基于有限元法的光子并矢格林函数重整化及其在自发辐射率和能级移动研究中的应用

赵运进; 田锰; 黄勇刚; 王小云; 杨红; 米贤武

doi:10.7498/aps.67.20180898

摘要

任意微纳结构中量子点的自发辐射率和能级移动均可用并矢格林函数表达.当源点和场点在同一位置时，格林函数的实部是发散的.为解决这一发散问题，可采用重整化格林函数方法.本文提出一种计算重整化格林函数和散射格林函数的方法.该方法利用有限元，计算点电偶极子的辐射场，将其在量子点体积内做平均得到重整化的并矢格林函数，减去均匀空间中解析的重整化格林函数，得到重整化的散射格林函数.在均匀空间情况下，本方法所得数值结果与解析解一致.将该方法应用到银纳米球系统，以解析的散射格林函数作为参考，结果表明该方法能准确处理散射格林函数的重整化问题.将该方法应用到表面等离激元纳米腔中，发现有极大的自发辐射增强和能级移动，且该结果不依赖于量子点的体积.这些研究在光与物质相互作用领域具有积极的意义.

关键词:

Abstract

The spontaneous emission rate and the energy level shift of a quantum dot in any micro-nanostructures can be expressed by the classical dyadic Green's function. However, the real part of the dyadic Green's function is divergent, when the source point and the field point are at the same position. This leads to an unphysical divergent level shift. Theoretically, the dyadic Green's function can be decomposed into a homogeneous part and a scattering part. Traditionally, the homogeneous field contribution is introduced into the definition of the transition frequency and the only need is to consider the effect of the scattering part which is non-divergent. Another renormalization method is to average the Green tensor over the volume of the quantum dot. In this work, a finite element method is proposed to address this problem. The renormalized dyadic Green function is expressed by the averaged radiation field of a point dipole source over the quantum dot volume. For the vacuum case, numerical results of the renormalized Green tensor agree well with the analytical ones. For the nanosphere model, the renormalized scattering Green tensor, which is the difference between the renormalized Green tensor and the analytical renormalized one in homogeneous space, agrees well with the analytical scattering Green tensor in the center of the quantum dot. Both of the above models clearly demonstrate the validity and accuracy of our method. Compared with the previous scattering Green function method where two different finite element runs are needed for one frequency point, our renormalization method just needs one single run. This greatly reduces the computation burden. Applying the theory to a gap plasmonic nano-cavity, we find extremely large modifications for the spontaneous emission rate and the energy level shift which are independent of the size of the quantum dot. For frequency around the higher order mode of the nano-cavity, spontaneous emission enhancement is about Г/Г0 2.02106 and the energy level shift is about △ 1000 meV for a dipole moment 24D. These findings are instructive in the fields of quantum light-matter interactions.

Keywords:

作者及机构信息

1.
吉首大学物理与机电工程学院, 吉首 416000;

2.
怀化学院电气与信息工程学院, 怀化 418000

通信作者: 黄勇刚, huang122012@163.com

基金项目: 国家自然科学基金（批准号：11464014，11347215，11364020，11564013，11464013）、湖南省自然科学基金（批准号：2016JJ4073）和湖南省研究生科研创新项目（批准号：CX2017B718）资助的课题.

Authors and contacts

1.
College of Physics, Mechanical and Electrical Engineering, Jishou University, Jishou 416000, China;

2.
College of Electrical and Information Engineering, Huaihua University, Huaihua 418000, China

Corresponding author: Huang Yong-Gang, huang122012@163.com

Funds: Project supported by the National Natural Science Foundation of China (Grants Nos. 11464014, 11347215, 11364020, 11564013, 11464013), the Natural Science Foundation of Hunan Province, China (Grant No. 2016JJ4073), and the Hunan Provincial Innovation Foundation for Postgraduate, China (Grant No. CX2017B718).

