the magic Fermi sea

electronic scattering, noise and entanglement

D. C. Glattli, SPEC CEA Saclay and LPA ENS Paris

INTRODUCTION

(disclaimer : these are just notes and many references to original works are missing)

electrons :

electrons interference (wave)

photons:

light interference

V

.elI2

2121 φφ ii eaea +

.

.

ph

el

I

I∝ or

)1(

)2( )2(

)1(.phIWhat is different between electrons and photons (Fermions and Bosons)

eV(i) (t)(r)

( D )

reservoir = detector

1 ( )D(i) (t)

(r)detector

( )D−1

electrons :

electrons interference (wave) scattering description

photons:

light interference

LaLb RaRbS(i) (t)

(r)

eV(i) (t)(r)

( D )

reservoir = detector

1 ( )D(i) (t)

(r)detector

( )D−1

electrons :

electrons interference (wave)

shot noise (particle)

photons:

light interference

photon noise

uncertainty due to the diffraction of the incoming wave into reflected and transmitted wave: ⇒ QUANTUM PARTITION NOISE

eV(i) (t)(r)

( D )

reservoir = detector

1 ( )D(i) (t)

(r)detector

( )D−1

electrons :

electrons interference (wave)

shot noise (particle)

photons:

light interference

photon noise

different quantum noise results for different quantum statistics (Bose versus Fermi)

Fermi sea gives noiseless electron generation while photons are fundamentally noisy

the magic Fermi sea

or εε dh

NddhedI elel 1.. == & ννε dNhdId

hNNd occphoccph .... )(== or&

⇒ quantum of conductance : (no equivalent, no known continuous generation of photon number states) h

e2heVeI .=

eVh=τI

1 1 111 1 1noiseless injection of electrons

noiseless electrons electron entanglement using linear ‘ electron optics ’

and equilibrium thermodynamic sources

noiseless electrons may generate low noise photons

εFE0 : sexcitation plus

0 : stateground FF EE cc <+> εε

Electron-hole pairs are naturally entangled.

Fermi sea rate of entanglement production eV/4h

the magic Fermi sea

(… from quantized conductance, sub-poissonian shot noise to entanglement...)

Introduction

I. electronic scattering1- one ideal mode (+ electron flux/photon flux analogy)2- one mode scattering3- multi-mode scattering: Landauer formula4- multi-terminal Büttiker formula (quantum Kirchhoff law)

II. Electronic analogs of optical systems1- ballistic chiral transport : Edge states in QHE. 2- the QPC electronic beam splitter3- electronic Mach-Zehnder interferometer 4- electronic Fabry-Pérot interferometer

III. Quantum Shot noise1 - Quantum partition noise

- one and two particle partitioning :electrons/ photons- electronic shot noise

2- scattering derivation of quantum shot noisea- S(ω) for an ideal one mode conductorb- quantum shot noise for a single modec-zero frequency shot noise and multimode case

3- experimental examples

(suite ⇒ )

(suite) Quantum Shot Noise (…)4- current noise cross-correlations

- scattering derivations- electronic analog of the optical Hanbury-Brown Twiss experiment- electronic quantum exchange

5- experimental techniques

Entanglement by the Fermi statistics- flying qubits

- difference with NMR like approach- free electron sources (biased contact, photo-assisted e-h pairs, single electron injection)

- two-particle interference and entanglement- electron-hole entanglement - linear optics versus ‘linear electronics’- quantifying entanglement- maximal rate of entanglement production by the Fermi sea

Shot noise: the tool to detect entanglement - Shot noise Bell’inequalities (cross-correlation relations, discussion on time-scale) - experimental proposals - electron-hole pairs entanglement

- two-way electron state teleportation

Combining electrons and photons- case where the photon statistics is not considered:

- ac quantum tranport - scattering derivations- the QPC- charge relaxation of the mesoscopic capacitor

- ac to dc - photo-current- photo-assisted dc shot noise

auto-correlationscross-correlations

- combining photon statistics and electron statistics- statistics of photons emitted by a quantum conductor

origine du cours:

