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, H_V the vacuum field, H_E the energy density of the classical field and H_D the dipole interaction.

My question is regarding H_E. 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?Niles.

 

[f95toli]

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

 

[Bill_K]

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*e^ik·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.