Recent advances and trends in quantum transport theory

Denis Lacroix (GANIL)lacroix@ganil.fr

LEA Workshop-Catane2008

Actual mean-field theories (TDHF)

Beyond mean-field : highlights

Stochastic Schrödinger Equation : introduction

Application to Open Quantum systems

Application to Closed N-body systems

Inclusion of nucleon-nucleon collisions

Pairing effects

Coll: M. Assié, B. Avez, S. Ayik, P. Chomaz, G. Hupin, C. Simenel, J.A. Scarpaci, K. Washyiama

Trends in dynamical mean-field theories (TDHF)

two-body

three-body

one-body

Mean-field: (DFT/EDF)

“Simple” Trial state:

Self-consistentMean-field

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).

Cou

rtesy t

o C

. S

imen

el

First applications : more than 30 years agoRenewal of interest: 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: illustrationWashiyama, Lacroix, PRC78 (2008).

Potential

Kinetic

Dissipation

Macroscopic reduction

Dynamical Reduction effect

Adamian et al.,

PRC56(1997)

Dissipation Internal Excitation

GQR in 208Pb

Missing physics and visible consequences on dynamics Single-nucleus dynamics : collective motion

RPA

|Coll>Two-body

Average energy is OKbut dissipation (damping) is missed

Miss tunneling

Di-nucleus dynamics: fusion/fission V(Q)

Q

Zero point motion (too small)No symmetry breaking

Cro

ss s

ectio

n

Center of mass En.

Simenel, Avez, Int. J. Mod. Phys. (2008).

Beyond mean-field transport models

Simenel, Avez, Lacroix, Lecture notes Ecole Joliot-Curie 2007, arXiv:0806.2614

What type of correlations / Which extension ?

Short range correlation: pairing

Long-Range correlations: configuration mixing

Statistical models: direct nucleon-nucleon collisions

TDHFB or

equivalent

ExtendedTDHF

Single Reference (SR) Multi- Ref. (MR)

QuantumMonte-Carlo

TDGCM

Beyond mean-field: strategy

Exact one-body dynamics of a correlated system

with

Simenel, Avez, Lacroix, Lecture notes Ecole Joliot-Curie 2007, arXiv:0806.2614

N-N collisions

Pairing

Higher order

Application of mean-field + pairing : (I) TDHFB

Response function18O±2n

Small amplitude dynamics

Avez, Simenel, Chomaz, arXiv:0808.3507

TDHFB

QRPAKhan et al.,

PRC69, (2004)

But…

N-N collisions

Pairing

Higher order

Dynamics with pairing (II)

Assié, Lacroix, Scarpaci, in preparation

Assume dominant coupling and correlations between time-reversed pairs

(1)M. Matsuo, NPA (2001)

TDDMP HFB(1)

22O -3,5 MeV -3,3 MeV

24O -3,1 MeV -3,4 MeV

Pairing Gap : comparison with HFB

Static properties

n

n

Some Intuition

Final relative angle

Attractive interaction

Repulsive interaction

Different initialcorrelations

probing correlations with nuclear break-up

Assié, Lacroix, Scarpaci, in preparation

cigare

di-neutron

Cor

réla

tion

rel (degree)

Break-up of 6He

M. Assié, PhD (2008)

One Body space

<A1>

<A2>

<B>

Exact evolution

Mean-field

Mis

sing

in

form

atio

n

Other correlations : Direct N-N collisions in the mediumY. Abe et al, Phys. Rep. 275 (1996)D. Lacroix et al, Progress in Part. and Nucl. Phys. 52 (2004)

Short time evolution

Approximate long time evolution+Projection

Correlation

with

Propagated initial correlation

Dissipation and fluctuation

Random initial condition

Dissipation

projected two-body effect

Semiclassical version for approaches in Heavy-Ion collisions

t t t t time

Vlasov

BUU, BNV

Boltzmann- Langevin

Chomaz,Colonna, Randrup, Phys. Rep. (2007).

Application in quantum systems

Col

lect

ive

ener

gies

RPA Coupling

to ph-phononCoupling

to 2p2h states

2p-2

h de

cay

chan

nels

D. Lacroix, S. Ayik and P. Chomaz, Prog. in Part. and Nucl. Phys. (2004)

mean-fieldmean-field+fluctuation+dissipation

Open systems

Self-interacting vs 0pen systems

One Body space

<A1>

<A2>

<B>

Exact evolution

Mean-field

Mis

sing

in

form

atio

n

Brownian motion

N-body

Towards Exact stochastic methods for N-body and Open systems

Environment

System(one-body)

(others)Environment

System

Standard Schroedinger equation:

Deterministic evolution

time

Stochastic Schroedinger equation (SSE):

Stochastic operator :

time

…Introducing the concept of Stochastic Schrödinger equation

Theorem of existence :One can always find a stochastic process for trial states such that

evolves exactly over a short time scale.

