Single-electron source: Adiabatic versus non-adiabatic emission

0.92 MB·English

by  Michael Moskalets

Single-electron source: Adiabatic versus non-adiabatic emission
Michael Moskalets,1,2 G´eraldine Haack,1,3 and Markus Bu¨ttiker1
1D´epartement de Physique Th´eorique, Universit´e de Gen`eve, CH-1211 Gen`eve 4, Switzerland
2Department of Metal and Semiconductor Physics,
NTU “Kharkiv Polytechnic Institute”, 61002 Kharkiv, Ukraine
3Dahlem Center for Quantum Complex Systems and Fachbereich Physik,
Freie Universita¨t Berlin, 14195 Berlin, Germany
(Dated: January 28, 2013)
We investigate adiabatic and non-adiabatic emission of single particles into an edge state using
an analytically solvable dynamical scattering matrix model of an on-demand source. We compare
3
adiabatic and non-adiabatic emissions by considering two geometries: a collider geometry where
1
two emitters are coupled to two different edge states and a series geometry where two emitters are
0
coupled to the same edge state. Most effects observed for adiabatic emitters also occur for non-
2
adiabatic emitters. In particular this applies to effects arising due to the overlap of wave-packets

n collidingataquantumpointcontact. SpecificallywecomparethePaulipeak(thefermionicanalog
a of the bosonic Hong-Ou-Mandel dip) for the adiabatic and non-adiabatic collider and find them
J to be similar. In contrast we find a striking difference between the two operating conditions in

5 the series geometry in which particles are emitted into the same edge state. Whereas the squared
2 averagechargecurrentcanbenullifiedforbothoperatingconditions,theheatcurrentcanbemade
to vanish only with adiabatic emitters.

]
l
l PACSnumbers: 72.10.-d,73.23.-b,73.50.Td,73.22.Dj
a
h
-
s I. INTRODUCTION a)Adiabaticemission
e
m
SES
Recent progress in the field of dynamical quantum 2�⌧
.
t transport1 opens new and fascinating perspectives for
a
m exploring and understanding mesoscopic and nanoscopic
conductors. With the implementation of an on-demand
-
d single-electronemitter2,3 notrelyingonelectron-electron b)Non-adiabaticemission
n interaction it is possible to address directly dynamic
o properties of a single-electron state in solids. The single- SES
c
particle nature of emitted wave-packets was demon-
[
strated using the noise measurements.4–7 To investigate ⌧
D
1 the coherence properties of emitted wave-packets an ap-
v proachbasedonthemeasurementofcurrentcorrelations
1 atabeamsplitter8,9 andanapproachbasedonthemea- FIG. 1: (Color online) Single-electron sources (SES) emit a
3
surement of current at the output of an interferometer10 train of electrons (black pulses) alternating with holes (red
0
have already been proposed. pulses) into an edge states (serving as an electronic waveg-
6
uide). Electron and hole pulses are well separated in time
. The state of an electron depends crucially on the way
1 and space. The shape of the single-particle wave-packets de-
0 it is emitted, see Fig. 1. In most experiments with such pendscruciallyonthewaythesourceisdriven: a)Adiabatic
3 high-speed single-electron sources – see, e.g., Refs. 11– emission: the SES is driven by a smooth periodic potential,
1 15, also the theoretical proposal in Ref. 16 and the the pulse as a function of time has a Lorentzian shape with
v: analysis of a single-electron capture in Ref. 17 – elec- width2Γτ. b)Non-adiabaticemission: theSESisdrivenbya
i trons are emitted from the quantum dot with energy pulsed periodic potential, the pulse as a function of time has
X far above the Fermi level. On the other hand in the- an exponential shape characterized by the dwell time τD.
r ory many effects were predicted for electrons emitted
a
adiabatically almost at the surface of the Fermi sea:
Theshot-noisequantization18–21;theshot-noisesuppres-
sion effect18,22; a two-particle interference and entangle- tally in Ref. 33. The properties of a single-electron state
ment generation23–25 interesting for quantum informa- generated by such a source are similar to those of the
tion applications26; a particle reabsorption27,28; the sup- state emitted adiabatically by a single lead mesoscopic
pression of a single-particle interference by collisions.29 capacitor.2
Recentlyalsosingleandfew-electronsourcesbasedonthe Our aim here is to answer the question of whether one
generation of Lorentzian voltage pulses applied to a bal- can expect similar effects with particles emitted non-
listicconductorasproposedinRefs.30,31,anddiscussed adiabatically or not. For this purpose we analyze the
in detail in Ref. 32, have now been realized experimen- single-electron source of Ref. 2 because it can operate
2
inbothadiabaticandnon-adiabaticemissionconditions.
