Quantum Shot noise

probing interactions and magic

properties of the Fermi sea

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

D.C. Glattli, NTT-BRL School, 03 november 05 INTRODUCTION

(disclaimer : these are just notes and some references to original works may be missing) D.C. Glattli, NTT-BRL School, 03 november 05 WHY STUDY SHOT NOISE ?

(D)

(r) (i) (t) ( ) eV 1 D reservoir = detector (i) (t) detector (r) (1− D )

(i) (t) ˆ aˆ L bR S bˆ L aˆ R electrons : (r) photons: electrons interference (wave) light interference scattering description

from de 70’s to 90’s (and beyond) mesoscopic physics addressed single particle coherence properties via conductance measurements

D.C. Glattli, NTT-BRL School, 03 november 05 (D)

(r) (i) (t) ( ) eV 1 D reservoir = detector (i) (t) detector (r) (1− D )

electrons : photons: electrons interference (wave) light interference shot noise (particle) photon noise

while in the 60’s, optics addressed two photons properties (Hanbury-Brown Twiss correlations) this is only beginning of the 90’s (mid 90’s for experiments) that two-electron correlations where considered via quantum shot noise D.C. Glattli, NTT-BRL School, 03 november 05 (D)

(r) (i) (t) ( ) eV 1 D reservoir = detector (i) (t) detector (r) (1− D )

electrons : photons: electrons interference (wave) light interference shot noise (particle) photon noise

different quantum noise results for different quantum statistics (Bose versus Fermi) Fermi sea gives noiseless electron generation while photons are fundamentally noisy

electronic quantum shot noise studies revealed yet unregarded beautiful properties of the Fermi sea D.C. Glattli, NTT-BRL School, 03 november 05 the magic Fermi sea

e & 1 = ε or = ε & Nocc. dIel. d dNel. d dN = dε or dI =(hν )N dν h h ph. h ph. occ. e2 ⇒ quantum of conductance : (no equivalent, no known continuous h generation of photon number states)

τ = h I eV eV I = e. noiseless injection of electrons h 1 111 1 1 1

noiseless electrons electron entanglement noiseless electrons may using linear ‘ electron optics ’ generate low noise photons ε and equilibrium thermodynamic sources

E F ground state : 0 Electron-hole pairs are + naturally entangled. plus excitations : cε > cε < 0 EF EF Fermi sea rate of

D.C. Glattli, NTT-BRL School, 03 november 05 entanglement production eV/4h more with shot noise :

current spectral density :

= ∆ 2 ∆ SI ( f ) I / f

proportional to the charge of the quasi-particle carrying current (...but only in the Poissonian regime) = SI 2 q I

q = e already in the 20’s attempt to determine the electron charge in vacuum diodes (but less accurate that Millikan’s experiments, due to space charge effect)

(repulsive interactions reduce shot noise)

q = e / 3 in 1997, the Laughlin fractional charge of the Fractional Quantum Hall Effect was unambiguously established via shot noise. The last (but not least) proof definitely establishing the FQH effect .

later :

q = 2e the Cooper pair charge observed at mesoscopic superconducting-normal interfaces.

Future :

q = g e in Luttinger liquids, such as long single wall carbon nanotubes (requires f > THz) D.C. Glattli, NTT-BRL School, 03 november 05 Introduction

I. Electronic scattering ( a brief introduction) 1- one ideal mode (+ electron flux/photon flux analogy) 2- one mode scattering 3- multi-mode scattering: Landauer formula 4- multi-terminal Büttiker formula (quantum Kirchhoff law) 5- chiral transport with QHE edge states

II. Quantum Shot noise 1 - Quantum partition noise - one and two particle partitioning :electrons/ photons - electronic shot noise 2- scattering derivation of quantum shot noise a- S( ω) for an ideal one mode conductor b- quantum shot noise for a single mode c-zero frequency shot noise and multimode case 3- experimental examples 4- current noise cross-correlations

