Integer and fractional charge Lorentzian voltage pulses analyzed in the frame of Photon-assisted Shot Noise

0.68 MB·English

by  Julie Dubois

Integer and fractional charge Lorentzian voltage pulses analyzed in the framework of
Photon-assisted Shot Noise
J. Dubois1, T. Jullien1, C. Grenier2,3, P. Degiovanni2, P. Roulleau1, and D. C. Glattli1
1 CEA, SPEC, Nanoelectronics group, URA 2464,
F-91191 Gif-Sur-Yvette, France.
2 Universit´e de Lyon - F´ed´eration de Physique Andr´e Marie Amp`ere
CNRS - Laboratoire de Physique de l’Ecole Normale Sup´erieure de Lyon
46 Alle d’Italie, 69364 Lyon Cedex 07, France
3
3 Centre de Physique Th´eorique (CHPT),
1
0 Ecole Polytechnique, 91128 Palaiseau Cedex, France
2 (Dated: January 22, 2013)

n Theperiodicinjectionnofelectronsinaquantumconductorusingperiodicvoltagepulsesapplied
a on a contact is studied in the energy and time-domain using shot noise computation in order to
J makecomparisonwithexperiments. WeparticularlyconsiderthecaseofperiodicLorentzianvoltage

9 pulses. When carrying integer charge, they are known to provide electronic states with a minimal
2 number of excitations, while other type of pulses are all accompanied by an extra neutral cloud of
electronandholeexcitations. Thispaperfocusesonthelowfrequencyshotnoisewhichariseswhen

] the pulse excitations are partitioned by a single scatterer in the framework of the Photo Assisted
l
l Shot Noise (PASN) theory. As a unique tool to count the number of excitations carried per pulse,
a
shot noise reveals that pulses of arbitrary shape and arbitrary charge show a marked minimum
h
when the charge is integer. Shot noise spectroscopy is also considered to perform energy-domain
-
s characterization of the charge pulses. In particular it reveals the striking asymmetrical spectrum
e of Lorentzian pulses. Finally, time-domain information is obtained from Hong Ou Mandel like
m
noisecorrelations whentwotrainsofpulsesgenerated onoppositecontactscollide onthescatterer.
. As a function of the time delay between pulse trains, the noise is shown to measure the electron
t
a wavepacketautocorrelationfunctionforintegerLorentzianthankstoelectronantibunching. Inorder
m tomakecontactwithrecentexperimentsallthecalculationsaremadeatzeroandfinitetemperature.
-
d PACSnumbers: 73.23.-b,73.50.Td,42.50.-p,42.50.Ar
n
o
c This paper addresses the noiseless injection of a small statistics [16] with a finite number of electrons [17]. The
[ finite number of electrons in a quantum conductor. In- injection of single or of few electrons injected in a bal-

2 deed, quantum effects become more and more accessible listic one dimensional conductor lets envisage to paral-
v whenonly few degreesoffreedomarecontrolled. During lel the flying qubit approach developed for photons [18–
1 the last thirty years, research in this direction has lead 20]. As opposed to Bose statistics, the Fermi statis-
2
tothepossibilitytomanipulatequantumstateswithsev- tics makes entanglement of several electrons injected in
9
eraldegrees of freedom and to entangle particles making a quantum conductor more favorable and the Coulomb
3
. possible simple quantum information processing. Up to interaction, when not screened, the possibility to make
2
now most advances have been obtained in quantum op- non-linear interaction more easier than the Kerr effect
1
tics with the manipulation of single photons emitted by used for photons. The ballistic conductors can be real-
2
1 atoms or semiconductor quantum dots, and in atomic ized using high mobility 2D electrons confined in high
: physicswithopticalarraysoftrappedcoldatomsorions quality III-V semiconductor heterojunctions. In addi-
v
More recently the manipulation of quantum states has tion, applying a perpendicular magnetic field gives rise
i
X become available in condensed matter systems using su- to one dimensional chiral propagation along the edge of
r perconductingcircuitsandsemiconductorquantumdots. the samples in the Quantum Hall regime. This, com-
a
bined with nano-lithographied split gates, allows to im-
Arecentapproachisthemanipulationofsinglecharges plement quantum gates similar to that used in quantum
injected in a quantum ballistic conductors. Realizations optics: electronic beam splitters, electronic Fabry-Perot
of time controlled single charge sources have been re- and Mach-Zehnder interferometers [21, 22].
portedin[1–5]andconsideredtheoreticallyin[6–13]with
a particular focus on the energy resolved single electron Buthowto inject singleto few electronsin aquantum
source based on a quantum dot [1]. Injecting more than conductor? SingleelectronpumpsbasedontheCoulomb
oneelectronrequiresadifferentpracticalapproachwhich blockade of tunneling have been first realizedusing ordi-
is the subject of this paper. We will consider the more nary metallic systems [23, 24]. They have potential ap-
general case of coherent trains of few undistinguishable plication in quantum metrology but are not suitable for
electrons [14, 15] which opens the way to entangle sev- our purpose. First the conductor is usually not ballis-
eral quasi-particles but also to probe the full counting tic. Second the electrons are sequentially injected. The
2
lack of quantum coherence between electron tunneling In the following, we explore some properties of the co-
eventsmakesthe injectionoftrainsoffewundistinguish- herent injection of several indistinguishable electrons by
able electrons impossible. More recently itinerant quan- periodicvoltagepulsesappliedonanOhmiccontactand
tumdotsobtainedusingsurfaceacousticwaves[25]have weconsidertheircharacterizationinenergyandtimedo-
beenshownabletotransportsingletotwoelectronsalong mainusingshotnoise. Wecomparedifferentpulseshapes
a depleted electron channel [4, 5]. These systems may andalsothedeparturefromintegerchargevaluesandwe
also have application in metrology. Because of the long includefinitetemperatureeffectswiththeaimtoprovide
coherence time of the spin, they could be used for spin directcomparisontoexperiments[31]. Thegoalbeingto
based qubit operation and to carry quantum informa- produce injection with minimal excitation, we will only
tion between distant dots [4, 5]. Finally, the on-demand consider the injection of chargeswith same signas alter-
injection of a single electron emitted from a single en- nate sign charge injection was found in general not suit-
ergy level suddenly risen at a definite energy above the able(see ref. [14]). Consideringperiodic pulses allowsto
Fermiseaoftheleadshasbeenrealizedusingaquantum use the powerful Floquet scattering theory approach of
mesoscopiccapacitor[26]in a non-linearregime [1]. The Ref. [32]. While Lorentzian shape pulses with n charges
system can be viewed as the electronic analog of a fre- per period correspond to exactly n excitations [14, 15],
quency resolvedsingle photon source and opens the way pulses of arbitrary shape contain more excitations [28]
for new quantum experiments where single electrons are whose number is calculated for sine, square and rectan-
injected at tunable energy well above the Fermi energy. gularwaveshape. Wealsoaddressthecaseofnon-integer
However this kind of on-demand emission is not gener- charge pulses which was shown in ref. [14, 33] to be the
alizable to a coherent train of several undistinguishable dynamical analog of the Anderson orthogonality catas-
electrons. trophe problem [34]. Here the periodicity introduces of
cut-offin the long time log-divergentincreaseofelectron
We consider here a powerful and simpler way to in- andhole excitationsat longtime. The Andersonphysics
ject a coherent train of n electrons in a single quantum reflects in a marked minimum of neutral excitations for
channelofaballisticconductor. Herespinisdisregarded integerchargeperperiodastheoreticallyobservedinRef.
