User:Martin van Horn/sandbox

= Light--Matter Interactions Beyond the Electric-Dipole Approximation =

Introduction
A semi-classical treatment invoking the electric-dipole (ED) approximation is a common starting point for a theoretical description of light–matter interactions. The latter approximation assumes that the spatial extent of the molecular system is small compared to the wavelength of the electromagnetic field such that the molecule effectively sees a uniform electric field, while the magnetic field component is neglected. Formally, it corresponds to retaining only the zeroth-order term of an expansion of the interaction operator in orders of the length of the wave vector. While this is often well- justifiable for the most commonly used optical laser sources and intensities, the availability of (i) high-energy x-ray photons, with wavelengths comparable to the molecular target,7–9 and (ii) intense laser sources, creating high-energy electrons strongly influenced by the magnetic component of the Lorentz force,10–12 motivates investigations into the effects of going beyond this simplification.

Theoretical Background
To model light--matter interactions, expressions are needed for the electromagnetic fields. For electromagnetic plane waves, the fields read

$$\mathbf{E}(\mathbf{r},t) = E_\omega \mathbf{\epsilon}\sin[\mathbf{k}\cdot\mathbf{r}-\omega t + \delta]$$$$\mathbf{B}(\mathbf{r},t) = \frac{E_\omega}{\omega}(\mathbf{k}\times\mathbf{\epsilon})\sin[\mathbf{k}\cdot\mathbf{r} - \omega t + \delta ]$$,

where $\omega$ is the frequency, $E_\omega$  the field strength, $\mathbf{k}$  and $\mathbf{\epsilon}$  the wave- and polarisation vector and $\delta$  a phase of the wave. Note that the wave vector and frequency are related through the dispersion relation $k=\frac{\omega}{c}$. The influence of these fields a quantum mechanical system can be found through the principle of minimal substitution

$$\hat{\mathbf{p}} \rightarrow \hat{\mathbf{p}} - q\mathbf{A}$$

$$\hat{H}\rightarrow \hat{H} + q\phi$$,

where appears the momentum operator $\hat{\mathbf{p}}$, the Hamiltonian $\hat{H}$ ,the vector potential $\mathbf{A}$ , the scalar potential $\phi$  and $$q$$ the charge of the particle under consideration. By imposing Coulomb gauge, i.e. $\mathbf{\nabla}\cdot\mathbf{A} = 0$, the scalar potential can be set to zero, whereas the vector potential reads

$$\mathbf{A}(\mathbf{r},t)= -\frac{E_\omega}{\omega}\mathbf{\epsilon}\cos[\mathbf{k}\cdot\mathbf{r} - \omega t + \delta];\quad \phi=0$$.

In a non-relativistic framework, the Hamiltonian obtained from minimal substitution is the Pauli Hamiltonian

$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{e}{2m}\mathbf{A}\cdot\hat{\mathbf{p}} + \frac{e^2A^2}{2m} + \frac{e\hbar}{2m}(\mathbf{\sigma}\cdot\mathbf{B})$$.

If we then assume that the system is close-shell, the spin-Zeeman term (fourth term) can be neglected. Furthermore, the diamagnetic term (third term) can be ignored for small field strengths. Therefore, the light-matter interaction is given by the second term

$$\hat{V}(t) = \frac{eE_\omega}{2m\omega}(\mathbf{\epsilon}\cdot\hat{\mathbf{p}})\cos[\mathbf{k}\cdot\mathbf{r}-\omega t + \delta] $$.

For time-periodic interaction operators, such as the one shown above, the absorption intensity is given by Fermi's golden rule

$$\sigma_{i\rightarrow f}(\omega) = \frac{\pi}{\varepsilon_0\hbar\omega c}|\langle 0|\frac{e}{m} e^{i\mathbf{k}\cdot\mathbf{r}}(\mathbf{\epsilon}\cdot\hat{\mathbf{p}})|i\rangle|^2 f(\omega,\omega_{fi},\gamma_{fi}) $$.

The transition moments are written in terms of the full light--matter interaction operator, which takes into account all multipolar effects, from zero to infinity. The familiar expression known from the electric-dipole approximation can be retrieved by imposing the limit $kr\ll 1 $

$$\sigma_{i\rightarrow f}(\omega) = \frac{\pi}{\varepsilon_0\hbar\omega c}|\langle 0|\frac{e}{m} (\mathbf{\epsilon}\cdot\hat{\mathbf{p}})|i\rangle|^2 f(\omega,\omega_{fi},\gamma_{fi}) $$.

The classical fields of electromagnetic plane waves

= Theoretical = In the simulation of X-ray absorption spectroscopy, the validity of the electric dipole approximation comes into question. Three different schemes exist to go beyond this approximation: the first scheme is based on the full semi-classical light--matter interaction, whereas the latter two schemes, referred to as the generalized length and velocity representation, are based on truncated multipole expansions.