III Shot Noise and Interactions: 1. Fractional Quantum Hall effect 2.. Superconducting-Normal mesoscopic interfaces 3. Interactions in a QPC : 0.7 structure

IV. Shot noise: the tool to detect entanglement

V. Shot noise and high frequencies D.C. Glattli, NTT-BRL School, 03 november 05 Introduction

I. Electronic scattering ( a brief introduction)

II. Quantum Shot noise

III Shot Noise and Interactions:

IV. Shot noise: the tool to detect entanglement 1. Entanglement with the Fermi statistics 2. Coincidence measurements using shot noise correlations V. Shot noise and high frequencies 1. Photo-assisted Shot Noise 2. High frequency Shot Noise 3. Photon Noise emitted by a Conductor

Conclusion

D.C. Glattli, NTT-BRL School, 03 november 05 - quantum point contact : shot noise, edge states, co-tunneling of Q.-Dots - ballistic qubits - mesoscopic capacitor - carbone nanotube B - Fractional Quantum Hall effect

- high frequency (40 GHz) - ultra low noise measurements - high magnetic field (18T) and low T (20mK) - lithography - cryo-electronics CEA-Saclay ENS Paris Patrice Roche Bernard Plaçais J. Segala Jean-Marc Berroir F. Portier Adrian Bachtold Preden Roulleau Julien Gabelli (L-H. Bize-Reydellet) Gwendal Fève (V. Rodriguez) Gao Bo (L. Saminadayar) Betrand Bourlon (A. Kumar) Adrien Mahe

D.C. Glattli, NTT-BRL School, 03 november 05 Julien Chaste Introduction

I. electronic scattering (a brief introduction) 1- one ideal mode (+ electron flux/photon flux analogy) 2- one mode scattering 3- multi-mode scattering: Landauer formula 4- multi-terminal Büttiker formula (quantum Kirchhoff law) 5- chiral transport with QHE edge states.

II. Quantum Shot noise

IV. Shot noise: the tool to detect entanglement

V. Shot noise and high frequencies

D.C. Glattli, NTT-BRL School, 03 november 05 ε ε

eV X EF X EF electron electron reservoir X reservoir 1 1 ε ε fL ( ) fR ( )

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

 aˆ   ˆ   L 1,   b R 1,   aˆ   ˆ  L 2, b R 2,  ...       ...      aˆ L,M  bˆ  ((( )))  R,N   ˆ  S  aˆ   b L 1,   R 1,   ˆ   aˆ  b L 2, R 2,    ...   ...     ˆ  aˆ  b L,M   R,N 

D.C. Glattli, NTT-BRL School, 03 november 05 I . 1 the ideal 1D wire (… or one mode conductor)

−−− i k L x i k R x reservoir e e reservoir µ µ R L T = 0 T = 0

ε ( k ) ( length L → ∞ ) ε ( k )

µ L µ eV R

kR kL

2 2 π h ± 2 kn 1 ikL( R ) x k = n and ε = , ϕε (x) = e n,L(R) L n 2m n ,L( R) L

D.C. Glattli, NTT-BRL School, 03 november 05 −−− i k L x i k R x reservoir e e reservoir µ µ R L T = 0 T = 0

ε ( k ) ( length L → ∞ ) ε ( k )

µ L µ eV R ε

kR kL

2 2 h k 1 ±ik (ε ) x k (ε ) and ε = L(R) , ϕ (x) = e L( R ) L(R) ε ,L( R) πh ε 2m 2 vL(R) ( )

ϕ ϕ =δ δ ε −ε density of state ε ',β ε ,α α ,β ( ' ) per unit of energy 1 per unit length D(ε ) = 2πhv(ε ) energy is a natural parameter for elastic scattering. (… and all results are simpler) D.C. Glattli, NTT-BRL School, 03 november 05 −−− i k L x i k R x reservoir e e reservoir µ µ R L T = 0 T = 0