for simplicity. The principle, proposed by Levitov et al. [28].
in [14, 15] uses voltage pulses V(t) applied to one con-
∞
tact. For a quantized action e V(t)dt = nh, n in- In order to quantify the creationof extra electron and
−∞
teger and h the Planck constanRt, exactly n charges are hole excitations, we need a physical quantity counting
injected. As the current I(t) at the injecting contact is them. The rightquantity is the low frequency shotnoise
at all time equal to I(t) = e2V(t) this is equivalent to of the electrical current already considered for dc trans-
h
∞
write I(t)dt = ne. This is the simplest injection of port [35–40] where electrons injected by a dc voltage bi-
−∞
chargeRthat we can imagine. On the experimental level, ased contact are partitioned by a scatterer playing the
itdoes notrequirethe implementationofa quantumdot role of an electronic beam splitter. In a quantum optics
northecombinationoftunnelbarriersorofseveralQuan- language,this canbe viewed as a Hanbury-BrownTwiss
tum Point Contacts. For quantum bit applications this experiment (HBT) with electrons. As remarked in the
will improve the reliability as the delicate circuitry sim- pioneering works or refs. [14, 15, 33], shot noise is also
plifies and reduces only to that necessary for quantum expected when the electron pulse trains pass through a
logic gate implementation. On the conceptual level, this beam splitter. Indeed all excitations are partitioned ir-
apparently naive approachinvolves a non trivial Physics respective to their charge, and the shot noise counts the
related to the deep properties of the Fermi sea. Indeed, sum of electron and hole excitations while the current
injecting n charges in general does not mean injecting n counts their difference. The current noise is then pro-
electrons alone but also involves a collective excitation portional to the number n of injected electrons, plus the
of the whole Fermi sea: a neutral cloud of electrons and number N and N of extra electron and hole excita-
e h
holes. However the beautiful theoretical observation of tions created per period, with N = N by definition.
e h
ref.[14]isthatavoltagepulsewithLorentzianshapecar- Using the separation of the voltage pulse into its mean
rying one electron is a minimal excitation state free of value, or d.c. part <V(t)>=V and its ac component
dc
these neutral electron-hole excitations. More generally a V (t)=V(t) V as in ref. [28], the physics of voltage
ac dc
−
combination of n Lorentzianhaving arbitrary shape and pulsescanbe convenientlydiscussedinthe frameworkof
position in time but all carrying a unit charge of same Photo-AssistedShot Noise (PASN) where both theoreti-
signisaminimalexcitationstate. Theremarkableresult cal [41–43] and experimental results [45, 48] are already
of ref. [14, 15] has triggered several relevant theoretical available. At zero temperature, the excess noise ∆S
I
contributions[9,17,27–30]inwhichthepropertyandpo- characterizingthe extraexcitationsisthenthedifference
tentialuseoftheLorentzianvoltagepulsesarediscussed. betweenthe PASNnoiseSPASN duetoV(t)=V +V
I ac dc
An experimentalimplementation has been recently done and the (transport) shot noise (TSN) STSN that we
I
[31]. would have with only the dc voltage V(t) = V [35–
dc
3
(cid:12)(cid:13)(cid:12)(cid:14)(cid:15)(cid:16)
40, 49, 50]. As ∆SI (Ne + Nh) and the current is ε+(cid:6)(cid:12)ν
I =eν(Ne−Nh+n)=∝n,ν beingthefrequency,minimiz- (cid:8)ε(cid:7) (cid:11)(cid:9)(cid:10)(cid:3)ε(cid:2)(cid:1)(cid:11)−(cid:9)ε(cid:5)(cid:5)(cid:1)(cid:11)−(cid:9)φ(cid:3)(cid:5)(cid:2) εε+(cid:12)ν
ing the noise at constant current implies Nh = Ne = 0. εε−−(cid:6)(cid:12)ν(cid:12)ν
In terms of photo-assisted effect we will see below that (cid:6)(cid:3)(cid:5)(cid:2)
this implies only absorption or only emission of photon,
depending on the sign of the n charges injected. (cid:17)(cid:7)(cid:18)(cid:19)(cid:20)(cid:5)(cid:7)(cid:8)(cid:9)(cid:4)(cid:5)(cid:3)(cid:8)(cid:5)(cid:7)(cid:21) (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:2)(cid:6)(cid:7)
(cid:8)(cid:9)(cid:4)(cid:10)(cid:2)(cid:8)(cid:5)(cid:9)(cid:11)
The paper is organized as follows. In section I we in-
troduce the basic Physics of the photo-assistedeffects in (cid:3)(cid:4)(cid:2)(cid:1)(cid:3)ε(cid:2) (cid:3)(cid:4)(cid:2)(cid:3)ε(cid:2)
a quantum conductor with a contact driven by a peri- (cid:4)
odic voltage source in the Floquet scattering approach. (cid:6)(cid:3)(cid:5)(cid:2) (cid:1) (cid:1)
WethenconsiderthePASNandthecompetitionbetween
PASNandTSNwhenbothadcbiasandanacvoltageare
applied. We recall the remarkable result of a singularity FIG. 1: Under the effect of an ac voltage, electrons emitted
appearinginthederivativeofthenoisewheneV =nhν, farinsidethecontactacquireatimedependentphaseontheir
dc
i.e. for exactlyn electrons inaverageinjected per period way to the scattering region (the quantum conductor). For
periodic voltage, frequency ν, the incoming electrons can be
T = 1/ν making direct connection with the problem of
described by a quantum superposition of states at different
periodic voltagepulses carryingintegercharges. We also
energies ε+lhν.
give the expression of the photo-assisted current.
