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Quantum Shot noise
probing interactions and magic
properties of the Fermi sea
D. Christian Glattli,
Nanoelectronics Group
Service de Physique de l’Etat Condensé, DSM,
CEA Saclay, F-91191 Gif-sur-Yvette France
Nanoelectronics Group
D.C. Glattli, NTT-BRL School, 03 november 05
WHY STUDY SHOT NOISE ?
(D)
eV
(r)
(i)
(t)
(D)
1
reservoir =
detector
(i)
(t)
detector
(r)
(i)
(t)
â L
b̂L
electrons :
electrons interference
S
(r)
(1 − D )
b̂R
â R
(wave)
scattering description
photons:
light interference
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)
eV
(r)
(i)
(t)
(D)
1
reservoir =
detector
(i)
(t)
detector
(r)
photons:
electrons :
electrons interference
shot noise
(1 − D )
(wave)
(particle)
light interference
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)
eV
(r)
(i)
(t)
(D)
1
reservoir =
detector
(i)
(t)
detector
(r)
photons:
electrons :
electrons interference
shot noise
(1 − D )
(wave)
(particle)
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
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
dI el . = dε or dN& el . = dε
h
h
⇒ quantum of conductance :
dN& ph . =
e2
h
(no equivalent, no known continuous
generation of photon number states)
I
eV
I = e.
h
noiseless electrons
D.C. Glattli, NTT-BRL School, 03 november 05
N occ .
dε or dI ph . = ( hν ) N occ . dν
h
τ=
h
eV
noiseless injection of electrons
1 1 1 1 1 1 1
electron entanglement
using linear ‘ electron optics ’
and equilibrium thermodynamic sources
noiseless electrons may
generate low noise photons
more with shot noise :
current spectral density :
S I ( f ) = ∆I 2 / ∆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=ge
in Luttinger liquids, such as long single wall carbon nanotubes (requires f > THz)
D.C. Glattli, NTT-BRL School, 03 november 05
I. Electronic scattering ( a brief introduction)
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: a 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
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: a 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
B
- 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
CEA-Saclay
Patrice Roche
J. Segala
F. Portier
Preden Roulleau
(L-H. Bize-Reydellet)
(V. Rodriguez)
(L. Saminadayar)
(A. Kumar)
D.C. Glattli, NTT-BRL School, 03 november 05
ENS Paris
Bernard Plaçais
Jean-Marc Berroir
(Adrian Bachtold)
Takis Kontos
Julien Gabelli
Gwendal Fève
Gao Bo
Betrand Bourlon
Adrien Mahe
Julien Chaste
II. 1 Quantum Partition Noise
single -particle partitioning :
assume only one particle in a wave incident on a scatter
⇒ the wave is diffracted
- scattering
- diffraction+reservoirs ⇒ the particle is randomly partitioned
⇒ quantum partition noise is diffraction (there is no classical analog) + particle-wave duality
( quantum noise exemplifies particle-wave duality )
n i = 1, 1, 1, 1, 1, 1, 1, 1, ...
n t = 1, 1,0, 1,0, 1,0,0, ...
0
n r = 0,0, 1,0, 1,0, 1, 1, ...
D 1 +i 1− D 0
nt =D
1
(i)
n r =1 − D
(t)
( ∆ n t ) 2 = n t (1 − n t ) = D (1 − D ) = ( ∆ n r ) 2
(r)
D 0 +i 1− D 1
D.C. Glattli, NTT-BRL School, 03 november 05
binomial = Poisson X reduction factor (1 - D )
∆ n t . ∆ n r = − D (1 − D )
assume only one particle in a wave incident on a scatter
⇒ the wave is diffracted
- scattering
- diffraction+reservoirs ⇒ the particle is randomly partitioned
⇒ quantum partition noise is diffraction (there is no classical analog)
( quantum noise exemplifies particle-wave duality )
0
this applies also to electronic waves:
(D)
D 1 +i 1− D 0
1
(i)
(t)
eV
(r)
(i)
(t)
(r)
D 0 +i 1− D 1
D.C. Glattli, NTT-BRL School, 03 november 05
⇒ responsible for current fluctuations
or quantum shot noise
two-particle partitioning : difference between Bosons and Fermions
1
â 2
 bˆ1 
 aˆ 
  = S  1 
 bˆ 
 aˆ2 
 2
b̂1
â 1


S =



1
2
1
2
−
1 

2
1 

2 
1
 bˆ1+ 
 aˆ1+ 
 +  = S  +  , as S + S = 1
 aˆ 
 bˆ 
 2 
 2 
⇒
b̂ 2
initial state :
+
+
i = aˆ1 aˆ2 0
final state:
f =
in
(
)
(
)
1 ˆ+ ˆ+ ˆ + ˆ +
1 + +
+ +
b1 b1 − b2 b2 + bˆ1 bˆ2 − bˆ2 bˆ1 0
2
2
= 0 (Fermion)
Fermions:
= 0 (Boson)
Bosons:
no bunching, no noise
D.C. Glattli, NTT-BRL School, 03 november 05
out
or
bunching, binomial two-particle partition noise
electron sources versus photon source : reservoir
electrons
dI el . =
photons
e
f F . D . (ε )dε
h
N τ = f F . D . (ε )
δε
h
dI ph . = hν f B . E . ( hν )dν
N τ = f B .E . (ε )δν τ
τ
(∆N )2 τ = (N − N τ ) τ
2
(∆N )2 τ = f F . D. (1 −
f F .D. )
δε
h
τ = N τ (1 − f F . D. )
(∆N )2 τ = f B. E . (1 +
sub - poissonian
(anti-bunching)
f B. E . ) δν τ = N τ (1 +
Nτ
δν τ
super - poissonian
(thermal photon bunching)
1
δν
h
δε
t
in particular : noiseless Fermi sea at T=0
t
)
simple derivation of the electronic quantum shot noise for a single mode conductor
ε
incoming current :
(1 − D )
ε
(D)
eV
I 0 = e ( eV / h )
(noiseless thanks
to Fermi statistics)
transmitted current :
e2
I = D I0 = D V
h
f L (ε )
I0
f R (ε )
∆t =
h
eV
I (t )
current noise in B.W. ∆f :
(∆I )
2
= 2eI 0 ∆f ⋅ D (1 − D )
(t)
(r)
Variance of partioning
binomial statistics
S I = 2eI (1 − D ) = 2eI . F
Poisson (Schottky)
(i)
reduction factor
(Fano)
anti-correlation of transmitted
and reflected current fluctuations
V. A. Khlus, Zh. Eksp. Teor. Fiz. 93 (1987) 2179 [Sov.
Phys. JETP 66 (1987) 1243].