ENS ParisBernard Plaçais

Jean-Marc BerroirAdrian BachtoldJulien GabelliGwendal Fève

Gao BoBetrand BourlonAdrien MaheJulien Chaste

CEA-SaclayPatrice RocheJ. SegalaF. Portier

Preden Roulleau(L-H. Bize-Reydellet)

(V. Rodriguez)(L. Saminadayar)

(A. Kumar)

- quantum point contact : shot noise, edge states, co-tunneling of Q.-Dots- ballistic qubits- mesoscopic capacitor- carbone nanotube- Fractional Quantum Hall effect

- high frequency (40 GHz)- ultra low noise measurements- high magnetic field (18T) and low T (20mK)- lithography- cryo-electronics

(MIP → 04 ; DEA 05 → ;Les Houches Summer School 03 on ‘Quantum information’)

2-Delectron

gas

Gate

Gate

Quantum Point Contact

Introduction

I. electronic scattering1- one ideal mode (+ electron flux/photon flux analogy)2- one mode scattering3- multi-mode scattering: Landauer formula4- multi-terminal Büttiker formula (quantum Kirchhoff law)

II. Electronic analogs of optical systems

III. Quantum Shot noise

IV. Entanglement by the Fermi statistics

V. Shot noise: the tool to detect entanglement

VI. Combining electrons and photons

Ve

ε

FE

1

)(εLf

ε

FE

1

)(εRfelectronreservoir

electronreservoir

X

X

X

electron reservoirs = - “black-body sources” of electrons- emit electrons according to Fermi distribution- absorb perfectly all electrons

incoming electron ( i ) → superposition of transmitted ( t ) and reflected ( r ) state

MLLLaaa , 2,1,ˆ ...ˆ NRRRaaa , 2,1,ˆ ...ˆ NRRRbbb , 2,1,ˆ...ˆ

MLLLbbb , 2,1,ˆ ...ˆ (((( ))))S

Rk Lk)(kε )( kεLµ RµVe0=T

Lµreservoir

0=TRµ

reservoir

I . 1 the ideal 1D wire (… or one mode conductor)

xki Lexki Re

−−−−

) length ( ∞→L

xikRLnnRLn RLn eL

xmk

Lnk )(1)(22 )(,22)(, ±=== εϕεπ

, and h

Rk Lk)(kε )( kεLµ RµVe0=T

Lµreservoir

0=TRµ

reservoirxki Le

xki Re−−−−

) length ( ∞→L

xikRLRLRLRL RLev

xm

kk )()()(,2)(2)( )(

)(21)(2)( ε

ε επϕεε ±==

h

h , and

)'(,,', εεδδϕϕ βααεβε −=

)(21)(

επε

vD

h=

density of stateper unit of energyper unit length

ε

energy is a natural parameter for elastic scattering.(… and all results are simpler)

Rk Lk)(kε )( kεLµ RµVe0=T

Lµreservoir

0=TRµ

reservoirxki Le

xki Re−−−−

) length ( ∞→L

xikRLRLRLRL RLev

xm

kk )()()(,2)(2)( )(

)(21)(2)( ε

ε επϕεε ±==

h

h , and

εϕε

ϕ εε dm

kedI RLRLRLRL )(,,)(,, )(h±=

εdhedI RL ±=)(

ε

… is the current carried by one mode within energy range dε(independent on specific energy band dispersion relation)

Rk Lk)(kε )( kεLµ RµVe

TLµ

reservoir

TRµ

reservoir

second quantification representation

(to be ready to go further than simple scattering: … shot noise, ac transport, entanglement …)xki Lexki Re

