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Recent advances and trends in quantum transport theory

Recent advances and trends in quantum transport theory. Denis Lacroix (GANIL) lacroix@ganil.fr. Coll: M. Assié, B. Avez, S. Ayik, P. Chomaz, G. Hupin, C. Simenel, J.A. Scarpaci, K. Washyiama. Actual mean-field theories (TDHF). Beyond mean-field : highlights.

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Recent advances and trends in quantum transport theory

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  1. Recent advances and trends in quantum transport theory Denis Lacroix (GANIL) lacroix@ganil.fr Coll: M. Assié, B. Avez, S. Ayik, P. Chomaz, G. Hupin, C. Simenel, J.A. Scarpaci, K. Washyiama Actual mean-field theories (TDHF) Beyond mean-field : highlights Pairing effects Inclusion of nucleon-nucleon collisions Application to Open Quantum systems Application to Closed N-body systems Stochastic Schrödinger Equation : introduction LEA Workshop-Catane2008

  2. three-body one-body “Simple” Trial state: two-body Trends in dynamical mean-field theories (TDHF) Mean-field: (DFT/EDF) Courtesy to C. Simenel Self-consistent Mean-field First applications : more than 30 years ago Renewal of interest: Kim, Otsuka, Bonche, J. Phys.G23, (1997). Nakatsukasa and K. Yabana, PRC71, (2005). Maruhn, Reinhard, Stevenson, Stone, Strayer, PRC71 (2005). Umar and Oberacker, PRC71, (2005). Simenel, Avez, Int. J. Mod. Phys. E 17, (2008). Now full 3D calculations With the complete Energy Density Functional In the near future, the predicting power of TD-EDF has to be benchmarked

  3. Predicting power of TDHF: illustration Washiyama, Lacroix, PRC78 (2008). Macroscopic reduction Kinetic Dynamical Reduction effect Dissipation Potential Adamian et al., PRC56(1997) Dissipation Internal Excitation

  4. Two-body |Coll> Missing physics and visible consequences on dynamics Single-nucleus dynamics : collective motion Average energy is OK but dissipation (damping) is missed GQR in 208Pb RPA Di-nucleus dynamics: fusion/fission V(Q) Miss tunneling Cross section Zero point motion (too small) No symmetry breaking Center of mass En. Q Simenel, Avez, Int. J. Mod. Phys. (2008).

  5. Multi- Ref. (MR) Single Reference (SR) Beyond mean-field transport models Simenel, Avez, Lacroix, Lecture notes Ecole Joliot-Curie 2007, arXiv:0806.2614 What type of correlations / Which extension ? TDHFB or equivalent Short range correlation: pairing Long-Range correlations: configuration mixing TDGCM Quantum Monte-Carlo Extended TDHF Statistical models: direct nucleon-nucleon collisions

  6. Beyond mean-field: strategy Simenel, Avez, Lacroix, Lecture notes Ecole Joliot-Curie 2007, arXiv:0806.2614 Exact one-body dynamics of a correlated system with N-N collisions Pairing Higher order Application of mean-field + pairing : (I) TDHFB Small amplitude dynamics Avez, Simenel, Chomaz, arXiv:0808.3507 Response function 18O±2n TDHFB QRPA Khan et al., PRC69, (2004) But…

  7. Dynamics with pairing (II) Assié, Lacroix, Scarpaci, in preparation N-N collisions Pairing Assume dominant coupling and correlations between time-reversed pairs Higher order Static properties Pairing Gap : comparison with HFB TDDMP HFB(1) 22O-3,5 MeV -3,3 MeV 24O-3,1 MeV -3,4 MeV (1)M. Matsuo, NPA (2001)

  8. di-neutron cigare Corrélation rel (degree) probing correlations with nuclear break-up Assié, Lacroix, Scarpaci, in preparation Different initial correlations Some Intuition n n Attractive interaction Repulsive interaction Break-up of 6He Final relative angle M. Assié, PhD (2008)

  9. <B> Missing information Short time evolution Exact evolution <A2> <A1> Mean-field One Body space Correlation Approximate long time evolution+Projection Dissipation with Dissipation and fluctuation projected two-body effect Propagated initial correlation Random initial condition Other correlations : Direct N-N collisions in the medium Y. Abe et al, Phys. Rep. 275 (1996) D. Lacroix et al, Progress in Part. and Nucl. Phys. 52 (2004)

