# Interaction of an electromagnetic wave with a two level system in the domain of quantum field theory

Suppose I shine an electromagnetic wave on a two-level system. I need to describe how the system evolves in context of quantum field theory i.e. using a quantized EM field in the problem. The first step would be to write down the interaction Hamiltonian. What would it be?

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Hi Venu, I added the homework tag to your question because it seems to apply here. See our homework policy for details. – Brandon Enright Jun 1 '13 at 16:49
oh I am not asking for a solution. any clues to how to go about with it is just fine. I was asked this in an interview and didn't know the answer or way to go about with it. – venu Jun 1 '13 at 17:52

A single free-space field mode of the quantized electric field in the dipole approximation can be written as: $$\vec{\hat{E}}(t) = i\left(\frac{\hbar \omega}{2\epsilon_0 V}\right)^{1/2}\vec{e}(\hat{a}e^{-i\omega t} - \hat{a}^{\dagger}e^{i\omega t}),$$ with $\omega$ the frequency of the field, V the volume in which the field 'lives' and $a$ and $a^{\dagger}$ the usual lower and raising operators. (See for instance Chapter 2 Gerry & Knight - Introductory Quantum Optics). The Hamiltonian for the atom interacting with the quantized field would be: $H = H_{atom} + H_{field} + H_{int}$, where $$H_{int} = -\hat{d}\cdot\hat{E} = -i\left(\frac{\hbar \omega}{2\epsilon_0 V}\right)^{1/2}(\hat{d}\cdot{e})(\hat{a}-\hat{a}^{\dagger}).$$ The operator $d$ is the dipole moment operator. (Note that the E field in the last equation has been taken in the Schrödinger picture!)