In section II we address the comparison between sev-
eral types of integer charge pulses : the square, the sine,
charge Lorentzian pulses the noise is directly linked to
the rectangular and the Lorentzian. Using the PASN
the wavepacket overlap of the injected charges while for
results of section I, we calculate the number of electron-
other pulse shapes the cloud of neutral excitations also
hole pairs excitations via the excess noise. For compar-
contributes to HOM noise.
ison with experiments the computation is done for both
zeroand finite temperature. The hierarchyofthe charge
I Floquet scattering description of periodic
pulses regarding noise production is compared with the
voltage pulses applied on a contact
hierarchy of the pulses based on photo-current produc-
tion. We consider a quantum conductor with one contact,
Section III addresses the case of non-integer charge say the left (L), periodically driven by a voltage Vac(t)
pulses. We show that for all type of pulses, the number of frequency ν = 1/T, see Fig.1 . Here, without loss
of electron-hole pairs rises for non-integer charges but is of generality we choose the approach of refs. [42, 51]
alwaysminimalfor integercharges(andevenzeroforin- where the periodic potential is assumed to be screened
teger Lorentzian pulses). For large non integer charge in all other regionsof the quantum conductor. The volt-
number, the number of neutral excitations quickly de- age drop is assumed sharp on the electron wavepacket
creasesfor Lorentzianwhile it remainsfinite orincreases length but smooth on the Fermi wavelength. The quan-
for other type of pulses. tumconductorhasasmallwidthcomparedtothatofthe
SectionIVprovidesanenergydomaincharacterization leads and a small length L compared with the electron
of the charge pulses using shot noise spectroscopy. We phase coherence length lϕ such that electrons can prop-
compute the PASN as a function ofanarbitrarydc volt- agate coherently overa length lϕ L on the leads. The
≫
age bias which provides a direct measure of the photo- regionwhere electron loose coherence is called the reser-
absorptionprobabilities[28,41,45,48,54]. Inparticular voirorcontact, followingthe standarddescriptionof the
the Lorentzian pulses show an asymmetric PASN versus scatteringtheoryofquantumtransport. Weassumethat
dc bias characteristic of the absence of hole type exci- theelectroncoherencetimelϕ/vF islargecomparedwith
tations while sine and square waves lead to symmetric theperiodT. Weconsideranelectronemittedbytheleft
PASN. reservoiratenergyεinastate exp(ik(ε)x)exp( iεt/~)
∼ −
withoccupationprobabilityf (ε)=1/(exp(ε/k T )+1)
Finally, section V gives a time domain characteriza- L B e
whereT istheelectronictemperatureandtheFermien-
tion of the pulses by looking at the shot noise generated e
ergy is the zero energy reference. From the left reservoir
by trains of electrons colliding on the scatterer. In anal-
to the left entrance of the conductor, the electron expe-
ogy with Hong Ou Mandel (HOM) experiments [60], a
riences the potential V (t) and acquires an extra term
time delay between the pulse trains emitted by opposite ac
exp( iφ(t)) in its amplitude probability with time de-
contacts controls the overlap of electronic wavefunctions
−
pendent phase:
which reflects in noise suppression. Here, the gauge in-
variance maps the HOM experimental scheme discussed
here to a simpler Hanbury Brown Twiss (HBT) prob- e t
φ(t)=2π V (t′)dt′ (1)
lem with only one driven contact. For colliding single hZ ac
−∞
4
The Fourier transform of: (cid:2)(cid:10)(cid:1)
(cid:1)(cid:2)(cid:1)(cid:3)(cid:4)(cid:5)
+∞ (cid:4)(cid:3) εε++(cid:3)(cid:5)(cid:5)νν (cid:4)(cid:3) (cid:5)ν
exp(−iφ(t))= plexp(−i2πlνt) (2) ε εεε−−(cid:5)(cid:3)ν(cid:5)ν (cid:2)(cid:2)(cid:2)(cid:1)(cid:1) (cid:5)ν
l=X−∞
(cid:4)
gives the probability amplitude p for an electron to ab- (cid:6)(cid:7)(cid:8)(cid:2)(cid:1)(cid:1) (cid:9)(cid:2)ε(cid:1)
l
sorb (l >0) or emit (l <0) l photons. Eq. (2) expresses
thatanelectronemittedatenergyεenterstheconductor
FIG.2: Electronsemittedbytheleftreservoirandpumpedby
inasuperpositionofquantumstatesatdifferentenergies
theac voltage in a superposition of states of different energy
ε+lhν. The knowledge of the pl completely defines the arescattered. Therandompartitioningbetweenreflectedand
statesoftheincomingelectrons. Themagnitudeofthepl transmittedscatteringstatesleadstocurrentnoise. Thenoise
depends onthe reducedquantityα=eV /hν whereV measuresthesumofthenumberofholesandelectronswhich
ac ac
is the characteristic amplitude of the ac voltage. Com- are photo-created. On the right is represented the energy
distributionfunction f˜(ε). Itshouldnot beconfusedwith an
bined with the scattering properties of the conductor,
incoherentdistributionfunction asthenon-diagonaltermsof
allphoto-assistedeffectsresultingfromtheabsorptionor
thedensity matrix are non-zero.
emission of energy quanta hν such has the ac current,
the photo-currentand the photo-assistedshotnoise, etc,
can be calculated. Noise occurs only whenan electronincoming fromthe
Thepropertiesofthep arebestexpressedintheframe left finds no incoming electron from the right or if a
l
oftheFloquetscatteringtheory[32]inwhichthecontin- hole incoming from the left finds an incoming electron
uous energy variable describing states in the reservoir is from the right. Indeed, because of the Pauli principle,
slicedintoenergywindowsofwidthhν, i.e. ε ε+l′hν orFermionicanti-bunching,thecasewheretwoelectrons
→
withεnowrestrictedto[ hν,0]. Wecanviewthephoto- or two holes are simultaneously incoming gives no noise.