ε ( k ) ( length L → ∞ ) ε ( k )

µ L µ eV R ε

kR kL

2 2 h k 1 ±ik (ε ) x k (ε ) and ε = L(R) , ϕ (x) = e L( R ) L(R) ε ,L( R) πh ε 2m 2 vL(R) ( ) h ε kL,R ( ) dI = ± e ϕε ϕε dε L,R ,L(R) m ,L(R) e = ± ε ε dI L(R) d … is the current carried by one mode within energy range d h (independent on specific energy band dispersion relation) D.C. Glattli, NTT-BRL School, 03 november 05 second quantification representation

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

−−− i k L x i k R x reservoir e e reservoir µ µ R L T T

ε ( k ) ε ( k )

µ L µ eV R

kR kL

e 1 i (k x −εt) ψ = ε ε L e 1 i (−k x−εt) ˆ L (x,t) ∫ d aˆL ( )e ψˆ (x,t) = dε aˆ (ε )e R h 2πhv (ε ) R ∫ πh ε R L h 2 vR ( )

∂(ψˆ +ψˆ ) ∂ e + Iˆ =0 ∂t ∂x { ε + ε }= δ δ ε −ε aˆβ ( ),' aˆα ( ) α ,β ( ' )

{}aˆβ (ε ),' aˆα (ε ) = 0 aˆα (β ) , act on the Fock space of the reservoirs + + + ε ε = ε δ δ ε −ε L = ∏ε = µ aˆ L (ε ) 0 and R = ∏ε = µ aˆ R (ε ) 0 aˆα ( ).aˆβ ( )' fα ( ) α ,β ( ' ) ,0 L L ,0 R R D.C. Glattli, NTT-BRL School, 03 november 05 −−− i k L x i k R x reservoir e e reservoir µ µ R L T T

ε ( k ) ε ( k )

µ L µ eV R

kR kL

e 1 i (k x −εt) ψ = ε ε L e 1 i (−k x−εt) ˆ L (x,t) ∫ d aˆL ( )e ψˆ (x,t) = dε aˆ (ε )e R h 2πhv (ε ) R ∫ πh ε R L h 2 vR ( )

e Iˆ (x t), = I = dε f (ε) L L h ∫ L D.C. Glattli, NTT-BRL School, 03 november 05 the quantum of conductance

−−− i k L x i k R x reservoir e e reservoir µ µ R L T T

ε ( k ) ε ( k )

µ L µ eV R

kR kL

∞ = ()ε − ε e ε I ∫ fL ( ) fR ( ) d 2 0 h e 2 the conductance : = e µ = µ + ∀ h I V L R eV V , T h is independent on temperature and voltage

eV I = e Pauli ( 1 × e ) + Heisenberg ( eV / h ) h

D.C. Glattli, NTT-BRL School, 03 november 05 I . 2 ONE MODE SCATTERING

conductor reservoir reservoir µ L µ R T T

0 0 xL xR  s s  r t'  ˆ = LL LR  ≡   aˆ L b S     R s s t r' S  RL RR    bˆ L aˆ R + + scattering states in the reservoirs: S S = S S =1

Idea: write the total fermion operator as: S contains all the transport information on the conductor.

e 1 − ψˆ(x ) = dε (aˆ (ε)e ki L xL +bˆ (ε)e ki L xL ) L ∫ πh ε L L h 2 vL( ) links the outgoing amplitudes to the incoming wave amplitudes e 1 − ψ = ε ( ε ki R xR + ˆ ε ki R xR ) ˆ(xR ) ∫d aˆR ( )e bR ( )e h 2πhv (ε) R amplitude are replaced by the annihilation operator acting on the D.C. Glattli, NTT-BRL School, 03 november 05 Fock space of each reservoir L,R

a simple example:

ε ( k ) ε ( k ) ε 1 t(ε) µ e i kL x e i kRL x L πh πh 2 v L 2 v R µ eV R r(ε) U U

