Gisin N 2004 J. Mod. Opt. 51 1637
[15]	Nagata T, Okamoto R, O'Brien J L, Sasaki K, Takeuchi S 2007 Science 316 726
[16]	Ding D, He Y Q, Yan F L, Gao T 2017 arXiv:1705.00392[quant-ph]
[17]	Kwiat P G 1997 J. Mod. Opt. 44 2173
[18]	Barreiro J T, Langford N K, Peters N A, Kwiat P G 2005 Phys. Rev. Lett. 95 260501
[19]	Vallone G, Ceccarelli R, de Martini F, Mataloni P 2009 Phys. Rev. A 79 030301
[20]	Du K, Qiao C F 2012 J. Mod. Opt. 59 611
[21]	Ding D, He Y Q, Yan F L, Gao T 2015 Acta Phys. Sin. 64 160301 (in Chinese) [丁东, 何英秋, 闫凤利, 高亭 2015 64 160301]
[22]	Deng F G, Ren B C, Li X H 2017 Sci. Bull. 62 46
[23]	Walborn S P, Pádua S, Monken C H 2003 Phys. Rev. A 68 042313
[24]	Wei T C, Barreiro J T, Kwiat P G 2007 Phys. Rev. A 75 060305
[25]	Sheng Y B, Deng F G, Long G L 2010 Phys. Rev. A 82 032318
[26]	Ren B C, Wei H R, Hua M, Li T, Deng F G 2012 Opt. Express 20 24664
[27]	Li X H, Ghose S 2016 Phys. Rev. A 93 022302
[28]	Xia Y, Chen Q Q, Song J, Song H S 2012 J. Opt. Soc. Am. B 29 1029
[29]	Sheng Y B, Deng F G 2010 Phys. Rev. A 81 032307
[30]	Ren B C, Long G L 2014 Opt. Express 22 6547
[31]	He Y Q, Ding D, Yan F L, Gao T 2015 J. Phys. B: At. Mol. Opt. Phys. 48 055501
[32]	Ren B C, Du F F, Deng F G 2014 Phys. Rev. A 90 052309
[33]	Du F F, Li T, Long G L 2016 Ann. Phys. 375 105
[34]	Liu H J, Xia Y, Song J 2016 Quantum Inf. Process. 15 2033
[35]	Ren B C, Wang H, Alzahrani F, Hobiny A, Deng F G 2017 Ann. Phys. 385 86
[36]	Ren B C, Wang G Y, Deng F G 2015 Phys. Rev. A 91 032328
[37]	Li T, Long G L 2016 Phys. Rev. A 94 022343
[38]	Wei H R, Deng F G, Long G L 2016 Opt. Express 24 18619
[39]	Wang T J, Song S Y, Long G L 2012 Phys. Rev. A 85 062311
[40]	Sheng Y B, Zhou L 2015 Sci. Rep. 5 7815
[41]	Jiang Y X, Guo P L, Gao C Y, Wang H B, Alzahrani F, Hobiny A, Deng F G 2017 Sci. China: Phys. Mech. Astron. 60 120312
[42]	Wu F Z, Yang G J, Wang H B, Xiong J, Alzahrani F, Hobiny A, Deng F G 2017 Sci. China: Phys. Mech. Astron. 60 120313
[43]	Nemoto K, Munro W J 2004 Phys. Rev. Lett. 93 250502
[44]	Munro W J, Nemoto K, Beausoleil R G, Spiller T P 2005 Phys. Rev. A 71 033819
[45]	Barrett S D, Kok P, Nemoto K, Beausoleil R G, Munro W J, Spiller T P 2005 Phys. Rev. A 71 060302
[46]	Lin Q, He B, Bergou J A, Ren Y H 2009 Phys. Rev. A 80 042311
[47]	Ding D, Yan F L 2013 Phys. Lett. A 377 1088
[48]	Ding D, Yan F L, Gao T 2014 Sci. China: Phys. Mech. Astron. 57 2098
[49]	He Y Q, Ding D, Yan F L, Gao T 2015 Opt. Express 23 21671
[50]	Zhou L, Sheng Y B 2015 Phys. Rev. A 92 042314
[51]	Sheng Y B, Pan J, Guo R, Zhou L, Wang L 2015 Sci. China: Phys. Mech. Astron. 58 060301
[52]	He Y Q, Ding D, Yan F L, Gao T 2017 Sci. Rep. 7 15356

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