−
assisted processes as the coherent scattering of electrons In the former case, the incoming charges are randomly
between the different energy windows. The scattering partitionedbytheconductorwithbinomialstatistics. At
matrix S(ε) = Sll′ relates the set of annihilation op- zerotemperatureandforzeroacvoltagethenoiseiszero.
erators a0(ε) ={ a0}(ε+l′hν) operating on the states Inpresenceofanacvoltageelectronandholeexcitations
L { L }
of the left contact to the set of annihilation operators are created in the left lead. The photon-created elec-
a (ε)=ba (ε+lhbν) of electrons incoming on the con- trons span energies above the right lead Fermi sea and
L L
{ }
ductor : the photon-createdholes below the right lead Fermi sea.
b b
Alltheexcitationscontributetopartitionnoise,whichis
aL(ε)=S(ε)×a0L(ε) (3) called the Photon-Assisted Shot Noise (PASN). Accord-
ingtoref.[41,42]thelowfrequencycurrentnoisespectral
wIigtihve{sSul′ls}ef=ulprle−lal′bt.ioTnhsefournitthaeriatymbrpelliatutidoenspSro†bSab=ilSitSie†s=: density due to photo-assisted process SIPASN is, includ-
ing the Fermi distributions f (ε) of the left and right
L,R
reservoirs:
+∞
p∗lpl+k =δk,0 (4) dε +∞
l=X−∞ SIPASN =SI0Z hν Pl{fL(ε−lhν)(1−fR(ε))
l=X−∞
Inparticular,thesumofthe probabilitiesP = p 2 to
l | l| +(1 fL(ε lhν))fR(ε) (5)
absorboremitphotonsortodonothingisequaltounity. − − }
AsshownbelowandusedinSectionIV,theprobabilities where S0 = 2e2D(1 D)hν is the typical scale of the
Pl canbeinferredfromShotNoisespectroscopy,whenin PASN. TIhe cohmplete−noise expression is obtained by
additiontothe acvoltageatunabledc voltageisapplied adding the thermal noise of the reservoirs: 4k T D2e2.
betweenthecontactsofthe conductor. Notethattheset B e h
Its origin is the thermal fluctuation of the population in
ofP does notcontainallthe informationonthe system.
l the reservoir and is not related to partitioning nor to
Indeed,theproductsp∗p , k =0,i.e. thenon-diagonal
l l+k 6 photo-assisted processes. In absence of ac voltage, Pl =
part of the matrix density, enters in the calculation of δ , the full noise reduces to thermal noise 4k T De2
the coherence [54] as discussed in [53]. l,0 B e h
and vanishes at zero temperature, as discussed above.
In order to best extract the Physics, we first consider
Photo-assisted shot noise : the zero temperature limit :
In the following we calculate the photo-assisted shot
+∞
noise which occurs when the conductor elastically scat-
SPASN =S0 l P (6)
ters the electrons. For simplicity we will consider a sin- I I | | l
l=X−∞
gle mode (or one-dimensional) quantum conductor with
transmission probability D as shown in Fig.2. The sum in the right hand side is directly proportional
5
tothenumberofelectronsandholescreatedrespectively excitation and by varying the transmission. Motivated
above and below the Fermi energy (chosen as the zero by this experiment, Rychkovet al [43] havetheoretically
of energy). To understand this, let us consider for sim- shown that the electron and hole excitations contribut-
plicitythatonlyP andP areimportantandfirstcon- ing to noise in Eq.(9) are not statistically independent.
0 ±1
centrate on the absorption process. Electrons below the Eq.(9) was derived by Lee et al. [44], Levitov et al. [14]
Fermi surface at energy ε khν, k positive integer, can andlaterbyKeelingetal.[15]. Inthedifferentcontextof
−
bepromotedtoenergyε (k 1)hν withprobabilityP . periodic injection of energy resolved single electron and
1
− −
This globalupwardshiftofthe Fermisealeavesthe elec- single hole from a quantum dot, a similar equation has
tronpopulationunchangedbelow E but fills the empty been derivedand experimentally tested by Bocquillonet
F
sates of energies ε [0,hν] with occupation probability al. [3] for the partitioning of single charges. Finally, as
∈
P , above E . Similarly, in the emission process, elec- discussed below it is important to note that Eq.(9) mea-
1 F
trons are displaced to energies ε (k+1)hν with prob- sures the number of electron and hole excitations only
−
ability P . The downward shift of the Fermi sea gives accurately at zero temperature.
−1
no net change of the population of states with energy
< hν, while for the energy range [ hν,0] the popula- Photo-assisted and Transport Shot noise :
− −
tion is now 1−P−1. More generally the l-photon pro- We consider now a dc voltage bias Vdc > 0 added to
cesses give electron excitations above the Fermi sea with theperiodicacvoltageV (t)ontheleftcontactwhilethe
ac
population Pl in the energy range [0,lhν], l > 0 and right lead Fermi energy is kept to zero. Let q =eVdc/hν
holesexcitationsbelowtheFermiseaintheenergyrange be the number of electrons emitted per period due to
[ lhν,0], see Fig.2. As the current of electrons emitted the dc bias. The total number of left electrons partic-
−
by the leftcontactandableto createal-photonelectron ipating to noise is q + +∞lP = q +N . The num-
l=1 l e
excitations is (e/h)lhν = leν [48], the number of corre- ber of holes generatingPpartition noise is however re-
sponding electrons created per period is lPl. The total duced by the positive shift of the left Fermi energy. For
number of electron excitations generated per period is (n 1)hν < q < nhν the number of holes participat-
thus: ing−to noise is reduced to −n ( l q)P < N . The
l=−∞ − − l h
+∞ shot noise expression (9)Pis then changed by replacing
Ne = lPl (7) the |lhν| terms by |lhν +eVdc| as originally derived by
Xl=1 Lesoviketal. andlaterbyPedersenetal. [41,42]. Inab-
sence of ac voltage one recoversthe transport shot noise
and similarly the number of holes : STSN =2e2D(1 D)eV .
I h − | dc|
−1 The mixed situation with both V and V leads to
ac dc
Nh = ( l)Pl (8) interestingeffects due to the competitionbetweenPASN
−
l=X−∞ andTSN.Theyarebestdisplayedusingtheexcessnoise
where the pure TSN is subtracted from the total noise:
and, from Eq.(6), the PASN is :
SPASN =S0(N +N ) (9) e2 +∞
I I e h ∆S =2 D(1 D)[ P lhν+eV eV ] (10)
I l dc dc
h − | |−| |
To conclude this part we must emphasize that the en- l=X−∞
ergy distributionfunction fL(ε)= +l=∞−∞PlfL(ε−lhν) This would correspond to an experimental situation
depicted in Fig.2 should not be cPonfused with an in- wherethe noisemeasuredwithV onissubtractedfrom
f ac
coherent non-equilibrium population. If this were the the dc transport shot noise measured with V off while
ac
case, even for a perfect lead (D = 1) one would expect keeping the dc voltage on [28, 31].
a current noise associated with the population fluctua- Finally, we canwrite reduced units for the excess shot
tion dεf (ε)(1 f (ε)) as for thermal noise. In the noise ∆S =S0∆N where :
∝ L − L I I eh
presentRcasefthe termfs p∗lpl+k contributing to the non- +∞
diagonalpartofthedensitymatrixshouldnotbe forgot-
∆N = l+q P q (11)
ten. For example they are important for the multiparti- eh | | l−| |
l=X−∞
cle correlationsconsideredby Moskalets et al. [52] or for
theconceptofelectroncoherencedefinedinanalogywith represents at zero temperature the total number ∆N
eh
quantumopticsbyGrenieretal. [53,54]. Theseinterfer- ofphotoncreatedelectronsandholesnotcontributingto
ence terms contribute to make the low frequency Photo the TSN.
