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
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