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Light and atom interaction hamiltonian

  1. Jan 27, 2015 #1

    naima

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    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)?
     
  2. jcsd
  3. Jan 27, 2015 #2

    mfb

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    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.
     
  4. Jan 27, 2015 #3

    naima

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    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?
     
    Last edited: Jan 27, 2015
  5. Jan 27, 2015 #4

    naima

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    Has an interference hamiltonian to commute with the free hamiltonian?
     
  6. Jan 27, 2015 #5

    Andy Resnick

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    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).
     
  7. Jan 27, 2015 #6

    naima

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    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?
     
    Last edited: Jan 27, 2015
  8. Jan 27, 2015 #7

    Andy Resnick

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  9. Jan 28, 2015 #8

    naima

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    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!
     
    Last edited: Jan 28, 2015
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