G. B. Lesovik, Pis'ma Zh. Eksp. Teor. Fiz. 49 (1989)
513 [JETP Lett. 49 (1989) 592].
electronic shot noise for a multi-mode conductor
S I 1 = 2 e I 1 (1 − D1 )
e2
G1 = D1
h
S I 2 = 2 e I 2 (1 − D2 )
S I = 2eI
∑D
n
(1 − D n )
= 2eI . F
n
∑D
n
n
e2
G 2 = D2
h
S I λ = 2 e I λ (1 − Dλ )
e2
G λ = Dλ
h
Fano
factor
Poisson
1
noiseless
V. A. Khlus, Zh. Eksp. Teor. Fiz. 93 (1987) 2179 [Sov.
Phys. JETP 66 (1987) 1243].
G. B. Lesovik, Pis'ma Zh. Eksp. Teor. Fiz. 49 (1989)
513 [JETP Lett. 49 (1989) 592].
0
1
2
∑
n
D.C. Glattli, NTT-BRL School, 03 november 05
D
n
Introduction
I. Electronic scattering ( a brief introduction)
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
- scattering derivations
- electronic analog of the optical Hanbury-Brown Twiss experiment
- electronic quantum exchange
III Shot Noise and Interactions:
IV. Shot noise: the tool to detect entanglement
V. Shot noise and high frequencies
D.C. Glattli, NTT-BRL School, 03 november 05
II. 2 Scattering derivation of quantum shot noise
I (t )
some classical definitions to start:
t1 + T
t2 + T
t3 + T
t1
I (t ) =
+∞
∑ xn e
−i 2π n
+∞
−i 2π n
t
T
T →∞ ;
;
n = −∞
I (t ) =
∑ xn e
t
T
I = x0
and
n = −∞
I (t ) = ∑
2
∑x x
n m
e
−i 2π
( n +m )t
T
t2
xn = x − n
*
t3
( ... )
(ensemble averaging over thermodynamically equivalent realizations)
xn xm = 0 if m ≠ − n
(stationary condition)
m
n
∞
= x0 + 2 ∑ xn xn
2
*
n =1
∞
∆I ( t ) = ( I − I ) = 2 ∑ x n x n
2
2
n =1
S I (ν ) = lim 2.T . xn xn
T →∞
*
∞
∆I 2 = ∫ S I (ν ) dν
with
ν≡
0
2
spectral density of the current
n
T
and
dν ≡
1
T
I (t ) I ( t + τ ) = ∑
∑x x
n m
e
−i 2π
( n +m )t
T
e
−i 2π
nτ
T
xn xm = 0 if m ≠ − n
m
n
∞
nτ
= 2 ∑ xn xn cos 2π
= ∫ dν S I (ν )cos 2πντ
T 0
n =1
∞
*
∞
S I (ν ) = 2 ∫ dτ I (t ) I (t + τ ) ei 2π ντ
−∞
this classical expression will be used to define the quantum noise spectral density
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 …)
e
reservoir
ik L x
e
−ik
R
x
reservoir
µL
µR
T
T
(ideal lead : no scatter, one channel)
ε (k )
ε (k )
µL
µR
eV
kR
ψˆ L ( x, t ) =
e
dε
h∫
kL
1
aˆ L (ε ) ei ( k L x −εt )
2πhvL (ε )
ψˆ R ( x, t ) =
∂ (ψˆ +ψˆ ) ∂ ˆ
+ I =0
e
∂t
∂x
{aˆ
{aˆ
aˆα ( β ) , act on the Fock space of the reservoirs
L = ∏ ε =0,µ L aˆ + L (ε ) 0
e
dε
h∫
L
and
D.C. Glattli, NTT-BRL School, 03 november 05
R = ∏ ε =0,µR aˆ + R (ε ) 0
R
β
β
1
aˆ R (ε ) ei ( − k R x −εt )
2πhv R (ε )
+
}
( ε ' ), aˆα ( ε ) = δ α , β δ ( ε ' − ε )
( ε ' ), aˆα ( ε )}= 0
+
aˆα ( ε ). aˆ β ( ε ' ) = f α ( ε ) δ α , β δ ( ε ' − ε )
e
reservoir
ik L x
e
−ik
R
x
reservoir
µL
µR
T
T
ε (k )
ε (k )
µL
µR
eV
kR
ψˆ L ( x, t ) =
e
dε
h∫
1
aˆ L (ε ) ei ( k L x −εt )
2πhvL (ε )
kL
ψˆ R ( x, t ) =
e
dε
h∫
1
aˆ R (ε ) ei ( − k R x −εt )
2πhv R (ε )
e
IˆL ( x, t ) = I L = ∫ dε f L (ε )
h
D.C. Glattli, NTT-BRL School, 03 november 05
e
reservoir
ik L x
e
−ik
R
x
reservoir
µL
µR
T
T
ε (k )
ε (k )
µL
µR
eV
kR
I =∫
∞
0
( f L (ε ) − f R (ε ) ) e d ε
e2
I = V
h
I =e
kL
eV
h
h
µL = µ R + eV
the conductance :
∀ V, T
Pauli ( 1 × e ) + Heisenberg ( eV / h )
D.C. Glattli, NTT-BRL School, 03 november 05
e2
h
does not depend on temperature and voltage
II. 2.a. quantum noise of an ideal one mode wire
reservoir
e
ik L x
e
−ik
R
x
reservoir
µL
µR
T
T
- no scattering ( no shot noise, here only reservoir noise is considered)
- useful to point out some specific and general properties of quantum noise
- first consider the noise contribution coming from the left reservoir : ψˆ L ( x , t ) =
v (ε ' ) + vL (ε ) +
e
IˆL ( x, t )= ∫ dε dε ' L
aˆ L (ε ' )aˆ L (ε ) ei (k L (ε )−k L (ε ') ) x e −i (ε −ε ') t
h
2 v L (ε ' )v L (ε )
IL =
e
dε f L (ε )
h∫
+
aˆ L (ε ' )aˆ L (ε ) = f L (ε ) δ (ε '−ε )
∞
we want to compute :
S I (ν ) = 2 ∫ dτ Iˆ(0) Iˆ(τ ) ei 2π ντ
−∞
D.C. Glattli, NTT-BRL School, 03 november 05
e
dε
h∫
1
aˆ L (ε ) ei ( k L x −εt )
2πhvL (ε )
e
Iˆ( x L ,0) Iˆ( xL ,τ ) =  
h
...
