Quantum Brownian Motion Seminar1

8/6/2019 Quantum Brownian Motion Seminar1
1/40
16/08/54 1,2,3 1
Quantum Brownian Motion
Samak Boonphan51641256
By
26 August 2009Department of Physics, Faculty of Science, Kasetsart University
Physics Seminar 420597

2/40
2
Out line
- Classical Brownian Motion
- Quantum Brownian Motion
- Density matrix for a single system- Time evolution of the system
- Caldeira - Leggett Model
- Influence functional- Reduced density matrix
- Master equation- Conclusions

3/40
3
Classical Brownian Motion

4/40
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

5/40
5
Consider a Brownian particle, which is the free particle,
)t(f)t(x)t(xm K
Friction force
System
Environment

6/40
6
Consider a Brownian particle, which is the free particle,
)t(f)t(x)t(xm K
Random force
System
Environment

7/40
7

8/40
8
System
Environment
Interaction

9/40
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
tHi
b
S[x(t)]i
ab xexeD[x],0)xt;,K(xJJ
||

10/40
10
Density matrix for a single system
|1//.
.
2222
1111
0
0
]]
]]
C
C
kkC ]]V ! kk
)Atr(A !
Define
Expectation Value

11/40
11
Time 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

12/40
12
xexx(0)xxexxdxd
xe(0)ext),x(x,
tHi
iiii
tHi
ii
tHi
tHi
dddd
dd
JJ
JJ
t)i
exp(( )t)i
exp(-(t)JJ
V!
)xK(x, i )x,x(K idd

13/40
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
tHiiiii
tHiii
tHi
tHi
dddd!
dddd!
d!d
JJ
JJ

14/40
14
t),x,x(K,x,xt),xK(x,xdxdt),x(x, iiiiii dddd!d
bxa
x
axd
bxd
x
t
t),xK(x, i
t),x,x(K idd
,0x,x ii d
t),x(x, d
t
x

15/40
15
Caldeira-Leggett Model
A.O. Caldeira and A.J. Leggett, Ann. Phys. (N.Y.) 149, 374(1983)

16/40
16
Modelof system + environment
M
m
m
m
m
m
SystemorBrownian particle
Environment
Interaction

17/40
17
IBAt t LLLL !
! !
!N
1n
N
1nnn
2n
2n
2n
n222tot )xqC()qq(
2
m)xx(
2
ML
Lagrangian of system +environment

18/40
18
! 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 Propagatorof system+environment
!! !
t
1
1
)xqC()qq()xx(dsq][x,
X, q
t
System Environment+Interaction

19/40

20/40
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

21/40
21
Time evolution operator of density matrix
),x,x(, )x,xt;x,(xxdxdt),x,(x abAababrababr dddddd! 0
- Time evolution of reduced density matrix
? A ? A ? Axx,FxSxSi
]expxD[D[x],0)x,xt;x,(xJb
a
b
a
x
x
x
xababr d
dd!dd
d
d J
where
Influence functional

22/40
22
Time 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
dd!ddt),x,(x
,)x,(xiir
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

23/40
23
)()cosI(d
kernelndissipatio)s-(s
0
21
21 ss !
!
g
[[
L
Memory effect
2s!
!
g
1n
0
21
s)cos2
cot(Id
kernelnoies)s-(s
J
R
- memory kernel
1s 2s Spectral density
of environment !
N
1n nn
2n
n 2m
C)(

24/40
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)

25/40
25
Master equationtime evolution of reduced density matrix
,0xx,(t,0)Jt),x'(x, rrr d!
t),x(x,)x(x21)
xx(
2Mt),x(x,
t r222
2
2
2
22
r d ddxxxx
!dxxJ
Without Environment

26/40
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

27/40
27
Derivation of the master equation
step1
]~,~,,[]~,~[],[]',[ xxxxAxxAxxAxxAi dddd!
path integral
path
mx
step2
dt
ts)x(xx(s)x~ mfm
!
dt,0)(tJr
(t,0)Jr
xix fxmx x(s)
t
s
dtt
dts-t
]x[ ]x~[
t
]x~[

28/40
28
Derivation of the master equation
dt, )(tJr
dt(t, )J-dt, )(tJ
t(t, )J rrr !x
x
step3 step4 step5
,0)x,(x iird
Multiply by
And integrateover coordinate
Subtract by
and take limitdt 0(t,0)Jr

29/40
29
Frequencyshift
Dissipation
Diffusion andDecoherence
The 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,xxM2
1
xx2Mt),x(x,
ti
rr2
rr222
r222
2
2
2
22
r
d
dxx
xx
ddd+
d
dxx
xx
ddd
d
d
dx
x
xx
!dxx
J
J
JJ
H

30/40
30
Frequency shift

31/40
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 densityA.J. Leggett[4]
!
2
21
exp
2I
0= cut-offrequency
A.J. Leggett et al., Rev. Mod. Phys. 59, 1(1987)

32/40
32
Dissipation

33/40
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

Quantum Brownian Motion Seminar1

Documents

Transcript of Quantum Brownian Motion Seminar1

Lider Forex - Web seminar1

ADA SEMINAR1

Seminar1: Inbound Marketing For Business

Geoapp seminar1

Free Flaps Seminar1

Ektf Seminar1 With Notes

TECHNICAL SEMINAR1

Seminar1 - Introducere in Semiotica

Evaluation seminar1

Android Seminar1

Credit Seminar1

FIDIC Seminar1 Hong

Micro Air Vehicles Seminar1

Consumers seminar1 summer1_2013

Employee Engagement Seminar1

Doctoral Seminar1

372 seminar1 michaelclarke

331 seminar1 duncanmacrae

Seminar1 Econometrie

Random walk on mated-CRT planar maps and Liouville ... ... -Liouville Brownian motion, the natural quantum time parametrization of Brownian motion on a -Liouville quantum gravity (LQG)