we have used a value of the cut-off frequency between 10 and 20 MeV  which gives realistic dissipation and fluc- tuation for the fusion or fission mechanism[14  15  18]. Our study clearly points out that a proper treatment of memory requires to include higher order effects. (iv) Fi- nally  in all cases  TCL4 could not be distinguished from the exact result. As we will see  the efficiency of TCL4 is similar for the inverted parabola. Since NZ method is not competitive  only the quan- tum Monte-Carlo and TCL methods are considered in the following application. Several approaches have been recently developed to de- scribe fusion and fission reactions [6  14  15  17  18  53]. In these mechanisms  few collective degrees of freedom couple to a sea of internal excitations while passing an inverted barrier. At very low energy  both quantum and non-Markovian effects are expected to play a significant role. Most of the theory currently used start from quan- tum master equations deduced from TCL2. The quan- tum Monte-Carlo method offers a practical alternative which has similarities with path integrals theory. Path integrals are known to provide a possible framework to include dissipation while passing barriers (see for instance [54]). However  due to their complexity  only few appli- cations have been made so far [2  55]. We compare here the different approaches for inverted potential ( ε = − 1). C. Quantum Monte-Carlo method applied to inverted oscillators The accuracy of different methods is illustrated in fig- ure 5 where evolutions of 〈 Q 〉   〈 P 〉   Σ$_{QQ}$ and Σ$_{PP}$ are shown as a function of time. Values of parameters re- tained for this figure are typical values generally taken in the nuclear context[16]. In all cases  including TCL2  sec- ond moments are well reproduced. However  only TCL4 and the stochastic simulation provides a correct descrip- tion of first moments. Calculations are shown here for Δ E = 0 MeV. TCL2 provides a better and better ap- proximation when Δ E increases while the disagreement increases below the barrier. This will be further illus- trated below. It is tempting to group trajectories into those passing the barrier and those reflected by the potential to get information on the passing probability or passing time  however  it should be kept in mind that the present the- ory is fully quantal. Since each trajectories are associ- ated with densities with quantum widths  both trajecto- ries presented in Fig. 4 contribute to the transmission probability. which is nothing but the difference between total initial energy and barrier high. Both trajectories shown in fig- ure 4 correspond to Δ E = 0 MeV. where Q ( t ) and P ( t ) denote the real part of 〈 Q ( t ) 〉 and 〈 P ( t ) 〉 along the trajectory. An illustration of two trajec- tories  one passing the barrier and one reflected is shown in figure 4. As illustrated in the following  it is conve- nient to group trajectories according to the quantity Δ E defined by Contrary to the classical theory of Brownian motion  the notion of trajectories is not so easy to tackle in the present Monte-Carlo framework. First  observables are complex. As mentioned in section III A  this difficulty can be overcomed by grouping trajectories by pairs which is equivalent to replace expectation of observables by their real parts. Second  it should be kept in mind that the present theory is a purely quantum theory where den- sities associated to wave-packets are evolved. Therefore  each trajectory should be interpreted in the statistical sense of quantum mechanics and contains many classical paths. Nevertheless  to visualize the trajectory we define the following energies: Initially  we consider a Gaussian density with quantum width σ$_{QQ}$ (0) = 0 . 16 fm 2 and σ$_{PQ}$ (0) = 0 MeV.fm/c and positioned on one side of the potential (here taken arbitrarily at 〈 Q (0) 〉 = Q 0 > 0 while the barrier height is is located at 0 fm and is by convention taken as V$_{B}$ = 0 MeV). The initial kinetic energy  denoted E$_{K}$ (0) is set by boosting the density with an initial momentum 〈 P (0) 〉 = P 0 < 0. 1. Initial conditions  trajectories and mean evolution 7 E ( t ) = P ( t ) 2 2 m − 1 2 mω 2 $_{0}$Q ( t ) 2 (26) Δ E = E (0) − V$_{B}$ (27) FIG. 3: (Color online) Evolution of Σ PP for different approx- imations: NZ2 (open triangles)  NZ4 (filled triangles)  TCL2 (open squares) and TCL4 (filled squares). The exact evo- lution is displayed with solid line. In all cases  ℏ ω 0 = 14 MeV  and k B T = ℏ ω 0 are used. The left side  corre- sponds to different cut-off frequencies: ℏ Ω = 20 ℏ ω 0 (top) and ℏ Ω = 5 ℏ ω 0 (bottom). In both cases  η = 0   5 ℏ ω 0 . In the right side  ℏ Ω = 10 ℏ ω 0 and different coupling strengths are used: η = ℏ ω 0 (top) and η = 0 . 1 ℏ ω 0 (bottom). 0.2 0.3 0.4  0  3  6 PP h 2 / 2 $_{o}$ t 0.2 0.4 0.6 PP h 2 / 2 0.15 0.2 0.25 0 3 6 PP h 2 / 2 $_{o}$ t 0.3 0.6 0.9 PP h 2 / 2