I am looking for a relatively clean expression for matrix elements for states of the form $$\rho_{\alpha,r,\bar{n}} = \hat{D}(\alpha)\hat{S}(r)\rho_{th}(\bar{n})\hat{S}^\dagger(r)\hat{D}^\dagger(\alpha),$$ where $\hat{D}(\alpha)$ is the displacement operator with a complex parameter, $\hat{S}(r)$ is the squeeze operator with a real parameter, and $\rho_{th}(\bar{n})$ is a thermal state with mean photon number $\bar{n}$. That is, I would like to express $\langle m|\rho_{\alpha,r,\bar{n}}|n\rangle$ in terms of $m,n,\alpha,r,\text{ and }\bar{n}$.
I have tried using the following three facts, that$$\rho_{th}=\frac{1}{\bar{n}+1}\sum\limits_{k=0}^\infty(\frac{\bar{n}}{\bar{n}+1})^k|k\rangle\langle k|,$$$$\hat{D}(\alpha)=e^{\alpha \hat{a}^\dagger - \alpha^*\hat{a}} = \sum\limits_{k=0}^\infty\frac{1}{k!}(\alpha \hat{a}^\dagger - \alpha^* \hat{a})^k,$$$$\hat{S}(r)=e^{\frac{r}{2}(\hat{a}^2 - \hat{a}^{\dagger2})} = \sum\limits_{k=0}^\infty\frac{1}{k!}(\frac{r}{2}(\hat{a}^2-\hat{a}^{\dagger2}))^k,$$ to make use of the creation and annihilation operators' actions on the number states, but the infinite sums become unwieldy.
If anyone could offer a better approach or point me toward some relevant resources, it would be greatly appreciated. Thanks!