−−−−

∫ −= )()(ˆ)(21),(ˆ txkiLLL Leav

dhetx εε

επεψ

h ∫ −−= )()(ˆ)(21),(ˆ txkiRRR Reav

dhetx εε

επεψ

h

{ }{ }

)'()()'(ˆ).(ˆ0)(ˆ),'(ˆ

)'()(ˆ),'(ˆ

,,εεδδεεε

εεεεδδεε

βααβα

αβ

βααβ

−=

=

−=

+

+

faaaaaa

( ) 0ˆˆˆ=

∂∂+

∂∂ +

Ixt

e ψψ

RRL aRaL

a RL 0)(ˆ0)(ˆˆ ,0,0)(

εε µεµε

βα

+=

+= ∏=∏=

,

and

reservoirs the of space Fock the on act

L

Rk Lk)(kε )( kεLµ RµVe

TLµ

reservoir

TRµ

reservoirxki Le

xki Re−−−−

∫ −= )()(ˆ)(21),(ˆ txkiLLL Leav

dhetx εε

επεψ

h ∫ −−= )()(ˆ)(21),(ˆ txkiRRR Reav

dhetx εε

επεψ

h

)(),(ˆ εε LLL fdheItxI ∫==

Rk Lk)(kε )( kεLµ RµVe

TLµ

reservoir

TRµ

reservoir

the quantum of conductance

xki Lexki Re

−−−−

( )

TVeVVheI

dheffI RRL

,

)()( L20∀+==

−= ∫∞

µµ

εεε

) / ( erg Heisenb ) 1 ( Pauli heV e h

eVeI +×=

the conductance :

is independent on temperature and voltagehe2

TLµ

reservoir

electron-photon analogy

xki Le

∫ −= )()(ˆ)(21),(ˆ txkiLLL Leav

dhetx εε

επεψ

h ∫ −=Ε )(0 )(ˆ2),(ˆ txkiLL Leahdtx νεενν)/)('(. )(ˆ)'(ˆ)()'(2

)()'('),(ˆ FL vxtiLLLL LLLel eaavvvdd

hetxI −−++= ∫ εεεε

εεεεεε ( ) )/)('(. )(ˆ)'(ˆ'),(ˆ cxtiLLLph LeaaddhtxI −−+∫= ννννννν

εε dfhedI DFel )(... = ννν dhfhdI EBph )(... =

xki Le

1≤ ∞→−

0:1

1TkhBe ν

TLµ

reservoir

TR

reservoirµ

1

''

==

≡

=

++ SSSS

rttr

ssss

SRRRL

LRLL

0 0Lx RxLaLb RaRbS

( )∫ −+= LLLL xkiLxkiLLL ebeav

dhex )(ˆ)(ˆ)(2

1)(ˆ εεεπ

εψh

( )∫ −+= RRRR xkiRxkiRRR ebeav

dhex )(ˆ)(ˆ)(2

1)(ˆ εεεπ

εψh

scattering states in the reservoirs:

conductor

S contains all the transportinformation on the conductor.

links the outgoing amplitudes to the incoming wave amplitudes

amplitude are replaced by the annihilation operator acting on the Fock space of each reservoir L,R

Idea: write the total fermion operator as:

I . 2 ONE MODE SCATTERING

k

)(kε Lµ

k

)(kε

RµVe

U U

TL ,µreservoir

TR ,µreservoir

ε xkiR RLev

thπε

2)(xkiL Le

vhπ21

)(εr

0 00

a simple example:

1. scattering states: electrons emitted by the left reservoir

( )

0)()(2)(ˆ

0)()(2)(ˆ),(ˆ ,

<=

<+=

−−

−−

∫∫ RtixikRL LtixikxikLLL

xeetv

ad

xeerev

adtx RR LLLL

ε

εε

εεπ

εε

εεπ

εεψ

h

h

mkk LL 2)(

22h=ε m

kUk RR 2)(22

h+=ε

(describes electronsemitted by the left reservoir)

(current probability amplitude)

similarly, for electrons emitted from the right reservoir

( )

0)()(2)(ˆ

0)()(2)(ˆ),(ˆ ,

<=

<+=

−−

−−

∫∫ RtixikRL LtixikxikLLL

xeetv

ad

xeerev

adtx RR LLLL

ε

εε

εεπ

εε

εεπ

εεψ

h

h

( )

0)(')(2)(ˆ

0)(')(2)(ˆ),(ˆ ,

<=

<+=

−−

−−

∫∫ LtixikLR RtixikxikRRR

xeetv

ad

xeerev

adtx LL RRRR

ε

εε

εεπ

εε

εεπ

εεψ

h

h

total state in the left lead

( )