Assisted Shot Noise strikingly vanishing at unit trans- The derivative of the excess shot noise ∆S (or of
I
mission. This was experimentally shownby Reydellet et ∆N ) with respect to q shows remarkable singularities
eh
al. [48]. In this work, the theory of quantum partition eachtimeq =eV /hν isanintegern. Atthesingularity,
dc
noiseofphoton-createdelectron-holepairs,equations(6- the change of slope of the variationof ∆S with positive
I
9), was experimentally checked from weak to large ac (negative)V is proportionalto 2P ( 2P ), for n=0
dc −n n
6
6
and(2P 1)forn=0. VaryingV thusprovidesdirect 0,35
0− dc e
information on the Pl. Their direct measure is provided ol 0,3
by the second derivative of the noise [45–47] : h
n eh0,25
oN
∂2∆Ne−h/∂q2 =(2P0−1)δq,0+Xl6=02Plδq,−l (12) electrber ∆00,1,25 square wave (exact)
Applying a monochromatic sine wave to a contact, the ess num 0,1 ssqinuea wrea wveave (asymptotic)
noise singularities at dc voltage multiple of the fre- xc 0,05
quency has been observed in a diffusive metallic wire by e
0
Schoelkopf et al. [45] via the second derivative of the
0 2 4 6 8 10
noise, giving the P spectroscopy. Later the controlled
l electron number q
suppression by a dc voltage bias of hole or electron exci-
tation contribution to PASN has been discussed and ob-
FIG. 3: Excess electron and hole particle for sine and square
served in a Quantum Point Contact by Reydellet et al.
wavepulsescarryingq=nintegerchargesperperiodatzero
[48]. The extractedP quantitatively agreeswith the ex-
l temperature. The asymptotic log divergence of the square
pected Bessel function. Inferring the energy distribution
for large number, as defined in the main text, is shown as a
ofphotoexcitedelectronsfromshotnoisespectrocopyhas dashed line
been discussed in [46] and was comparedto tunnel spec-
troscopy by Shytov [47]. In the different context of the
energy resolved single electron and hole source using a with thermal excitations. This reduces the shot noise
mesoscopic capacitor, a similar shot noise spectroscopy which therefore misses these excitations [55].
wasproposedbyMoskaletsandBu¨ttiker[9]. Inthissame
context,theconceptofshotnoisespectroscopyandequa- Photo-assisted Current : Finally, we consider another
tion(12)hasbeenextendedtoproposeafulltomography photon-assisted effect, the photo-current, which also de-
of the electron and hole quantum states by Grenier et pends on the probabilities P . We consider a weakly
l
al. [54]. The Shot Noise spectroscopy is used in Section energy dependent transmission probability D(ε) D+
≃
IV to analyze the excitation content of periodic charge ε∂D/∂εandneglectthe energydependence ofthe trans-
voltage pulses. mission amplitude phase for simplicity. This situation
Finally, directly relevant to the topics of charge in- is known to give a rectification effect characterized by a
jection, the singularity at eV = nhν (q = n) corre- quadratictermin the low frequency I-Vcharacteristicof
dc
sponds exactly to the condition required for injecting n theconductor. Intermsofphoto-assistedeffectthisleads
electronsper period. Thisiswhythepresentationofperi- to a dc photo-current whose expression is :
odic charge injection using voltage pulses is particularly
relevant and enlightening in the frame of PASN. This e2 +∞
I = (hν)2∂D/∂ε l2P (14)
singularity is also a useful tool to characterize the car- ph h l
l=X−∞
rierchargeininteractingsystems. Thesuperconducting-
normaljunctionwhereconjugatedelectron-holeAndre’ev OnecanseethatI givesinformationontheP andcan
ph l
pairs carry twice the electron charge have been studied discriminate between different types of ac signal. How-
by Torres et al [56]. The finite frequency PASN noise of ever as shown in the next sections it is not as useful
chargee and chargee/3partitionedby a QPCwasstud- as shot noise as it can not distinguish between electron
iedinrespectivelytheintegerandthefractionalquantum and hole excitations and in particular can not identi-
Hallregimeat1/3LandaulevelfillingfactorbyCr´epieux fies the minimal excitation Lorentzian pulses. Indeed,
et al [57] and later by Chevallier et al. [58]. +∞ l2P = 1 T dϕ(t)/dt2 = (e/~)2 V (t)2 and
For comparison with realistic experimental situation, l=−∞ l T 0 | | h ac i
PIph gives the sameRinformation than a classical time av-
we provide the excess noise formula at finite electron eragingofV(t)2. Thisquantityisalsoproportionaltothe
temperature Te. The excess noise is ∆SI(Vac,Vdc,Te)= productionrateofheatinthecontactsI(t)V(t) V(t)2.
S0∆N (α,q,θ ) where : ∝
I eh e
II Integer periodic charge injection
+∞
l+q q
∆N (α,q,θ )= (l+q)coth( )P (α) qcoth( ) In this part, we give the expression of the probabil-
eh e l
2θ − 2θ
l=X−∞ e e ities Pl associated with various type of pulses carrying
(13) q = n charges per period: the sine, the square and the
andθ =k T /hν isthetemperatureinfrequencyunits. Lorentzian. From the shot noise we establish the hierar-
e B e
Atfinitetemperature∆N (α,0,θ )nolongerrepresents chy of the pulses in terms of number of e-h excitations
eh e
adirectmeasureofN +N astheexcitationscreatedin ∆N . Wealsocalculatethephoto-currentandconclude
e h eh
theenergyrangek T aroundtheFermienergyinterfere that shot noise is the right quantity able to character-
B e
7
(cid:13)(cid:14) (cid:20)(cid:15) (cid:26)(cid:3)(cid:7)(cid:8)(cid:9)(cid:4)(cid:27)(cid:28)(cid:12)(cid:9) of Lorentzian pulses where each pulse carry n electrons
(cid:6) and the Full Width at Half Maximum (FWHM) is 2W
(cid:12)(cid:13) (cid:19) (cid:29)(cid:30)(cid:31)(cid:10) (cid:15)(cid:13)(cid:20) writes:
(cid:11) (cid:26)(cid:3)(cid:7)(cid:8)(cid:9)(cid:4)(cid:27)(cid:28)(cid:12)(cid:9)
(cid:10)
(cid:9)(cid:4) (cid:18) (cid:29)(cid:30)(cid:31)(cid:10) (cid:15)(cid:13)(cid:15)! V +∞ 1
(cid:8) "(cid:24)(cid:6)(cid:12)(cid:7)(cid:8) V(t)= ac (16)
(cid:7) π 1+(t kT)2/W2
(cid:6)(cid:7) (cid:17) k=X−∞ −
(cid:5) "(cid:28)(cid:9)(cid:8)
(cid:3) where eV =nhν.