参考文献

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[2]	Tannoudji C C, Roc D J, Grynberg G 1997 Photons and Atoms:Introduction to Quantum Electrodynamics (New York:John Wiley Sons) pp197-200
[3]	Milonni P W 1993 The Quantum Vacuum:An Introduction to Quantum Electrodynamics (San Diego:Academic Press) pp78-107
[4]	Wang X H, Wang R Z, Gu B Y, Yang G Z 2002 Phys. Rev. Lett. 88 093902
[5]	Zhou Y S, Wang X H, Gu B Y, Wang F H 2006 Phys. Rev. Lett. 96 103601
[6]	Wang X H, Gu B Y (in Chinese) [王雪华, 顾本源 2005 物理 34 18]
[7]	Xing R, Xie S Y, Xu J P, Yang Y P 2014 Acta Phys. Sin. 63 094205 (in Chinese) [邢容, 谢双媛, 许静平, 羊亚平 2014 物理学报 63 094205]
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[10]	Zhu S Y, Chen H, Huang H 1997 Phys. Rev. Lett. 79 205
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[14]	Kinkhabwala A, Yu Z F, Fan S H, Avlasevich Y, Mllen K, Moerner W E 2009 Nature Photon. 3 654
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[19]	Khajavikhan M, Simic A, Katz M, Lee J H, Slutsky B, Mizrahi A, Lomakin V, Fainman Y 2012 Nature 482 204
[20]	Xu H X, Bjerneld E J, Kll M, Brjesson L 1999 Phys. Rev. Lett. 83 4357
[21]	Imada H, Miwa K, Imai-Imada M, Kawahara S, Kimura K, Kim Y 2017 Phys. Rev. Lett. 119 013901
[22]	Liu R M, Zhou Z K, Yu Y C, Zhang T W, Wang H, Liu G H, Wei Y M, Chen H J, Wang X H 2017 Phys. Rev. Lett. 118 237401
[23]	Zhang Y, Meng Q S, Zhang L, Luo Y, Yu Y J, Yang B, Zhang Y, Esteban R, Aizpurua J, Luo Y, Yang J L, Dong Z C, Hou J G 2017 Nat. Commun. 8 15225
[24]	Gonzlez-Tudela A, Huidobro P A, Martn-Moreno L, Tejedor C, Garca-Vidal F J 2014 Phys. Rev. B 89 041402
[25]	Delga A, Feist J, Bravo-Abad J, Garcia-Vidal F J 2014 Phys. Rev. Lett. 112 253601
[26]	Zhao Y J, Tian M, Wang X Y, Yang H, Zhao H P, Huang Y G 2018 Opt. Express 26 1390
[27]	van Vlack C, Kristensen P T, Hughes S 2012 Phys. Rev. B 85 075303
[28]	Yaghjian A D 1980 Proc. IEEE 68 248
[29]	Huttner B, Barnett S M 1992 Phys. Rev. A 46 4306
[30]	Scheel S, Knll L, Welsch D G 1999 Phys. Rev. A 60 4094
[31]	Scheel S, Knll L, Welsch D G, Barnett S M 1999 Phys. Rev. A 60 1590
[32]	de Vries P, van Coevorden D V, Lagendijk A 1998 Rev. Mod. Phys. 70 447
[33]	Dung H T, Buhmann S Y, Knll L, Welsch D, Scheel S, Kstel J 2003 Phys. Rev. A 68 043816
[34]	Chaumet P C, Sentenac A, Rahmani A 2004 Phys. Rev. E 70 036606
[35]	van Vlack C, Hughes S 2012 Opt. Lett. 37 2880
[36]	Martin O J F, Piller N B. 1998 Phys. Rev. E 58 3909
[37]	Tannoudji C C, Roc D J, Grynberg G 1992 Atom-Photon Interactions:Basic Processes and Applications (New York:John Wiley Sons) pp165-205
[38]	Agarwal G S 1974 Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches (Berlin, Heidelberg:Springer) pp17-23
[39]	Jin J M 2014 The Finite Element Method in Electromagnetics (3rd Ed.) (New York:Wiley-IEEE Press) pp1-188
[40]	Benjamin G, Jrmy B, MartF, Olivier J 2015 Laser Photon. Rev. 9 577
[41]	Chen Y T, Nielsen T R, Gregersen N, Lodahl P, Mrk J 2010 Phys. Rev. B 81 125431
[42]	https://www.comsol.com/[2018-5-6]
[43]	Bai Q, Perrin M, Sauvan C, Hugonin J P, Lalanne P 2013 Opt. Express 21 27371
[44]	Zhang Y, Luo Y, Zhang Y, Yu Y J, Kuang Y M, Zhang L, Meng Q S, Luo Y, Yang J L, Dong Z C, Hou J G 2016 Nature 531 623
[45]	Halas N J, Lal S, Chang W S, Link S, Nordlander P 2011 Chem. Rev. 111 3913
[46]	Yang C J, An J H 2017 Phys. Rev. B 95 161408