( ) 1

tixikLxikLLtixikRxikLxikLLLL

eebeav

d

eetaeraeav

adtx LLLL LLLLLLε

εε

εεεπ

ε

εεεεεεπ

εεψ

−−

−−−

+=

++=

∫∫

)(ˆ)(ˆ)(2

)(')(ˆ)()(ˆ)(ˆ)(2)(ˆ),(ˆ

h

h

… and similarly in the right lead

=

R

L

R

L

aa

rttr

bb

ˆˆ

''

ˆˆ

( check using the specific example: )

this ensures that :

current conservation:

implies unitarity of S :

2222RLRL bbaa +=+

U

ε xkiR RLev

thπε

2)(xkiL Le

vhπ21

)(εr

0

{ }{ } 0)(ˆ),'(ˆ

)'()(ˆ),'(ˆ ,=

−=+

εε

εεδδεε

αβ

βααβ

bb

bb

current conservation

( ){ }( ))()('

)(')()(2 22εεε

εεεε RL RLLfftdh

eftfrfdh

eI

−=

+−=

∫∫

{ }∫ ++ −= )(ˆ)'(ˆ)(ˆ)'(ˆ' εεεεεε LLLL bbaaddheI

{ } )/)('()()'(2)()'()'(ˆ)'(ˆ)'(ˆ)'(ˆ'),(ˆ FL vxtiLL LLLLLLL e

vvvbbaadd

hetxI −−++ +−= ∫ εε

εεεεεεεεεε

current conservation relations are useful to calculate the expectation value of the current operator

)'()()'(ˆ).(ˆ , εεδδεεε βααβα −=+ faausing : , the average current in the left reservoir is

))()'(( εε LL vv −∝

factor a plus ...

(expression to keep: … will be used later for noise or ac transport)

1== ++ SSSS using

current operator:

TLµ

reservoir

TR

reservoirµ

1==

=

++ SSSS

ssss

SRRRL

LRLL0 0Lx RxLaLb RaRbS

( ){ }∫ +−= )()()( 22 εεεε RLRLLLL fsfsfdheI

( )∫ −= )()(2 εεε RLLR ffsdheI )(

2ε

εε Dfdh

eG ∫

∂∂−=

to summarize:

0T @ == )(2 FDheG εRSSSSD RRLLRLLR −=−=−=== 111 2222Landauer formula

Introduction

I. Electronic scattering1- one ideal mode (+ electron flux/photon flux analogy)2- one mode scattering3- multi-mode scattering: Landauer formula4- multi-terminal Büttiker formula (quantum Kirchhoff law)

II. Electronic analogs of optical systems

III. Quantum Shot noise

IV. Entanglement by the Fermi statistics

V. Shot noise: the tool to detect entanglement

VI. Combining electrons and photons

modeling of the reservoirs for a two-terminal conductor: (2D here for simpler notations)

Ve

ε

Rµ

1

)(εRfX

X

X

TL ,µreservoir

TR ,µreservoir

Lµε

1

)(εLfLy

LxLw

0

Incoming wave : emitted by (L) mode (m)

ideal multi-mode lead

=),;(, LLmL yxεϕ

I . 3 Multiple mode scattering : Landauer fromula

Ve

ε

Rµ

1

)(εRfTL ,µreservoir

TR ,µreservoir

Lµε

1

)(εLfX

X

X

MLLLaaa , 2,1,ˆ ...ˆ NRRRaaa , 2,1,ˆ ...ˆ NRRRbbb , 2,1,ˆ...ˆ

MLLLbbb , 2,1,ˆ ...ˆ (((( ))))S

scattering matrix for many modes:

= )()()()( RRRL LRLL

SSSS

S

∫ ∑ ∑∑= ==

−

++=)( 1 )( 1' ',',)( 1' ',',,,, ˆˆ22ˆ),(ˆ ,,ε εε

πχ

πχ

εψMm Nm mnLRnRMm mmLLmLxikLmmLxikLmmLmLLL sasaevevadyx LmLLmL

hh

(m)

(m’)(n’)

