# Derivation of the refractive index dependency of the light-matter interaction hamiltonian

I tried to derive a relation of the light-matter interaction Hamiltonian in vacuum and a medium that is characterized by a linear real refractive index but I am not sure if the following is correct or if this approach is valid at all. My result is given in a boxed equation at the bottom but I thought it is sensible to show my approach instead of simply asking whether the result is correct.

The time averaged (over 1 period) energy density $$u$$ of a electromagnetic wave in a homogenous medium characterized by its permittivity $$\varepsilon=\varepsilon_r\varepsilon_0$$ is

$$u = \frac{1}{2}\varepsilon |E_0|^2$$ where the electric field was assumed to be $$\vec E(r,t)=E_0\cos(\vec k\cdot \vec r-\omega t)\hat e$$.

Let the electric field amplitude be connected to the vector potential via $$\vec E(r,t) = -\frac{\partial }{\partial t} \vec A(r,t)$$ Asuming for the vector potential the form $$\vec A(r,t) = -A_0\sin(kr-\omega t)\hat e$$, we obtain for the amplitudes the relation $$E_0 = \omega A_0.$$ Inserting this int the expressoin for the energy density yields $$u=\frac{1}{2}\varepsilon\omega ^2|A_0|^2$$ Now we connect this equation to the an expression for the energy density based on $$N$$ quantized photons in Volume $$V$$, $$u = N\frac{\hbar \omega}{V} =\frac{1}{2}\varepsilon\omega ^2|A_0|^2$$ and rearrange for $$A_0$$, $$A_0 = \sqrt \frac{2N\hbar }{V\omega \varepsilon}$$

Now I want to establish a dependency on the refractive index. We have $$n^2 = \varepsilon_r\mu_r$$. We assume that $$\mu_r \approx 1$$ such that $$n^2\approx \varepsilon_r$$

Inserting this into the equation via $$\varepsilon = \varepsilon_r \varepsilon_0$$ for the vector potential amplitude yields, $$A_0 = \sqrt \frac{2N\hbar }{V\omega \varepsilon_0} \cdot \frac{1}{n}$$

Can I conclude from this the following equations for the light-matter interaction Hamiltonian \boxed{\begin{aligned} \hat V_{medium} \propto \frac{e}{m} \hat p\cdot A_{medium} &= \frac{e}{m} \hat p\cdot A_{vac} \frac{1}{n}\\ \approx -\hat \mu \cdot E_{medium} &= - \frac{1}{n} \hat \mu \cdot E_{vac} \end{aligned}}

where the second line is the light-matter interaction Hamiltonian within common approximations typically used when describing the interaction of molecules with electromagnetic waves.

This is a follow up question to this.

• You may check up treatments of Kubo formula, especially in many-body texts. Also Balslev & Stahl's book may be useful. Jul 28, 2021 at 8:47

The derivation is valid but it misses one very important aspect. The electric field at the interacting molecule/system is different from the macroscopic field in the isotropic medium characterized by $$n$$. A way to model this is the so called cavity factor. This factor establishes a proportionality between the macroscopic field and the local field responsible for the transition. The relation in an isotropic medium is $$\vec E^{loc} = f \vec E^{macr}$$ The cavity factor is itself a function of the refractive index. There are many models for the exact functional form. In the case of a freely rotating molecule in solution, one can use the so called Empty Spherical Cavity Model, which defines $$f(n)=\frac{3n^2}{2n^2 +1}$$
Assuming Fermi's Golden rule to calculate a transition rate, one typically has terms of the form $$\Gamma \propto \langle |\mu\cdot E^{local}|^2\rangle$$ Here we need to replace the local field by the macroscopic field which yields $$\Gamma \propto f(n)^2\langle |\mu\cdot E^{macr}|^2\rangle$$ in addition to any other factors of $$n$$ due to the field depenendency as derived in the question.