# Semiclassical descriptions of atom-light interaction

Niles
Hi

Say I want to describe the interaction between a free atom atom and a classical EM field. The full Hamiltonian for the problem must be
$$H = (H_0 + p^2/2m) + (H_V + H_E) + H_D$$
where H0 denotes the internal levels of the atom, HV the vacuum field, HE the energy density of the classical field and HD the dipole interaction.

My question is regarding HE. I have never seen it written explicitly and have not been able to find a reference where they do so. How would one write this? Is it QM-version of something similar to Poyntings vector?

Best regards,
Niles.

Gold Member
The Hamiltonian for a classical drive is something along the line of

$$E(e^{-i\omega t}a^\dagger+e^{i \omega t} a)$$

I would suggest you look up some info about the driven Jaynes-Cummings model

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The usual form for the classical Hamiltonian for the electromagnetic field is the energy density, H = ½(E2 + B2). The easiest way to get to the quantum form is to use the radiation gauge, Φ = 0 and ∇·A = 0, allowing H to be written in terms of the vector potential as H = ½((A·)2 + (∇ x A)2).

Now Fourier transform, A = ∫d3k/√(2ω) ∑ε(k)[a(k)e-ik·x + a*eik·x], where ε(k) are polarization vectors. In terms of a(k) the Hamiltonian reduces to H = ∫d3k ω ∑a*(k)a(k). This is still classical.

Quantum mechanics comes in when you now say that a*(k)a(k) = ħ N(k) where N(k) is the number operator.

Niles
Ah, I see. Thanks for taking the time to explain that. I will study your reply in depth.

Best wishes,
Niles.