the partial Wigner transform of a product of operators, $$\begin{aligned}
\left(\hat\chi_1\hat\chi_2\right)_{\rm W}(X)
&=&\hat\chi_{1,\rm W}(X)
e^{\frac{i\hbar}{2} \overleftarrow{\mbox{\boldmath$\nabla$}}
\mbox{\boldmath${\cal B}$}
\overrightarrow{\mbox{\boldmath$\nabla$}}}\hat\chi_{2,\rm W}(X) \;,\end{aligned}$$ and truncated the exponential after first order in $\hbar$, [i.e.]{}, $$e^{\frac{i\hbar}{2} \overleftarrow{\mbox{\boldmath$\nabla$}}
\mbox{\boldmath${\cal B}$}
\overrightarrow{\mbox{\boldmath$\nabla$}}}
\approx
1+ \frac{i\hbar}{2} \overleftarrow{\mbox{\boldmath$\nabla$}}
\mbox{\boldmath${\cal B}$}
\overrightarrow{\mbox{\boldmath$\nabla$}} \;.
\label{eq:linexp}$$

It should be noted that Equation (\[eq:linexp\]) is exact for Hamiltonians with quadratic bath terms and bilinear coupling between the $\hat x$ and $X$ DOF. In Ref. [@kapra-cicco], it is shown how the linear expansion can be performed in terms of the parameter $\mu=\sqrt{m/M}$, which is small in cases where the bath DOF are much more massive than those of the subsystem. Equation (\[eq:qcle\]) is exactly equivalent to Equation (\[eq:qclm\]).

Integration Algorithm
---------------------

A number of algorithms, which depend on the basis representation, exist for approximately solving the QCLE [@wan-schofield; @wan-schofield2; @wan-schofield3; @schutte2; @schutte3; @roman-martens; @sergi-theor; @sstp; @trotter; @asfp; @dauasfp; @dauasfp2; @fmgh; @ddagh; @ddagh2; @ddagh3; @ddagh4; @jlgh]. Herein, we illustrate the so-called Sequential Short-Time Propagation (SSTP) algorithm [@sergi-theor; @sstp], which offers a good compromise between accuracy and simplicity of implementation. The SSTP algorithm is based on the representation of the QCLE in the adiabatic basis, which is defined by the eigenvalue equation $$\hat h_{\rm W}|\alpha;Q\rangle 