Light and atom interaction hamiltonian

[naima]

I found this in a Phd thesis

consider a two level atom interacting with the electromagnetic field.
The atom is described by
$H_{at} = \hbar ω_0 J_z$
a monomode electric field is described by
$H_{em} = \hbar \omega (a^\dagger a + 1/2)$
We have $E = E_0(a^\dagger + a)$ and the dipolar moment is $D = d_o(J_+ - J_-)$
We have then $H_{int} = -DE$

So this interference hamitonian contains four terms
a) $a J_+ and a^\dagger J_-$
but also
b) $a J_- and a^\dagger J_+$
the author writes later a formula without the b) terms.
How can we make them disappear (mathematically)?

 

[mfb]

Physically, they are zero because they correspond to transitions to non-existing states (as we just have two states). There is also some more mathematical argument that I forgot, but they are really zero.

 

[naima]

I have no doubt about it. But as he begins with a hamiltonian with the b) terms i think that there is a mathematicall trick to erase them.

Take (g,n). Does his hamiltonian permit to get (e,n+1) which exists?

 

[naima]

Has an interference hamiltonian to commute with the free hamiltonian?

 

[Andy Resnick]

I have no doubt about it. But as he begins with a hamiltonian with the b) terms i think that there is a mathematicall trick to erase them.

Take (g,n). Does his hamiltonian permit to get (e,n+1) which exists?

If I understand your notation, the terms in (b) either lower the state and absorb a photon (which cannot happen) or raise the state and emit a photon (which also cannot happen). As for the above, the interaction Hamiltonian connects (g,n) with (e, n-1).

 

[naima]

My question is about the DE hamiltonian (D is the dipole,E is the electric field)
How is it quantized?
The answer may be in the fact that DE is a scalar product and that E is transverse so it gives 2 components?

 

[Andy Resnick]

Most derivations I have seen that quantize the electromagnetic field use a 'canonical quantization method'. Similarly, quantizing the dipole moment can be done several different ways, but generally start with the Jaynes-Cummings model:

http://mina4-49.mc2.chalmers.se/~gojo71/KvantInfo/QuantizedField_JC.pdf

 

[naima]

Thank you for the link. It says that two of the four terms can be omitted according to the Rotating Frame Approximation (RWA)
So in a full correct calculus we would have a small term corresponding to an electron emitting a photon and getting a higher energy!