Moreover it seems that its properties are well described
by a non-interacting theory.2,34,35 That makes it possi- τD. (3)
T (cid:29)
bletodeveloparelativelysimpleanalyticaltheorywhich
describes both operating conditions.
Note that to operate the source periodically, the energy
We use a non-interacting model2,36,37, in which the
level needs to be returned back to its initial position by
source consists of a single circular edge state, a Fabry-
applying the opposite potential ∆θ(t t ). Here, t
P´erotcavity,weaklycoupledtoalinearedgestate,which − − + +
denotes the time at which the emission of a hole starts.
plays the role of an electron waveguide. This analytical
The delay between subsequent potential steps should be
model is in good agreement with actual experiments. In
longerthanthedurationofacurrentpulse,t t τ ,
the weak coupling limit, the transparency of the quan- +− − (cid:29) D
to allow an electron emission to be completed: the emit-
tumpointcontactconnectingthecavityandtheelectron ted charge q =(cid:82)t+I(t)dt should be equal to an electron
waveguide is small, T 1. All relevant energies are t−
smaller than the Fermi(cid:28)energy, µ and the energy spec- charge, q =e. At time t+ a hole can be emitted.
trum of electrons can be linearized in the vicinity of µ. Second, the dwell time, τD, defines the condition of
That results in the equidistant spectrum of the cavity adiabaticornon-adiabaticemission. IfthepotentialU(t)
with level spacing ∆=h/τ defined by the time of flight, changes fast on the scale of τD, then we speak about
τ, around the circular edge state of the cavity. A metal- a non-adiabatic emission. In this case the shape of an
lic top-gate with potential U(t) periodically changes the emittedcurrentpulseisasymmetricandgivenbyEq.(1).
position of the quantum levels in the cavity. We assume In contrast, if U(t) changes smoothly compared to τD,
optimaloperatingconditions2,5,35 whichrequirethatthe thecurrentpulseispredictedtobesymmetric.18,20 Close
Fermi level is positioned in the middle of two levels of to t−, the corresponding current pulse Iad reads:
the cavity and the gate potential changes with ampli-
tude ∆. In this case only one level crosses the Fermi eΓ /π
τ
energy: When it raises above the Fermi level an electron Iad(t)= (t t )2+Γ2 . (4)
is emitted from the cavity into the waveguide, whereas − − τ
when the level sinks below the Fermi level an electron is
Nowtheduration2Γ ofacurrentpulseisdefinedbythe
absorbedbythecavityhenceaholeappearsinthestream τ
time of crossing,
of electrons within the waveguide. Such a source gener-
atesnoDCcurrentandisoftenreferredtoasaquantum
capacitor.34,38 δ
duTraotigoent aonf ainstiunigtilve-epeasrttiimclaetweaovfeb-potahcktehtewshealpooekanadt tthhee Γτ = (cid:12)(cid:12)edU/dt|t=t−(cid:12)(cid:12), (5)
current pulse emitted by the cavity. The sudden change
where 2δ is the width of a quantum level in the weakly
of a potential, eU(t)=∆θ(t t ), results in a transient
− coupled cavity. In the model used δ = T∆/(4π). For
current pulse (an expectation−value),2,20,37
eU(t)=(∆/2)cos(Ωt),whereΩ=2π/ ,andt =3 /4
−
T T
we find
e −t−t−
Ina(t)=θ(t t−) e τD , (1)
− τD T
Γ = . (6)
τ T 4π2
with highly asymmetric shape (we ignore a fine
structure37,39 on the scale of τ). Here e is the electron Remarkably, it was shown in Ref. 40 that the pulse du-
charge and θ(t) the Heaviside theta-function. The time ration 2Γ also sets the single-particle coherence time of
τ
t− denotesthetimeatwhichthepotentialchanges,lead- an electron emitted adiabatically. This shows that the
ing to the emission of an electron and the label “na” source, described by this analytical model, has no in-
stands for non-adiabatic. Indeed, as explained below, trinsic dephasing processes.10,41 This makes the emitted
such an emission process corresponds to non-adiabatic singleelectronstatesofparticularinterestforfurtherap-
emission conditions. The time τD, plications in quantum information processing.
