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## Recent advances and trends in quantum transport theory

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**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**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**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**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).**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**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…**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)**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)**<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)**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**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**time**Stochastic Schroedinger equation (SSE): Stochastic operator : … time Introducing the concept of Stochastic Schrödinger equation Standard Schroedinger equation: Deterministic evolution**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**<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**V(Q)**Q Under development: applications to system with potential energy surface Environment Benchmark : The Caldeira-Leggett model System + heat-bath Coupling**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**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**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**Preliminary Results**Position and momentum evolution Exact T = hw0 T = 0.1 hw0 This work TCL**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**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.**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**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). …**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)**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)