Recent advances and trends in quantum transport theory
Denis Lacroix (GANIL)[email protected]
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. !