( reservoir ‘L’) ( reservoir ‘R’)( conductor : S )

mLb ,

ˆ

ε

Rµ

1

)(εRf

ε

1

)(εLf MLLLaaa , 2,1,ˆ ...ˆ

NRRRaaa , 2,1,ˆ ...ˆ NRRRbbb , 2,1,ˆ...ˆ MLLLbbb , 2,1,ˆ ...ˆ (((( ))))S

average current on the right side:

(… and other useful unitarity relations) ∑=+ nm nmRLRLRL sSSTr , 2,)( :Note

Several expressions of the Landauer formula:

finite voltage V and temperature T

conductance ( V ~ 0) at kT=0

Introduction

I. electronic scattering1- one ideal mode (+ electron flux/photon flux analogy)2- one mode scattering3- multi-mode scattering: Landauer formula4- multi-terminal Büttiker formula (quantum Kirchhoff law)

II. Electronic analogs of optical systems

III. Quantum Shot noise

IV. Entanglement by the Fermi statistics

V. Shot noise: the tool to detect entanglement

VI. Combining electrons and photons

I . 4 multi-terminal conductors: Landauer-Büttiker formula.the quantum Kirchhof law:

2eV

1eV

3eV

1I2I

3I

αβ VI the and the between relation linear the find

Electrochemical potential of the contacts: ααµ eVEF +=

ααββ VGI =

1w1x0

2x 0

3x0

reflection

transmission

add the quantum probablities with the right sign

and the thermodynamical wheight

Recipe:

current calculated in lead α:

2eV

1eV

3eV

1I2I

3I

incoming from α(occupation fα )

emitted by andreflected into α(occupation fα )

emitted by β andtransmitted into α

(occupation fβ )

quantum Kirchhof law:

current in lead α:

2eV

1eV

3eV

1I2I

3I

∑=β

βαβα VGI(expression useful later for current cross-correlations )

( )

−−= ∑∫β

βαβααααα εεε )()( fTRMfdheI

1== ++ SSSS using

(number of occupied channels at energy ε in lead α )

Introduction

I. Electronic scattering

II. Electronic analogs of optical systems1- ballistic chiral transport : Edge states in QHE. 2- the QPC electronic beam splitter3- electronic Mach-Zehnder interferometer 4- electronic Fabry-Pérot interferometer

III. Quantum Shot noise

IV. Entanglement by the Fermi statistics

V. Shot noise: the tool to detect entanglement

VI. Combining electrons and photons

II . 1 ballistic chiral transport : edge states

WHY ? - Fermi statistics properties are best seen with ballistic electrons- ballistic electrons allow a direct comparison with most quantum optics experiments- edge states in high field provide ‘super’-ballistic transport.

example of the beam splitter:source (b)

source (a) D

DR −−−−====1

tN

rN GV−

possible beam splitterusing a ballistic conductor

(2DEG in zero field)

(technical ) problems:

- several incoming electronic modes ~ W /( λF / 2 )

- smooth disorder : no electron mirror with λF / 4 ( or better) “polishing”

D

DR −= 1

top gate

contact (a)

contact (b)

contact (a)

contact (b)

source (b)

source (a) D

DR −−−−====1

tN

rN

ballistic 2DEG in high field (QHE regime)

advantage:

- 1D (chiral) modes : QHE edge states

- no backscattering (robust to disorder)

D

DR −= 1

GV−

top gate

gate control of the transparency D

B

the mesoscopic quantum Hall effect:

FEε

Cωh21 Cωh23 Cωh25(Landau levels)0ε

1ε2ε

y

electrons form highly degenerated Landau levels with energies en , n = 0, 1, 2, ...

frequency cyclotron the : meBωn CCn =

+= ωε h21

( ) ( ) kpByAAepmH x hrrr =−=−= 0,0,2

1 2

2 2)(21, )(1 c kl yyknikxXkn eyyHeL−−

−=Φ

LheBlLy Ck ==∆ 22π

( ) ; 2/12222

.21

2

=−+= eBllkymmpH CCCy hω

(spin disregarded)

the (macroscopic) Integer Quantum Hall Effect

1980

K. von KlitzingG. DordaM. PepperIntegere

hR Hall 12 ⋅=

ehn =Φ=Φ 0 quantumflux ofdensity

Φ= nehB sHall n

nehR Φ⋅= 2

Φ×= nIntegern s )(

zBvE Hall ˆ×−= r

r sHall neBR =

Edwin Hall 1879

VHallI

I

V xx

I

Ohms )6(805.258122 =eh

Exactness : gauge invariance ( B. Laughlin 1981)

