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I would like to ask when we describe coherent light interacting with atoms, what's the difference in hot atom case and BEC regime?

In hot atom vapor, I know we could start with a simple interaction Hamiltonian with rotating wave approximation

$H=\hbar(a\sigma^++a^{\dagger}\sigma^-)$

and we usually neglect the atom kinetic part in the Hamiltonian, (or did we?). However, it seems to me that in BEC regime, we have to consider the recoil momentum effect.

1.Is there any substantial difference between the hot and BEC case?

2.Is there a similar interaction Hamiltonian in second quantization form in BEC regime?

Thanks in advance!

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A few scattered thoughts....

-The Hamiltonian you have written looks rather like a Jaynes-Cummings type Hamiltonian which only couples one EM mode to an atom. This is fine, but it is most relevant to an experiment in an optical cavity. In free space a more natural starting point would be something like the EM dipole Hamiltonian: $$H_I=-\vec{d}\cdot\vec{E}(x,t)$$, where $\vec{d}$ is the electron dipole operator, $\vec{d}=e\hat{x}$. This is more general in that it isn't assuming that only two modes are coupled.

-In general you certainly need to take the momentum transfer into account. There will be some cases when it is negligible, which you can roughly determine by asking what the wavelength of the light is relative to the spatial extent of the atomic wavefunction (which is, of course, generally larger for a BEC than for a thermal atom). For example, shining RF light onto atoms confined to tens or hundreds of microns will cause negligible recoil.

-On the other hand, for visible light the field changes so quickly that the atom typically only reacts to the time-averaged field, which might mean again that recoil is not important.

-For a more general case, it is probably best to go back to the general dipole expression and start taking matrix elements.

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