1. Stochastic methods beyond the independent particle picture Denis Lacroix IPN-Orsay Collaboration: S. Ayik, D. Gambacurta, B. Yilmaz, K. Washiyama, G. Scamps Outline: Mean-field dynamics: advantages, limitations and recent progress Stochastic mean-field with initial fluctuations Two-body effects through quantum jump Quantum Monte-Carlo approach to the N-body problem Deterministic approach beyond the single-particle picture

2. Nuclear Time-dependent Density Functional Theory two-body three-body one-body Mean-field: (DFT/EDF) “Simple” Trial state: Self-consistent Mean-field -Kim, Otsuka, Bonche, J. Phys.G23, (1997). -Nakatsukasa and Yabana, PRC71, (2005). -Maruhn, Reinhard, Stevenson, Stone, Strayer, PRC71 (2005). -Umar and Oberacker, PRC71, (2005). -Simenel, Avez, Int. J. Mod. Phys. E 17, (2008). -Washiyama, Lacroix, PRC78 (2008) Current status Typical examples of application Courtesy C. Simenel Ion-Ion collisions Collective motion System size particles Interaction (hard-core, spin orbit, tensor, 2, 3-body… )

3. Recent progress Relevant space irrelevant space Mean-field exact Selection of few degrees of freedom Find closed equation for relevant DOF Mean-Field strategy and Actual trend: Inclusion of pairing Y. Hashimoto and K. Nodeki, arXiv:0707.3083. B. Avez et al Phys. Rev. C 78, 044318 (2008). S. Ebata et al, Phys. Rev. C 82, 034306 (2010). I. Stetcu et al, Phys. Rev. C 84, 051309 (2011). G. Scamps et al, Phys. Rev. C 85, 034328 (2012). G. Scamps et al Phys. Rev. C 87, 014605 (2013). (see G. Scamps talk)

4. Missing quantum effects No tunneling Potential Energy Surface Collective variable Absence of quantum tunneling in collective space Missing quantum fluctuation In collective space Wrong dynamics close to a symmetry breaking point Mean-field stays there Mean-Field Underestimation of dissipation DL, Ayik, Chomaz, Prog. Part and Nucl. Phys. (2004) Collective motion Mean-field

5. Assié and Lacroix, PRL102 (2009) Some deterministic Beyond Mean-Field Approach Relevant space irrelevant space Mean-field exact Enlarging the space of relevant DOF Time Dependent Density Matrix Time Dependent Coupled Cluster Illustration: BBGKY Difficulty

6. Some deterministic Beyond Mean-Field Approach Relevant space irrelevant space Mean-field exact Enlarging the space of relevant DOF Time Dependent Density Matrix Time Dependent Coupled Cluster Reproject the effect of irrelevant DOF Extended TDHF

7. Theories beyond the mean-field One Body space Exact evolution Mean-field Missing information Y. Abe et al, Phys. Rep. 275 (1996) D. Lacroix et al, Progress in Part. and Nucl. Phys. 52 (2004) Short time evolution Approximate long time evolution+Projection Correlation with Propagated initial correlation Quantum zero point motion Dissipation projected two-body effect MF Fluctuations

8. Dynamics beyond mean-field Non-Markovian effects with Non-Markovian master equation Average position Occupation number evolution Occupation numbers DL, Chomaz, Ayik, Nucl. Phys. A (1999). 1D Example: two interacting fermions in 1dimension Difficulty: memory effect! Starts to look like TDDMFT with memory

9. Strategy of stochastic methods tackling the N-body problem Question:Is it possible to recover some of the quantum mechanics aspects by considering an ensemble of independent mean-field trajectories? Quantum Monte-Carlo Stochastic TDHF Stochastic Mean-Field Correlations that built up in time Direct NN collisions Initial fluctuations All Correlations D. Lacroix and S. Ayik EPJA Review (in preparation)

10. Correlations that built-up in time: in medium collisions

11. We assume that the residual interaction can be treated as an ensemble of two-body interaction: Statistical assumption in the Markovian limit : Weak coupling approximation : perturbative treatment Residual interaction in the mean-field interaction picture Reinhard and Suraud, Ann. of Phys. 216 (1992) GOAL: Restarting from an uncorrelated state we should: 2-interpret it as an average over jumps between “simple” states 1-have an estimate of Markovian limit, quantum-diffusion and stochastic Schrödinger Equation

12. { t t+Dt Replicas Collision time Average time between two collisions Mean-field time-scale Hypothesis : Average Density Evolution: Time-scale and Markovian dynamic

13. with Initial simple state One-body density Master equation step by step 2p-2h nature of the interaction with Separability of the interaction Dissipation contained in Extended TDHF is included The master equation is a Lindblad equation Associated SSE Lacroix, PRC73 (2006) Dissipation: link between Extended TDHF and Lindblad Eq.