0 k 0 0 k reservoir reservoir µ µ L , T R , T

2 2 2 2 h h k ε = + kR ε (k ) = L (kR ) U L 2m 2m

1. scattering states: electrons emitted by the left reservoir

aˆ (ε ) − − ε ψ = ε L ( ikL xL + ε ikL xL ) ti < ˆε ,L (x,t) d e r( )e e xL 0 ∫ πh ε (describes electrons 2 vL ( ) emitted by the left reservoir) aˆ (ε ) − − ε = dε L t(ε )e ikR xR e ti x < 0 ∫ πh ε R 2 vR ( )

D.C. Glattli, NTT-BRL School, 03 november 05 (current probability amplitude) aˆ (ε ) − − ε ψˆ (x,t)= dε L (e ikL xL + r(ε )e ikL xL )e ti x < 0 ε ,L ∫ πh ε L 2 vL ( )

aˆ (ε ) − − ε = dε L t(ε )e ikR xR e ti x < 0 ∫ πh ε R 2 vR ( )

similarly, for electrons emitted from the right reservoir

aˆ (ε ) − − ε ψˆ (x ,t)= dε R (e ikR xR + r (' ε )e ikR xR )e ti x < 0 ε ,R ∫ πh ε R 2 vR ( )

aˆ (ε ) − − ε = dε R t (' ε )e ikL xL e ti x < 0 ∫ πh ε L 2 vL ( )

total state in the left lead

aˆ (ε ) − − − ε ψˆ (x ,t)= dε L (aˆ (ε )e ikL xL + aˆ (ε ) r(ε )e ikL xL + aˆ (ε t (') ε )e ikL xL )e ti ε L ∫ πh ε L L R 2 vL ( )

1 − − ε = dε ()aˆ (ε ) e ikL xL + bˆ (ε )e ikL xL e ti ∫ πh ε L L 2 vL ( )

… and similarly in the right lead bˆ  r t' aˆ   L  =   L   ˆ    ˆ  bR   t r'aR 

D.C. Glattli, NTT-BRL School, 03 november 05 to summarize:

reservoir reservoir µ L µ R T T  s s  S = LL LR   s s  0 0  RL RR  xL xR

aˆ bˆ + + L R S S = S S =1 S bˆ L aˆ R

e 2 2 I = dε {f (ε ) − (s f (ε ) + s f (ε ))} Landauer formula h ∫ L LL L LR R

2 ∂ = e ε 2 ()ε − ε = e ε − f  ε I ∫ d s LR f L ( ) f R ( ) G d   D( ) h h ∫  ∂ε 

e2 D = S 2 = S 2 =1− S 2 =1− S 2 =1− R G = D(ε ) @ T = 0 LR RL LL RR h F

D.C. Glattli, NTT-BRL School, 03 november 05 Introduction

I. electronic scattering (a brief introduction) 1- one ideal mode (+ electron flux/photon flux analogy) 2- one mode scattering 3- multi-mode scattering: Landauer formula 4- multi-terminal Büttiker formula (quantum Kirchhoff law) 5- chiral transport with QHE edge states.

II. Quantum Shot noise

IV. Shot noise: the tool to detect entanglement

V. Shot noise and high frequencies

D.C. Glattli, NTT-BRL School, 03 november 05 ) ε ( 1 R f R ε µ T V e , R

(2D here for simpler notations) µ reservoir

: Landauer fromula Landauer : Incoming wave : emitted by (L) mode (m) mode (L) by emitted : wave Incoming = X ) L y , L x ; ε ( X m , L ϕ X L x Multiple mode scattering mode Multiple 0 L y I . 3 3 . I T , L

µ reservoir ) ε ( 1 L f L ε µ L w ideal multi-mode lead multi-mode ideal modeling of the reservoirs for a two-terminala conductor: for reservoirs of the modeling

