How can one get the density operator from the characteristic function? To solve analytically the master equation of two qubits interacting with a cavity  mode through their environment we use the charactristic function,
$$\chi (\beta)=\operatorname{tr}[\rho D(\beta)],$$  with $D(\beta)$  being the displacement operator.
How do we get the $\rho$ from the $\chi$ expression?
 A: The fundamental reconstruction theorem in phase-space quantum mechanics inverts the Wigner transform of an arbitrary operator $\hat G$ through the Weyl transform,
$$
\hat G= \frac {\hbar}{2\pi} \int\!\! d\tau d\sigma~~ e^{i(\tau \hat p + \sigma \hat x)}   ~~\operatorname{tr} [\hat G~ e^{-i(\tau \hat p + \sigma \hat x) } ] .
$$
(If you blink in a right-brain hemisphere vision, your might dream of a collapsing 2d operator δ-function there, but don't worry if this does not evoke anything. The trivial formal proof is in our Concise Treatise of Quantum Mechanics in Phase Space, Ch 0.18. It first appeared in Groenewold's 1946 breath-taking  dissertation.)
Mindful of the connection to optical phase space, 
$$
\hat D(\beta)= e^{\beta a^\dagger -\beta ^* a}= e^{\hat x (\beta -\beta^*)/\sqrt{2}-i\hat p (\beta + \beta^*)/\sqrt{2}},
$$
and using $\hat ρ$ as your operator, you translate this to a linear combination of translation operators, your building blocks,
$$
\hat\rho= \int \frac{d^2\beta }{\pi}  ~~  \hat D(-\beta)   ~~   \operatorname{tr} [ \hat\rho \hat D(\beta) ]=  \int \frac{d^2\beta }{\pi}  ~~   \hat D(-\beta)   ~~ \chi(\beta) .
$$
A: Although the expression you give is quite correct, it may help to think of the calculation of the characteristic function from the density operator as a two-step process. First one would compute the Wigner function of the density operator. Then one computes the Fourier transform of the Wigner function to get the characteristic function.
Now the inverse process is easier to understand. With the inverse Fourier transform one can recover the Wigner functions from the characteristic function. And the Weyl transform then reproduces the density operator from the Wigner function.
