# Second order correlation function of a quantized electric field

I'm analysing an open quantum system where the optical cavity is interacting with a quantum dot. I modelled the system using cavity QED and used Lindblad master equation to model the system.

Now I have obtained an equation for the cavity field operators $\hat{a}^\dagger$ and $\hat{a}$. Now I want to calculate first-order correlation function of field operators.

$$g^{(1)}= \langle \hat{a}^\dagger(t)\hat{a}^\dagger(t+\tau) \hat{a}(t+\tau)\hat{a}(t) \rangle$$

Firstly, I am new to quantum optics and can't figure out how to calculate above analytically or plot it numerically. Secondly, I don't know how I can get the time dependence of Schrodinger operators $\hat{a}^\dagger$ and $\hat{a}$

If you can provide me with useful tools or some hints it will be useful.

It very much depends on what states you are working with. A rather simplistic example is with coherent states $|\alpha\rangle$, which are eigenstates of the annihilation operator. In that case $$\langle\alpha|\hat{a}^{\dagger}(t)\hat{a}^{\dagger}(t+\tau) \hat{a}(t+\tau)\hat{a}(t)|\alpha\rangle = \alpha^*(t)\alpha^*(t+\tau) \alpha(t+\tau)\alpha(t) = |\alpha(t+\tau)|^2|\alpha(t)|^2 .$$ Here, $|\alpha(t)|^2$ could represent the intensity of a laser beam as a function of time.
• thank you. I understand this now. Can you help me a bit more? I know the frequency spectrum of $\hat{a}$. Can you please explain to me how to get $\alpha(t+\tau)$. As there are infinite possibilities for $\tau$, I don't understand how to get it for all values of $\tau$. Commented Jul 6, 2018 at 13:44
• Yes, you'll get the result as a funcion of $\tau$. Usually one would integrate over $t$ so that the result only depends on $\tau$. This then gives you the correlation function. Commented Jul 7, 2018 at 13:27