ε ε µ L eV µ X R X reservoir reservoir µ L , T X µ R , T 1 1 ε ε fL ( ) fR ( ) scattering matrix for many modes:

 aˆ   ˆ   L 1,   b R 1,   aˆ L 2,   ˆ  b R 2,  ...       ...   ˆ    a L,M  bˆ  (S ) (S )  ((( )))  R,N  = LL LR   ˆ  S  aˆ  S  b L 1,   R 1,    (SRL) (SRR)  ˆ   aˆ R 2,    b L 2,    ...   ...       ˆ  aˆ R,N b L,M   

D.C. Glattli, NTT-BRL School, 03 november 05 (m)

(n’) (m’) ( ) ( ) sLR,mn' sLL,mm'

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

e   2 2  = ε ε −  ε + ε  I ∫ d ∑ f L ( ) f L ( ) ∑sLL,mm' f R ( ) ∑ sLR,mn'  h m  m' n 

 aˆ   ˆ   L 1,   b R 1,   aˆ L 2,   ˆ    b R 2, (sLL ) (sLR )  ...    =      ...  S    ˆ    a L,M  bˆ  (sRL ) (sRR ) ((( )))  R,N   ˆ  S  aˆ   b L 1,   R 1,   ˆ   aˆ  b L 2, R 2,    ...  + +  ...    S S = S S =1  ˆ  aˆ  b L,M   R,N 

D.C. Glattli, NTT-BRL School, 03 november 05 I . 4 multi-terminal conductors: Landauer-Büttiker formula. the quantum Kirchhof law:

µ = + Electrochemical potential of the contacts : α EF eVα

I 2

eV2

I1

eV1 I 3

eV3

= find the linear relation between the I β and the Vα I β GαβV α

D.C. Glattli, NTT-BRL School, 03 november 05 0

x2

w1 x3 0

0 x1

Recipe: 2 add the quantum probablitiessαβ ,mn with the right sign and the thermodynamical weight f β (ε )

incoming from α emitted by β and emitted by and (occupation f α ) transmitted into α reflected into α D.C. Glattli, NTT-BRL School, 03 november 05 (occupation fβ ) (occupation f α ) (number of occupied channels at energy ε in lead α )

+ + Tr(sαα sαa ) Tr(sαβ sαβ ) I   2 = e ε ε ()− − ε Iα ∫ d  fα ( ) M α Rαα ∑ Tαβ f β ( )  eV h  β  2

+ = + = using S S SS 1 I1 quantum Kirchhof law:

eV = 1 I I α ∑ GαβV β 3 β

2 e eV3 Gαα = ()M α − Rαα h e2 Gαβ = Tαβ h

D.C. Glattli, NTT-BRL School, 03 november 05 Introduction

I. electronic scattering (a brief introduction) 1- one ideal mode (+ electron flux/photon flux analogy) 2- one mode scattering 3- multi-mode scattering: Landauer formula 4- multi-terminal Büttiker formula (quantum Kirchhoff law) 5- chiral transport with QHE edge states.

II. Quantum Shot noise

IV. Shot noise: the tool to detect entanglement

V. Shot noise and high frequencies

D.C. Glattli, NTT-BRL School, 03 november 05 I . 5 chiral transport : edge states WHY ? - Fermi statistics properties are best seen with ballistic electrons - ballistic electrons allow a direct comparison ballistic 2DEG in high field (QHE regime) with most quantum optics experiments - edge states in high field provide ‘super’-ballistic transport.

example of the beam splitter: B source (b)

R = 1− D N contact (a) source (a) D t

D contact (b) R === 1−−− D top gate

N r − VG advantage:

- 1D (chiral) modes : QHE edge states gate control of the - no backscattering (robust to disorder) transparency D D.C. Glattli, NTT-BRL School, 03 november 05 (macroscopic) Quantum Hall Effect