2
∫ dε ' ' 'dε ' ' dε 'dε
v L (ε ' ' ' ) + v L (ε ' ' ) vL (ε ' ) + v L (ε ) i (kL (ε ''')−kL (ε '') ) x i (kL (ε )−k L (ε ') ) x
...
e
e
2 v L (ε ' ' ' )v L (ε ' ' ) 2 vL (ε ' )v L (ε )
aˆ L+ (ε ' ' ' )aˆ L (ε ' ' ) aˆ L+ (ε ' )aˆ L (ε ) e −i (ε −ε ')τ
normal pairing :
ε ''' = ε"
gives the contribution:
→ the fluctuations :
ε'= ε
and
Iˆ( xL ,0) Iˆ( xL ,τ ) = I L
2
∆Iˆ( xL ,0) . ∆Iˆ( xL ,τ ) = Iˆ( xL ,0) Iˆ( xL ,τ ) − Iˆ( xL ,0) Iˆ( xL ,τ )
only come from the
exchange pairing term :
ε'''= ε
aˆL+ (ε ' ' ' )aˆL (ε ' ' ) aˆL+ (ε ' )aˆL (ε )
D.C. Glattli, NTT-BRL School, 03 november 05
and
exchange
ε'= ε''
≡ f L (ε ) δ (ε ' ' '−ε ) (1 − f L (ε ' ) )δ (ε ' '−ε ' )
e
Iˆ( x L ,0) Iˆ( x L , τ ) =  
h
2
2
(
v L (ε ' ) + v L (ε ) )
∫ dε 'dε
4v L (ε ' )v L (ε )
f L (ε )(1 − f L (ε ' ) ) e i 2 (k L (ε )−k L (ε ') ) x e −i (ε −ε ' )τ / h
∞
spectral density of the current fluctuations:
S I (ν ) = 2 ∫ dτ Iˆ(0) Iˆ(τ ) ei 2π ντ
−∞
e2
S I L (ν ) ≅ 2 ∫ dε f L (ε )(1 − f L (ε − hω ) )
h
( hω << EF and vF / ω >> L)
adding contribution of right the reservoir:
2
e
h
e2
=4
h
S I (ν ) = 4
=4
f L (ε )(1 − f L (ε − hω ) )
∫ dε
hω
e
hω
k BT
−1
2
sI (ν )
4 G k BT
k BT
hν
e
hω N (ω )
h
no (detectable) reservoir noise at zero temperature
D.C. Glattli, NTT-BRL School, 03 november 05
e2
S I (ν ) = 4 ∫ dε f L (ε )(1 − f L (ε − hω ) )
h
e2 hω
=4
h khBωT
e −1
e2
= 4 hν N (ν )
h
hω
only fluctuations corresponding to electronic transitions
down in energy are considered in this definition of SI
∞
S I (ν ) = 2 ∫ dτ Iˆ(0) Iˆ(τ ) ei 2π ντ
−∞
(‘spontaneous’ fluctuations )
∞
S I (ν ) = S I ( −ν ) = 2 ∫ dτ Iˆ(0) Iˆ(τ ) e
'
does not commute
−i 2π ντ
−∞
e2
S I (ν ) = 4 ∫ dε f L (ε )(1 − f L (ε + hω ) )
h
e2
= 4 hν (N (ν ) + 1)
h
'
(‘stimulated’ fluctuations )
∞
= 2 ∫ dτ Iˆ(τ ) Iˆ(0) e +i 2π ντ
−∞
hω
transitions up in energy: these fluctuations are revealed when
connecting to an ‘active’ detector able to excite the Fermi sea
-fluctuation -dissipation (here calculated in the frame of the scattering approach)
S I (− ν ) − S I ( v ) = 2 G hν
(Kubo)
(see poster : Pierre Billangeon
also : Deblock/Kouwenhoven : noise of a
supercond. charge qubit using SIS junctions)
- meaning of SI(ν) :
T
quantum
conductor
R=1/G
T =0
I (t )
ZC = R
hν
(detector)
R det. = R
P = R S I (ν )dν
( noise detectable with detector in ground state ) (also Nyquist 1928)
(see more in last part of the talk, if time permits)
For a detector at finite temperature
P ∝ S I (ν ).( N Det . + 1) − S I ( − ν ). N Det .
