关键词: 有限元法/
重整化格林函数/
自发辐射率/
能级移动
English Abstract
Renormalization of photon dyadic Green function by finite element method and its applications in the study of spontaneous emission rate and energy level shift
Zhao Yun-Jin1,Tian Meng1,
Huang Yong-Gang1,
Wang Xiao-Yun1,
Yang Hong1,
Mi Xian-Wu2
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
Fund Project: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).Received Date:06 May 2018
Accepted Date:19 July 2018
Published Online:05 October 2018
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: finite element method/
renormalized Green function/
spontaneous emission rate/
energy level shift
