1 high order moments of shot noise in mesoscopic systems michael reznikov, technion experiment: g....
TRANSCRIPT
![Page 1: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/1.jpg)
1
High order moments of shot noisein mesoscopic systems
Michael Reznikov, Technion
Experiment: G. Gershon, Y. Bomze, D. ShovkunTheory: E. Sukhorukov
Whether noiseWhether noiseis noisence is noisence
or signalor signalmay depend on may depend on
whom you askwhom you ask
![Page 2: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/2.jpg)
2
Classical Shot Noise
( ) probability
!for electrons to be transmitted
m nn eP m
mm
n m
ff
tt
ff
II S(S())
2
(0)J
S e J
![Page 3: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/3.jpg)
3
Noise in mesoscopic systemsscattering approach
2 2
2
2
2
=|t | , + =1
2
(0) (1 ), 2
for T=0 and eV
nn
n nn
t r
eJ V
eS e V
Khlus (1987), Lesovik (1989), Yurke and Kochansky (1989)
( )S
/eV
![Page 4: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/4.jpg)
Magnetic Field (T) J. Smet, V. Umansky
Rxy
(h
/e2 )
Rxx
(k
)Fractional Quantum Hall Effect
- Experimental Results
![Page 5: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/5.jpg)
5
The QPC
![Page 6: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/6.jpg)
Expected Noise…..(intuitively)
= 1/3
e/3
e/3
e/3
(0)i rS qI
q = e ; whole electrons
q = e/3 ; quasi particles
quasi particles partition
whole electrons partition
e
partitioningbarrier
Both, e or e/3 lead to the same conductance !
t
t
(0)iS eI
![Page 7: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/7.jpg)
Quantum Shot Noise in QPC- Experimental Results -
0
2
4
6
0 1 2 3
Cu
rren
t N
oise
, Si (
10 -
28 A
2 /Hz)
T=57 mK=0.37
I
2(0) 4 2 (1 ) coth
2B
i BB
k TeVS k Tg eI
k T eV
Total Current (nA)
![Page 8: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/8.jpg)
ns=1.1x1011 cm-2 ; B=13 T
Current Noise Measurements at bulk
• preamp noise subtracted
• calibration at each point
• averaging time 4 s
Lesovik’s formula, q=e/3
I=tVg0 /3
See also :Saminadayar et. al. 1997
Cur
rent
Noi
se, S
[10
-29
A2 /H
z]
Back-scattered Current, Ir [pA]
2
3
4
5
6
7
0 200 400
=0.82
e e/3
=0.73
2(0) 4 2 (1 ) coth
2B
i BB
k TqVS k Tg qI
k T qV
Ir=V(g0/3-g)
![Page 9: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/9.jpg)
Quantum Shot Noise at =2/5 - Weak Back Scattering -
70
72
74
76
78
0.33
0.34
0.35
0.36
0.37
0.38
0.39
0.4
0 10 20 30 40 50 60 70 80Cur
rent
Noi
se, S
[10-3
0 A2 /
Hz]
Conductance, g/g
0
Back-Scattered Current, Ir [pA]
B=2/5
t=0.86
T=85 mK
e/3
e/5
=2/5 q=e/5 !
Ir=V(2g0 /5-g)
![Page 10: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/10.jpg)
10
High-order cumulants - motivation
1 2 3 4 50
1
2
3
4
5
6
eV/kB
T
J
2 /
Gk B
T
2 40
0.5
1
1.5
2
2.5
J
3 /
eGk B
T
eV/kB
T
<<1=0.5
![Page 11: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/11.jpg)
12
Is this what is really measured?
0
( ) (0) ( ) ( ) (0) i tsS I I t I t I e dt
At least not always S(ω)!
( ) (0) ( ) i tS I I t e dt
( 0) - emission
( 0) - absorption
S
S
Lesovik, Loosen (1997)
![Page 12: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/12.jpg)
13
Naïve calculations
ˆ ˆ( ) ( ) ( )
k
k
o o
q J t dt q J t dt
+2
2
( b )
t irt aI a
birt t
a b11
r
it
2t
![Page 13: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/13.jpg)
14
Naïve calculations
ˆ( ) ( )
ˆ( )
o
k
k
o
q I t dt
q I t dt
3 20 (1 )q g V
For ~0 does not reproduce Poisson result q3=g0V =eI !