E

degenerateΦnCωh

Landau levels0Φ

the mesoscopic quantum Hall effect:

y

)( yU FEconfining potential

ε

Cωh21 Cωh23 Cωh25(Landau levels)

0ε

1ε

2ε

y

( ) ( )0,0,)(21 2 ByAyUAepmH −=+−=

rr

r

( )2/1 2222)(.2

12

=

+−+=

eBl

yUlkymmpH

C CCyh

ω

)2CkCkn lkyUn .(21, =+

+≈ ωε h

Landau level degeneracy is lifted by the confining potential for each εεεεn , n = 0, 1, 2, ...

( no current in the bulk )

( edge current )

y )( yU ε

FE

Cωh21 Cωh23 Cωh250ε 1ε

(Landau levels)

confining potential

x

2 edge modes 2ε

electrons drift along along the edgewith velocity:

)2CkCkn lkyUn .(21, =+

+≈ ωε h

bend Landau levels give p chiral 1D modes ( edge channels) if p Landau level are filled.

yU

eBkU

kkv nnx ∂∂=

∂∂≈

∂∂= 111)()(

hh

εG(Landauer) bulk the in

filled level Landauif phepG2

=

I I21−V I

ehV

G221

11

−=− νν

obtained using :

∑=β

βαβα VGI

II . 2 the QPC beam splitter

-1,4 -1,2 -1,0 -0,8 -0,6 -0,4 -0,2 0,0

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

tran

smis

sion

/ re

flect

ion

gate voltage Vg ( Volt )

transmitted current reflected current total current

contact (a)

contact (b)D

DR −= 1

GV−

top gateB

V

transmittedcurrent

reflected current

D

DR −= 1

( © from P. Roche/ F. Portier, nanoelectronics group Saclay)

.transI

VheII

VheRI

VheDI

refltransrefltrans2.. 2. 2.

.2

)1(

=+

=

+=

possible expressions for the scattering matrix of a QPC (or beam splitter in general):

1cossinsincos

==

−

= ++−−

−

SSSSeeeeS ii ii

; θθθθ

ϕψ

ψϕ

time reversal symmetry implies the relation:

)()( BSBS t=−

for zero magnetic field the scattering matrix is symmetric

: ; example for

±=⇒irttir

2πψ

In high magnetic field (edge states), there is no such a condition

: example for

−θθθθ

cossinsincos

expression often used later (when initial phases are irrelevant)

(can also be represented as a point on the Bloch sphere)

II . 3 electronic Mach - Zehnder interferometer (from Yang Ji et al, M. Heiblum’s group, Weizmann)

0221211

2

~

ΦΦπφ

φ

====

++++ iD errttI 2222221211

1

1

rtT

rtT

−−−−========

−−−−========edge states of Quantum Hall effect= chiral 1D ideal conductors.

Area of the loop = 45 µm2

electron temperature : 20mK maximum visibility for t1=t2=1/2

II . 4 the electronic Fabry - Pérot resonator

)(tI tphotonsource

Fabry - Perotresonator

zB ˆ.

Lt Rt

100 % fringe visibility iscommonly observed in

the conductancein high field B

Quantum dot (artificial atom)

mµ1

(edge states regime)

Lµ Rµ

K 1 to 0.1 spacing levelenergy °°°°≈≈≈≈∆

Breit-Wigner resonance

zB ˆ.

Lt Rt)2(

)1(V

V

difference between the Fabry-Perot (FP) and the Mach-Zehnder (MZ)

FP : - multiple interferences due to multiple reflections- transmission visibility depends on energy and temperature (thermal broadening)

MZ :- single interference, no reflection- if electronic lengths (1) and (2) are equal, no thermal or voltage broadening to affect fringe visibility