Second order correlation function of a quantized electric field

Question

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.

Answer 1

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.

As for the time dependence of the creation and annihilation operators, usually these operator are associated with specific frequencies. So to get a time dependent signal, one needs to consider a spectrum of such frequencies and then perform an inverse Fourier transform.