content_image/403666.jpg content_image/403667.jpg content_image/403665.jpg content_image/403668.jpg content_image/403671.jpg content_image/403670.jpg content_image/403669.jpg
Starting from Eq. (3), the evolution of Λ$_{stat}$ over a small time step reads 

where 〈 Q $^{2}$〉 S ≡ Tr S (Q $^{2}$ρ S (t)) and 〈 B $^{2}$〉 E ≡ Tr E (B $^{2}$ρ E (t)). Statistical errors associated with Eq. (3) have been estimated numerically and turn out to grow very fast in time [46]. As a consequence, the stochastic process in the present form is useless to simulate physical situations and methods to reduce statistical errors should be used. 

To do so, it is worth to note that the stochastic equa- tion of motion is not unique. Indeed, any stochastic pro- cess of the form: 

<img src='content_image/403666.jpg'>

where Δ$_{S}$( leads to the same average evolution. These stochastic equations also provide a reformulation of the initial sys- tem+environment problem. Indeed, we have 

<img src='content_image/403667.jpg'>

Therefore, terms appearing in the deterministic part are exactly compensated by equivalent terms coming from the average over the noise. Accordingly, the evolution of the average density identifies with the exact equation of motion (2). 

Up to now, the flexibility has been essentially exploited by using [35, 36] 

This choice is justified by the fact that it directly appears when the Ehrenfest theorem is applied to separable to- tal density for system or environment observables. By modifying the stochastic evolution, part of the coupling 

<img src='content_image/403665.jpg'>

<img src='content_image/403668.jpg'>

is already contained in the deterministic evolution. Ac- cordingly, we do expect that the amount of coupling to be treated by the noise is significantly reduced as well as the statistical errors. In the latter case, statistical fluctuations are given by: 

d Λ$_{stat}$ = 2 dt ℏ { ( 〈 Q $^{2}$〉$_{S}$ − 〈 Q 〉 2 $_{S}$) + ( 〈 B $^{2}$〉$_{S}$ − 〈 B 〉 2 $_{S}$) } , (10) and are always smaller than the original ones (7). As shown numerically in ref. [35], introduction (9) signifi- cantly reduces statistical fluctuations and opens new per- spectives for the application of the present framework. 

The modified stochastic theory has other advantages. For instance, the traces of densities are constant and remain equal to their initial values, i.e. d Tr( ρ$_{S/E}$ ) = 0. This greatly simplifies expectation values of system and/or environment observables. Indeed, denoting by X a system operator, along a trajectory, we have 

<img src='content_image/403671.jpg'>

For stochastic processes with varying trace of densi- ties, the observable evolution will contain terms com- ing from d Tr( ρ$_{S/E}$ ) and cross terms coming from d Tr( ρ$_{S/E}$ ) d Tr( Xρ$_{S}$ ( t )). In the case considered here, we simply have 

The QMC theory with centered noise overcomes the dif- ficulty (ii) but does not help for (i) since the environment degrees of freedom should still be followed in time. In the next section, we show how irrelevant degrees of freedom can be projected out to obtain a closed stochastic master equation for the system only. 

## B. Reduced system density evolution and link with influence-functional theory 

The stochastic formulation suffers a priori from the same difficulty as the total dynamics: the environment is in general rather complex and has a large number of degrees of freedom which can hardly be followed in time. In Eq. (8), the influence of the environment on the sys- tem only enters through 〈 B ( t ) 〉$_{E}$ . Therefore, instead of following the full environment density evolution, one can concentrate on this observable only. As shown in ref. [43], the second equation in Eqs. (8) can be integrated in time to give: 

<img src='content_image/403670.jpg'>

<img src='content_image/403669.jpg'>

