# Density equations (light considered as reservioir)

1. Jun 18, 2014

### Robert_G

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.

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

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. Jun 26, 2014