Equation (4) is calculated assuming that27
τ
τ = , (2)
D
T
Γ τ . (7)
τ D
(cid:29)
thedwelltimeofanelectroninthecavity. Thereforethe
dwelltimesetsarelevanttime-scaleoftheproblemunder It means that the level of the cavity crosses the Fermi
consideration. First, the period of the gate voltage, sea level so slowly that an electron has enough time to
T
U(t) = U(t+ ), should be long enough for the driven leave the cavity once his energy becomes larger than the
cavity to workTas a single-particle source,2 Fermi energy.
3
From Eq. (7) it also follows that the width of a wave- Ie(t) = Ih(t) I(t) = Ie(t)+Ih(t) = 0. To clarify
− →
packet emitted adiabatically is much larger than the whetheritismerelyacompensationeffectorareabsorp-
width of a wave-packet emitted non-adiabatically. Ap- tion effect, additionally the heat generated by the two
parently with decreasing crossing time Γ , keeping the cavities was analyzed.28 It was shown that each particle,
τ
period large compared to τ , the shape of the pulse eitheranelectronorahole,carriesanexcessenergy(over
D
T
evolvesfromadiabatic,Eq.(4),tonon-adiabatic,Eq.(1). the Fermi energy)
For a level driven with a constant speed, an analysis de-
scribing this crossover can be found in Ref. 20.
¯h
The current pulses Ina(t), Eq. (1), and Iad(t), Eq. (4), Ead = 2Γ . (8)
have both similar and different features. On one hand, τ
they both carry a quantized charge. Therefore, we an-
This energy can be understood as the work done by the
ticipate that they both should show similar quantization
potential U(t) on the particle during its escape from the
effects18–21 andeffectsarisingduetotheoverlapofwave-
cavity. The particle starts to escape when its energy
packets18,22–25,29. We use below the shot noise suppres-
becomes equal to the Fermi energy. The time it takes to
sion effect as an example.
escape is the dwell time, τ =h/(T∆), given in Eq. (2).
On the other hand for some effects the shape of a D (cid:12) (cid:12)
wave-packet is crucial. As an example below we use We use Eq. (5) and find Ead = τD(cid:12)edU/dt|t=t−(cid:12). Notice
the energy of a particle in the cavity has an uncertainty
the effect of reabsorption27,28 predicted for the adia-
δ (the level width). This results in the uncertainty Γ of
τ
batic regime: If two cavities are coupled to the same
thetimewhenaparticlestartstoescapethecavity. That
edge state, then the electron emitted adiabatically by
in turns defines the width of the current pulse, Eq. (4).
one cavity can be reabsorbed by another cavity emit-
If two cavities emit an electron and a hole at different
ting a hole at the same time, see Fig. 2 (a). First of
times, then these two particles together carry the energy
all, in this regime the time-dependent current is zero,
2 . However, if an electron and a hole are emitted at
I(t) = 0.27 This current consists of two parts, electron, Ead
the same time (the time of flight between the cavities
Ie(t), and hole, Ih(t), which compensates each other:
should be trivially taken into account) then the extra
energy flowing out of the system is zero.28 Clearly this
means that an electron emitted by the cavity A and car-
a)Adiabaticemission ryinganenergy wasreabsorbedbythecavityB. This
A B I(t)=0 Ead
effectisparadoxical: Ononehand,infact,theholeemis-
IQ =0
DC sion is an electron absorption. On the other hand, the
cavityB canabsorbanyelectronsinthewaveguidepass-
ing it. Why does it absorb the electron emitted by the
cavity A?
