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 :
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: