\label{eq:prop_sstp_W}$$ When $\tau$ is infinitesimal, the right-hand side of Equations (\[eq:prop\_sstp\]) and  (\[eq:prop\_sstp\_W\]), become essentially equal to the left-hand side, as can be seen from the Dyson identity [@sstp].

The transition operator is purely off-diagonal. Its action generates quantum transitions in the subsystems and changes the bath momenta accordingly. Upon setting the transition operator to zero, we obtain an adiabatic expression for the propagator. If the non-adiabatic effects are not too strong, they may be treated in a perturbative fashion by sampling the action of the transition operator in a stochastic fashion. Typically, researchers have used  [[@b4; @mqc-prot; @isotope; @h-complex; @2spins; @spectra; @etransf; @opto; @prot-el; @plasmon; @prot-el2; @pump-probe; @vibr; @helix; @sergi-theor; @rev4; @rev7; @rev9; @rev10; @rev11; @sergi-spin1; @sergi-spin2; @sstp; @fmgh; @ddagh; @ddagh2; @ddagh3; @ddagh4]]{} the following expressions for the probabilities of making a transition (jump) and not-making a transition, respectively: $$\begin{aligned}
{\cal P}_{\rm J} &=& \frac{\vert \tau \frac{P}{M}\cdot d_{\alpha\beta}\vert }
{1 + \vert \tau \frac{P}{M}\cdot d_{\alpha\beta}\vert }\;,
\\
Q_{\rm NO-J} &=& \frac{ 1}
{1 + \vert \tau \frac{P}{M}\cdot d_{\alpha\beta}\vert }\;.\end{aligned}$$ Another important technical ingredient of the algorithm is the approximation of the transition operator in Equation (\[eq:T\_aa’bb’\]) with its momentum-jump form: $$\begin{aligned}
{\cal T}_{\alpha\alpha',\beta\beta'}^{\rm MJ}
&=&
\delta_{\alpha'\beta'}
\frac{P}{M}\cdot d_{\alpha\beta}
e^{(E_\alpha-E_\beta)M\partial/\partial(P\cdot\hat 