  10. Application in quantum systems time Coupling to ph-phonon Coupling to 2p2h states RPA Vlasov mean-field +fluctuation +dissipation mean-field BUU, BNV Collective energies Boltzmann- Langevin 2p-2h decay channels D. Lacroix, S. Ayik and P. Chomaz, Prog. in Part. and Nucl. Phys. (2004) Chomaz,Colonna, Randrup, Phys. Rep. (2007). Semiclassical version for approaches in Heavy-Ion collisions Catane Group Milano Group

  11. Environment Environment <B> Missing information Exact evolution System System <A2> <A1> Mean-field One Body space Self-interacting vs 0pen systems N-body Open systems Brownian motion (others) (one-body) Towards Exact stochastic methods for N-body and Open systems

  12. time Stochastic Schroedinger equation (SSE): Stochastic operator : … time Introducing the concept of Stochastic Schrödinger equation Standard Schroedinger equation: Deterministic evolution

  13. Stochastic one-body evolution time with Trial states Exact state { two-level system Bosons Occupation probability time Stochastic quantum mechanics from observable evolution D. Lacroix, Ann. of Phys. 322 (2007). Philosophy: Theorem of existence : One can always find a stochastic process for trial states such that evolves exactly over a short time scale. Application to Many-body problems D(t0) Application to Bosonic systems Observables Fluctuations

  14. <B> Exact evolution Environment <S2> Relevant degrees of freedom: system <S1> System space Recent advances : Combining SSE with projection technique Lacroix, Phys. Rev. E77 (2008). H = HS + HE+ Q×B Use SSE Project the effect of the Environment Exact Stochastic master equation for open quantum systems Indept .evol. drift Mean-field Non-local in time noise

  15. V(Q) Q Under development: applications to system with potential energy surface Environment Benchmark : The Caldeira-Leggett model System + heat-bath Coupling

  16. x x x More insight in the stochastic process G. Hupin, D. Lacroix in preparation Observables evolution Complex noise on both P and Q Fluctuations Quantum Statistical Exact time Quantum+Stat Quantum

  17. Quantum + Statistical fluctuations T = 0.1 hw0 T = hw0 Exact This work TCL Correct asymptotic Behavior Next: -Non-harmonic potential -Tunneling+dissipation -Decoherence … Next: application to N-body problem

  18. Summary 3D TDHF dynamics with full Skyrme forces are now possible Beyond mean-field theories will be necessary Pairing like correlations Configuration mixing (long range correlations) Approximate or Exact stochastic methods can be very useful

  19. Preliminary Results Position and momentum evolution Exact T = hw0 T = 0.1 hw0 This work TCL

  20. Environment System Zoology in the theory open quantum systems: approximations S+E Hamiltonian : Exact S+E evolution: Reduced System evolution : Standard Approximations Separable interaction Weak coupling (Born approximation) + Stationary Env. Master equation: Memory effect Markov approximation Gardiner and Zoller, Quantum noise (2000) Breuer and Petruccione, The Theory of Open Quant. Syst. t-s

  21. Hamiltonian Environment Exact dynamics System time At t=0 A stochastic version { with Average evolution + + The dynamics of the system+environment can be simulated exactly with quantum jumps (or SSE) between “simple” state. Average density Interesting aspects related to the introduction of Stochastic Schröd. Eq.

  22. Environment <B> Complex self-interacting System Missing information Exact evolution System <A2> Relevant degrees of freedom <A1> Mean-field One Body space Hamiltonian splitting Good part: average evolution More insight in mean-field dynamics: Exact state Trial states Environment System { exact Ehrenfest evolution Missing part: correlations The approximate evolution is obtained by minimizing the action: The idea is now to treat the missing information as the Environment for the Relevant part (System) Mean-field from variational principle

  23. Theorem : One can always find a stochastic process for trial states such that evolves exactly over a short time scale. with Valid for or In practice Mean-field level Exact evolution <A1A2>-<A1A2>MF Mean-field + noise <A2> <A1> Mean-field Existence theorem : Optimal stochastic path from observable evolution D. Lacroix, Ann. of Phys. 322 (2007). …

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

  25. Application : spin-boson model + heat bath Leggett et al, Rev. Mod. Phys (1987) System + bath D0 e Coupling sz=-1 sz=+1 Comparison with related work : Path integrals + influence functional Result (2000 trajectories) Zhou et al, Europhys. Lett. (2005) strong coupling 224 traj. ! weak coupling Stockburger, Grabert, PRL (2002)

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