First things firstFirst things first

40

100

24,980

OutlineOutline

Physics = surprisePhysics = surpriseElectron counting = field theoryElectron counting = field theoryFull Counting Statistic of a scattererFull Counting Statistic of a scattererTwo conductors togetherTwo conductors togetherSaddle pointSaddle pointQuantum effects = Coulomb blockadeQuantum effects = Coulomb blockadeOhmic Renormalization Ohmic Renormalization Quantum renormalizationQuantum renormalization

Full counting statistics of a Full counting statistics of a coherent conductorcoherent conductor

Coherent conductor = scatterer = Coherent conductor = scatterer = {T{Tpp} } (Landauer-Buttiker)(Landauer-Buttiker) Probability to transfer N electrons during time interval Probability to transfer N electrons during time interval t t = =

characteristic function (of characteristic function (of ). ).

Levitov ’93Levitov ’93

Electrons gambleElectrons gamble

1 ( 1) (1 )ln ( ) 2 ln

2 ( 1) (1 )

ip L R

s ip p R L

T e f fdEt

T e f f

( ) exp( ) ( )t tN

i N P N

Charge quantization

L R

Two conductors togetherTwo conductors together

Connect ‘m in seriesConnect ‘m in series

Ohm’s lawOhm’s law

Noises add asNoises add as

FCS?FCS?

A BR R R

2 2

2A A B BS R S R

SR

make a field theorymake a field theory

FixedFixed field(s) in the terminals (Keldysh = field(s) in the terminals (Keldysh = 2)2)

FluctuatingFluctuating field(s) in the node field(s) in the nodeComposition law for the actionComposition law for the action

( ) ( ); ( ) ,2

tet d V t

1 1( ), ( )t t

FCS!!!

Predictions and experimentsPredictions and experiments

Slow realizations, saddle-point solutionSlow realizations, saddle-point solution A-noisy, B- not(environment), shot noise limitA-noisy, B- not(environment), shot noise limit

Beyond noiseBeyond noise Pioneering measurement Pioneering measurement Of the 3Of the 3rdrd cumulant cumulant

M. Kindermann, Yu. V. Nazarov and C. W. J. Beenakker, Phys. Rev. Lett. 90, 246805 (2003).

Reulet B, Senzier J, Prober DE PHYS. REV. LETT. 91, 196601 (2003)  

22

4 43 3

4 5

3A A B A

A B AA

QR R RQ Q

R Q R

Properties of the field theoryProperties of the field theory

Low frequencies: always saddle-point solutionLow frequencies: always saddle-point solution Physically: I-V curves, Full Counting StatisticsPhysically: I-V curves, Full Counting Statistics

High frequencies: quantum effects High frequencies: quantum effects Coulomb blockadeCoulomb blockade RRA+BA+B ≠ R ≠ RAA + R + RBB

Renormalization loopRenormalization loop High frequency fluctuations renormalize the properties of low-High frequency fluctuations renormalize the properties of low-

frequency fluctuationsfrequency fluctuations

How big are renormalizations?How big are renormalizations? Universally, GUniversally, GQQ/G /G

2

Q

eG

Coulomb blockade Coulomb blockade (G (G ≪≪GGQQ))

2( )g gC

C VE E N

e 1

2g gNC V

e

Shifts potential= induces charge

CgVg

0

0.1

0.2

0.3

0.4

0.5

0.6

-4 -3 -2 -1 0 1 2 3 4

E

CgVg/e

N

CgVg/e

-4

-2

0

2

4

-4 -3 -2 -1 0 1 2 3 4

Vg

C Cg

N

Where’s the capacitance?Where’s the capacitance?

Never goneNever gone

Defines high-energy cut-offDefines high-energy cut-off

Traditional Coulomb blockade = at the cut-offTraditional Coulomb blockade = at the cut-off

Island

Renormalizations by Ohmic connectorRenormalizations by Ohmic connector

Complicated action – but in terms of Complicated action – but in terms of transmission eigenvaluestransmission eigenvalues

ZGZGQQ – small, but log-divergent at low energies – small, but log-divergent at low energies Program: renormalization of transmission Program: renormalization of transmission

eigenvalues eigenvalues

M. Kindermann and Yu. V. Nazarov, Phys. Rev. Lett. 91, 136802 (2003)

Transmission flowTransmission flow

Key to environmentKey to environment

Consequencies: Consequencies: two classes of coherent conductorstwo classes of coherent conductorsuniversality of resonant tunnelinguniversality of resonant tunneling

(1 )ln Qp

p pZGdT

T Td E

Renormalizations in a quantum dotRenormalizations in a quantum dot

G G ≫≫GGQQ

Is there Coulomb Is there Coulomb blockade?blockade?

Extra parameterExtra parameterSS –mean level –mean level

spacingspacingEECC ≫≫SS

D. A. Bagrets and Yu. V. Nazarov, Phys. Rev. Lett 94, 056801 (2005)

Transmission flowTransmission flow

Very similar, with its own conductanceVery similar, with its own conductance

Conductance itself subject to renormalizationConductance itself subject to renormalization That stops at That stops at

2(1 )

ln k

pp p

k

dTT T

d E T

( / )Th Q SE E G G

Two scenariosTwo scenarios

Blockade with reduced Blockade with reduced Coulomb energyCoulomb energy

Alternative : finite Alternative : finite renormalizationrenormalization