D.C. Glattli, NTT-BRL School, 03 november 05
Lesovik and Loosen, JETP. Lett. 65, 295 (1997)
Y. Gavish, Y. Imry and Y. Levinson, Phys.Rev.B.
62, 10637 (2000)
see poster Marjorie CREUX
II . 2.b. quantum shot noise for a single mode
conductor
reservoir
reservoir
µL
µR
T
T
0
xL xR
0
â L
b̂L
b̂R
S
S S + = S + S =1
ψˆ ( xR ) =
(
)
(
)
e
1
ε
d
aˆL (ε ) ei kL xL + bˆL (ε ) e−i kL xL
∫
h
2πhvL (ε )
e
1
i kR xR
ˆ (ε ) e−i kR xR
ˆ
ε
ε
d
a
(
)
e
b
+
R
R
h∫
2πhvR (ε )
D.C. Glattli, NTT-BRL School, 03 november 05
sLR 
 r it 
 ≡ 

sRR 
 it r 
â R
scattering states in the reservoirs:
ψˆ ( xL ) =
 sLL
S = 
 sRL
t2 = D = 1 − R
r2 = R
to summarize:
reservoir
reservoir
µL
µR
T
T
0
xL xR
0
â L
S
b̂L
I =
e
h
e
I =
h
{
(
∫ d ε f L ( ε ) − s LL
∫ dε
s LR
2
2
( f L (ε ) −
f L ( ε ) + s LR
f R (ε ) )
D = S LR = S RL = 1 − S LL = 1 − S RR = 1 − R
2
2
D.C. Glattli, NTT-BRL School, 03 november 05
b̂R
2
2
s
S =  LL
 sRL
sLR 

sRR 
S S + = S + S =1
â R
2
f R (ε )
)}
Landauer formula
e2
 ∂f 
G = ∫ dε  −  D (ε )
h
 ∂ε 
e2
D (ε F )
G=
h
@T =0
Current fluctuations will be calculated in the left reservoir lead
ψˆ ( xL , t ) =
(
)
1
e
ε
d
aˆL (ε ) ei kL xL + bˆL (ε ) e−i kL xL e−iεt / h
∫
h
2πhvL (ε )
bˆL = r aˆL + it aˆR
{
}
v (ε ' ) + v L (ε ) i ( k L −kL ') xL i (ε ' −ε )t
e
Iˆ( x L , t )= ∫ dε dε ' aˆ L+ (ε ' )aˆ L (ε ) − bˆL+ (ε ' )bˆL (ε ) L
e
e
h
2 L (ε ' )v L (ε )
...
plus a factor
∝
( vL (ε ' ) − vL (ε ) ) +
+
aL bL ...bL a L
1/ 2
(vL (ε ' )vL (ε ) )
(
)
also : ~ 1
negligeable if ε’-ε∼ hν << EF
contribute to fluctuations within reservoirs
 t 2 (aˆ L+ (ε ' )aˆ L (ε ) − aˆ R+ (ε ' )aˆ R (ε ) ) − ... i ( k L −kL ') xL i (ε ' −ε ) t
e
Iˆ( x L , t )= ∫ dε dε ' 
e
 e
+
+
h
...irt (aˆ L (ε ' )aˆ R (ε ) − aˆ R (ε ' )aˆ L (ε ) ) 
contribute to fluctuations between reservoirs
(partitionning)
D.C. Glattli, NTT-BRL School, 03 november 05
[
2
]
e
Iˆ(0) Iˆ(τ ) =   ∫ dε ' ' 'dε ' ' dε 'dε t 2 (aˆ L+ (ε ' ' ' )aˆ L (ε ' ' ) − aˆ R+ (ε ' ' ' )aˆ R (ε ' ' )) − irt (aˆ L+ (ε ' ' ' )aˆ R (ε ' ' ) − aˆ R+ (ε ' ' ' )aˆ L (ε ' ' )) ....
h
... × t 2 (aˆ L+ (ε ' )aˆ L (ε ) − aˆ R+ (ε ' )aˆ R (ε )) − irt (aˆ L+ (ε ' )aˆ R (ε ) − aˆ R+ (ε ' )aˆ L (ε )) e−i (ε −ε ')τ
[
]
Iˆ( x L ,0) Iˆ( x L ,τ ) = ??
normal pairing :
ε ''' = ε"
ε'= ε
Iˆ(0) Iˆ(τ ) = I L
gives the contribution:
→ the fluctuations :
and
2
∆Iˆ( xL ,0) . ∆Iˆ( xL ,τ ) = Iˆ( xL ,0) Iˆ( xL ,τ ) − Iˆ( xL ,0) Iˆ( xL ,τ )
again come from the exchange term :
ε'''= ε
and
ε'= ε''
first part:
D 2 aˆ L+ (ε ' ' ' )aˆ L (ε ' ' ) aˆ L+ (ε ' )aˆ L (ε ) ≡ f L (ε ) (1 − f L (ε ' ) )δ (ε ' ' '−ε ) δ (ε ' '−ε ' )
D 2 aˆ R+ (ε ' ' ' )aˆ R (ε ' ' ) aˆ R+ (ε ' )aˆ R (ε )
≡ f R (ε ) (1 − f R (ε ' ) )δ (ε ' ' '−ε ) δ (ε ' '−ε ' )
= emission noise of each reservoir. the D2 term indicates two-particle emitted by a reservoir and transmitted
D.C. Glattli, NTT-BRL School, 03 november 05
[
2
]
e
Iˆ(0) Iˆ(τ ) =   ∫ dε ' ' 'dε ' ' dε 'dε t 2 (aˆ L+ (ε ' ' ' )aˆ L (ε ' ' ) − aˆ R+ (ε ' ' ' )aˆ R (ε ' ' )) − irt (aˆ L+ (ε ' ' ' )aˆ R (ε ' ' ) − aˆ R+ (ε ' ' ' )aˆ L (ε ' ' )) ....
h
... × t 2 (aˆ L+ (ε ' )aˆ L (ε ) − aˆ R+ (ε ' )aˆ R (ε )) − irt (aˆ L+ (ε ' )aˆ R (ε ) − aˆ R+ (ε ' )aˆ L (ε )) e−i (ε −ε ')τ
[
]
∆Iˆ( xL ,0) . ∆Iˆ( xL ,τ ) = Iˆ( xL ,0) Iˆ( xL ,τ ) − Iˆ( xL ,0) Iˆ( xL ,τ )
→ the fluctuations :
ε'''= ε
again come from the exchange term :
and
ε'= ε''
second part:
( −irt) 2 aˆ L+ (ε ' ' ' )aˆ R (ε ' ' ) (− aˆ R+ (ε ' )aˆ L (ε ) ) + (− aˆ R+ (ε ' ' ' )aˆ L (ε ' ' ) )aˆ L+ (ε ' )aˆ R (ε )
⇒
RD [ f L (ε ) (1 − f R (ε ' ) ) + f R (ε ) (1 − f l (ε ' ) )]δ (ε ' ' '−ε ) δ (ε ' '−ε ' )
= partition
shot noise..
