Quantum dynamics of two Brownian particles

A. O. CaldeiraIFGW-UNICAMP

Outline

a) Introduction

b) Alternative model and effective coupling

c) Quantum dynamics

d) Results

e) Conclusions

Introduction

tfqVqqm ' where

0tf and

'2' ttTktftf B

Equation of motion of a classical Brownian particle

qVm

pH S

2

2

; kk

kqqCH int ; 222

2

1

2 kkkk k

kR qm

m

pH ;

k kk

kCT m

qCH 2

22

2

Phenomenological approach

Defining the spectral function

kk kk

k

m

CJ

2

2

one shows that the condition for ohmic dissipation q is

if

ifJ

0

Strategy: trace over the variables of R on the time evolution of the density operator of the entire system S+R

Effective dynamics depends only on

Other forms of the same model

k

kk

kkk

kk

kk p

mC

pqm

Cq

22and

2222

)(22

)(2

qqp

qVm

pH k

kk

k k

k

)()(2

)(2

)(

3

2

4

2

kk

kk

kk kk

k

kk

kk m

CJ

mC

where

Manifestly translational invariant if V(q)=0!

V(q)

Mechanical analogue

Manifestly translational invariant if V(q)=0!

If we write the Lagrangian of the whole system as

IRS LLLL ~

where 1~with

~~ kkkkk

kI CCqqCL

and go over to the Hamiltonian formalism, we recover the original model (with the appropriate counter – term) , after the canonical transformation

(notice there is no counter – term! )

kk

kkkkkk m

pqqmpqqpp

and,,

Two free Brownian particles (classical)

)(4)()(and0)(

;2

where)(2

111

111

ttkTmtftftfm

tfqmqm

)(4)()(and0)(

where)(2

222

222

ttkTmtftftf

tfqmqm

Two independent particles immersed in a medium, if acted by no external force obey

Two free Brownian particles (classical)

2and2,,

2if 21

21 mmMqqu

qqq

)(4)()(and0)(

;2

)()()(where)(2 21

ttkTMtftftf

tftftftfqMqM

CMCMCM

CMCM

)(4)()(and0)(

;)()()(where)()(2)( 21

ttkTtftftf

tftftftftutu

RRR

RR

Alternative model and effective coupling

Single particle

Going over to the Hamiltonian formulation + canonical transformation

O.S.Duarte and AOCPhys. Rev. Lett 97250601 (2006)

Alternative model and effective coupling

Single particle

modelling

counter -term becomes a constant

Equation of motion

Damping kernel

Fluctuating force

Alternative model and effective coupling

Single particle(0)

2

0

Im ( )( , ) 2 cos cosk

k kk

K r t d k kr t

Assumption

Resulting equation

Alternative model and effective coupling

Two particles

next page

Alternative model and effective coupling

Two particles

modelling

For the center of mass and relative coordinates

1 21 2and

2

x xq u x x

Alternative model and effective coupling

Two particles

Quantum dynamics O.S.Duarte and AOCTo appear PRB 2009

Tracing the bath variables from the time evolution of the fulldensity operator one gets

for the reduced density operator of the system

Quantum dynamics

Results

Initial reduced density operator

and2

i ii i i i

x yq x y

1 21 2and

2v

1 2

1 2and2

q qr u q q

New variables are defined in terms of

as and

reduced density operator at any time

z is the squeeze parameter

Results

Characteristic function

Covariance matrix

Eigenvalues of the PT density matrix

Logarithmic negativity

Results

401.0, 10 , 0, 10kT k L

Results

01.0, 10, 0, 10kT k L

Results

00, 5, 0, 10z kT k L

Results

41.0, 10 , 0, 10kT z

Results

41.0, 10 , 0.3, 10kT z

Conclusions

1) Generalization of the conventional model properly describes the dynamics of two Brownian particles.

2) Novel possible effects: static and dynamical effective interaction between the particles.

3) Possibility of two-particle bound states. Analogy with other cases in condensed matter systems; Cooper pairs, bipolarons etc.

4) Dynamical behaviour of entanglement for limiting cases.