2 32 3 (3)(0) and , 1/ max( , )
q qJ S J S eV T
11
r
it
![Page 14: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/14.jpg)
15
“Gentle” electron counting Spin 1/2 as a galvanometerSpin 1/2 as a galvanometerL.S. Levitov and G.B. Lesovik (1993) L.S. Levitov and H. Lee (1996)
200
3 2 200
(1 ) (1 )
(1 )(1 2 ) (1 )(1 2 )
T
T
J eg V eI
J e g V e I
![Page 15: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/15.jpg)
16
“Gentle” electron counting Spin 1/2 as a galvanometerSpin 1/2 as a galvanometerL.S. Levitov and G.B. Lesovik (1993) L.S. Levitov, H. Lee, G. Lesovik (1996)
2 20
30
2 T coth 12
sinhT (1 ) 6 (1 2 )
cosh 1
B
B
UJ g K eV
U UJ eg K U
U
U=eV/T
![Page 16: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/16.jpg)
17
Gaussian vs. Poisson distributions
n=20
In our measurementsn~1000
![Page 17: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/17.jpg)
19
Intrinsic cumulants for a single channel conductor 0.5)
-15 -10 -5 0 5 10 15
-2
0
2
4
S3(1
0-46 A
3/H
z2)
U
3
3.5
4
S2(1
0-26 A
2/H
z)
(3) 2
1
1
U
S e I
(2)
1
(1 )
U
S eI
Khlus (1987), Lesovik (1989), Khlus (1987), Lesovik (1989), Yurke and Kochansky (1989)Yurke and Kochansky (1989)
L.S. Levitov and G.B. Lesovik (1993) L.S. Levitov and G.B. Lesovik (1993) L.S. Levitov and H. Lee (1996)L.S. Levitov and H. Lee (1996)
(2)
0
2 Re( ) B
U
S K T Z
B
eVU
K T
![Page 18: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/18.jpg)
20
Experimental results from Yale
-20 -10 0 10 20
-0.5
0.0
0.5
T0~<i02>
<S
V3>
/e2 (
A
3 )
eV/kBT
Noise of env.
voltage bias
result: -RD
3I
feedbackof env.
Sample AR=50 T=4.2KR
0=42
T0=7 K
B. Reulet, J. Senzier and D. E. Prober, 2002
![Page 19: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/19.jpg)
21
and in QPC2J 3J
Filling factor Filling factor =4=4T=4.2 KT=4.2 K0.30.3
![Page 20: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/20.jpg)
22
How to measure?
Opening and closing of the barrier
I. Klich, 2001
VV00 Z Z samplesample V
RRll CCVV CCstst
What is actually measured?
11
(1- )
i
![Page 21: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/21.jpg)
23
Zs>>Zl – voltage bias
VV00 Z Z samplesample V
RRll CCVV CCstst
0number of attempts fixed: n V e hmeasured: fluctuations of q em
statistics of charge: Binomia
( (1 )
l
) m n mn
mP m
n
K. Nagaev – cascade correctionsKindermann, Nazarov, Beenakker (2002)
![Page 22: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/22.jpg)
24
Zs<<Zl – current bias
VV00 Z Z samplesample V
RRll CCVV CCstst
Current and therefore transmitted charge is fixedq me
0
Measured: fluctuations of attempts ( )e
n dtV th
1
Flux distribution is Pascal: ( ) (1 )1
m n mm
nP n
m
![Page 23: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/23.jpg)
25
General case for a tunneling junction<<1, T<<eV
23 3Voltage bias: J q e I
23 3Current bias: 2V R e I
In general:In general:
2
23 3 23d J
V R e I R JdV
Kindermann, Nazarov, Beenakker (2002)
![Page 24: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/24.jpg)
26
and in QPC2J 3J
Filling factor Filling factor =4=4T=4.2 KT=4.2 K0.30.3
![Page 25: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/25.jpg)
27
Experimental Setup
I
Vg
QPCN
RlCv
Low temperature
Cst
Cc
Networkanalyzer
A/D
![Page 26: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/26.jpg)
29
“Intrinsic” contribution
tt3 t2 t1
J(t) A(t)
31 2 3 1 1 2 2 3 3( ) ( ) ( ) ( ) ( ) ( ) ( )V t dt dt dt A t t J t A t t J t A t t J t
3 3 31 1( )
VV J dt A t t
“Intrinsic” (constant voltage) contribution
![Page 27: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/27.jpg)
30
Corrections, “environmental” and nonlinear
2
3 2 21 2 1 2 1 23 ( ) ( ) ( )
env
d JV J dt dt A t t A t t Z t t
dV
t2 t1 t
J(t) A(t) Z(t)
2 23 2
1 2 1 2 1 223 ( ) ( ) ( )nl
d IV R J dt dt A t t A t t Z t t
dV
![Page 28: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/28.jpg)
31
Environmental correction is not small!