â L
b̂L
S
partition noise
D.C. Glattli, NTT-BRL School, 03 november 05
b̂R
â L
â R
b̂L
S
b̂R
â R
‘Pauli blocking’ of partition noise
∞
S I (ν ) = 2 ∫ dτ I (t ) I (t + τ ) ei 2π ντ
−∞
complete finite frequency, finite temperature and voltage formula:
e2
S I (ν ) = 2 ∫ dε { D 2 [ f L (ε ) (1 − f L (ε − hν ) ) + f R (ε ) (1 − f R (ε − hν ) ) ] + RD [ f L (ε ) (1 − f R (ε − hν ) ) + f R (ε ) (1 − f l (ε − hν ) ) ] }
h
reservoir emission noise
EQUILIBRIUM :
e2
S I (ν ) = 4 D
h
e
S I (ν ) = 4 G k B T
shot noise
( fR = fL = f )
hω
hω
k BT
−1
e2
G=D
h
hν << k B T
D = D + D (1 − D )
2
thermal noise
of reservoirs
partition noise of
thermal electrons
equilibrium noise
=
thermal noise
+
partition noise
(equal amount)
S 1 (ω )
NON EQUILIBRIUM, and T=0 :
( µ L = µ R + eV )
(non-equilibrium noise)
S1 ( 0 )
e2
S I (ν ) = 2. D (1 − D ) (eV − hν ) eV ≥ hν
h
=0
eV < hν
eV
h
ω
2π
II. 2. C. zero frequency shot noise and multimode case
shot noise at low frequency and zero temperature
(one mode )
S I = 2eI (1 − D ) = 2eI . F
Poisson (Schottky)
for
ν →0
reduction factor
(Fano)
generalization to multiple modes:
[
+
+
Tr S 21 S 21 (1 − S 21 S 21 )
S I = 2eI
∑D
n
(1 − D n )
n
∑D
]
Fano
factor
Poisson
1
= 2eI . F
noiseless
n
n
0
1
2
∑
n
G. B. Lesovik, Pis'ma Zh. Eksp. Teor. Fiz. 49 (1989)
513 [JETP Lett. 49 (1989) 592].
D.C. Glattli, NTT-BRL School, 03 november 05
D
n
Introduction
I. Electronic scattering ( a brief introduction)
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
- scattering derivations
- electronic analog of the optical Hanbury-Brown Twiss experiment
- electronic quantum exchange
III Shot Noise and Interactions:
IV. Shot noise: the tool to detect entanglement
V. Shot noise and high frequencies
D.C. Glattli, NTT-BRL School, 03 november 05
EXPERIMENTAL EXAMPLES
numbers:
100mK ≈ 10 µV
2e 2
× 10 µV ≈ 0.8 nA
h
S I = 2eI
(
(
≈ 2.610−38 A 2 / Hz = 16 fA/ Hz
SV = 200pV / Hz
)
)
2
noise power added by
the amplifier reffered
to resistor R:
detection noise:
k BTN ∆f =
δ VN
CL
low T
< °K
optimal resistor:
+
nano-conductor
under test R
room temperature
300 °K
D.C. Glattli, NTT-BRL School, 03 november 05
δ IN
1
R
(
δ V N )2 + (δ I N )2
4R
4
(δ V )
(δ I )
2
Ropt . =
N
2
N
lowest TN:
k BTN ∆f =
1
2
(δ V ) (δ I )
2
N
2
N
excellent room temperature commercial LNA (100kHz range) :
(δVN )2
≈ 1.3 nV/ Hz
(δI N )2
≈ 13 fA/ Hz
Ropt . ≈ 100kOhms
TN
opt
≈700mK
( LI75A from NF)
well adapted to quantum point contacts,quantum
dots, STM,etc,... provided microphonic noise sources
in the audio range are carefully eliminated
(one meter coax limits to f < 16 kHz for 100kOhm sample)
good room temperature commercial 80MHz range LNA:
(δVN )2
≈ 0.45 nV/ Hz
(δI N )2
≈ 130 fA/ Hz
Ropt . ≈ few kOhms
TN
opt
≈2.5K
microwave LNAs ( GHz range):
( 220 FS from NF)
well adapted to diffusive wire in semi-conductors,
superconducting/2DEG hybride junctions, ...
Higher frequency (up to few MHz without appreciable
capacitive shunting)
bad coupling : 50 Ohms versus 13kOhms
but fast:
(δVN )2
≈ 0.30 nV/ Hz (room temperature) TN = 30° K on 50 Ω
(δVN )2
≈ 100 − 80 pV/ Hz (cooled < 20Kelvin) TN = 3 − 2° K on 50 Ω
δTN ≡
TN
∆f .τ
cross-correlations
δ VN2
δ V N1
+
(1)
+
δ IN
R
2
room temperature
300 °K
low T
< °K
(
δ I N1
(2)
room temperature
300 °K
SV1V2 ∆f = R 2 . S I ∆f + (δI N1 ) + (δI N2 )
physical sample noise
to be measured
2
2
)
(white) current noise of
each amplifier added
eliminates uncorrelated amplifier voltage noise, thermal noise of leads, reduces microphonic noise.
Improvement in reliability (not noise sensitivity):
(like four-point resistance measurements).
