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Extensions of mean-field with stochastic methods

Mapping the nuclear N-body dynamics into a open system problem. Stochastic one-body mechanics applied to nuclear physics. Quantum jump approach to the many-body problem. Extensions of mean-field with stochastic methods. Denis Lacroix Laboratoire de Physique Corpusculaire - Caen, FRANCE.

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Extensions of mean-field with stochastic methods

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  1. Mapping the nuclear N-body dynamics into a open system problem. Stochastic one-body mechanics applied to nuclear physics Quantum jump approach to the many-body problem Extensions of mean-field with stochastic methods Denis Lacroix Laboratoire de Physique Corpusculaire - Caen, FRANCE One Body space TDHF and beyond … -Saclay 2006

  2. Assuming an initial uncorrelated state : Evolution in time Deg3 Mean-field approximation: Environment Deg2 One-body subspace Deg1 Mapping the nuclear dyn. to a system-environment problem One can improve the mean-field approximation by considering one-body degrees of freedom as a system coupled to an environment of other degrees of freedom.

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

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

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

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

  7. Basic idea of the wavelet method Observation D. Lacroix and Ph. Chomaz, PRC60 (1999) 064307. Recent extensions : +1 D. Lacroix et al, PLB 479, 15 (2000). A. Shevchenko et al, PRL93, 122501 (2004). -1 Intermezzo: wavelet methods for fine structure

  8. Success Results on small amplitude motions looks fine The semiclassical version (BOB) gives a good reproduction of Heavy-Ion collisions Numerical Implementation of Stochastic methods for large amplitude motion are still an open problem (No guide to the random walk) Instantaneous reorganization of internal degrees of freedom? Theoretical justification of the introduction of noise ? Discussion on one-body evolution from projection technique Critical aspects

  9. Environment Exact dynamics System Dissipative dynamics At t=0 If waves follow stochastic eq. At t=0 { Projection technique Weak coupling approx. Markovian approx. with Then, the average dyn. identifies with the exact one Lindblad master equation: For total wave 1 For total density 2 Can be simulated by stochastic eq. on |F>, The Master equation being recovered using : In fermionic self-interacting systems Breuer, Phys. Rev. A69, 022115 (2004) Lacroix, Phys. Rev. A72, 013805 (2005) 1 Stochastic mean-field Juillet and Chomaz, PRL 88 (2002) Gardiner and Zoller, Quantum noise (2000) Breuer and Petruccione, The Theory of Open Quant. Syst. Stochastic BBGKY Lacroix, PRC 71 (2005) 2 Quantum jump method -introduction

  10. we should: GOAL: Restarting from an uncorrelated state 1-have an estimate of 2-interpret it as an average over jumps between “simple” states Weak coupling approximation : perturbative treatment R.-G. Reinhard and E. Suraud, Ann. of Phys. 216, 98 (1992) Residual interaction in the mean-field interaction picture Statistical assumption in the Markovian limit : Quantum jump in the weak coupling regime We assume that the residual interaction can be treated as an ensemble of two-body interaction:

  11. Mean-field time-scale t+Dt t { Collision time Replicas Average time between two collisions Hypothesis : Two strategies have been considered: Considering waves directly (philosophy of exact treatment) Considering densities directly (philosophy of dissipative treatment) Time-scale and Markovian dynamics

  12. Interpretation of the equation on waves as an average over jumps: Let us simply assume that We consider densities and focus on one-body density: Additional hypothesis: We end with: Mean-field like term D. Lacroix, arXiv:quant-ph/ 0509038 Simplified scenario for introducing fluctuations beyond MF with Matching with a quantum jump process between “simple states” ?

  13. Important properties r remains a projector At all time with time Average evolution One-body Correlations beyond mean-field, denoting by similar to Ayik and Abe,PRC 64,024609 (2001). Numerical implementation : flexible and rather simple. Nature of the Stochastic one-body dynamics

  14. Stoch. Schrödinger Equation (SSE) on single-particle states: t<0 Mean-field part : Assuming and All the information on the system is contained in the one-body density Residual part : Associated quantum jumps on single particle states: Application : 40Ca nucleus l = 0.25 MeV.fm-2 Root mean-square radius evolution: Monopole vibration in nuclei TDHF Average evol. rms (fm) time (fm/c) Application

  15. Standard deviation No constraint Compression Dilatation l = 0.25 MeV.fm-2 Similar to Nelson quantization theory Nelson, Phys. Rev. 150, 1079 (1966). Ruggiero and Zannetti, PRL 48, 963 (1982). The stochastic method is directly applicable to nuclei It provide an easy way to introduce fluctuations beyond mean-field It does not account for dissipation. In nuclear physics the two particle-two-hole components dominates the residual interaction, but !!! Diffusion of the rms around the mean value Summary and Critical discussion on the simplified scenario

  16. Second Philosophy Contains an additional term Master equation for the one-body evolution and its one-body density Starting from with Matching with the nuclear many-body problem The residual interaction is dominated by 2p-2h components Generalization: quantum jump with dissipation Equivalent to the collision term of extended TDHF

  17. All interaction of 2p-2h nature can be decomposed into a sum of separable interaction, i.e. with Koonin, Dean, Langanke, Ann.Rev.Nucl.Part.Sci. 47 (1997). Juillet and Chomaz, PRL 88 (2002). We can use standard quantum jump methods to simulate this equation Again The equation can be interpreted as the feedback action of the On operators on the one-body density time Existence and nature of the associated quantum jump ?

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

  19. Variational QJ Mean-field Simplified QJ Generalized QJ Exact QJ Partially everything Fluctuation  Fluctuation  Fluctuation   Everything Dissipation  Dissipation Dissipation Flexible Fixed Fixed O. Juillet (2005) Summary Quantum Jump (QJ) methods to extend mean-field Numerical issues Flexible

  20. Giant resonances

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

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

  23. Linear response Extended mean-field Mean-field Notations for RPA equations Response to harmonic vibrations Using Mean-field Extended mean-field +

  24. Fourier transform and coupling to decay channels Coherent damping Incoherent damping S. Ayik and Y. Abe, PRC 64, 024609 (2001). Ph. Chomaz, D. Lacroix, S. Ayik, and M. Colonna PRC 62, 024307 (2000) Coupling to 2p-2h states Coupling to ph-phonon states

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

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

  27. Calculated strength Main peaks energies , comparison with experiment Experiments Systematic improvement of the GQR energy

  28. N-body exact

  29. S. Levit, PRCC21 (1980) 1594. S.E.Koonin, D.J.Dean, K.Langanke, Ann.Rev.Nucl.Part.Sci. 47, 463 (1997). General strategy 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 ... Functional integral and stochastic quantum mechanics

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

  31. D. Lacroix , Phys. Rev. C71, 064322 (2005). The state of a correlated system could be described by a superposition of Slater-Determinant dyadic time Generalization to stochastic motion of density matrix Stochastic evolution of non-orthogonal Slater determinant dyadics : Quantum jump in one-body density space Quantum jump in many-body density space with

  32. Stochastic evolution of many-body density Many-Body Stochastic Schrödinger equation Stochastic evolution of one-body density One-Body Stochastic Schrödinger equation 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 : Discussion of exact quantum jump approaches Actual applications : -Bose-condensate (Carusotto et al, PRA (2001)) -Two and three-level systems (Juillet et al, PRL (2002)) -Spin systems (Lacroix, PRA (2005))

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

  34. Stochastic mean-field from statistical assumption (approximate) Stochastic mean-field from functional integral (exact) One Body space Dab Dac Dde time Applications: Stochastic mean-field in the perturbative regime Sub-barrier fusion : Vibration : Violent collisions : Summary

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