If we ignore peculiarities of the circuit
2
3 23env
d JJ R J
dV
2 2 BkJ
R
T
-mostly determined by the load thermal noise
2
3 6 Benv
d JJ k T
dV Not small even when R! 0
![Page 29: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/29.jpg)
32
QPC characterization
-1.3 -1.2 -1.1 -10
0.5
1
1.5
Gate voltage [V]
Con
duct
ance
G [
2e2/h
]
0 0.5 1 1.70
0.5
1
F(2
)
G [2e2/h]
T=1.5K
![Page 30: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/30.jpg)
33
QPC ~0.3
-3 -2 -1 0 1 2 3
-4
-2
0
2
4
Current I [10-8 A]
S3 [
10-4
6 A
3/H
z2]
-2 -1 0 1 22.8
2.9
S2 [
10
-24 A
2 /Hz]
Current I [10-8 A]
0.28
0.32
Current I [10-8 A]
Noise
Transmission
0
5
Ig V
T K
![Page 31: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/31.jpg)
34
Two different amplifiers
0 2 4 6 8Current I [10-8 A]
0 2 4 6 8-6
-4
-2
0
2
4
6
Current I [10-8 A]
S(3
) [10-4
6 A
3/H
z2]
a b
= 0.6 = 0.7
![Page 32: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/32.jpg)
35
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
F
3"Fano" factor for at eV TJ
3
, 0J
F JJ
![Page 33: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/33.jpg)
36
Calculation of the statistics
,
( exp) )( ) ( )( ) (f ii f
i q q P f i P i
*,
,,) ( )
ex
( exp( ( ))
( )) (0)) ( ( )p( (0exp( e p( )x )
f i f ii
f if
i q q U U
q
i
i it q qT ti q
T-ordering is to put q(0) to the right of q(t)
Using e.g. wave packet approach one can get the statistics(Levitov, Lesovik, 1993)
![Page 34: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/34.jpg)
37
How to express it through the integral of the currents (Bachman, Graf, Lesovik, 2009)
3 3( ( ) (0))T q t qq
31 2 3 1 2 3( ( ( ) (0))( ( ) (0))( ( ) (0)) ), ,Q t T q t q q t q q tt t q
Consider a slightly different object
Properties: Q3=0 if one of ti=0. Therefore it can be expressed as:
3 33
1 2 3 1 2 31 2 30
,( , )t
t t dt dt dtdt dt d
d Q
tQ t
Time ordering is crucial to ensure Q3=0 for ti=0 !!!
![Page 35: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/35.jpg)
38
“Contact” terms
3
1 2 33
( ( ) (0))( ( ) (0)) ( )dt
dQT q t q q t q j t
3
1 2 32 3
1 2 2 2 3
)( ( ) (0)) ( ) (
( ( ) (0))[ ( ), ]( ) ( )
dQT q t q j t j t
T q t q q t j t
dt dt
t t
Differentiation ovet t1 would generate 2 more -functions,provided [q,j] 0. So, there are additions to the term accountedfor in naïve calculations: h j(t1) j(t2) j(t3)i
![Page 36: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/36.jpg)
39
My favorable choice of j
a11
r
itb
L2
2
2
2
0 0
0 1
0
( ) ( )
( ) (0
0 1)
t itrq
itr r
t itrj
x L x L
x L x Litr r
2
2( )
t itrj
itr tx
Compare with:
![Page 37: 1 High order moments of shot noise in mesoscopic systems Michael Reznikov, Technion Experiment: G. Gershon, Y. Bomze, D. Shovkun Theory: E. Sukhorukov](https://reader031.vdocument.in/reader031/viewer/2022013101/56649ca75503460f94969f8a/html5/thumbnails/37.jpg)
40
Conclusions and questions
•Prediction for <<J3>> in QPC is verified•Effect of interactions on <<J3>>.•Charge statistics under FQHE?•Charge statistics in HTC superconductors•<<J3>> in diffusive systems with interactions•Frequency dependence of <<J3>>.