Computational Nanophotonics

6.4.3.2.3  Interaction Hamiltonian

The interaction energy between atom and radiation field can be approximated by the electric-dipole interaction energy as

$H_i=-d\cdot E$

(6.35)

where

$\widehat{E}$ is the electric field operator

$\widehat{d}$ is the electric-dipole moment of the atom, given by

$d=-eD=-e{\displaystyle {\sum }_n{r}_n}$

summed over all electrons of the atom. In this approximation, it is assumed that the contributions from electric quadruple moment and magnetic dipole moment are much smaller than that of electric-dipole moment. Now, the operator $\widehat{D}$ can be expressed by

$D=IDI={\displaystyle \sum _i|i\rangle \langle i|D{\displaystyle \sum _j|j\rangle \langle j|}}$

(6.36a)

where

I is the identity matrix

$|i\rangle$ and $|j\rangle$ are the atomic energy eigenstates

By reorganizing bras and kets in Equation 6.36a, it is found that

$D={\displaystyle \sum _{i,j}{\overrightarrow{D}}_{ij}|i\rangle \langle j|}$

(6.36b)

where

${\stackrel{}{D}}_{}$