effects. In the NZ case, the evolution of the system den- sity at time t depends on its full history (i.e. on ρ$_{S}$ ( s ) for all s ≤ t ). In the TCL case, the master equation is local in time and non-Markovian effects are treated in time-dependent transport coefficients. To illustrate the differences between our new QMC method and TCL, we remind below the corresponding local master equation, for V = Q ⊗ B . Δ( t ), λ ( t ), D$_{PQ}$ ( t ) and D$_{PP}$ ( t ) are time-dependent transport coefficients that contain memory effects. Simi- larly to the QMC case, the solution of the master Eq. (25) is equivalent to follow first and second moments In Figure 3, the evolution of Σ$_{PP}$ for different cut-off frequencies ℏ Ω and coupling strengths η are compared to the exact evolution (solid line). Explicit forms of the equation of motion in the NZ and TCL case can be re- spectively found in ref. [19, 38, 39, 52] and [19, 40, 41, 52]. Several important remarks could be drawn from this com- parison: (i) In all cases, when the coupling strength is considered up to second order, NZ2 (open triangles) pro- vides a better approximation than TCL2 (open squares). This might indeed be expected since NZ2 and TCL2 are respectively equivalent to the Born and Redfield master equation and the former contains a priori less approxima- tions than the latter. (ii) While the TCL4 (filled squares) leads to a clear improvement compared to the TCL2, NZ4 (filled triangles) is in general worse than NZ2. This is a known difficulty of NZ approach and was one of the mo- tivation for the introduction of TCL method (see discus- sion in ref. [19, 40, 41]). This stems from the fact that the order in perturbation in NZ cannot be identified. For in- stance, NZ2 (resp. NZ4) contains orders in coupling con- stant greater than 2 (resp. 4). As a result, the NZ theory does not lead to better results when the ”apparent” order in the coupling increases. The TCL method essentially cures this pathology and precise orders in the coupling can be selected. (iii) Rather large deviations between the exact and TCL2 are observed for different cut-off fre- quencies and coupling strengths. This issue is important since several theories have been recently developed along the line of TCL2 to include memory effects in fusion and fission reactions [14, 15, 17]. Note that, the accuracy of TCL2 depends on different parameters used in the spec- tral density, in particular of the parameter ℏ Ω. Here, with ω $_{p}$( t ) = ω 0 + Δ( t ). In practice, the exact NZ or TCL theory cannot be exactly solved and an expansion in powers of the coupling constant is made. In the following, NZ2 (or TCL2) will refer to the expansion up to second order while NZ4 (or TCL4) will refer to the expansion is made up to fourth order. By neglecting higher orders in the coupling in NZ2 (resp. TCL2) or NZ4 (resp. TCL4), both theory are not exact anymore. In the following, the efficiency of each method is systematically discussed. given by:         d 〈 Q 〉 dt = 〈 P 〉 M d 〈 P 〉 dt = − Mω 2 $_{p}$( t ) 〈 Q 〉 − 2 λ ( t ) 〈 P 〉 d Σ$_{PP}$ dt = − 2 Mω 2 $_{p}$( t ) Σ$_{PQ}$ − 4 λ ( t )Σ$_{PP}$ + 2 D$_{PP}$ ( t ) d Σ$_{QQ}$ dt = 2 Σ$_{PQ}$ M d Σ$_{PQ}$ dt = − Mω 2 $_{p}$( t ) Σ$_{QQ}$ − 2 λ ( t )Σ$_{PQ}$ + Σ$_{PP}$ M + 2 D$_{PQ}$ ( t ) , 2 2 6 0.1 0.2 0.3  0  3  6 $_{o}$ t  0.1 0.2 0.3 PP h 2  / $^{γ}$ 2 0.2 0.6 1 1.4 2 2.6 0 3 6 $_{o}$ t  1.5 3 4.5 QQ 2 7 14 21 -0.2 0 0.2 0 3 6 $_{o}$ t  -0.3 0 0.3 PQ h 0 1 2 FIG. 2: (Color online) Evolution of Σ PP (left), Σ QQ (middle) and Σ PQ (right) obtained with 10 5 trajectories are displayed with red filled circles as a function of time and systematically compared with the exact evolution (solid line). k B T = 5 ℏ ω 0 , ℏ ω 0 and 0 , 1 ℏ ω 0 are respectively shown from top to bottom. In all cases, η = 0 , 5 ℏ ω 0 , ℏ Ω = 5 ℏ ω 0 and ℏ ω 0 = 14 MeV . ℏ d dt ρ$_{S}$ ( t ) = − i [ H$_{S}$, ρ$_{S}$ ( t )] − i 2 Δ( t )[ Q, { Q, ρ$_{S}$ ( t ) } ] − 2 iλ ( t )[ Q, { P, ρ$_{S}$ ( t ) } ] − D$_{PP}$ ( t ) ℏ [ Q, [ Q, ρ$_{S}$ ( t )]] + 2 D$_{PQ}$ ( t ) ℏ [ Q, [ P, ρ$_{S}$ ( t )]] . (25)