V xx

V Hall I I I Edwin Hall 1879

r r = − × = B E Hall v B zˆ R Hall e ns

= h ⋅ 1 K. von Klitzing R Hall 2 G. Dorda e Integer M. Pepper 1980

h h nΦ B = nΦ = ⋅ E R Hall 2 e e ns Landau levels h nΦ = density of flux quantum Φ = 0 e Φ 0 = × n s (Integer) nΦ

h hω = 25812.805 )6( Ohms C e 2

Exactness : gauge invariance ( B. Laughlin 1981) n Φ degenerate r r 1 r 2 H = (p − eA) A= ()− yB 0,0, p = h k 2m x

2 2/1 p 1 2  h  H = y + mω 2 ()y −k.l 2 ; l =  2m 2 C C C  eB 

− 2 −1 ( y yk ) 1 2 2 electrons form highly degenerated Φ = ikx − lc n,k e H n ( y yk )e Landau levels with energies en , n = 0, 1, 2, ... L X

ε =  + 1  hω = eB 2π eB n n  C ωC : the cyclotron frequency ∆y = l 2 =  2  m k C ε L h L (spin disregarded)

5 hω C ε 2 2 3 hω E F 2 C ε 1 1 hω (Landau levels) C ε 2 0 y

D.C. Glattli, NTT-BRL School, 03 november 05 the mesoscopic quantum Hall effect:

r r 1 r 2 H = (p − eA) +U( y) A= ()− yB 0,0, 2m

2 p 1 2 H = y + mω 2 ()y −k.l 2 +U( y) 2m 2 C C h 2/1 =  lC    eB  Landau level degeneracy is lifted by the εεε confining potential for each n , n = 0, 1, 2, ...

 1  ε ≈ n + hω +U( y =k.l 2 ) ε n,k  2  C k C ε 2 confining potential 5 ε hω 1 2 C U( y) 3 hω E F C ε 2 0 1 hω (Landau levels) 2 C

y y

D.C. Glattli, NTT-BRL School, 03 november 05  1  ε ≈ n + hω +U( y =k.l 2 ) n,k  2  C k C

bend Landau levels give p chiral 1D modes ( edge channels) if p Landau level are filled. electrons drift along along the edge e2 with velocity: G = p if p Landau level filled h ∂ε ∂ ∂ (n) 1 n 1 U 1 U in the bulk (Landauer) Gv (k) = ≈ = x h ∂k h ∂k eB ∂ y

y confining 1 hω 3 hω 5 hω ε potential U( y) C C C 2 edge modes 2 2 2

ε ε ε 0 1 2

( no current in the bulk ) (Landau levels)

( edge current )

x D.C. Glattli, NTT-BRL School, 03 november 05 E F  1 1  h V =  −  I V − 1−2 ν ν  2 1 2  G  e

obtained using : I = I Iα ∑GαβV β β

D.C. Glattli, NTT-BRL School, 03 november 05 the QPC beam splitter

reflected current

2,0 B 1,8 1,6 D R = 1− D 1,4 1,2 contact (a) 1,0

0,8 R = 1− D I transV. 0,6 contact (b) 0,4 D top gate transmitted current 0,2 reflected current transmission/ reflection 0,0 total current -1,4 -1,2 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 transmitted gate voltage Vg ( Volt ) current − VG ( © from P. Roche/ F. Portier, nanoelectronics group Saclay) e2 I = 1( + D) V trans. h e2 I = R V refl. h 2 + = e I trans. I refl. .2 V D.C. Glattli, NTT-BRL School, 03 november 05 h electronic Mach - Zehnder interferometer (from Yang Ji et al, M. Heiblum’s group, Weizmann)

2 +++ i φ 2 2 I D1 ~ t1 t 2 r1 r2 e T === t ===1−−− r edge states of Quantum Hall effect 1 1 1 = chiral 1D ideal conductors. Φ === 2 === −−− 2 φ === 2π T2 t 2 1 r2 2 Φ Area of the loop = 45 µm 0 electron temperature : 20mK maximum visibility for t1=t 2=1/2 D.C. Glattli, NTT-BRL School, 03 november 05