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Quantum Brownian Motion

Samak Boonphan51641256

By

26 August 2009Department of Physics, Faculty of Science, Kasetsart University

Physics Seminar 420597



2

Out line

- Classical Brownian Motion

- Quantum Brownian Motion

- Density matrix for a single system- T ime evolution of the system

- Caldeira - Leggett Model

- Influence functional - Reduced density matrix

- Master equation- Conclusions



3

Classical Brownian Motion



4

How the environment affects on the Brownian particle?

Consider a Brownian particle, which is the free particle

interacting with the environment.

)t(f)t()t(m K!

System

Environment



5

Consider a Brownian particle, which is the free particle,


)t(f)t(x)t(xm K

Friction force

System

Environment




6

Consider a Brownian particle, which is the free particle,


)t(f)t(x)t(xm K

Random force

System

Environment




7




8


System

Environment

Interaction



9

§|pat

sall¡

v¢ r

£ [ ¤ (t)]i

ab et),,(J

)L( ,t[ (t)]t

¥

´!

X

t

xa

xb

Feynman path Integration

Feynman had showed that the path integral give as

where

is action

a

tH×i

b

S[x(t)]i

ab xexeD[x],0)xt;,K(x JJ

|| ´



10

Density matrix for a single system

¼¼

¼

½

»

¬¬

¬«

|1//.

.

2222

1111

0

0

] ]

] ]

C

C

k k C ] ] V§! k

k ×

)A××tr(A !

Define

Expectation Value



11

T ime evolution of the system

Coordinate representation

t)×i

exp(( )t)×i

exp(-(t)JJ

!

xe(0)exx(t)xt),x(x,tH×

itH×

i

d!d|d

JJ

iii xxdx´ iii xxxd ddd´



12

xexx(0)xxexxdxd

xe(0)ext),x(x,

tH×i

iiii

tH×i

ii

tH×i

tH×i

dddd

dd

´´ JJ

JJ



t)×i

exp(( )t)×i

exp(-(t)JJ

V!

)xK(x, i )x,x(K idd



13



t)×i

exp(( )t)×i

exp(-(t)JJ

V!

t),x,x(K,0x,xt),xK(x,xdxd

xexx(0)xxexxdxd

xe(0)ext),x(x,

iiiiii

tH×iiiii

tH×iii

tH×i

tH×i

dddd!

dddd!

d!d

´´

´´ JJ

JJ



14


t),x,x(K,x,xt),xK(x,xdxdt),x(x, iiiiiidddd!d

´´

bxa

x

axd

bxd

x

t

t),xK(x, i

t),x,x(K idd

,0x,x iid

t),x(x, d

t·

x·



15

Caldeira-Leggett Model

A.O. Caldeira and A.J. Leggett, Ann. Phys. (N.Y.) 149, 374(1983)



16

Model of system + environment

M

m

m

m

m

m

System orBrownian particle

Environment

Interaction



17

IBAt t LLLL !

§ §! !

!N

1n

N

1nnn

2n

2n

2n

n222tot )xqC()qq(

2

m)x�x(

2

ML

Lagrangian of system + environment



18

À¿¾

°®̄ ¹

º ¸©

ª¨ ! ´ ´ q][x,[q][x]iexpD[x]D[q])q,x;q,K(x

IBA

x

x

q

qaabb

b

a

b

a J

Action and Propagator of system+environment

´ § §À¿¾

°¯®

!! !

t

¦

§

1 ̈

§

1 ̈

̈ ̈

©

̈

©

̈

©

̈

̈

© © © )xqC()qq()x�x(dsq][x,

X, q

t

System Environment+Interaction



20

Reduced Density matrix

-Elimination environment coordinate out of density matrix

q)(x, ? Atrq q(t)qdq´

q)(x, ? Atrq

-So that after trace out of environment we have

t),x'(x,r

Reduce density matrix



21

T ime evolution operator of density matrix

),x,x(, )x,xt;x,(xxdxdt),x,(x abAababrababr ´ dddddd! 0

- T ime evolution of reduced density matrix

? A ? A ? Axx,FxSxSi

]expxD[D[x],0)x,xt;x,(xJb

a

b

a

x

x

x

xababr dÀ¿¾

¯ dd!dd ´ ´

d

d J

where

Influence functional



22

T ime evolution operator of density matrix

t

t·

bx

x·

ax

axd

bxd

x

]xA[x,]xS[S[x]]xA[x, dd!d

´ ´

´ ´((

7(!d

t

0 022121

t

0

s

022121

1

1

)()()(dd

)()()(dsds],[s

ssssssi

ssssxxA

R

LH

2

1

2

´d

d À¿¾

°¯®

dd!ddt),x,(x

, )x,(xii

r

ii

]xA[x,i

]expxD[x]D[, )x,xt,x,(xJJ

xx d!(

2

xx d!§

Effective action

Influence effective action

|

|

Relative path

Center of relative path



23

)()cosI(d

kernelndissipatio)s-(s

0

21

21 s s !

!

´

g

[ [

L

Memory effect

2s !

!

´g

1n

0

21

s)cos2

cot(Id

kernelnoies)s-(s

J

R

- memory kernel

1s 2

s Spectral densityof environment §

!

