Density equations (light considered as reservioir)

  1. Hi, there

    I am reading the book called "Atom-Photon Interaction", the chapter of " Radiation considered as a Reservoir". My question is actually short, but I have to describe the background.

    Following is the density equation which describes the interaction between the damped harmonic oscillator and the radiation.

    [itex]\frac{d \sigma}{dt}=-\frac{\Gamma}{2}[a, b^\dagger b]_+ - \Gamma'[\sigma, b^\dagger b]_+-i(\omega_0+\Delta)[b^\dagger b, a]+\Gamma b \sigma b^\dagger + \Gamma'(b^\dagger \sigma b + b \sigma b^\dagger)[/itex].

    Here, the ##\sigma## is the density operator for the harmonic oscillator, and ##b## (##b^\dagger##) is the annihilation (creation) operator of the harmonic oscillator, and all the properties of the radiation is contained in the paremeters ##\Gamma## and ##\Gamma'##. Now we want to see how the population evolves, and this is about the calculation ##\langle n| \cdot \cdot \cdot|n \rangle##. So we need to calculate the term ##\langle n|b \sigma b^\dagger|n \rangle##. The following is how I did it, and it actually can lead to the answer that printed in the book.

    ##\langle n| b \sigma b^\dagger|n \rangle=(b^\dagger |n\rangle)^\dagger \; \sigma \; b^\dagger|n \rangle##

    Using ##b^\dagger |n \rangle = \sqrt{n+1}|n+1\rangle## can bring us

    ##(n+1)\sigma_{n+1,n+1}##

    -------------------------------------------------------------------
    My question is how about do it the other way.

    ##\langle n| b \sigma b^\dagger|n \rangle=\langle n | b \sigma (b^\dagger | n \rangle)##

    ##=\sqrt{n+1}\langle n | b \sigma|n+1\rangle##

    Now, If I knew the commuter of ##[\sigma, b]## or, what's ##\sigma |n+1 \rangle##, I can go on with the calcuation, But I don't. Does anyone know how to do it in this way? Do not calculate from the left to right.




    PS: It 's correct in the first way, right?
    PPS: This is not a stupid question, I hope.
     
  2. jcsd
  3. Greg Bernhardt

    Staff: Admin

    I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
     
  4. What does the variable 'a' stand for?
     
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