Stochastic quantum dynamics beyond mean-field.

Denis LacroixLaboratoire de Physique Corpusculaire - Caen, FRANCE

Introduction to stochastic TDHF

Application to collective motions

Functional integrals for dynamical Many-body problems

Alternative exact stochastic mechanics

One Body space

Introduction to stochastic theories in nuclear physics

Mean-field

Bohr picture of the nucleus

n

N-N collisions

n

Statistical treatment of the residual interaction(Grange, Weidenmuller… 1981)

-Random phases in final wave-packets (Balian, Veneroni, 1981)

-Statistical treatment of one-body configurations (Ayik, 1980)

-Quantum Jump (Fermi-Golden rules) (Reinhard, Suraud 1995)

Historic of quantum stochastic one-body transport theories :

if

Introduction to stochastic mean-field theories :

The correlation propagates as :

where

{ Propagated initial correlation

Two-body effect projected on the one-body space

Starting from :

D. Lacroix, S. Ayik and Ph. Chomaz, Progress in Part. and Nucl. Phys. (2004)

{

The initial correlations could be treated as a stochastic operator :

where

{Link with semiclassical approaches in Heavy-Ion collisions

t t t t time

Vlasov

BUU, BNV

Boltzmann- Langevin

Adapted from J. Randrup et al, NPA538 (92).

Molecular chaos assumption

{Incoherent nucleon-nucleon collision term.

Coherent collision term

Evolution of the average density :

One Body space Fluctuations around the mean density :

Average ensemble evolutions

Application to small amplitude motion

Standard RPA states Coupling

to ph-phononCoupling

to 2p2h states

Average GR evolution in stochastic mean-field theory

Full calculation with fluctuation and dissipations

RPA response

D. Lacroix, S. Ayik and Ph. Chomaz, Progress in Part. and Nucl. Phys. (2004)

Mean energy variation

fluctuation

dissipationRPA

Full

Effect of correlation on the GMR and incompressibility

Incompressibility in finite system

in 208Pb MeVE 10 MeVK RPA

A 156

MeVK ERPAA 135{

Evolution of the main peak energy :

More insight in the fragmentation of the GQR of 40Ca

EWSR repartition

Intermezzo: wavelet methods for fine structure

Observation

E

-1

+1

D. Lacroix and Ph. Chomaz, PRC60 (1999) 064307.

Basic idea of the wavelet method

Recent extensions : A. Chevchenko et al, PRL93 (2004) 122501.

Discussion on approximate quantum stochastic theories based on statistical assumptions

Results on small amplitude motions looks fine

The semiclassical version (BOB) gives a good reproduction of Heavy-Ion collisions

Success

Critical aspects

Which interaction for the collision term

Stochastic methods for large amplitude motion are still an open problem(No guide to the random walk)

Theoretical justification of the introduction of noise

Instantaneous reorganization of internal degrees of freedom?

Functional integral and stochastic quantum mechanics

Given a Hamiltonianand an initial State

Write H into a quadratic form

Use the HubbardStratonovich transformation

Interpretation of the integral in terms of quantum jumpsand stochastic Schrödinger equation

t time

Example of application: -Quantum Monte-Carlo Methods -Shell Model Monte-Carlo ...

General strategy S. Levit, PRCC21 (1980) 1594.S.E.Koonin, D.J.Dean, K.Langanke, Ann.Rev.Nucl.Part.Sci. 47, 463 (1997).

Carusotto, Y. Castin and J. Dalibard, PRA63 (2001).O. Juillet and Ph. Chomaz, PRL 88 (2002)

Recent developments based on mean-field

Nuclear Hamiltonian applied to Slater determinant

Self-consistent one-body part

Residual partreformulated stochastically

Quantum jumps between Slater determinant

Thouless theorem

Stochastic schrödinger equation in one-body space

Stochastic schrödinger equation in many-body space

Fluctuation-dissipation theorem

Stochastic evolution of non-orthogonal Slater determinant dyadics :

Quantum jump in one-body density space

Quantum jump in many-body density space

with

Generalization to stochastic motion of density matrix D. Lacroix , Phys. Rev. C (2005) in press.

The state of a correlated system could be described bya superposition of Slater-Determinant dyadics

t time

DabDac

Dde

Discussion of exact quantum jump approaches

Many-Body Stochastic Schrödinger equation

Stochastic evolutionof many-body density

One-Body Stochastic Schrödinger equation

Stochastic evolutionof one-body density

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 :

Actual applications : -Bose-condensate (Carusotto et al, PRA (2001)) -Two and three-level systems (Juillet et al, PRL (2002)) -Spin systems (Lacroix, PRA (2005))

Exact stochastic dynamics guiding approximate quantum stochastic mechanics

Weak coupling approximation : perturbative treatment

Residual interaction in the mean-field interaction picture

We assume that the residual interaction can be treated as An ensemble of two-body interaction:

Statistical assumption in the quasi-Markovian limit :

Time-scale and Markovian dynamics

{t t+t

Rep

lica

s

Collision time

Average time between two collisions

Mean-field time-scale

Hypothesis :

Interpretation in terms of average evolution of quantum jumps :

with

Stochastic term

From stochastic many-body to stochastic one-body evolution

We need additional simplification

Following the exact stochastic dynamics

We introduce the density

Following approximate dynamics One Body space

We focus on one-body degrees of freedom

Gaussian approximation for quantal fluctuations

We obtain a new stochastic one-body evolution in the perturbative regime:

Mean-field like term D. Lacroix, in preparation (2005)

Perturbative/Exact stochastic evolution

Perturbative Exact

Many-body density

Properties

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 : Flexible: one stoch. Number or more… Fixed :

“s” determines the number of stoch. variables

Application to spherical nuclei

2rt<0

Residual part :

Mean-field part :

Application : 40Ca nucleus = 0.25 MeV.fm-2

Root mean-square radius evolution:

rms

(fm

)

time (fm/c)

TDHFAverage evol.

time (fm/c)

D=179

D=260

D=340

D=1040

0=100

0=3000=200

0=500

Lifetime of the determinant:

Summary

One Body space

Stochastic mean-field from statistical assumption

(approximate)

t time

DabDac

Dde

Stochastic mean-field from functional integral

(exact)

Stochastic mean-field in the perturbative

regime

Sub-barrier fusion : Violent collisions :Vibration :

Applications: