Extensions of mean-field with stochastic methods

Denis LacroixLaboratoire de Physique Corpusculaire - Caen,

FRANCE

Mapping the nuclear N-body dynamicsinto a open system problem.

Quantum jump approach to the many-body problem

One Body space

TDHF and beyond … -Saclay 2006

Stochastic one-body mechanics applied to nuclear physics

Mapping the nuclear dyn. to a system-environment problem

Assuming an initial uncorrelated state :

Evolution in time

One can improve the mean-field approximation by considering one-body degrees of freedom as a system coupled to an environment of other degrees of freedom.

Mean-field approximation:

Deg1

Deg2

Deg3

One-body subspace

Environment

Illustration:

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

Application to small amplitude motion

Standard RPA states Coupling

to ph-phononCoupling

to 2p2h states

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 : D. Lacroix et al, PLB 479, 15 (2000). A. Shevchenko et al, PRL93, 122501 (2004).

Discussion on one-body evolution from projection technique

Results on small amplitude motions looks fine

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

Success

Critical aspects

Numerical Implementation of 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?

Quantum jump method -introduction

Environment

System

{If waves follow stochastic eq.

with

Exact dynamics

At t=0

Breuer, Phys. Rev. A69, 022115 (2004)Lacroix, Phys. Rev. A72, 013805 (2005)

Then, the average dyn. identifies with the exact one

1 For total wave

For total density2

Projection technique

Weak coupling approx.

Markovian approx.

At t=0

Dissipative dynamics

Lindblad master equation:

Gardiner and Zoller, Quantum noise (2000) Breuer and Petruccione, The Theory of Open Quant. Syst.

Can be simulated by stochastic eq. on |>, The Master equation being recovered using :

1

In fermionic self-interacting systems

2

Stochastic mean-field Juillet and Chomaz, PRL 88 (2002)

Stochastic BBGKY Lacroix, PRC 71 (2005)

Quantum jump in the weak coupling regime

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

Statistical assumption in the Markovian limit :

Weak coupling approximation : perturbative treatment

Residual interaction in the mean-field interaction picture

R.-G. Reinhard and E. Suraud, Ann. of Phys. 216, 98 (1992)

GOAL: Restarting from an uncorrelated state we should:

2-interpret it as an average over jumps between “simple” states

1-have an estimate of

Time-scale and Markovian dynamics

{t t+t

Rep

licas

Collision time

Average time between two collisions

Mean-field time-scale

Hypothesis :

Two strategies have been considered:

Considering densities directly (philosophy of dissipative treatment)

Considering waves directly(philosophy of exact treatment)

Simplified scenario for introducing fluctuations beyond MF

Additional hypothesis:

We end with:

Mean-field like term

D. Lacroix, arXiv:quant-ph/ 0509038

Interpretation of the equation on waves as an average over jumps:

Let us simply assume that with

Matching with a quantum jump process between “simple states” ?

and focus on one-body density:We consider densities

Nature of the Stochastic one-body dynamics

Important properties remains a projector

Numerical implementation : flexible and rather simple.

t timeAverage evolution

One-body

Correlations beyond mean-field, denoting by

similar to Ayik and Abe,PRC 64,024609 (2001).

At all time

with

Application

Root mean-square radius evolution:

rms (

fm)

time (fm/c)

TDHFAverage evol.

Stoch. Schrödinger Equation (SSE) on single-particle states:

Assuming and

All the information on the system is contained in the one-body density

2rt<0

Residual part :

Mean-field part :

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

Monopole vibration in nuclei

Associated quantum jumps on single particle states:

Diffusion of the rms around the mean value

Standard deviation

No const

rain

t

Compression Dilatation

= 0.25 MeV.fm-2

Similar to Nelson quantization theoryNelson, Phys. Rev. 150, 1079 (1966).Ruggiero and Zannetti, PRL 48, 963 (1982).

Summary and Critical discussion on the simplified scenario

The stochastic method is directly applicable to nuclei

It provide an easy way to introduce fluctuations beyond mean-field

It does not account for dissipation.

In nuclear physics the two particle-two-hole components dominates the residual interaction, but !!!

Generalization: quantum jump with dissipation

Second Philosophy

Contains an additional term

Master equation for the one-body evolution

Starting from and its one-body density

Matching with the nuclear many-body problem

The residual interaction is dominated by 2p-2h components

with

Equivalent to the collision term of extended TDHF

Existence and nature of the associated quantum jump ?

with

All interaction of 2p-2h nature can be decomposed into a sum of separable interaction, i.e.

Koonin, Dean, Langanke, Ann.Rev.Nucl.Part.Sci. 47 (1997). Juillet and Chomaz, PRL 88 (2002).

time

Again

We can use standard quantum jump methods tosimulate this equation

The equation can be interpreted as the feedback action of the On operators on the one-body density

SSE on single-particle state :

with

time (arb. units)

wid

th o

f th

e

con

den

sate

mean-field

average evolution

Condensate size

Application to Bose condensate

N-body density:

1D bose condensate with gaussian two-body interaction

The numerical effort is fixed by the number of Ak

r

(r)

(arb

. u

nit

s)

t=0t>0

mean-field

average evolution

Density evolutio

n

SummaryQuantum Jump (QJ) methods to extend mean-field

Simplified QJ

FluctuationDissipation

Generalized QJ

FluctuationDissipation

Exact QJ

Everything

Mean-field

FluctuationDissipation

Variational QJ

Partially everything

Numerical issues

FlexibleFlexible Fixed Fixed

O. Juillet (2005)

Giant resonances

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

{Incoherent nucleon-nucleon collision term.

Coherent collision term

Evolution of the average density :

One Body space Fluctuations around the mean density :

Average ensemble evolutions

Linear response

Mean-field Mean-field Extended mean-fieldExtended mean-field

Response to harmonic vibrations

Notations for RPA equations

Using

+

Mean-field Mean-field Extended mean-fieldExtended mean-field

Fourier transform and coupling to decay channels

Incoherent damping

Ph. Chomaz, D. Lacroix, S. Ayik, and M. Colonna PRC 62, 024307 (2000)

Coherent damping

S. Ayik and Y. Abe, PRC 64, 024609 (2001).

hphp E

CollVhp

22

2

22

22

phph E

CollVph

2

Coupling to 2p-2h states Coupling to ph-phonon 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 :

Systematic improvement of the GQR energy

Calculated strength Main peaks energies , comparison with experiment

Experiments

N-body exact

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. C71, 064322 (2005).

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

t time

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))

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

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: