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A calculation about coherent state

  1. Jul 6, 2014 #1
    Hi, there. The following is copied from a book " atom-photon interaction " by Prof. Claude Cohen-Tannoudji, Page 415.

    I do not understand it at all. I do know some thing about the coherent state.
    such as

    ##a |\alpha\rangle = \alpha |\alpha\rangle##

    and

    ##|\alpha\rangle = e^{-|\alpha|^2/2} \sum_n \frac{\alpha^n}{\sqrt{n!}}|n\rangle##

    But I don't understand what's going on here.

    (1) Why the coherent state is ##|\alpha \exp(-i\omega_L t)\rangle##?

    (2) ##\langle N \rangle = \alpha^2## means ##\langle \alpha \exp(-i\omega_L t)| N |\alpha \exp(-i\omega_L t)\rangle = \alpha^2##?

    (3) Where does the ##\cos(\omega_L t)## come from in the first equation?
     
  2. jcsd
  3. Jul 6, 2014 #2

    mfb

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    2016 Award

    Staff: Mentor

    The exponential is just the time-evolution with the laser frequency.

    Write N in terms of a and ##a^\dagger## and let them operate on the different sides, that should work.

    It is composed of the two exponentials on the left side, but I didn't check in detail which part comes from what.
     
  4. Jul 6, 2014 #3
    In a book called quantum optics by M.O.Scully, the eigen-state of the operator ##a## is written as ##|\alpha\rangle##, Here, in the book by Claude. the eigen-state is ##|\alpha \exp(-i\omega_L t)\rangle##. This is the same thing. The latter is just using the form of ##\rho e^{i \theta}## to represent the complex eigen-value of the ##a##.

    am i right?
     
  5. Jul 8, 2014 #4

    mfb

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    2016 Award

    Staff: Mentor

    Probably, yes.
     
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