b)Non-adiabaticemission
I(t)=0
A B Possibly this effect can be understood using time-
IDQC >0 reversal symmetry arguments. First, let us take only
one cavity and let it emit an electron. After that let us
reversetime. Apparentlytheemittedelectronwillbere-
absorbed. Importantly, the portion of the wave-packet
emittedlastwillbereabsorbedfirst. Nowletustaketwo
identical cavities and let us drive them with potentials
FIG. 2: The two-particle emitter consists of two cavities U1(t) and U2(t) related by the time-reversal symmetry,
specified by same parameters and coupled to the same chiral U2(t) = U1( t). Note with such potentials if the first
−
edge state. Electrons propagate along edge states shown as cavity emits an electron the second cavity emits a hole
blue solid lines. The cavity B is tuned to emit a hole at the andviceversa. Wecanexpectthesecondcavitytobean
timewhentheelectronemittedbycavityAreachescavityB. analogue of the time-reversal twin of the first cavity. To
(a)Adiabaticemission: whenthedrivenpotentialisslowand make such an analogy complete, the shape of the wave-
smooth, an electron emitted by the cavity A is completely
packet does matter. Because the second cavity will first
reabsorbed by the cavity B. The re-absorption process is the
reabsorb (if possible) the part of the wave-packet, which
time reversed emission process which is possible because of
wasemittedfirst. Incontrast,thetruetime-reversaltwin
the symmetric shape of the single-particle states. Both the
charge current I(t) and the DC heat flow IQ nullify. (b) will first absorb what was emitter last. If the shape of
DC a wave-packet is symmetric, as in the adiabatic emission
Non-adiabaticemission: thecavitiesaredrivenbypulsedpo-
tentialsandelectron-holepairsareemitted. Sincethispairis regime,(forthecorrespondingcurrentpulseseeEq.(4)),
neutral, the time-dependent current is zero, I(t) = 0. How- then there is no difference between what was emitted
ever,boththeelectronandtheholecarryenergy. Becauseof first and what was emitted last. Consequently the sec-
theasymmetricshapeofthepulsesreabsorptioncannotbea ondcavitycanplaytheroleofthetime-reversaltwinand
time-reversedemissionprocess. Thereisnoabsorptioneffect, reabsorb what was emitted by the first cavity.
the generated DC heat flow is not zero: IDQC >0. However, if the shape of a wave-packet is non-
symmetric, as in the non-adiabatic emission regime, (for
4
the corresponding current pulse see Eq. (1)), then there liesexactlyinthemiddlebetweenthetwoquantumlevels
is a striking difference between what was emitted first of the cavity:
and last. As a consequence what the second cavity sees
is different from what the time-reversal twin would see.
Tfreehrgeeinmrteelfyob.reoA,tshadwcieaavbsihatotieiwcsbaemneldoitwnto,onign-eattdhhieearbnaoatnni-caeldceaicavtbirtaoitenis-chweomolerikspsdaioiirfn-, Uj(t)=UU00,+∆/e, −tj−T</2t<<ttj+<,tj−, (9)
which carries no charge, I(t)=0, but carries a non-zero
eneTrhgey.paper is organized as follows: In Sec. II we dis- U0, tj+ <t<T/2.
cuss the shot noise quantization and the shot noise sup-
pressioneffectforelectronsemittednon-adiabatically. In
We are interested in the zero-frequency correlation
Sec. III the DC heat flow generated by the two-particle function44, , for currents outgoing to contacts 3 and
34
emitter working in the non-adiabatic regime is analyzed. P
4. These currents are due to the particles, electrons
WeconcludewithabriefdiscussioninSec.IV.Detailsof and holes, emitted by the sources at times tj +a and
thecalculationsareinappendices. InAppendixAwede- − T
tj +a , respectively. Here a is an integer.
rive the Floquet scattering amplitude for a cavity driven + T
To calculate we use the theory of Ref. 46 with the
by the pulsed potential. In Appendix B the zero fre- P34
centralobjectbeingtheFloquetscatteringmatrixofthe
quency correlation function for currents flowing through
source j, Sˆj. In our model this is a matrix in the energy
the electron collider circuit is found. In Appendix C we F
space with elements Sj(E ,E), where E = E +n¯hΩ,
discuss the DC heat flow generated by the two-cavity F n n
which are photon-assisted amplitudes for an electron in
emitter.
thewaveguidetopassthroughthecavityj andtoabsorb
(for n>0) or emit (for n<0) n energy quanta h¯Ω.
It is convenient to introduce the scattering ampli-
II. PAULI SUPPRESSION OF SHOT NOISE
tude Sj (t,E) whose Fourier coefficients are Sj (E) =
in in,n
Sj(E ,E),seeEq.(A3a). Forthemodelofthesourcein-
A mesoscopic electron collider is a circuit in which F n
troduced above we have (see Eq. (A19) and Appendix A
electrons incident from different leads can meet and
collide42–45 at a wave-splitter (a mesoscopic quantum for details of calculations):10
point contact). We consider a collider with two single-
electron emitters, A and B, weakly coupled to the two sin(cid:0)πnh¯Ω(cid:1)
chiral edge states, the electron waveguides, which are in Sj (E) = S(E) ∆ eiπn∆h¯Ω
in,n πn
turn coupled via a quantum point contact C (QPC C)
(10)
with transmission probability T , Fig. 3. For simplicity
C
wtineguspeartthicelecsavpiotisessibwlyitahtiddeiffnetirceanltptaimraems.etFerosurbumteetmaliltic-  1e−inΩnh¯tj−Ω + 1e+innΩh¯tj+Ω 
δ ∆ ∆ .
reservoirs, 1 to 4, are kept at the same electrochemical × n,0− ρ∗(E)ρ(En) 
potential µ and zero temperature.
