Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.

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Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP

Transcript of Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.

Page 1: Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.

Quantum dynamics of two Brownian particles

A. O. CaldeiraIFGW-UNICAMP

Page 2: Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.

Outline

a) Introduction

b) Alternative model and effective coupling

c) Quantum dynamics

d) Results

e) Conclusions

Page 3: Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.

Introduction

tfqVqqm ' where

0tf and

'2' ttTktftf B

Equation of motion of a classical Brownian particle

Page 4: Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.

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

Page 5: Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.

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

Page 6: Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.

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!

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V(q)

Mechanical analogue

Manifestly translational invariant if V(q)=0!

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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,,

Page 9: Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.

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

Page 10: Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.

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

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Alternative model and effective coupling

Single particle

Going over to the Hamiltonian formulation + canonical transformation

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

Page 12: Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.

Alternative model and effective coupling

Single particle

modelling

counter -term becomes a constant

Equation of motion

Damping kernel

Fluctuating force

Page 13: Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.

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

Page 14: Quantum dynamics of two Brownian particles A. O. Caldeira IFGW-UNICAMP.

Alternative model and effective coupling

Two particles

next page

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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

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Alternative model and effective coupling

Two particles

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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

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Quantum dynamics

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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

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Results

Characteristic function

Covariance matrix

Eigenvalues of the PT density matrix

Logarithmic negativity

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Results

401.0, 10 , 0, 10kT k L

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Results

01.0, 10, 0, 10kT k L

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Results

00, 5, 0, 10z kT k L

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Results

41.0, 10 , 0, 10kT z

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Results

41.0, 10 , 0.3, 10kT z

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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.