In Equation (\[eq:qcsd-dyna-ad\]), the phase $\omega_{\alpha\alpha'}$ has a dynamical nature while the phase $\phi_{\alpha\alpha}^{\bf S}$ is of a geometric origin and it can be considered an instance of the famous Berry phase [@berry; @qphases; @mead]. Interestingly, Equation (\[eq:pWS\]) predicts that the geometric phase $\phi_{\alpha\alpha}^{\bf S}$ can be non-zero also for open paths of the classical spins of the bath (open-path Berry phases were discussed in Ref. [@pati]). Moreover, the phase factor $\phi_{\alpha\alpha}^{\bf S}-\phi_{\alpha'\alpha'}^{\bf S}$ is purely off-diagonal (off-diagonal Berry phases for environments described by canonically conjugate variables were discussed in Refs. [@filipp; @englman; @manini]). It is worth mentioning that the geometric phase $\phi_{\alpha\alpha}^{\bf S}$ is predicted also for non-adiabatic dynamics.

When the total Hamiltonian is time-independent, as the one in Equation (\[eq:tot-hamS\]), the adiabatic evolution of the matrix elements of the spin-bath operator-valued quasi-probability function, given by Equation (\[eq:qcsd-dyna-ad\]), can be rewritten as $$\begin{aligned}
\frac{\partial}{\partial t}W^{\bf S}_{\alpha\alpha'}({\bf S},t)
&=&\left[-i\omega_{\alpha\alpha'}
-\left( \langle\alpha,S|\frac{d}{dt}|\alpha,S\rangle -\langle\alpha',S|\frac{d}{dt}|\alpha',S\rangle \right)
- H^{\alpha\alpha'}_{\rm S}
\overleftarrow{\nabla}^{\bf S}
\mbox{\boldmath$\cal B$}^{\bf S}
\overrightarrow{\nabla}^{\bf S}
\right]
W^{\bf S}_{\alpha\alpha'}({\bf S},t) \;. \nonumber\\
\label{eq:qcsd-dyna-ad2}\end{aligned}$$

Using the Dyson identity, one can obtain the following form for $\hat W^{\bf S}({\bf S},t)$ in terms of the adiabatic propagator: $$\begin{aligned}
W^{\bf S}_{\alpha\alpha'}({\bf S},t)
&=&\exp\left[-i\int_{t_0}^t dt'\omega_{\alpha\alpha'}(t')\right]
\exp\left[
-\int_{t_0}^t dt'\left( 