(cid:4) (cid:16) ac
(cid:3) ThetotalphaseΦ(t),includingthedcvoltagepartand
(cid:2)
(cid:1) (cid:15) using reduced time units u=t/T and η =W/T, gives:
(cid:15) (cid:20) (cid:16) (cid:25) sin(π(u+iη)) n
exp( iΦ(t))= (17)
(cid:8)(cid:21)(cid:8)(cid:5)(cid:4)(cid:7)(cid:3)(cid:9)(cid:10)(cid:9)(cid:6)(cid:22)(cid:23)(cid:8)(cid:7)(cid:10)(cid:24) − (cid:18)sin(π(u iη))(cid:19)
−
Remarkably, the above expression has only simple poles
FIG. 4: Photocurrent in arbitrary unit versuscharge per pe-
intheuppercomplexplane. ThisimpliesthatitsFourier
riod. Weseethatthisphoto-assistedeffectcannotprobethe
hierarchyofexcitationcontentofdifferenttypeofpulses. The transformcontains only positive frequencies. In physical
dashed and doted curvesare guidefor theeyes. terms, the total voltage only allows electrons to absorb
photons,i.e. itinducesaglobalupwardshift oftheFermi
sea with no hole creation. Going back to the decompo-
ize the cleanliness of the charge pulses. We consider a sition into dc and ac part of the voltage:
periodic excitation carrying a charge q = ne per period
T = 1/ν. This occurs if 1/T t+T V(t′)dt′ = nhν. It is eV(t)=nhν sinh(2πη) (18)
convenientto decompose the vRotltage into its mean value 2 sinh(πη)2+sin(πu)2
V =nhν and its ac part V [28, 31].
dc ac with eV = nhν, the amplitude probabilities associ-
dc
For the sine wave V(t) = V (1 cos(2πνt)), we
ac − ated with the only ac part are the Fourier transform of
have V = V = nhν/e. The emission and absorp-
dc ac exp( i(Φ(t) nνt)) is:
tion probabilities are given by integer Bessel functions − −
with Pl = Jl(n)2 where Pl is calculated putting in the 1 sin(π(u+iη)) n
time dependent phase only the Vac term and not the dc pl =Z du(cid:18)sin(π(u iη))(cid:19) exp(i(l+n)u) (19)
voltage part. Then, we can directly use the PASN shot 0 −
noise formula with dc bias (11) to calculate the number and the probabilities P = p 2. From the above consid-
l l
| |
of excitations (n > 0). Similarly, for the square wave eration on the analyticity of Φ we see that P = 0.
(l<−n)
V(t) = 2Vdc = 2hν/e for t [0,T/2] and V(t) = 0 for Note that this is consistent with vanishing negative fre-
∈
t [T/2,T] (mod T). The probabilities are given by quency Fourier component for the total phase Φ as the
P∈= 4 n2 for odd l n, P = 0 for even l n, p differs from them by of shift of l into l n.
l π2(l2−n2)2 − l − l −
andPn(n)=1/4. Boththesineandthesquarewavehave Forn=1wefindPl =0forl<−1,P−1 =exp(−4πη),
symmetric variation around zero voltage which implies andPl =exp( l4πη)(1 exp( 4πη))2. Theexpressionof
− − −
symmetric electron and hole excitation creation. With the Pl for integer charge number > 1 involves the same
P =P , equation (11) now writes : exponential factor exp( l4πη) and a Laguerre polyno-
l −l −
mial. The complete expression is given in the next part
+∞ as a special case of fractional charges q when q =n. Us-
∆Neh =2 (l−n)Pl (15) ing ∞l=−nlPl =0 and equation (11) gives:
l=Xn+1 P
∆N =0 (20)
eh
The aboveexpressionisusefulto providethe asymptotic
expression of ∆N at large n for square wave pulses: The excess shot noise vanishes and only the transport
eh
∆N (n)sq 1 (ln(n)+γ +2ln(2) 1) where γ is the shot noise of n electrons remains. The Lorentzianpulses
eh ≃ π2 −
Eulerconstant. Forthesinewavecase,nologdivergence arethereforeminimal excitation states as no excess elec-
occurs as P 1 (en/2l)l for l n. Figure 3 gives tron and hole excitations are created. The n electronic
l ≃ 2πl ≫
∆N versus the number n of injected electrons for the excitations are however not concentrated on the energy
eh
square and the sine wave. Clearly the square wave pulse window [E ,E + eV ] but they occupy states above
F F dc
creates more excitations than the sine wave. the Fermi energy with a weight exponentially decaying
WenowconsidertheLorentzianpulsewhosebehaviour on the scale ~/2W.
∼
is quite different. Strikingly we will see below that the We also compare the photo-current for different types
P associated to the ac part of the voltage pulse are ofpulses,seeFig.4. Forthesquareandthesinewavewith
l
zero for l < n. The expression for the periodic sum nelectronsthesum +∞ l2P is1and1/2respectively.
