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

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>>.