14. SSE on single-particle state : with time (arb. units) width of the condensate mean-field average evolution Condensate size N-body density: 1D bose condensate with gaussian two-body interaction The numerical effort is fixed by the number of A k r  (r) (arb. units) t=0 t>0 mean-field average evolution Density evolution Application to Bose-Einstein condensates

15. Correlations that are here initially and propagates Collective space

16. Strategy to construct a stochastic mean-field theory Ayik, Phys. Lett. B 658, (2008). MF Collective phase-space Quantum fluctuations The dynamics is described by a set of mean-field evolutions with random initial conditions Mean-Field theory at all time Stochastic Mean-Field at all time Constraint:

17. The stochastic mean-field (SMF) concept applied to many-body problem Ayik, Phys. Lett. B 658, (2008). MF Collective phase-space Quantum fluctuations The dynamics is described by a set of mean-field evolutions with random initial conditions The average properties of initial sampling should identify with properties of the mean-field. SMF in density matrix space SMF in collective space

18. Description of large amplitude collective motion with SMF The case of spontaneous symmetry breaking Lipkin Model  See for instance : Ring and Schuck book Severyukhin, Bender, Heenen, PRC74 (2006) p=1p=2… p=N N=40 particles Exact dynamics Mean-field is stationary

19. Description of large amplitude collective motion with SMF The stochastic mean-field solution Initial condition One-body observables Exact SMF Lacroix, Ayik, Yilmaz, PRC 85 (2012) Formulation in quasi-spin space

20. Description of large amplitude collective motion with SMF The stochastic mean-field solution Formulation in quasi-spin space Initial condition Fluctuations Lacroix, Ayik, Yilmaz, PRC 85 (2012)

21. The stochastic mean-field (SMF) with pairing Following the general strategy: TDHFB with initial fluctuations Setting the initial fluctuations Quasi-particle occupation Lacroix, Gambacurta, Ayik, PRC 87 (2013) Requires to fix: and

22. Illustration with the pairing Hamiltonian Quasi-spin operators : degeneracy Occupation probabilities Pair creation/annihiliation operators TDHFB equations

23. Illustration with the pairing Hamiltonian =2 Occupation number evolution and dissipation Exact SMF

24. Illustration with the pairing Hamiltonian Departure from the independent Particle picture Departure from the independent Particle picture Quantum Fluctuations weak coupling strong coupling

25. Comparison between deterministic and stochastic methods Strong coupling Exact Stochastic MF Evolution of both One- and two-body densities SMF might even be better than BBGKY hierarchy truncation

26. Open systems One Body space Exact evolution Mean-field Missing information Brownian motion N-body Towards Exact stochastic methods for N-body and Open systems Environment System (one-body) (others) Environment System Exact stochastic formulation

27. More insight in mean-field dynamics: Exact state Trial states { The approximate evolution is obtained by minimizing the action: Included part: average evolution exact Ehrenfest evolution Missing part: correlations Environment System Complex self-interacting System Hamiltonian splitting System Environment One Body space Exact evolution Mean-field Missing information Relevant degrees of freedom The idea is now to treat the missing information as the Environment for the Relevant part (System) Mean-field from variationnal principle

28. Exact evolution - MF with D. Lacroix, Ann. of Phys. 322 (2007). … Mean-field Mean-field level Mean-field + noise Theorem : One can always find a stochastic process for trial states such that evolves exactly over a short time scale. Valid for or In practice Existence theorem : Optimal stochastic path from observable evolution

29. t>0 Mean-field evolution: x t>0 Reduction of the information: I want to simulate the expansion with Gaussian wave- function having fixed widths. t=0 with Relevant/Missing information: Relevant degrees of freedom Missing information Trial states Coherent states illustration: simulation of the free wave spreading with “quasi-classical states”

30. Stochastic c-number evolution from Ehrenfest theorem Densities with Nature of the stochastic mechanics with the quantum wave spreading can be simulated by a classical brownian motion in the complex plane x x time x fluctuations mean values Guess of the SSE from the existence theorem

31. The method is general. the SSE are deduced easily Ehrenfest theorem BBGKY hierarchy D. Lacroix, Ann. Phys. 322 (2007) Starting point: with Observables Fluctuations with Stochastic one-body evolution The mean-field appears naturally and the interpretation is easier extension to Stochastic TDHFB DL, arXiv nucl-th 0605033 Occupation probability time two-level system Bosons but… the numerical effort can be reduced by reducing the number of observables unstable trajectories SSE for Many-Body Fermions and bosons

32. Stochastic MF Fluctuation Dissipation Stochastic TDHF Fluctuation Dissipation Exact QD Everything Mean-field Fluctuation Dissipation variational QD Partially everything Numerical issues Flexible Fixed Approximate evolution Summary, stochastic methods for Many-Body Fermionic and bosonic systems Numerical instabilities