− l=−∞ l
P
8
h
e
N
(cid:1)(cid:2)(cid:3)(cid:3) ∆
r
e0,2
(cid:11) b
(cid:10) m
(cid:4)(cid:6)(cid:7)(cid:6)(cid:8)(cid:9) e nu square wave
(cid:4)(cid:5) ticl0,1 sine wave
(cid:3) ar Lorentzian W/T=0.1
(cid:1)(cid:2) s p
s
e
c
x
(cid:1) 0
(cid:1) (cid:2) (cid:3) (cid:4)
charge per period q
FIG. 6: Excess electron and hole particle for square, sine
FIG. 5: absorption l > 0 and emission l < 0 probabilities
and Lorentzian pulses carryingq charges perperiod. Forthe
corresponding respectively to electron and hole particle cre-
Lorentzian theratio W/T =0.1
ation for periodic Lorentzian pulses of width W/T =0.1 and
various charge q per pulse. The asymmetric spectrum for
q=0.99 reflectingthelackofholecreation quicklyleadstoa (cid:1)(cid:2)(cid:4) (cid:17)(cid:14)(cid:8)(cid:10)(cid:18)(cid:19)(cid:20)(cid:13)(cid:7)(cid:18) (cid:1)(cid:2)(cid:29) (cid:8)(cid:10)(cid:5)(cid:19)(cid:7)(cid:18)(cid:9)(cid:30)(cid:31)(cid:7)(cid:8)(cid:11)(cid:12)(cid:30)(cid:31) (cid:10)
symmetricspectrumforq≪1. Linesconnectingthediscrete (cid:21)(cid:22)(cid:23)(cid:11)(cid:24)(cid:11)(cid:1)(cid:25)(cid:1)(cid:26) (cid:21)(cid:22)(cid:23)(cid:11)(cid:24)(cid:11)(cid:1)(cid:25)(cid:1)(cid:26)
Pl valuesare guide for theeyes. (cid:1)(cid:1)(cid:25)(cid:25)(cid:3)(cid:4) (cid:1)(cid:2)(cid:28) (cid:1)(cid:1)(cid:25)(cid:25)(cid:3)(cid:4)
(cid:1)(cid:25)(cid:26)
(cid:1)(cid:2)(cid:3) (cid:1)(cid:2)(cid:27)
This is consistent with the hierarchy found using shot (cid:1)(cid:2)(cid:4)
noise which showed that the square contains more ener-
geticexcitations. FortheLorentzian,thesumishowever (cid:1)(cid:1) (cid:1)(cid:2)(cid:3) (cid:4) (cid:4)(cid:2)(cid:3) (cid:5) (cid:5)(cid:2)(cid:3) (cid:6) (cid:1)(cid:1) (cid:1)(cid:2)(cid:3) (cid:4) (cid:4)(cid:2)(cid:3) (cid:5) (cid:5)(cid:2)(cid:3) (cid:6)
non zero and given by coth(2πη) 1. It strongly in- (cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:10)(cid:8)(cid:11)(cid:12)(cid:10)(cid:8)(cid:13)(cid:14)(cid:15)(cid:11)(cid:11)(cid:16) (cid:5)(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:10)(cid:8)(cid:11)(cid:12)(cid:10)(cid:8)(cid:13)(cid:14)(cid:15)(cid:11)(cid:11)(cid:16)
−
creases with the sharpness of the Lorentzian shape. We
FIG. 7: Excess electron and hole particle number for
conclude that the photocurrent can not characterize the
Lorentzian (left) and rectangular (right) pulses carrying q
neutral excitation content of charge pulses.
charges per period and for different width to period ratio
W/T.
III Periodic injection of arbitrary charges
In this part, we calculate ∆N for arbitrary charge q
eh
carried per period. As in the previous part, we consider erty of the Fermi sea. A similar figure can be found in
thesquare,sineandLorentzianpulseshapesandalsoin- Ref. [28].
clude rectangular pulses. We show that ∆N oscillates Figure 7, left, shows how the excess particle number
eh
with q and is locally minimal for integer q = n. For the evolves for Lorentzian pulses of different width. We see
sine wave P = J (q)2 while for the Lorentzian case the that for W/T 0.1 the particle number becomes expo-
l l ≥
calculation of the P for non integer charges is less triv- nentially weak. Indeed the potential becomes close to a
l
ial. Physically one may expect that carryingnon integer constantvoltageV(t) Vdc. Forsmallwidthandhalfin-
≃
charges will involve more complex excitations. Mathe- teger charge the electron hole excitation number is large
matically, we immediately see that the term in the right but quickly decreases with q which contrast with the al-
hand side of equation (17) is no longer analytic in the most constant value found for the sine wave and the in-
lower plane when q, replacing n, is not integer. We thus creasing value for the square wave. For comparison, the
expectaproliferationofholeexcitationscontrastingwith right graph of figure 7 shows the excess particle content
theintegercase. Thecalculationisdoneintheappendix. ofdiscrete Dirac pulses (or rectangularpulses) ofsimilar
Figure5showsthe P forperiodicLorentzianpulsesof width W (the voltage pulses have amplitude h/eW for
l
width W/T =0.1. The absence of components for l 1 the time duration W and zero otherwise). Here again
when q = 0.99 1 strikingly contrasts with the case≤of the excitationcontentismuchlarger. Itincreaseswithq
q < 1. For sma≃ll q the P spectrum is almost symmet- whichcontrastswiththeLorentzianpulsebehaviorwhich
l
rical with l signaling nearly equal electron and hole pair definitely shows the lowest noise.
excitation creation. Figure 6 shows the evolution of the
excess particle number (or excess noise) versus q from 0 It is interesting to see how the temperature affects the
to 3 for square and sine wave pulses and for Lorentzian excess noise. As mentioned previously ∆N no longer
eh
pulses of width η = W/T = 0.1. We observe that, even measures the number of excess electron and hole quasi-
if the square and the sine do not provide minimal exci- particles with good fidelity. For simplicity we will keep
tations states, they do provide a minimum of excitations the same notation but call it now the effective excess
for integer charges. This seems to be a remarkableprop- particle number. For sine and squarewavesthere is only
9
1,6
sine wave
cle Neh1,4
ective excess partinumber ∆Neh d excess noise ∆ 001,,,6821 αα== eeVVaacc//hhνν== 12
eff ce 0,4
u
d
e 0,2
r
0
−5 −4 −3 −2 −1 0 1 2 3 4 5
q = eV / h v
FIG. 8: Effective excess electron and hole particle for sine dc
pulses carrying q charges per period and for different values
FIG. 10: Zero temperature excess partition noise versus dc
of the electron temperature Te .
voltagein reducedunitsforsineamplitudesα=1andα=2
corresponding to single and double charge pulses when re-
0,14 kBT / hν spectively q=1,2.
e 0.30
ctive excess particlnumber ∆Neh00000,,,,0001,14682 000000.....2211050505 leefiuxlnenpkccletatrdiioonntneodoatfnihnVdedIhc,a.octlhHeaiesmerxaepclltliiohttwauetdsdieoctnopbsaimarasaanmkqde=eitaneefrseVprαdectc/=hthreoνesPVicslaocnpf/royohlmνoof.nttgAhheeesr
e
eff 0,02 excessnoisevariationwithVdc orfromthefirstorsecond
0 noisederivativewithrespecttoVdc. Thecalculationsand
(cid:1) (cid:1)(cid:2)(cid:3) (cid:4) (cid:4)(cid:2)(cid:3) (cid:5) (cid:5)(cid:2)(cid:3) (cid:6)
graphsaredone anddisplayedatzeroandfinite temper-
charge per period q
ature.