施引文献

[1]	Berestetskii V B, Lifshitz E M, Pitaevskiǐ L P 1982 Quantum Electrodynamics (2nd Ed.) (England:Butterworth-Heinemann) pp159-165
[2]	Tannoudji C C, Roc D J, Grynberg G 1997 Photons and Atoms:Introduction to Quantum Electrodynamics (New York:John Wiley Sons) pp197-200
[3]	Milonni P W 1993 The Quantum Vacuum:An Introduction to Quantum Electrodynamics (San Diego:Academic Press) pp78-107
[4]	Wang X H, Wang R Z, Gu B Y, Yang G Z 2002 Phys. Rev. Lett. 88 093902
[5]	Zhou Y S, Wang X H, Gu B Y, Wang F H 2006 Phys. Rev. Lett. 96 103601
[6]	Wang X H, Gu B Y (in Chinese) [王雪华, 顾本源 2005 物理 34 18]
[7]	Xing R, Xie S Y, Xu J P, Yang Y P 2014 Acta Phys. Sin. 63 094205 (in Chinese) [邢容, 谢双媛, 许静平, 羊亚平 2014 物理学报 63 094205]
[8]	Xing R, Xie S Y, Xu J P, Yang Y P 2017 Acta Phys. Sin. 66 014202 (in Chinese) [邢容, 谢双媛, 许静平, 羊亚平 2017 物理学报 66 014202]
[9]	Yang Y P, Zhu S Y 2000 Phys. Rev. A 61 043809
[10]	Zhu S Y, Chen H, Huang H 1997 Phys. Rev. Lett. 79 205
[11]	Xie S Y, Yang Y P, Lin Z X, Wu X 1999 Acta Phys. Sin. 48 1459 (in Chinese) [谢双媛, 羊亚平, 林志新, 吴翔 1999 物理学报 48 1459]
[12]	Yang Y P, Lin Z X, Xie S Y, Feng W G, Wu X 1999 Acta Phys. Sin. 48 603 (in Chinese) [羊亚平, 林志新, 谢双媛, 冯伟国, 吴翔 1999 物理学报 48 603]
[13]	Huang Y G, Chen G Y, Jin C J, Liu W M, Wang X H 2012 Phys. Rev. A 85 053827
[14]	Kinkhabwala A, Yu Z F, Fan S H, Avlasevich Y, Mllen K, Moerner W E 2009 Nature Photon. 3 654
[15]	Okamoto K, Niki I, Shvartser A, Narukawa Y, Mukai T, Scherer A 2004 Nature Mater. 3 601
[16]	Li M, Cushing S K, Wu N Q 2015 Analyst 140 386
[17]	Taylor A B, Zijlstra P 2017 ACS Sens. 2 1103
[18]	Lu Y J, Kim J, Chen H Y, Wu C, Dabidian N, Sanders C E, Wang C Y, Lu M Y, Li B H, Qiu X G, Chang W H, Chen L J, Shvets G, Shih C K, Gwo S J 2012 Science 337 450
[19]	Khajavikhan M, Simic A, Katz M, Lee J H, Slutsky B, Mizrahi A, Lomakin V, Fainman Y 2012 Nature 482 204
[20]	Xu H X, Bjerneld E J, Kll M, Brjesson L 1999 Phys. Rev. Lett. 83 4357
[21]	Imada H, Miwa K, Imai-Imada M, Kawahara S, Kimura K, Kim Y 2017 Phys. Rev. Lett. 119 013901
[22]	Liu R M, Zhou Z K, Yu Y C, Zhang T W, Wang H, Liu G H, Wei Y M, Chen H J, Wang X H 2017 Phys. Rev. Lett. 118 237401