D.C. Glattli, NTT-BRL School, 03 november 05
trick: make use of chirality, when possible to eliminate current noise of the amplifiers
Edge states of the Quantum
Hall effect
V
g
6
5
1
4
(0)
I
B
I
2
3
V
g
I
0
(V)
+
-
High frequency (microwaves): isolators (also called circulators)
50 Ohm load
in both cases
chirality helps!!
From
sample
D.C. Glattli, NTT-BRL School, 03 november 05
conductance ( 2e / h )
quantum point contact
Gate
2
2-D
electron
gas
Gate
F = ∑D
n
n
n
λ F ≈ 70 nm
2
1
0
-50
0
l elast . ≈ 10 − 20 µm
Vg
S I = F.2eI
T =
50
100
( mV )
2 e2
G=
.∑ D n
h n
(ballistic conductor)
SI
*
4 G kB
first mode :
slope ~ (1 - D1 )
Fano reduction factor
∑ Dn (1− Dn )
3
1
1,0
(
.6
.4
.2
)
1 − T1et al. PRL (1996)
Kumar
.8 0,8
0,6
T2 (1 − T2 )
1 + T2
0,4
0,2
0,0
0 0.
0.5
1.
1.5
2.
2.5
Conductance 2e² / h
D.C. Glattli, NTT-BRL School, 03 november 05
A. Kumar et al. Phys. Rev. Lett. 76 (1996) 2778..
M. I. Reznikov et al., Phys. Rev. Lett. 75 (1995) 3340.
Thermal to shot noise cross-over regime
 eV 
e2
e2
2

S I =2
k B T .∑ D n + 2
eV . ∑ D n (1 − D n ) coth 
h
h
n
n
 2 kB T 
thermal emission
noise of reservoirs
shot noise
M. Büttiker, Phys. Rev. Lett. 65 (1990) 2901.
R. Landauer and Th. Martin, Physica B 175 (1991)
Johnson Nyquist noise at V = 0
e2
S I =4
k B T .∑ D n = 4 G k B T
h
n
pure shot
noise
Tn
noise
temperature
slope = F
Fano factor
Def. : noise temperature Tn
T
Tn =
SI
4 G kB
∑D
T
D
∑
2
n
n
n
n
D.C. Glattli, NTT-BRL School, 03 november 05
thermal
noise floor
0
1
eV/2kT
conductance ( 2e2 / h )
thermal to shot-noise cross-over
checked using a QPC
3
2
1
0
D1 = 1 / 2
-50
0
50
100
Vg ( mV )
A. Kumar et al. Phys. Rev. Lett. 76 (1996) 2778..
no adjustable parameter
2
 eV
e2
e
2
SI =2
k B T . D1 + 2
eV . D 1 (1 − D 1 ) coth 
h
h
 2 kB T
D.C. Glattli, NTT-BRL School, 03 november 05



- electron shot noise reaches quantum partition noise limit
- in general quantum conductor show sub-poisonian noise
various Fano factor have been observed in agreement with theory
- F =1/3 : diffusive conductors
- F =1/4 : electron billards (quantum chaos)
- F =1/2 : quantum dots
- ….
∑ D (1 − D )
D (1 − D )
λ
λ
F= λ
≡
where :
is the average over the probability
∑D
D
λ
λ
distribution P ( {D})
diffusive :
D =
l el
L
of transmissions D λ
<< 1
chaotic :
l el
P(D)
P(D)
L
P(D) =
l el
1
2 L D 1− D
P(D) =
1
π
1
D (1 − D )
0
1 D
1 D
: what is quantum in the 1/3 Fano factor of diffusive electronic systems ?)
D.C.
D.C.
Glattli,
Glattli,
NTT-BRL
NTT-BRL
School,
School,
03 november
03 november
05(question
05
0
Chaotic cavity (electron billard)
cross-over from quantum to classical (no noise) regime.
noise is quantum !
FANO Factor
S I / 2e I
1/ τ D
( GHz)
Oberholzer et al. Nature (2002)
D.C. Glattli, NTT-BRL School, 03 november 05
S I = F .2eI
lower slope : diffusive regime
upper slope : hot electron regime
(see later )
F = 1/ 3
" F" =
3/4
M. Henny, S. Oberholzer, C. Strunk, and C. Schonenberger,
Phys. Rev. B 59 (1999) 2871.
(question : what is quantum in the 1/3 Fano factor of diffusive electronic systems ?)
heating effect : apparent shot noise
lelectron −electron
I
Rlead
•
•
Q
Q
T
Te
Te
I
Rlead
T
GQPC
•
Q=
•
Q=
1
GQPC V 2
2
heat produced
π 2  k B  Te 2 − T 2
heat flow
Wiedeman-Franz
2
 
3  e2 
Rlead
S I = 4 GQPC k B Te
( note : no heating effect if chiral system )
D.C. Glattli, NTT-BRL School, 03 november 05
2
3  e2 
2
2
Te = T + 2   GQPC RleadV 2
2π  k B 
S I = 2eI
6
π2
GQPC Rlead
for eV > k BT
not shot noise, just heating,
apparent fano factor F
Important : good, low resistive contacts
electron heating effect in a diffusive wire
Te (x )
T
S I = 2eI × " F "
diffusive wire
good
contact
L >>lel −el
" F" =
good
contact
A. H. Steinbach, et al.
Phys. Rev. Lett. 76 (1996) 3806.
D.C. Glattli, NTT-BRL School, 03 november 05
3/4
(just electron heating, not transport shot noise)
Also :
M. Henny, S. Oberholzer, C. Strunk, and C. Schonenberger,
Phys. Rev. B 59 (1999) 2871.
Introduction
I. Electronic scattering ( a brief introduction)
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
- scattering derivations
- electronic analog of the optical Hanbury-Brown Twiss experiment
- electronic quantum exchange
III Shot Noise and Interactions:
IV. Shot noise: the tool to detect entanglement
V. Shot noise and high frequencies
D.C. Glattli, NTT-BRL School, 03 november 05
II. 4 . current noise correlations.
Zero temperature expression:
cross-correlation between two different leads are always negative
Special case of a three terminal branch :
one finds :
∆I 2
2
1
∆I 2 ∆I 3 = S23 ∆f < 0
V
binomial partitioning is here replaced by multinomial partitioning
(just ‘gambling’ law!)
