1. Stochastic quantum dynamics beyond mean-field. Denis Lacroix Laboratoire de Physique Corpusculaire - Caen, FRANCE Introduction to stochastic TDHF Application to collective motions Functional integrals for dynamical Many-body problems Alternative exact stochastic mechanics One Body space

2. Introduction to stochastic theories in nuclear physics Mean-field Bohr picture of the nucleus n N-N collisions n Statistical treatment of the residual interaction (Grange, Weidenmuller… 1981) -Random phases in final wave-packets (Balian, Veneroni, 1981) -Statistical treatment of one-body configurations (Ayik, 1980) -Quantum Jump (Fermi-Golden rules) (Reinhard, Suraud 1995) Historic of quantum stochastic one-body transport theories :

3. Introduction to stochastic mean-field theories : The correlation propagates as : where { Propagated initial correlation Two-body effect projected on the one-body space Starting from : D. Lacroix, S. Ayik and Ph. Chomaz, Progress in Part. and Nucl. Phys. (2004) {

4. The initial correlations could be treated as a stochastic operator : where { Link with semiclassical approaches in Heavy-Ion collisions time Vlasov BUU, BNV Boltzmann- Langevin Adapted from J. Randrup et al, NPA538 (92). Molecular chaos assumption

5. { Incoherent nucleon-nucleon collision term. Coherent collision term Evolution of the average density : One Body space Fluctuations around the mean density : Average ensemble evolutions

6. Application to small amplitude motion Standard RPA states Coupling to ph-phonon Coupling to 2p2h states

7. Average GR evolution in stochastic mean-field theory Full calculation with fluctuation and dissipations RPA response D. Lacroix, S. Ayik and Ph. Chomaz, Progress in Part. and Nucl. Phys. (2004) Mean energy variation fluctuation dissipationRPA Full

8. Effect of correlation on the GMR and incompressibility Incompressibility in finite system in 208 Pb { Evolution of the main peak energy :

9. More insight in the fragmentation of the GQR of 40 Ca EWSR repartition

10. Intermezzo: wavelet methods for fine structure Observation +1 D. Lacroix and Ph. Chomaz, PRC60 (1999) 064307. Basic idea of the wavelet method Recent extensions : A. Chevchenko et al, PRL93 (2004) 122501.

11. Discussion on approximate quantum stochastic theories based on statistical assumptions Results on small amplitude motions looks fine The semiclassical version (BOB) gives a good reproduction of Heavy-Ion collisions Success Critical aspects Which interaction for the collision term Stochastic methods for large amplitude motion are still an open problem (No guide to the random walk) Theoretical justification of the introduction of noise Instantaneous reorganization of internal degrees of freedom?

12. Functional integral and stochastic quantum mechanics Given a Hamiltonian and an initial State Write H into a quadratic form Use the Hubbard Stratonovich transformation Interpretation of the integral in terms of quantum jumps and stochastic Schrödinger equation time Example of application: -Quantum Monte-Carlo Methods -Shell Model Monte-Carlo... General strategy S. Levit, PRCC21 (1980) 1594. S.E.Koonin, D.J.Dean, K.Langanke, Ann.Rev.Nucl.Part.Sci. 47, 463 (1997).

13. Carusotto, Y. Castin and J. Dalibard, PRA63 (2001). O. Juillet and Ph. Chomaz, PRL 88 (2002) Recent developments based on mean-field Nuclear Hamiltonian applied to Slater determinant Self-consistent one-body part Residual part reformulated stochastically Quantum jumps between Slater determinant Thouless theorem Stochastic schrödinger equation in one-body space Stochastic schrödinger equation in many-body space Fluctuation-dissipation theorem

14. Stochastic evolution of non-orthogonal Slater determinant dyadics : Quantum jump in one-body density space Quantum jump in many-body density space with Generalization to stochastic motion of density matrix D. Lacroix, Phys. Rev. C (2005) in press. The state of a correlated system could be described by a superposition of Slater-Determinant dyadics time D ab D ac D de

15. Discussion of exact quantum jump approaches Many-Body Stochastic Schrödinger equation Stochastic evolution of many-body density One-Body Stochastic Schrödinger equation Stochastic evolution of one-body density Generalization : Each time the two-body density evolves as : with Then, the evolution of the two-body density can be replaced by an average ( ) of stochastic one-body evolution with : Actual applications : -Bose-condensate (Carusotto et al, PRA (2001)) -Two and three-level systems (Juillet et al, PRL (2002)) -Spin systems (Lacroix, PRA (2005))

16. Exact stochastic dynamics guiding approximate quantum stochastic mechanics Weak coupling approximation : perturbative treatment Residual interaction in the mean-field interaction picture We assume that the residual interaction can be treated as An ensemble of two-body interaction: Statistical assumption in the quasi-Markovian limit :

17. Time-scale and Markovian dynamics { t t+  t Replicas Collision time Average time between two collisions Mean-field time-scale Hypothesis : Interpretation in terms of average evolution of quantum jumps : with Stochastic term

18. From stochastic many-body to stochastic one-body evolution We need additional simplification Following the exact stochastic dynamics We introduce the density Following approximate dynamics One Body space We focus on one-body degrees of freedom Gaussian approximation for quantal fluctuations We obtain a new stochastic one-body evolution in the perturbative regime: Mean-field like term D. Lacroix, in preparation (2005)

19. Perturbative/Exact stochastic evolution Perturbative Exact Many-body density Properties Many-body density Projector Number of particles Entropy Average evolution One-body Correlations beyond mean-field Numerical implementation : Flexible: one stoch. Number or more… Fixed : “s” determines the number of stoch. variables

20. Application to spherical nuclei t<0 Residual part : Mean-field part : Application : 40 Ca nucleus = 0.25 MeV.fm -2 Root mean-square radius evolution: rms (fm) time (fm/c) TDHF Average evol. time (fm/c)  D =179  D =260  D =340  D =1040  0 =100  0 =300  0 =200  0 =500 Lifetime of the determinant:

21. Summary One Body space Stochastic mean-field from statistical assumption (approximate) time D ab D ac D de Stochastic mean-field from functional integral (exact) Stochastic mean-field in the perturbative regime Sub-barrier fusion : Violent collisions : Vibration : Applications: