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Stochastic quantum dynamics beyond mean-field.

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

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Stochastic quantum dynamics beyond mean-field.

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

  2. 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)

  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 Introduction to stochastic mean-field theories :

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

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

  7. 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)

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

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

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

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

  12. 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

  13. 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

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

  15. 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))

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

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

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

  19. 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

  20. Application to spherical nuclei t<0 Mean-field part : Residual part : Application : 40Ca nucleus l = 0.25 MeV.fm-2 Root mean-square radius evolution: Lifetime of the determinant: TDHF Average evol. s0=500 tD=179 rms (fm) s0=300 s0=200 s0=100 tD=260 tD=340 tD=1040 time (fm/c) time (fm/c)

  21. 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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