- #1
thefacemyer
- 10
- 0
I am trying to verify the results of the hybrid polariton case with the following hamiltonian, but cannot seem to verify the results in various published papers. Can someone please explain what is wrong and how to get the a similar dispersion graph? I'm solving for the eigenenergies for the matrix below, then solve the cubic using the trigonometric method.
Where Ec is the photon/cavity energy, Ef is the Frenkel exciton energy, and Ew is the Wannier-Mott exciton energy.
The characteristic equation is:
Where n = {0,1,2} corresponding to the three hybrid polariton branches. I used the trigonometric formula to solve the cubic for the energy.
Where p and q are the coefficients of the transformed cubic
.
Code:
\begin{pmatrix}
E_c - \lambda & F & W \\
F & E_f - \lambda & 0 \\
W & 0 & E_w - \lambda
\end{pmatrix}
Where Ec is the photon/cavity energy, Ef is the Frenkel exciton energy, and Ew is the Wannier-Mott exciton energy.
The characteristic equation is:
Code:
\begin{equation}
0 = \lambda_n^3 - (E_c + E_f +E_w)\lambda_n^2 + (E_f E_w + E_c E_w + E_c E_f - F^2 - W^2)\lambda_n - E_c E_f E_w +F^2 E_w + W^2 E_f
\end{equation}
Where n = {0,1,2} corresponding to the three hybrid polariton branches. I used the trigonometric formula to solve the cubic for the energy.
Code:
\begin{equation}
\lambda_n = 2 \sqrt{- \frac{p}{3}} \cos \left( \frac{1}{3} \arccos \left( \frac{3q}{2p} \sqrt{- \frac{3}{p}} \right) - \frac{2 \pi n}{3} \right)
\end{equation}
Where p and q are the coefficients of the transformed cubic
Code:
\begin{equation}
\lambda_n^3 + p\lambda_n + q = 0
\end{equation}