[23]	Zhang Y, Meng Q S, Zhang L, Luo Y, Yu Y J, Yang B, Zhang Y, Esteban R, Aizpurua J, Luo Y, Yang J L, Dong Z C, Hou J G 2017 Nat. Commun. 8 15225
[24]	Gonzlez-Tudela A, Huidobro P A, Martn-Moreno L, Tejedor C, Garca-Vidal F J 2014 Phys. Rev. B 89 041402
[25]	Delga A, Feist J, Bravo-Abad J, Garcia-Vidal F J 2014 Phys. Rev. Lett. 112 253601
[26]	Zhao Y J, Tian M, Wang X Y, Yang H, Zhao H P, Huang Y G 2018 Opt. Express 26 1390
[27]	van Vlack C, Kristensen P T, Hughes S 2012 Phys. Rev. B 85 075303
[28]	Yaghjian A D 1980 Proc. IEEE 68 248
[29]	Huttner B, Barnett S M 1992 Phys. Rev. A 46 4306
[30]	Scheel S, Knll L, Welsch D G 1999 Phys. Rev. A 60 4094
[31]	Scheel S, Knll L, Welsch D G, Barnett S M 1999 Phys. Rev. A 60 1590
[32]	de Vries P, van Coevorden D V, Lagendijk A 1998 Rev. Mod. Phys. 70 447
[33]	Dung H T, Buhmann S Y, Knll L, Welsch D, Scheel S, Kstel J 2003 Phys. Rev. A 68 043816
[34]	Chaumet P C, Sentenac A, Rahmani A 2004 Phys. Rev. E 70 036606
[35]	van Vlack C, Hughes S 2012 Opt. Lett. 37 2880
[36]	Martin O J F, Piller N B. 1998 Phys. Rev. E 58 3909
[37]	Tannoudji C C, Roc D J, Grynberg G 1992 Atom-Photon Interactions:Basic Processes and Applications (New York:John Wiley Sons) pp165-205
[38]	Agarwal G S 1974 Quantum Statistical Theories of Spontaneous Emission and Their Relation to Other Approaches (Berlin, Heidelberg:Springer) pp17-23
[39]	Jin J M 2014 The Finite Element Method in Electromagnetics (3rd Ed.) (New York:Wiley-IEEE Press) pp1-188
[40]	Benjamin G, Jrmy B, MartF, Olivier J 2015 Laser Photon. Rev. 9 577
[41]	Chen Y T, Nielsen T R, Gregersen N, Lodahl P, Mrk J 2010 Phys. Rev. B 81 125431
[42]	https://www.comsol.com/[2018-5-6]
[43]	Bai Q, Perrin M, Sauvan C, Hugonin J P, Lalanne P 2013 Opt. Express 21 27371
[44]	Zhang Y, Luo Y, Zhang Y, Yu Y J, Kuang Y M, Zhang L, Meng Q S, Luo Y, Yang J L, Dong Z C, Hou J G 2016 Nature 531 623
[45]	Halas N J, Lal S, Chang W S, Link S, Nordlander P 2011 Chem. Rev. 111 3913
[46]	Yang C J, An J H 2017 Phys. Rev. B 95 161408

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