Stochastic quantum mechanics from observable evolution

D. Lacroix, Ann. of Phys. 322 (2007).Philosophy:

Exact state Trial states

{

t time

D(t0)

Application to Many-body problems

Observables

Fluctuationswith

Stochastic one-body evolutionO

ccu

pati

on

pro

bab

ilit

y

time

two-level systemBosons

Application to Bosonic systems

System space

<S1>

<S2>

<B>Exact evolution

Envi

ronm

ent

Relevant degrees of freedom: system

Recent advances : Combining SSE with projection techniqueLacroix, Phys. Rev. E77 (2008).

Exact Stochastic master equation for open quantum systems

Indept .evol.

Mean-fieldNon-local in

time

drift

noise

H = HS + HE+ Q×B

Use SSE

Project the effect of the Environment

Under development: applications to system with potential energy surface

V(Q)

Q

Environment

Benchmark : The Caldeira-Leggett model

Coupling

System + heat-bath

More insight in the stochastic process

x

xtim

e

x

Observables evolution Complex noise on both P and Q

Fluctuations

Quantum Statistical

Quantum

Quantum+Stat

Exact

G. Hupin, D. Lacroix in preparation

ExactThis work

TCL

Quantum + Statistical fluctuations

Correct asymptotic Behavior

Next: -Non-harmonic potential -Tunneling+dissipation -Decoherence …Next: application to N-body problem

T = 0.1 h0 T = h0

Summary

3D TDHF dynamics with full Skyrme forces are now possible

Approximate or Exact stochastic methods can be very useful

Beyond mean-field theories will be necessary

Pairing like correlations

Configuration mixing (long range correlations)

Preliminary Results

ExactThis work

TCL

Position and momentum evolution

T = 0.1 h0T = h0

Zoology in the theory open quantum systems: approximations

Environment

System

Exact S+E evolution:

Reduced System evolution :

Weak coupling (Born approximation)

+ Stationary Env.

Separable interaction

Standard Approximations

Master equation:

Memory effect

t-s

Markov approximation

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

S+E Hamiltonian :

Average density

The dynamics of the system+environment can be simulated exactly with quantum jumps (or SSE) between “simple” state.

Interesting aspects related to the introduction of Stochastic Schröd. Eq.

Environment

System

Hamiltonian

{ with

A stochastic version

Exact dynamics

At t=0

time

Average evolution

++

Mean-field from variational principle

More insight in mean-field dynamics:

Exact state Trial states

{The approximate evolution is obtained by minimizing the action:

Good part: average evolution

exact Ehrenfestevolution

Missing part: correlations

Environment

System

Complexself-interacting

System

Hamiltonian splitting

SystemEnvironment

One Body space

<A1>

<A2>

<B>

Exact evolution

Mean-field

Mis

sing

in

form

atio

n

Relevant degrees of freedom

The idea is now to treat the missing informationas the Environment for the Relevant part (System)

<A1>

Exact evolution

<A2>

<A1A 2

>- <

A 1A 2

> MF

Existence theorem : Optimal stochastic path from observable evolution

with

D. Lacroix, Ann. of Phys. 322 (2007).

…Mean-field

Mean-field level

Mean-field + noise

Theorem :One can always find a stochastic process for trial states such thatevolves exactly over a short time scale.

Valid for

orIn practice

The method is general.the SSE are deduced easily

Ehrenfest theorem BBGKY hierarchy

SSE for Many-Body Fermions and bosons

D. Lacroix, Ann. Phys. 322 (2007)

Starting point:

with

Observables

Fluctuations

with

Stochastic one-body evolution

The mean-field appears naturally and the interpretation is easier

extension to Stochastic TDHFB D. Lacroix, arXiv nucl-th 0605033

Occ

upati

on p

robabili

ty

time

two-level systemBosons

but…

the numerical effort can be reduced by reducing the number of observables

unstable trajectories

Application : spin-boson model + heat bath

z=+1 z=-1

Leggett et al, Rev. Mod. Phys (1987)

Coupling

System + bath

weak coupling

strong coupling

Result (2000 trajectories)

Stockburger, Grabert, PRL (2002)

Comparison with related work : Path integrals + influence functional

Zhou et al, Europhys. Lett. (2005)

224 traj. !