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

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.

Renormalization of photon dyadic Green function by finite element method and its applications in the study of spontaneous emission rate and energy level shift

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 ≈ 2.02×10⁶ 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.