∆I 3
3
Hanbury Brown & Twiss experiment with electrons
Oberholzer et al. ‘ 00
from sub-poissonian statistics
∆I t ∆I r < 0 = −∆I t 2 = −∆I r 2
to poissonian statistics
∆I t ∆I r = 0
D.C. Glattli, NTT-BRL School, 03 november 05
Four terminal lead :
(A)
V1 = V ; V3 = 0
(B)
V1 = 0 ; V3 = V
(A + B)
∆I 2 ( t ' )
2
1
V1 = V ; V3 = V
3
∆I 4 ( t )
S( A+ B ) − ( S( A) + S( B ) ) = − 2
2
[(
) (
e
*
*
*
*
eV s21s23 s43s41 + s23s21 s41s43
h
exchange terms : non classical
exchange terms : non classical
D.C. Glattli, NTT-BRL School, 03 november 05
4
)]
(M. Büttiker, Phys. Rev. B 46 (1992) 12485.
Introduction
I. Electronic scattering ( a brief introduction)
II. Quantum Shot noise
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. Combining electrons and photons
D.C. Glattli, NTT-BRL School, 03 november 05
III 1. Quantum Hall effect
Integer Q.H.E. (von Klitzing 80)
Energy
level
B > Tesla
R
Hall
h
1
= 2⋅
e integer
e
ωC =
eB
m
hω C
cyclotron
frequency
hω C
Fractional Q.H.E. (Tsui, Störmer,Gossard 1982)
( Laughlin 1983)
ν = 1/3 :
-e
R
Hall
=
h
1
⋅
e 2 fraction
e/3
e/3 e/3
D.C. Glattli, NTT-BRL School, 03 november 05
ν=
number of electrons
number of quantum states
1 electron for 3 quantum states
elementary excitation
≡ empty a quantum state
≡ carry fractional charge e/3
available current is at the edges of the sample
E
EF
r
r
v drift = E Confinement × B zˆ
( no current
in the bulk )
hω C
( edge current )
Transfer of hole trough
Q.H.E. fluid:
Tunneling trough
barrier :
I 0 =e 2 V / h
I (t )
e
V
ν = 1 QHE
liquid
I ≈ I0
I 0 =e 2 V / h
ν = 1 QHE
e
liquid
I B (t )
(∆I )
2
= 2eI ∆f
D.C. Glattli, NTT-BRL School, 03 november 05
D << 1
(∆I )
2
= 2eIB ∆f
D≈1
e
Tunneling trough
barrier :
1/3 - FQHE
liquid
1/3 - FQHE
liquid
q=e
(∆I )
2
= 2eI ∆f
D << 1
Transfer trough
1/3 FQHE fluid:
1/3 - FQHE
liquid
e/3
1/3 - FQHE
liquid
q = e/3
(∆I )
∆I
∆f
2
= 2 (e/ 3) I B ∆f
D≈1
2
= S I = 2q I
D.C. Glattli, NTT-BRL School, 03 november 05
measuring both quasiparticle shot noise (Poissonian regime!)
+ mean current gives the charge with no adjustable parameters.
DSM / DRECAM
ENS-Paris
current is carried by fractional charges
(direct evidence, no unknown parameters)
L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne,
Phys. Rev. Lett. 79, 2526 (1997).
4.2 °K
1.5 °K
e/3
0.7 °K
0.12 °K
0.007 °K
(∆I ) 2
= 2 ( e / 3) I R ∆f
charge q=e/3
D.C. Glattli, NTT-BRL School, 03 november 05
15 Tesla
superconducting coil
20 µm
6
Vg
5
(0)
IB
4
1
I
2
I0
(V)
3
Vg
+
-
e / 3 fractional charges are observed
at filling factor 1/3 ( Weizmann and Saclay 97 ) and
e / 5 charges at filling 2/5 (Weizmann 99 ).
L. Saminadayar, D. C. Glattli, Y. Jin, and B. Etienne,
Phys. Rev. Lett. 79, 2526 (1997).
V
0
g
+
-
R. de-Picciotto et al., Nature 389, 162 (1997).
M. Reznikov, R. de-Picciotto, T. G. Griths,
M. Heiblum, and V. Umansky, Nature 399, 238 (1999).
cross-over from thermal noise to fractional charge shot noise
SI = 2
eV ds
e
I B coth
3
3k B T
theory : 2 e/3 Ib coth (e/3Vds/2kT)
data
/ Hz)
0,24
2
1/3 - FQHE
liquid
current noise (10
1,0
2
differential conductance ( e / 3 h )
-28
e/3
A
1/3 - FQHE
liquid
0,8
0,6
charge e/3
0,20
0,18
0,16
0,14 VG
Tvraie
0,12
-0,3
0,2
-150 -100
-50
0
50
100
= 23.5 mV
= 40 mK
-0,2
Vg = 32.22 mV
Vg = 23.24 mV
Vg = 17.55 mV
Vg = 12.23 mV
Vg = 7.77 mV
Vg = 2.35 mV
0,4
0,0
0,22
-0,1
2 e/3 IB (10
e
D.C. Glattli, NTT-BRL School, 03 november 05
-28
0,1
A
2
/ Hz)
charge e
150
bias voltage V (µV)
V. Rodriguez et al (2000)
0,0
1/3 - FQHE
liquid
??
