In this paper : https://arxiv.org/abs/1206.3405
They consider a density matrix :
$$ \rho = \int P(\alpha) |\alpha\rangle \langle \alpha | $$
Where $|\alpha \rangle$ are coherent states.
Using it, we can easily prove their formula (5) :
$$ \langle ({a^{\dagger}})^m a^n \rangle =\operatorname{Tr}[\rho ({a^{\dagger}})^m a^n]=\int P(\alpha) \operatorname{Tr}[({a^{\dagger}})^m a^n|\alpha\rangle \langle \alpha |] $$
And we use the circular permutation of the trace + the fact that $\operatorname{Tr}(|\alpha \rangle \langle \alpha|)=1$, and we end up with :
$$\langle ({a^{\dagger}})^m a^n \rangle=\int \alpha^n {\alpha^{*}}^m P(\alpha)$$ that is their formula (5).
But I don't understand how we can find their formula (7) :
Indeed, we would have :
$$ \langle a^n ({a^{\dagger}})^m \rangle=\operatorname{Tr}[\rho a^n ({a^{\dagger}})^m ]=\int P(\alpha) \operatorname{Tr}[(a^n {a^{\dagger}})^m |\alpha\rangle \langle \alpha |] $$
But to continue I would need to either know the action :
$({a^{\dagger}})^m |\alpha\rangle$ (I don't remember why exactly, but I know it is not ${\alpha^{*}}^m |\alpha \rangle$)
Or I would need to do the big commutation of the power of creation/annihilation.
Thus I'm a little stuck : how can we prove the formula $(7)$ of the article ?
I have read the pages : https://en.wikipedia.org/wiki/Glauber%E2%80%93Sudarshan_P_representation https://en.wikipedia.org/wiki/Optical_equivalence_theorem and I am still stuck.
What I understand from the first page is that we can write the density matrix either :
$$ \rho = \int P(\alpha) |\alpha\rangle \langle \alpha | $$
or
$$ \rho_A = \sum_{jk} c_{jk} a^k (a^{\dagger})^k $$
And we have the relationship $P(\alpha)=\frac{1}{\pi} \rho_A(\alpha, \alpha^*)$
So I think the usefull thing to use is the formula with $ \rho_A$ but I'm stuck when using it to try to prove (7).