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Density equations (light considered as reservioir) 
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#1
Jun1814, 11:23 PM

P: 36

Hi, there
I am reading the book called "AtomPhoton 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 \cdotn \rangle##. So we need to calculate the term ##\langle nb \sigma b^\daggern \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^\daggern \rangle=(b^\dagger n\rangle)^\dagger \; \sigma \; b^\daggern \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^\daggern \rangle=\langle n  b \sigma (b^\dagger  n \rangle)## ##=\sqrt{n+1}\langle n  b \sigman+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
Jun2614, 10:36 PM

Admin
P: 9,553

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?



#3
Jul1214, 11:54 PM

P: 109

What does the variable 'a' stand for?



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