1. Quantum Monte-Carlo for Non-Markovian Dynamics Collaborator : Denis Lacroix Guillaume Hupin GANIL, Caen FRANCE  Exact  TCL2 (perturbation)  TCL4  NZ2  NZ4

2. Overview 1.Motivation in nuclear physics : Fission and Fusion. 2.Non-Markovian effects : Projective methods. 3.Quantum Monte-Carlo. 4.Applications : A.To the spin-star model. B.To the spin-boson model. C.In the Caldeira Leggett model. Environment System

3. Physics case : fission Environment  Nucleons response thermal bath.  Fast motion vs fission decay. —› Markovian H. Goutte, J. F. Berger, P. Casoli, Phys. Rev. C 71 (2005) System: and potential Environment: Nucleons (intrinsic) motion Coupling:

4. Physical case : fusion Different channels are opened in a fusion reaction : Microscopic evolution can be mapped to an open quantum system. K.Washiyama et al., Phys. Rev. C 79, (2009)  System : relative distance.  Environment : nucleon motion, other degrees of freedom (deformation…). Environment Relative distance. => Non-Markovian effects are expected.

5. Motivations Non-Markovian. Markovian. Environment

6. The total Hamiltonian written as  (S) is the system of interest coupled to (E).  (E) is considered as a general environment. Exact Liouville von Neumann equation. The environment has too many degrees of freedom. Framework

7. Standard approaches : projective methods Therefore, use projective methods to “get rid” of (E).  First, define two projection operators on (S) and (E).  Project the Liouville equation on the two subspace. System Environment Exact evolution System System only evolution Environment I = interaction picture

8. Standard approaches : projective methods H. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002) Nakajima-Zwanzig’s method (NZ). The equation of motion of the system is closed up to a given order in interaction. Time Convolution Less method (TCL). tt s Time non local method : mixes order of perturbation. (NZ2, NZ4..) Time local method : order of the perturbation under control. (TCL2, TCL4 …) s

9. Comparison between projective techniques TCL4 is in any case a more accurate method. In the Caldeira- Leggett model: an harmonic oscillator coupled to a heat bath. Using a Drude spectral density: G.Hupin and D. Lacroix, Phys. Rev. C 81, (2010)

10. New approach : Quantum Monte Carlo  The exact dynamics is replaced by a number of stochastic paths to simulate the exact evolution in average.  Noise designed to account for the coupling. D is complicated : The environment has too many degrees of freedom. t0t0 t

11. Exact Liouville von Neumann equation. Theoretical framework Stochastic Master Equations (SME). D. Lacroix, Phys. Rev. E 77, (2008) Other stat. moments are equal to 0.

12. Theoretical framework : proof SME Proof : Using our Stochastic Master Equations.

13. SME has its equivalent Stochastic Schrödinger Equation Deterministic evolution Stochastic Schrödinger equation (SSE): … Leads to the equivalence : Independence of the statistical moments : L.Diosi and W.T. Strunz Phys. Lett. A 255 (1997) M.B. Plenio and P.L. Knight, Rev. Mod. Phys. 70 (1998) J.Piilo et al., PRL 100 (2008) A. Bassi and L. Ferialdi PRL 103 (2009), Among many others…..

14. Application of the SSE to spin star model A Monte-Carlo simulation is exact only when the statistical convergence is reached. Statistical fluctuation : Noise optimized. 1000 Trajectories

15. Including the mean field solution  Take the density as separable :  Position & momentum : ok  Width : not ok  Then, take the Ehrenfest evolution :

16. Mean field + QMC Evolution of the statistical fluctuation have been reduced using an optimized deterministic part in the SME. D. Lacroix, Phys. Rev. A 72, (2005)

17. Projected Quantum Monte-Carlo + Mean-Field Exact evolution System Stochastic trajectories Environment The environment response is contained in :

18. Link with the Feynman path integral formalism J. T. Stockburger and H. Grabert, Phys. Rev. Lett. 88, (2002)

19. Application to the spin boson model This method has been applied to spin boson model. Exact (stochastic) TCL2 Second successful test. A two level system interacting with a boson bath D. Lacroix, Phys. Rev. E 77, (2008) Y. Zhou, Y. Yan and J. Shao, EPL 72 (2005)

20. Application to the problems of interest Environment Fission/fusion processes The potential is first locally approximated by parabola.

21. Benchmark in the Caldeira-Leggett model  QMC.  Exact. Convergence is achieved  for second moments for different temperatures.  with an acceptable time of calculation.  with a limited number of trajectories (≈10 4 ). Second moment evolution.

22. Observables of interest  Exact.  Quantum Monte Carlo.  Projection 2 nd order in interaction.  Projection 4 nd order in interaction.

23. Passing probability  Accuracy of MC simulations is comparable to the fourth order of projection.  Accuracy of such calculations are of interest for very heavy nuclei. Markovian approximation 2 nd order in perturbation TCL 4 th order in perturbation TCL Monte Carlo Exact

24. Summary  It has been pointed out that TCL4 should be preferred.  New theory based on Monte Carlo technique has been applied and tested for simple potentials.  This study shows that the new method is effective.  Now, the new technique Monte-Carlo+Mean-Field should be applied to more general potentials. Critical issue: diverging path Possible solutions :  Remove the diverging trajectories.  Semi-classic approximations : Initial Value Methods C. Gardiner, “A Handbook of Stochastic Methods” W. Koch et al., PRL 100 (2008)