1/3 - FQHE
liquid
0,2
0,3
/ Hz)
0,5
- 28
0,3
0,2
(a)
Vg = 2.07 mV
0,0
-30 -20 -10
0
10 20 30
Bias Voltage ( µ V )
excess current noise (10
0,4
0,1
T = 47.8 mK
T = 36 mK
0,4
A
2
T = 47.8 mK
T = 36.7 mK
0,3
0,2
0,1
0,0
(c)
-30 -20 -10
0
10 20
Bias Voltage ( µ V )
30
2
/ Hz)
differential conductance ( e
2
/ 3h )
integer charge in the strong backscattering regime
0,0
-0,1
(b)
-30 -20 -10
Vg = 2.07 mV
0
10
0,3
SI =2e I
0,2
Schottky noise
- 28
A
T = 47.8 mK
T = 36.7 mK
20
Bias Voltage ( µV )
30
excess current noise (10
current ( nA )
0,1
for charge e
0,1
e
0,0
(d)
-0,3 -0,2 -0,1 0,0 0,1 0,2 0,3
2 e I ( 10
-28
A
2
1/3 - FQHE
liquid
1/3 - FQHE
liquid
/ Hz )
In the same sample, same quantum point contact, one can go from the e/3 regime to the e regime just by changing coupling
III . 2. normal / superconductor interface:
no single particle current between
superconducting and normal metal
expected for sub-gap energies.
+∆
e
+ε
−ε
eV
second order process involving
two quasi-particle allows for finite
current: Andreev reflection.
h
−∆
Superconductor
normal conductor
ideal interface :
incoming electron ( energy + ε ) → outgoing hole ( energy - ε )
plus phase γ ≡ exp( − i cos −1 (ε / ∆ ) )
incoming
hole ( energy − ε ) → outgoing electron ( energy + ε )
plus phase γ ≡ exp( − i cos −1 (ε / ∆ ) )
D.C. Glattli, NTT-BRL School, 03 november 05
ˆ
b'
e
â e
b̂ e
S N (ε )
ˆ e
a'
γ ≈i
 r
S N ( ε ) =  11
 t 21
t12 

r22 
scattering matrix in the normal lead
â h
b̂ h
S N+ (−ε )
the complete scattering matrix including Andreev reflection and normal scattering is :
 bˆ e 
 aˆ 
 = S e
 aˆ h 
 bˆ h 
 
 
 se e
S = 
 she
se h 

s h h 
(
she (ε ) = t 21 (ε ) γ t12 ( −ε ) + t21 (ε ) γ r22 ( −ε ) γ r22 (ε ) t12 ( −ε ) + t 21 (ε ) γ r22 ( −ε ) γ r22 (ε )
*
*
*
*
)t
2
12
*
( −ε ) + ...
t 21 (ε ) γ t12 ( −ε )
=
*
1 − γ r22 ( −ε ) γ r22 (ε )
*
(Fabry-Pérot like multiple interferences)
D.C. Glattli, NTT-BRL School, 03 november 05
+∆
e
+ε
−ε
eV
h
−∆
Superconductor
normal conductor
( 2e) 2
( 2e) 2 D 2
2
G=
she =
h
h (1 + R ) 2
‘doubled’ shot noise :
e2
2
2
S I = 2 ( 2e) V she (1 − she )
h
- twice the electron charge
- binomial law of quantum partitioning
- noiseless property of the Fermi sea
D.C. Glattli, NTT-BRL School, 03 november 05
R =1− D
doubling of the shot-noise
for a diffusive S-N junction
A. A. Kozhevnikov, R. J. Schoelkopf, and D. E. Prober,
Phys.Rev.Lett. 84, 3398 (2000)
1
S I = 2.(2e ) I
3
Fano for
diffusive
regime
Doubled
charge
D.C. Glattli, NTT-BRL School, 03 november 05
T Noise =
SI
4G k B
( also M. Sanquer et al. 2001)
III. 3. interaction effects in a Quantum Point Contact :
the ‘0.7x2e2/h’ structure
long QPC: conductance plateau around 0.7X2e2/h
conductance ( 2e2 / h )
ultra-short QPC: no ‘0.7’
3
↑↓
↑↓
2
↑↓
↑
1
0
↑↓
-50
0
50
100
Vg ( mV )
↓
plateaus at:
e2
G = (integer ) × 2
h
K.J. Thomas et al., Phys.Rev.Lett. 77, 135 (1996)
- resonance in transmission? (single spin degenerate mode)
(1 + 1)
↑+↓
D.C. Glattli, NTT-BRL School, 03 november 05
- or spin degeneracy lifted by interaction? (two distinct modes)
↑↓
↑↑
III. 3. interaction effects in a Quantum Point Contact :
the ‘0.7x2e2/h’ structure
long QPC: conductance plateau around 0.7X2e2/h
conductance ( 2e2 / h )
ultra-short QPC: no ‘0.7’
↑↓
3
↑↓
(275mK to 800mK)
2
(0 Tesla)
↑↓
1
0
↑↓
-50
0
50
↑+↓
100
Vg ( mV )
(0 to 8 Tesla)
↓
plateaus at:
e2
G = (integer ) × 2
h
- resonance in transmission? (single spin degenerate mode)
(1 + 1)
↑+↓
D.C. Glattli, NTT-BRL School, 03 november 05
- or spin degeneracy lifted by interaction? (two distinct modes)
↑↓
↑↑
spin degenerate case
1,0
S I = 2eI (1 − D↑↓ ) = 2eI . F↑↓
F ano factor F
0,8
spin degeneracy fully lifted:
0,6
0,4
0,2
0,0
0,0
0,3
0,5
0,8
2
D↓ (1 − D↓ ) + D↑ (1 − D↑ )
= 2eI . F↑+↓
S I = 2eI
D↓ + D↑
D.C. Glattli, NTT-BRL School, 03 november 05
conductance (2 e /h)
1,0
conductance can not
distinguish between
one or two modes
but shot noise can
Here: the Fano factor
shows direct signature
of two modes
↓
↑↓
→ lifted spin degeneracy
scenario.
P. Roche, J. Segala, and D. C. Glattli,
J. T. Nicholls, M. Pepper, A. C. Graham,
M. Y. Simmons, and D. A. Ritchie,
Phys. Rev. Letters 2004
D.C. Glattli, NTT-BRL School, 03 november 05