Figures10and11showthevariationsof∆N versus
FIG. 9: Effective excess electron and hole particle for e−h
q at zero temperature for respectively a sine wave and a
Lorentzian pulses of width W/T = 0.1 carrying q charges
per period and for different electron temperature Te Lorentzian wave. For each case we show the curves for
twodifferent amplitudes V correspondingto α=1 and
ac
2. These amplitudes would correspond respectively to
one energy scale to compare with the temperature, i.e. single and double charge voltage pulses when q =α = 1
hν. For the Lorentzian case there are two energy scales or 2 respectively. For sine and square waves ∆Neh(α,q)
hν and ~/2W. Figure 8 shows the sine wave case. We is symmetric with q ( or Vdc ) as Pl = P−l. However
observe that the minima occur to higher q values. The it is asymmetric for the case of Lorentzian pulses. Such
effectisevenmorepronouncedforthe caseofLorentzian asymmetry is expected for pulses whose ac part of the
voltage pulses shown in figure 9 and is probably related voltage is not symmetrical with respect to zero voltage.
tothestrongerasymmetryof∆N withqaroundq =n. But more relevant and striking, for the Lorentzian case
eh
forη =W/T =0.1. Theoscillationsof∆Neh arequickly the excess noise is zero for q > α (or eVdc > nhν), a
dampedby the temperature andwhen kBTe >0.2hν are direct consequence of zero Pl for l <−n.
almost unobservable. The effect of finite temperature is shown on Figure 12
To end this section, it is worth to note that the non for a sine wave of amplitude α = 1. Finite temperature
integer charge considered here are injected in a non- calculations are also shown in figure 13 and figure 14 for
interacting Fermionic system (more precisely a good Lorentzian pulses of width W/T = 0.1 with respectively
Fermiliquidwhereinteractiongiveriseto Landauquasi- α=1 and 2. The relevant temperature needed to reveal
particles). Extension to fractional charges in Luttinger the asymmetry of ∆N with dc bias for a Lorentzian is
eh
liquids has been considered by [15]. The e/3 fractional givenbythewidth, thesmallerthe width,thehigherthe
charge of fractional quantum Hall edge states has been energy ~/W at which we find the contributions of the
considered by Jonckheere et al. [59]. positive P responsible for the long tail at negative volt-
l
ages. The other temperature scale is given by the ratio
IV Energy domain: spectroscopy of the e-h kBT/hν whichcontrolsthesmoothingofthesingularities
pairs excitations at integer q.
In this part we use the Shot Noise spectroscopy tool
discussed in I to analyze the periodic charged states in VTimedomain: shotnoisecharacterization us-
the energy domain. To do this, we compute ∆N as a ing collision of periodic charge pulses
e−h
10
1,6 1,6
Lorentzian wave h Lorentzian wave
∆Neh11,,24 α= eVac/hν= 1 e ∆Ne11,,24 kαB=Tee/Vhaνc/h=ν= 2
e α= eV /hν= 2 s 0
ois 1 ac noi 1 0.05
uced excess n 000,,,468 uced excess 000,,,468 00000.....1122305050
d d 0,2
e 0,2 e
r r
0
0 −5 −4 −3 −2 −1 0 1 2 3 4 5
−5 −4 −3 −2 −1 0 1 2 3 4 5
q = eV / h v
q = eV / h v dc
dc
FIG.14: Finitetemperatureexcesselectronandholeparticle
FIG. 11: Zero temperature excess partition noise versus dc
noiseversusdcvoltagein reducedunitsfor Lorentzian waves
voltageinreducedunitsforLorentzianamplitudesα=1and
of amplitude α=2.
α=2correspondingtosingleanddoublechargepulseswhen
respectively q=1,2.
Thetimeperiodicityimposesthatonlyinformationon
a discrete energy spectrum is available and full charac-
(cid:15)(cid:16)(cid:22)
terization must be completed with a dual time domain
sine wave
Neh(cid:15)(cid:16)(cid:21) α=eVac/hν= 1 informationwithinaperiod. Hereagain,shotnoiseisthe
∆ useful tool to provide this information. Inspired by the
oise (cid:15)(cid:16)(cid:20) kBTe/0hν= opticalHongOuMandel(HOM)correlationexperiment,
n
s 0.05 an electronic HOM analog can be built as a useful tool
es (cid:15)(cid:16)(cid:19) 0.10 to infer the time shape of wave-packets where electrons
c
x 0.15
e emittedfromtwocontactswitharelativetimedelaycol-
d (cid:15)(cid:16)(cid:18) 0.20
e 0.25 lide onthe scatterer. Aclearrelationbetweenshotnoise
c
du (cid:15)(cid:16)(cid:17) 0.30 and wavepacket overlap can be made for a single charge
e
r Lorentzian pulse. For other pulse shapes, it is expected
0−5 −4 −3 −2 −1 0 1 2 3 4 5 thatthecontributionoftheneutralexcitationcloudgives
an extra contribution to the shot noise.
q = eV / h v
dc
In a optical experiment, single photons are emitted
FIG. 12: Finite temperature excess electron and hole parti- from two distinct sources in each of the two input chan-
cle noise versus dc voltage in reduced units for sine wave of nels of a semi-transparent beam-splitter. Photon de-
amplitudes α=1. tectors are placed on the two respective outputs and a
time-delay between them, sizable with the photon wave-
packet, is introduced. For zero delay, Bose statistics im-
plies a constructive two particle interference where the
0,6
h Lorentzian wave two photons bunch and exit, at random, in one of the
e
e ∆N 0,5 αkB=Tee/Vhaνc/h=ν= 1 tawndo tohuetppuatrtcichlaenflnueclst.uaTtihonesc(otihnecindoeniscee) iesvednotusbalerdewzeitrho
nois 0,4 00.05 respect to a Hanbury Brown-Twiss (HBT) experiment
ss 0,3 0.10 where only one photon at a time arrives on the beam-
ce 0.15 splitter. The lattersituationis recoveredwhenthe delay
d ex 0,2 00..2205 τ is much longer than the size of the wave-packets. For
ce 0.30 intermediate time delays the noise variation is directly
u 0,1
d related to the overlap of photon wavepackets [60, 61]. A
e
r
similar experiment could be done with electrons with an
0
−5 −4 −3 −2 −1 0 1 2 3 4 5
artificial scatterer in the form of a controllable beam-
q = eV / h v splitter, i.e. a Quantum Point Contact. For zero time
dc
delay, Fermi statistics leads to a destructive interference
FIG.13: Finitetemperatureexcesselectronandholeparticle for the probability of finding two electrons in the same
noiseversusdcvoltage inreducedunitsfor Lorentzian waves output channel. In terms of charges counted by the de-
of amplitude α=1. tector(herethecontactsofaconductor)thereisalwaysa
chargearrivingineachcontactand consequentlyno cur-