The nuclear many- body problem: an open quantum systems perspec6ve. Denis Lacroix GANIL- Caen
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1 The nuclear many- body problem: an open quantum systems perspec6ve Denis Lacroix GANIL- Caen Coll: M. Assié, S. Ayik, Ph. Chomaz, G. Hupin, K. Washiyama Trento, Decoherence -April 2010
2 The nuclear many- body problem as an open quantum object Generali;es: Reduc;on of informa;on Environment Few relevant degrees of freedom needs to be selected (System) System Illustrations discussed here Fusion reac;ons: the role of open channels (discrete and con;nuous) System: collec;ve space Env: intrinsic degrees of freedom The nuclear many- body problem System: one- body observables Env: two- body and higher correla;ons
3 The nuclear many- body problem as an open quantum object (i) Macroscopic reduc;on
4 The nuclear many- body problem as an open quantum object (i) Macroscopic reduc;on Other collective space: deformation, mass/charge asymmetry
5 The nuclear many- body problem as an open quantum object Open channels : discrete internal excita;ons Internal Excitation One of the difficulty is to treat both discrete and con;nuous channels in a common framework
6 The nuclear many- body problem as an open quantum object Stochas;c semi- classical treatment of discrete channels Excitation Collective Motion + Coupling Discrete Channels Esbensen et al, PRL 41 (1978) Ini;al Phase- space sampling of zero point mo;on Classical dynamics of system+environment With stochas;c ini;al condi;on
7 The nuclear many- body problem as an open quantum object Stochas;c semi- classical treatment of discrete channels relative dist. Classical dynamics Environment Ayik, Yilmaz,DL, PRC81 (2010)
8 three-body two-body one-body The nuclear many- body problem as an open quantum object Microscopic reduc;on Mean-field: (DFT) Self-consistent Mean-field Courtesy to C. Simenel Simple Trial state: Selection of few relevant degrees of freedom: Microscopic one-body space Macro. space
9 Fusion reac;ons: macroscopic vs microscopic dynamics Role of con;nuous channel: Disorder and Dissipa;on Expected One-body origin of dissipation -transfer of particle -reflection of particles R
10 Macroscopic reduc;on: dissipa;on Washiyama, DL, PRC78 (2008). Washiyama, DL, Ayik, PRC79 (2009). Kinetic R Dissipation Potential Dynamical Reduction effect Adamian et al., PRC56(1997) Dissipation Internal Excitation
11 Ayik, PLB 658, (2008). Fluctua;ons associated to dissipa;on Application to fusion R Mean-field MF 40 Ca+ 40 Ca Mean-field+Initial fluct. R B Ayik, Washiyama, DL, PRC (2009)
12 What next? Incoherent Channels Add standard Dissipation (Markovian/Non-Markovian) Microscopic one-body dynamics Semi-classical Phase-space dynamics Add quantum Fluctuations associated To discrete channels Coherent Channels
13 Part II Mapping many-body systems To Open quantum systems N-body Microscopic space Microscopic one-body space Macro. space
14 Y. Abe et al, Phys. Rep. 275 (1996) D. Lacroix et al, Progress in Part. and Nucl. Phys. 52 (2004) Short time evolution Dynamics beyond mean- field Projec;on technique <B> Exact evolution <A 2 > <A 1 > One Body space Mean-field Correlation Approximate long time evolution+projection (Nakajima-Zwanzig) Dissipation (Extended TDHF) with Dissipation and fluctuation projected two-body effect Propagated initial correlation Random initial condition
15 Dynamics beyond mean- field Non- Markovian effects with Occupation number evolution Non-Markovian master equation Occupation numbers Example: two interacting fermions in 1dimension 1D Average position DL, Chomaz, Ayik, Nucl. Phys. A (1999).
