\langle\alpha,S|\frac{d}{dt'}|\alpha,S\rangle
 -\langle\alpha',S|\frac{d}{dt'}|\alpha',S\rangle \right)\right]
\nonumber\\
&\times&
\exp\left[ -(t-t_0)
H_{\alpha\alpha'}^{\rm S}
\overleftarrow{\nabla}^{\bf S}
\mbox{\boldmath$\cal B$}^{\rm S}
\overrightarrow{\nabla}^{\bf S}
\right]
W^{\bf S}_{\alpha\alpha'}({\bf S},t_0)\;.
\label{eq:qcsd-dyna-ad-prop}\end{aligned}$$ Equation (\[eq:qcsd-dyna-ad-prop\]) provides a convenient starting point for devising numerical integration schemes based on the SSTP propagation scheme [@sstp].

In Ref. [@sergi-spin2], the following model Hamiltonian was considered: $$\begin{aligned}
\hat H({\bf S}) &=& -\Omega \hat\sigma_x-c_1b\hat\sigma_z
-\mu{\bf S}\cdot\mbox{\boldmath$\sigma$}-c_2bS_z+\frac{S_z^2}{2}
\\
&=&\hat h_{\bf S}({\bf S}) -c_2bS_z +\frac{S_z^2}{2}\;,\end{aligned}$$ where $\Omega$, $c_1$, and $c_2$ are real parameters, $b$ is the $z$ component of the magnetic field ${\bf B}=(0,0,b)$, while $\mbox{\boldmath$\sigma$}=(\sigma_x,\sigma_y,\sigma_z)$ is a vector having the Pauli matrices $\sigma_x$, $\sigma_y$, and $\sigma_z$ as components. The SSTP algorithm was applied to Equation (\[eq:qcsd-dyna-ad-prop\]) and the action of the classical like Liouville operator $H_{\alpha\alpha'}^{\rm S} \overleftarrow{\nabla}^{\bf S}
\mbox{\boldmath$\cal B$}^{\rm S} \overrightarrow{\nabla}^{\bf S}$ was evaluated using time reversible integration algorithms based on the symmetric break-up of the Liouville propagator [@tuck1; @tuck2; @ribes].

Stochastic Classical Baths {#sec:qcd-sb}
==========================

Consider a quantum-classical system comprising a quantum subsystem and a classical environment whose classical phase space coordinates are partitioned into two sets: one set $X=(Q,P)$ interacts directly with the quantum subsystem while the second set $X'=(Q^{\prime},P^{\prime})$ interacts only with the coordinates $X$ (and therefore is not directly coupled to the quantum 