A calculation about coherent state

[Robert_G]

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

If the laser mode is in a coherent state $|\alpha \exp(-i\omega_L t)\rangle$ with $\alpha$ being real, Then the average value

$\langle\alpha \exp(-i\omega_L t)|E(R)|\alpha \exp(-i\omega_L t)\rangle=\mathscr{E}_0 \cos(\omega_L t)$

with

$\mathscr{E}_0=2 \epsilon \sqrt{\frac{\hbar \omega_L}{2 \epsilon V}} \sqrt{\langle N \rangle}$

$\langle N \rangle = \alpha^2$

and

$E(R)=\sqrt{\frac{\hbar \omega_L}{2 \epsilon_0 V}}\epsilon_L(a+a^{\dagger})$

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?

 

[mfb]

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

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

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

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

(3) Where does the $\cos(\omega_L t)$ come from in the first equation?

It is composed of the two exponentials on the left side, but I didn't check in detail which part comes from what.

 

[Robert_G]

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.

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?

 

[mfb]

Probably, yes.