N

1n nn

2n

n 2m

C)(



24

Master Equation

B.L Hu, Juan Pablo Paz, and Yuhong Zhang, Phys.Rev. D 45, 2843(1992)

Subhasis Sinha and P.A. Sreeram ,Phys.Rev. E 79,051111(2009)



25

Master equationtime evolution of reduced density matrix

,0xx,(t,0)Jt),x'(x, rrr d!

t),x(x,)x(x�21)

xx(

2Mt),x(x,

t r222

2

2

2

22

r d¼½»¬« ddxxxx!dxx

J

Without Environment



26

Master equationtime evolution of reduced density matrix

(t,0)Jr

,0xx,(t,0)Jt),x'(x, rrr d!

dt,0)(tJr

.)???.......(.........t),x(x,

t

r !d

x

x

Environment



27

Derivation of the master equation

step 1

]~,~,,[]~,~[],[]',[ x x x x A x x A x x A x x Ai dddd!

path integral

path

mx

step 2

dt

ts)x(xx(s)x~ mfm

!

dt,0)(tJr

(t,0)Jr

xix f x

mx x(s)

t

s

dtt

dts-t

]x[ ]x~[

t

]x~[



28

Derivation of the master equation

dt, )(tJr

dt(t, )J-dt, )(tJ

t(t, )J rrr !x

x

step 3 step 4 step 5

,0)x,(x iird

Multiply by

And integrateover coordinate

Subtract by

and take limit dt 0

(t,0)Jr



29

Frequencyshift

Dissipation

Diffusion andDecoherence

T he Master equation

t),x(x,xx

xx(t)f(t)t),x(x,xx(t) (t)iM

t),x(x,xx

)x(t)(xit),x(x,xx(t)�M21

t),x(x,xxM�2

1

xx2Mt),x(x,

ti

rr2

rr222

r222

2

2

2

22

r

d¹ º ¸

©ª¨

dxx

xx

ddd+

d¹ º ¸

©ª¨

dxx

xx

ddd

d¼½

»¬«

d¹¹ º

¸©©ª

¨dx

x

xx

!dxx

J

J

JJ

H



30

Frequency shift



31

(frequency shift)

p;

ohmic(n=1, a) subohmic(n=1/2, b)

Supraohmic(n=3, c)2000,3.00

!0!K

(t)��(t)� 222p !

masterequation

spectral density A.J. Leggett[4]

¹¹ º

¸©©ª

¨¹

º ¸

©ª¨!

2

21

exp

�

2I

0= cut-of frequency

A.J. Leggett et al., Rev. Mod. Phys. 59, 1(1987)



32

Dissipation



33

dissipation,

2000,3.00

!0!K

)(t+

Dissipation constant ,ohmic(n=1, a)subohmic(n=1/2, b)Supraohmic(n=3,c) 200030

0!0! ,.K

)(

)()(ds(t)

t

0

tuM

sust

1

1

!+

´ L

Dissipation kernel

´g

!

s)-(t)sinI(ds)-(tL

A.J. Leggett et al., Rev. Mod. Phys. 59, 1(1987)



34

Diffusion and Decoherence



35

Example decoherence of two-slit experiment

1]

2]

21

2

2

2

1R e2 ] ] ] ]

Coherence state

¼½»¬«

22221221

21121111

] ] ] ]

] ] ] ]

C C C C

W.H. Zurek, Rev. Mod. Phys. 75, 715(2003)



36

example decoherence of two-slit

experiment

1]

2]

21

2

2

2

1R e2 ] ] ] ]

21

2

2

2

1R e2 ] ] ] ]

Coherencedecoherence

¼½

»¬«

2222

1111

0

0

] ]

] ]

C

C ¼½

»¬«

22221221

21121111

] ] ] ]

] ] ] ]

C C

C C

W.H. Zurek, Rev. Mod. Phys. 75, 715(2003)



37

(Decoherence)

Decay factor ,ohmic(n=1, a)subohmic(n=1/2, b)Supraohmic(n=3,c)

2000,3.00

!0!K

noise kernel

)s)cos(2

cot(Id(s)

0

J´

g

!R

T K

1

B

! F

B.L Hu, Juan Pablo Paz, and Yuhong Zhang, Phys.Rev. D 45, 2843(1992)



38

Conclusions

- reduced density matrix

- memory effect

- master equation=time evolution of reduced density matrix

- frequency shift

- dissipation

- diffusion and decoherence



39



40

1. Subhasis Sinha and P.A. Sreeram ,Phys.Rev. E 79,051111(2009)2. W.H. Zurek, Rev. Mod. Phys. 75, 715(2003)3. B.L Hu, Juan Pablo Paz, and Yuhong Zhang, Phys.Rev. D 45,2843(1992)4. A.J. Leggett et al., Rev. Mod. Phys. 59, 1(1987)5. A.O. Caldeira and A.J. Leggett, Ann. Phys. (N.Y.) 149, 374(1983)6. R.P. Feynman and F.L. Vernon, Jr., Ann. Phys. (N.Y) 24, 118(1963)

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