noise process.

The well-developed machinery of Keldysh quantum field theory provides a precise method for accomplishing our task [@Levitov1996; @Nazarov2003; @BelzigPRL2010; @Clerk2011; @Hofer2017]. While this approach is completely general, the simplest derivation is to imagine coupling an ancilla qubit to $\hat{\xi}$, such that the only qubit dynamics is from the interaction picture Hamiltonian $\hat H_{\mathrm{int}} (t) = \frac{1}{2} F(t) \hat{\xi}(t)  \hat \sigma_z$. We then use the dephasing of the qubit to [*define*]{} the MGF of the noise in the quantum case, exactly like we would if the noise were classical: $$\begin{aligned}
\label{Eq:MGFqu}
  &   \Lambda[F( t );t_f] \equiv 
         \langle \hat{\sigma}_{-}(t) \rangle / \langle \hat{\sigma}_{-}(0) \rangle,
         \\
                \label{Eq:MGFK}
         = &
        \textrm{Tr} \left[
            \mathcal{T} e^{- \frac{i}{2} \int_0^{t_f} dt' F(t') \hat{\xi}(t')}
            \hat{\rho}_{\rm B}
            \tilde{\mathcal{T}} e^{-\frac{i}{2} \int_0^{t_f} dt' F(t') \hat{\xi}(t')}
        \right].    
                $$ Here $\hat{\rho}_B$ is the initial bath density matrix, the trace is over bath degrees of freedom, and $\mathcal{T}$ ($\tilde{\mathcal{T}}$) is the time-ordering (anti-time ordering) symbol. Expanding $\Lambda$ in powers of $F(t')$ defines correlation functions at a given order with a particular time-ordering prescription (the so-called Keldysh ordering). We stress that this approach amounts to trying to ascribe the qubit evolution to an effective classical stochastic process; this correspondence then defines cumulants (and implicitly 