- #1

- 10

- 0

Code:

```
\begin{pmatrix}
E_c - \lambda & F & W \\
F & E_f - \lambda & 0 \\
W & 0 & E_w - \lambda
\end{pmatrix}
```

Where E

_{c}is the photon/cavity energy, E

_{f}is the Frenkel exciton energy, and E

_{w}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}
```