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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
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
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
Magnetic Field (T) J. Smet, V. Umansky
Rxy
(h
/e2 )
Rxx
(k
)Fractional Quantum Hall Effect
- Experimental Results
5
The QPC
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
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)
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)
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)
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
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)
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
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
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
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
17
Gaussian vs. Poisson distributions
n=20
In our measurementsn~1000
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
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
21
and in QPC2J 3J
Filling factor Filling factor =4=4T=4.2 KT=4.2 K0.30.3
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
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)
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
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)
26
and in QPC2J 3J
Filling factor Filling factor =4=4T=4.2 KT=4.2 K0.30.3
27
Experimental Setup
I
Vg
QPCN
RlCv
Low temperature
Cst
Cc
Networkanalyzer
A/D
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
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
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
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
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
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
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
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)
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 !!!
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
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:
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>>.
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