16 Non- Markovian dynamics beyond mean- field applica;on to collec;ve mo;on Quadrupole moment time (fm/c) Giant Quadrupole resonances GQR in lead Mean energy is OK Damping (dissipation) and fragmentation is missed DL, Ayik, Chomaz, Prog. Part and Nucl. Phys. (2004)
17 Non- Markovian dynamics beyond mean- field applica;on to collec;ve mo;on Mean-field Coupling to 2p2h states Coupling to ph-phonon Collective energies System 2p-2h decay channels Environment mean-field +fluctuation +dissipation mean-field Incorporate dissipation in many-body system Not so easy to use in Large amplitude Collective motion
18 Markovian limit, quantum- diffusion and stochas;c Schrödinger Equa;on GOAL: Restarting from an uncorrelated state we should: 1-have an estimate of 2-interpret it as an average over jumps between simple states Weak coupling approximation : perturbative treatment Reinhard and Suraud, Ann. of Phys. 216 (1992) Statistical assumption in the Markovian limit : Residual interaction in the mean-field interaction picture We assume that the residual interaction can be treated as an ensemble of two-body interaction:
19 Time- scale and Markovian dynamic Mean-field time-scale t t+dt { Replicas Collision time Hypothesis : Average time between two collisions Average Density Evolution:
20 Dissipa;on: link between Extended TDHF and Lindblad Eq. One-body density Master equation step by step Initial simple state with 2p-2h nature of the interaction with Separability of the interaction Dissipation contained in Extended TDHF is included The master equation is a Lindblad equation Associated SSE DL, PRC73 (2006)
21 Applica;on to Bose- Einstein condensates SSE on single-particle state : 1D bose condensate with gaussian two-body interaction N-body density: with t=0 t>0 The numerical effort is fixed by the number of A k ρ(r) (arb. units) mean-field average evolution width of the condensate average evolution mean-field r time (arb. units)
22 Self- interac;ng vs Open Quantum systems N-body Open systems <B> Exact evolution <A 2 > <A 1 > One Body space Mean-field Brownian motion Environment (others) Environment System (one-body) System Towards Exact stochastic methods for N-body and Open systems
23 Environment Self- interac;ng vs Open Quantum systems Approximate and exact Quantum jump System Projection Lindblad master Eq. +quantum Diffusion Lindblad master Eq. +quantum Diffusion Gardiner and Zoller, Quantum noise (2000) Breuer and Petruccione, The Theory of Open Quant. Syst. (2002). Quantum Monte-Carlo (Exact) Stoch. master Eq. + quantum Diff. Stoch. master Eq. + quantum Diff. (G. Hupin talk)
24 Mean- field from varia;onnal principle More insight in mean-field dynamics: Included part: average evolution Exact state { Trial states exact Ehrenfest evolution The approximate evolution is obtained by minimizing the action: Missing part: correlations <A 1 > <B> One Body space Exact evolution <A 2 > Relevant degrees of freedom Mean-field The idea is now to treat the missing information as the Environment for the Relevant part (System) Hamiltonian splitting System Environment Environment Complex self-interacting System System
25 Existence theorem : Op;mal stochas;c path from observable evolu;on D. Lacroix, Ann. of Phys. 322 (2007). with Theorem : One can always find a stochastic process for trial states such that evolves exactly over a short time scale. Valid for Mean-field level In practice or Exact evolution Mean-field + noise <A 2 > <A 1 > Mean-field
26 illustra;on: simula;on of the free wave spreading with quasi- classical states t=0 t>0 x with Mean-field evolution: Reduction of the information: I want to simulate the expansion with Gaussian wavefunction having fixed widths. t>0 Relevant/Missing information: Relevant degrees of freedom Missing information Trial states Coherent states
27 Guess of the SSE from the existence theorem Stochastic c-number evolution from Ehrenfest theorem Densities with mean values fluctuations x Nature of the stochastic mechanics time with x x the quantum wave spreading can be simulated by a classical brownian motion in the complex plane
28 SSE for Many- Body Fermions and bosons D. Lacroix, Ann. Phys. 322 (2007) Starting point: Observables with Fluctuations Ehrenfest theorem BBGKY hierarchy Stochastic one-body evolution with The method is general. the SSE are deduced easily extension to Stochastic TDHFB DL, arxiv nucl-th The mean-field appears naturally and the interpretation is easier the numerical effort can be reduced by reducing the number of observables but Occupation probability two-level system unstable Bosons trajectories time
29 Summary, stochas;c methods for Many- Body Fermionic and bosonic systems Approximate evolution Mean-field Simplified QJ Generalized QJ Exact QJ variational QJ Fluctuation Dissipation Fluctuation Dissipation Fluctuation Dissipation Everything Partially everything Numerical issues Flexible Fixed Fixed Flexible Numerical instabilities
30
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