Quantum Monte-Carlo method applied to Non-Markovian barrier transmission Guillaume Hupin 1 and Denis Lacroix 1 $^{1}$GANIL, Bd Henri Becquerel, BP 55027, 14076 Caen Cedex 5, France In nuclear fusion and fission, fluctuation and dissipation arise due to the coupling of collective degrees of freedom with internal excitations. Close to the barrier, both quantum, statistical and non- Markovian effects are expected to be important. In this work, a new approach based on quantum Monte-Carlo addressing this problem is presented. The exact dynamics of a system coupled to an environment is replaced by a set of stochastic evolutions of the system density. The quantum Monte-Carlo method is applied to systems with quadratic potentials. In all range of temperature and coupling, the stochastic method matches the exact evolution showing that non-Markovian effects can be simulated accurately. A comparison with other theories like Nakajima-Zwanzig or Time- ConvolutionLess ones shows that only the latter can be competitive if the expansion in terms of coupling constant is made at least to fourth order. A systematic study of the inverted parabola case is made at different temperatures and coupling constants. The asymptotic passing probability is estimated in different approaches including the Markovian limit. Large differences with the exact result are seen in the latter case or when only second order in the coupling strength is considered as it is generally assumed in nuclear transport models. On opposite, if fourth order in the coupling or quantum Monte-Carlo method is used, a perfect agreement is obtained. PACS numbers: 24.60-k,25.70.Jj,05.60.Gg Keywords: open quantum systems, Monte-Carlo methods, non-Markovian effect. I. INTRODUCTION To understand nuclear reactions, the dynamics of nu- clei is often replaced by few selected collective degrees of freedom expected to contain important information on the dynamic. This is for instance the case in fusion reac- tions where the relative distance and/or mass asymmetry is retained [1, 2]. Another example is provided by the fis- sion process which is often treated as a trajectory in an energy landscape function on different deformation pa- rameters [3, 4]. Although the evolution is projected onto few variables, other internal degrees of freedom may play an important role to understand the onset of dissipa- tion or fluctuation phenomena [5]. To treat these effects, the relevant degrees of freedom should be regarded as an Open Quantum System coupled to an environment which simulates the internal dynamics. To include dissipation in collective space, two impor- tant simplifications are often made. First, most of cur- rent models treating fusion/fission neglect quantum ef- fects and consider a classical treatment [6–9]. Such an approximation is however expected to be valid only if the internal excitation is high and therefore is not ex- pected to hold close to or below the Coulomb barrier. As it is discussed in ref. [10], a proper treatment of quan- tum and decoherence effects might be crucial in this re- gion. Second, when the time scale associated to collec- tive dynamics cannot be dissociated from the one of the environment, ”non-Markovian” (also called ”memory”) effect should also be properly treated [11, 12]. Large ef- fort is now devoted to account for both quantal and non- Markovian effects in nuclear reactions [13–18] and more generally in open quantum systems [19]. Recently, the description of open quantum systems by stochastic methods has received much attention [19– In this work, we apply the stochastic formulation pro- posed in ref. [36] to the case of quadratic potentials coupled to a heat-bath, the so-called Caldeira-Leggett model [37]. The case of inverted potential is the first step towards realistic situations like fusion or fission. The aim of the present work is threefold. First, to introduce the new QMC method and apply it for potentials with barriers similar to those appearing in fusion/fission pro- cesses. Second, we show that the exact quantum Monte- Carlo method can be rather accurate to treat the dis- sipative dynamics of a quantum system. Last, we also present a comparison of this theory with other meth- ods based on projection, namely Nakajima-Zwangig (NZ) and Time-ConvolutionLess (TCL) [38–41], which are ac- tually widely used to treat non-Markovian effects. Doing so, we show that only TCL up to at least fourth orders in the coupling constant can provide a competitive the- ory. The paper is organized as follows. In section II, the ingredients and properties of the quantum Monte-Carlo approach are discussed and the link with functional inte- gral is precised. In section III, the method is first illus- trated to the case of parabolic potential. Then, the pass- ing probabilities are estimated for the inverted parabola case. 21]. In the Markovian limit, several methods have been proposed to treat fluctuation and dissipation starting from a quantum master equation on the system density [19, 20, 22–28]. These methods have been extended also to treat non-Markovian effects like in Quantum State Dif- fusion (QSD) [29–32] or quantum Monte-Carlo (QMC) methods [33]. Several groups have shown that these effects could eventually be simulated using Feynman- Vernon influence functional [21, 34] or directly stochastic master equations [35, 36]. arXiv:1001.0652v1 [nucl-th] 5 Jan 2010