density characteristic of the environment [19, 50]. In the following, a Drude spectral density [43] is considered where M is the nucleon mass. As a first illustration, the harmonic case ( ε = +1) is considered. This case has been already studied in ref. [51] using the stochastic method proposed in ref. [21]. In the CL model, starting from an initial Gaussian density, the system density remains gaussian along the stochastic path. Therefore, the stochastic evolution of the system density reduces to the first and second moments evolution of 〈 P 〉 and 〈 Q 〉 given by [36]: These equations illustrate the differences between the new exact reformulation and standard methods to treat dissipation. Generally, the noise enters into the evolu- tion of 〈 P 〉 only and affects directly the second moment. Here, we see that second moments identify with the un- perturbed ones while the random forces enter in both 〈 Q 〉 and 〈 P 〉 . In addition, the noise is complex, which im- plies that observables make excursions into the complex plane. This stems from the specific noise used to design the exact formulation that leads to non-Hermitian den- sities along paths. Part of the conceptual difficulties in understanding the physical meaning of observable evolu- tions can be overcome by noting that if ρ$_{S}$ ( t ) belongs to the set of trajectories, by symmetry ρ † $_{S}$( t ) will also be- longs to the set. By grouping these two trajectories to estimate observables, real quantities are deduced. The exact evolution is obtained by averaging over dif- ferent trajectories. For second moments, this leads to where 〈 X 〉 denotes the statistical average of quantum expectation values 〈 X 〉 . It is a particularity of the CL model that total fluctuation is recovered by simply adding up quantum and statistical fluctuations. An example of Σ$_{QQ}$( t ) evolution obtained using Eq. (24) (red filled circles) is compared to the exact result (solid line) in figure 1. As an indication, the evolution of Conjointly to the benchmark of quantum Monte-Carlo approaches, we also tested projection method either based on the Nakajima-Zwanzig [19, 38, 39, 52] or Time ConvolutionLess [19, 40, 41, 52] formalisms. Both theo- ries provide a priori exact re-formulations of the initial problem and lead to a closed master equation for the sys- tem density. However, they differ completely in the strat- egy and equation of motion used to incorporate memory The accuracy of the quantum Monte-Carlo theory has been systematically investigated for various temperatures and coupling strengths. In all cases, averaged evolutions could almost not be distinguished from the exact evo- lution. This is illustrated in figure 2 where Σ$_{QQ}$ (left), Σ$_{PP}$ (middle) and Σ$_{PQ}$ (right) are displayed as a func- tion of time and compared to exact solutions for various temperatures. Figure 2, clearly shows that the stochas- tic method properly includes all non-Markovian effects. In particular at low temperature, typically k$_{B}$T < ℏ ω 0 , and medium coupling constant η , large memory effect is expected. quantum fluctuation σ$_{QQ}$ , which is identical for all tra- jectories, is also displayed (dotted line). Note that, Eq. (16) is already exact for the evolution of first moments 〈 P 〉 and 〈 Q 〉 , even if the noise is omitted. However, it completely fails to account for fluctuation. While the quantum evolution does not present any damping, the average evolution closely follows the exact solution. The harmonic oscillations in σ$_{QQ}$ are due to the fact that the width of the initial density differs from the width of the coherent state associated to the considered oscillator, i.e. σ$_{QQ}$ (0) = ℏ / (2 Mω 0 ). This is at variance with the simu- lation made in ref. [51]. FIG. 1: (Color online) Evolution of Σ QQ (filled circles) ob- tained by averaging over 10 5 trajectories. This evolution is compared to the exact result (solid line) and to the quan- tum fluctuation σ QQ evolution (dotted line). Parameters of the simulation are k B T = ℏ ω 0 , ℏ Ω = 5 ℏ ω 0 , η = 0 , 5 ℏ ω 0 and ℏ ω 0 = 14 MeV . The factor γ defined as γ 2 = ℏ / (2 Mω 0 ) equals here γ = 1 . 216 fm $^{$_{−}$1}$. 5 J ( ω ) = 2 Mη π ω Ω 2 ω 2 + Ω $_{2}$, (22) A. Quantum Monte-Carlo method for parabolic potentials       d 〈 Q 〉 = 〈 P 〉 M dt + 2 du$_{S}$σ$_{QQ}$ d 〈 P 〉 = − Mω 2 $_{0}$〈 Q 〉 dt − dt 〈 B 〉 + 2 du$_{S}$σ$_{PQ}$ − ℏ dv$_{S}$ dσ$_{QQ}$ = 2 dt $_{M}$σ$_{PQ}$ dσ$_{PP}$ = − 2 Mω 2 $_{0}$dtσ$_{PQ}$ dσ$_{PQ}$ = dt $_{M}$σ$_{PP}$ − Mω 2 $_{0}$σ$_{QQ}$ dt (23) These equations illustrate the differences between the  1  2  3  4  5 0 3 6 QQ 2 $_{o}$ t  Σ$_{QQ}$ ≡ 〈 Q $^{2}$〉 − 〈 Q 〉 2 = σ$_{QQ}$ + 〈 Q 〉 2 − 〈 Q 〉 2 (24) B. Application of NZ and TCL