Quantum Monte-Carlo for Non-Markovian Dynamics

Quantum Monte-Carlo for Non-Markovian Dynamics

Collaborator : Denis Lacroix

Guillaume Hupin

GANIL, Caen FRANCE

Overview

1. Motivation in nuclear physics : Fission and

Fusion.

2. Non-Markovian effects : Projective methods.

3. Quantum Monte-Carlo.

4. Applications :

A.To the spin-star model.B.To the spin-boson model.C. In the Caldeira Leggett model.

Environment

System

Physics case : fission

Environment

Nucleons response thermal bath. Fast motion vs fission decay.

—› Markovian

H. Goutte, J. F. Berger, P. Casoli, Phys. Rev. C 71 (2005)

System:

and

potential

Environment:

Nucleons (intrinsic)

motion

Coupling:

Physical case : fusion

Different channels are opened in a fusion reaction :

Microscopic evolution can be mapped to an open quantum system.

K.Washiyama et al., Phys. Rev. C 79, (2009)

System : relative

distance.Environment : nucleon

motion, other degrees of

freedom (deformation…).

Environment

Relative distance.

=> Non-Markovian effects are expected.

Motivations

Non-Markovian.Markovian.

Q Q

Environment

The total Hamiltonian written as

(S) is the system of interest coupled to (E). (E) is considered as a

general environment.

Exact Liouville von Neumann equation.

The environment has too many degrees of freedom.

Framework

Standard approaches : projective methods

Therefore, use projective methods to “get rid” of

(E).

First, define two projection operators on (S)

and (E). Project the Liouville equation on the two subspace.

System

Environment

Exact evolution

System

System only evolution

Envi

ronm

ent

I = interaction picture

Standard approaches : projective methods

H. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002)

Nakajima-Zwanzig’s method (NZ).

The equation of motion of the system is closed up to a given order in interaction.

Time Convolution Less method (TCL).t ts

Time non local method : mixes order of perturbation. (NZ2, NZ4..)

Time local method : order of the perturbation under control. (TCL2, TCL4 …)

s

Comparison between projective techniques

TCL4 is in any case a more accurate method.

Q

In the Caldeira-Leggett model: an harmonic oscillator coupled to a heat bath.

ExactNZ2NZ4ExactTCL2 (perturbation)

TCL4

Using a Drude spectral density:

G.Hupin and D. Lacroix, Phys. Rev. C 81, (2010)

New approach : Quantum Monte Carlo

The exact dynamics is replaced by a number of stochastic paths to simulate the exact evolution in average.

Noise designed to account for the coupling.D is complicated : The

environment has too many degrees of freedom.

t0 t

Exact Liouville von Neumann equation.

Theoretical framework

Stochastic Master Equations (SME).

D. Lacroix, Phys. Rev. E 77, (2008)

Other stat. moments are equal to 0.

Theoretical framework : proof

SME Proof :

Using our Stochastic Master Equations.

SME has its equivalent Stochastic Schrödinger Equation

Deterministic evolution

Stochastic Schrödinger equation (SSE):

…

Leads to the equivalence :

Independence of the statistical moments :

L.Diosi and W.T. Strunz Phys. Lett. A 255 (1997)M.B. Plenio and P.L. Knight, Rev. Mod. Phys. 70 (1998)J.Piilo et al., PRL 100 (2008)A. Bassi and L. Ferialdi PRL 103 (2009),Among many others…..

Application of the SSE to spin star model

A Monte-Carlo

simulation is exact

only when the

statistical

convergence is

reached.

Statistical fluctuation :

Noise optimized.

1000 Trajectories

Including the mean field solution

Take the density as

separable :

Position &

momentum : ok

Width : not ok

Then, take the

Ehrenfest evolution :

Mean field + QMC

Evolution of the

statistical fluctuation

have been reduced

using an optimized

deterministic part in

the SME.

D. Lacroix, Phys. Rev. A 72, (2005)

Projected Quantum Monte-Carlo + Mean-Field

Exact evolution

System

Stochastic trajectories

Envi

ronm

ent

The environment response is contained in :

Link with the Feynman path integral formalism

J. T. Stockburger and H. Grabert, Phys. Rev. Lett. 88, (2002)

Application to the spin boson model

This method has been applied to spin boson model.

Exact (stochastic)

TCL2

Second successful test.

A two level system interacting with a boson bath

D. Lacroix, Phys. Rev. E 77, (2008)

Y. Zhou , Y. Yan and J. Shao, EPL 72 (2005)

Application to the problems of interest

EnvironmentFission/fusion

processes

The potential is first locally approximated by parabola.

Benchmark in the Caldeira-Leggett model

QMC. Exact.

Convergence is achieved for second moments for different temperatures. with an acceptable time of calculation. with a limited number of trajectories (≈104 ).

Second moment evolution. Q

Observables of interest

Exact. Quantum Monte Carlo.

E

Projection 2nd order in interaction.Projection 4nd order in interaction.

Passing probability

Accuracy of MC

simulations is

comparable to the fourth

order of projection.

Accuracy of such

calculations are of

interest for very heavy

nuclei.

Markovian approximation

2nd order in perturbation TCL

4th order in perturbation TCL

Monte Carlo

Exact

E

Summary

It has been pointed out that TCL4 should be preferred.

New theory based on Monte Carlo technique has been

applied and tested for simple potentials.

This study shows that the new method is effective.

Now, the new technique Monte-Carlo+Mean-Field should

be applied to more general potentials.

Critical issue: diverging path

Possible solutions : Remove the diverging trajectories.Semi-classic approximations : Initial Value Methods

C. Gardiner, “A Handbook of Stochastic Methods”

W. Koch et al. , PRL 100 (2008)