Each source, j = A,B, is driven by the periodic,
Uj(t) = Uj(t+ ), potential with amplitude ∆/e > 0. Here S(E) is the stationary scattering amplitude of the
T
TheminimalvalueU ischosensuchthattheFermilevel cavity with potential U , see Eq. (A5) with U(t) = U ,
0 0 0
and ρ(E) = (cid:2)1 √1 Teiφ(E)(cid:3)/√T with φ(E) = π +
− −
2π(E µ)/∆aphasepickedupbyanelectronwithenergy
SESB −
4 E during one turn around the cavity.
SESA
A. Quantized noise of a single source
2
QPCC
1 Letusforamomentswitchoffoneofthesources. Then
we find, see Appendix B, Eq. (B7),
3
FIG. 3: (Color online) A mesoscopic electron collider. Two P3n4a =−2P0, (11)
single-electron sources, A and B, driven by pulsed potentials
emitelectronsintothewaveguides. Theemittedelectronscol-
lide at the quantum point contact C. The metallic reservoirs where =e2T (1 T )/ . This result coincides com-
0 C C
are labeled by the number from 1 to 4. pletelyPwith the one−foundTfor the cavity emitting parti-
clesadiabatically18 and, therefore, ittellsusthatatzero
5
temperature the zero frequency current correlation func- Dad(δt) = 1 4Γ2τ . (13b)
tion is independent of the parameters of both the cavity − (δt)2+4Γ2
τ
and driving potential as far as the cavity emits separate
particles. The quantity na, Eq. (11), at zero tempera- The reduction function D(δt) can also be calculated
P34
ture can be interpreted as due to the shot noise of two from the overlap of wave functions of colliding particles
indivisible quanta, one electron and one hole, emitted at the quantum point contact (the wave splitter).44,49
during each period and scattered at the quantum point If the two sources emit wave packets of the same shape
contact C to either the contact 3 or 4. Such a partition thentheoverlapcanbeformallyexpressedintermsofthe
noise was measured in Ref. 6. The deviation from the single-particle correlation function G(1)(t ,t ) discussed
e 1 2
theoretical prediction found is attributed to the effect of in Ref. 10. The reduction function then can be written
a non-zero temperature. as follows :
If the source emits N electron and N holes during the
period then the factor 2 in Eq. (11) is replaced by the
(cid:12)(cid:90) (cid:12)2
factor2N. Wealsonotethatthenoiseperparticle,−P0, D(δt)=1 v2 (cid:12)(cid:12) dtG(1)(t+δt,t)(cid:12)(cid:12) , (14)
is just the result of the partition noise of a dc-source − D(cid:12) e (cid:12)
biased with the voltage eV =h¯Ω, see e.g. Ref. 44.
Let us now consider the situation where the sources A
where the integral runs over the time interval when the
and B are both operating as shown in Fig. 3.
particles pass the QPC and v is the velocity of an elec-
D
tron evaluated at the Fermi energy µ. The factor v2 is
D
introduced to account for the wave-function normaliza-
B. Shot noise suppression effect
tion in such a way that at the complete overlap, δt = 0,
the reduction function D = 0. For the single-electron
If both sources work then the total shot noise de- source of Ref. 2 we found10 for the adiabatic emission :
pends crucially on whether two electrons (respectively
twoholes)emittedbythedifferentsourcespasstheQPC
Catdifferenttimesornot. Iftheparticlespassthequan- (1) 1 1
ttuhmenpthoientshcootntnaocitseCisa,t dniaffe=rent4tim,essi,n|cteA∓b−otthB∓|s(cid:29)ourτcDes, Ge,ad(t+δt,t)= πΓτvD (cid:16)1−it+Γτδt(cid:17)(cid:16)1+iΓtτ(cid:17), (15)
P34 − P0
together emit 4 particles, two electrons and two holes,
during each period. Due to the Pauli principle the and for the non-adiabatic emission
noise is reduced when particles arrive nearly simulta-
neously at the QPC. This leads to the Pauli peak for
−t+δt/2
the cross-correlator, see Fig. (4), or the Pauli dip in the G(1) (t+δt,t)=e−i∆2 δh¯tθ(t)θ(δt)e τD . (16)
auto-correlator. The Pauli peak is the fermionic analog e,ad τ v
D D
of the bosonic Hong-Ou-Mandel47 dip known in optics.
We describe the aforementioned reduction with a func-
ByinsertingEq.(15)andEq.(16)inEq.(14),werecover
tion D(δt) dependent on the difference of arrival times
respectivelythereductionfactorintheadiabaticregime,
δt = tA tB. The calculations presented in Appendix
∓ − ∓ Eq. (13b), and in the non-adiabatic regime, Eq. (12b).
B2 give for the non-adiabatic case: InFig.4weshow na and ad asfunctionofthetime
P34 P34
difference δt δt = δt (with δt = tA tB, χ = )
≡ − + χ χ − χ ∓
P3n4a = 2Dna(cid:0)tA tB(cid:1) 2Dna(cid:0)tA tB(cid:1),(12a) normalized on τD and Γτ, respectively. The Pauli peaks
− −− − − +− + are remarkably similar under the two limiting operating
0
P conditions despite the fact that the emitted states are
verydifferent. Howeverthetwooperatingconditionscan
Dna(δt) = 1 e−|τδDt|, (12b) be perhaps differentiated experimentally taking a closer
−
lookatthetopofapeak. Itissharpinthenon-adiabatic
caseandsmootherintheadiabaticcase. Weremarkthat
wherewechosethesourcestobeplacedthesamedistance
for different incident states the Pauli peak has an asym-
from the QPC C. Remember we assumed that the two metric shape in the non-adiabatic case48 but remains
cavities emit wave-packets of the same shape. For the symmetric in the adiabatic case18. Measurements on an
case of cavities emitting non-adiabatically wave-packets
electronic collider have now succeeded in detecting the
of different shape see Ref. 48. Pauli peak.50
Thebehavioroftheshotnoisediscussedabovequalita-
tivelyagreeswithwhatwepredictedforemittersworking
under adiabatic conditions:18
III. TWO-PARTICLE EMITTER
P3a4d = 2Dad(cid:0)tA tB(cid:1) 2Dad(cid:0)tA tB(cid:1),(13a) Inthissectionweconsiderthecircuitwithtwocavities
− −− − − +− + connected in series to the same edge state as shown in
0
P
6
0.0 forI(t)andEq.(8)for andfindfromEq.(18)thatthe
relevantresistance,R E R =h/(2e2),isthechargere-
P laxationresistancequaadn≡tumq38,51–54 whichappearsinthe
-0.2
linear response (admittance) of the cavity34 (a quantum
capacitor) at low-temperature. Under the non-adiabatic
-0.4
emission condition we use Eq. (1) for I(t) and Eq. (17)
for . From Eq. (18) we then find R = h/(e2T).
-0.6 E na
Thisisanordinarytwo-terminalresistanceofthe(single
channel)quantumpointcontactconnectingthecavityto
-0.8
the electron waveguide. This resistance was found ex-
perimentally in the optimal operation conditions2 and it
-1.0
appears in theory in both the high-temperature37 and
-10 -5 0 5 10
incoherent55 case. Therefore, we see that the factor R in
δt
Eq. (18) is not universal but it depends crucially on the
way how an electron is emitted.
FIG. 4: (Color online) The Pauli peak: The shot noise per NotealsothataccordingtoRef.56theadiabaticsource
particle of an electron collider, P = P34/(2NP0), as a func- isoptimalinthesensethatitdissipatestheminimalheat
tion of the time delay δt ≡ tA −tB = tA −tB normalized
− − + + per generated particle (an electron or a hole), Rad =Rq.
by Γ for an adiabatic emission, Eq. (13), (red dashed line)
τ In contrast the non-adiabatic source dissipates more en-
andbyτ foranon-adiabaticemission,Eq.(12),(blacksolid
D ergy, R R and thus it is non optimal. It is worth-
line). na (cid:29) q
while to mention that the criteria for an optimal pump
generating a DC quantized current56 works also in our
case for the emitter which produces a quantized AC cur-
rent, a sequence of alternating electrons and holes.
Fig. 2 (b). If both cavities emit particles at close times,
Now we come back to the two-particle emitter. If two
such a circuit serves as a two-particle emitter. Its work
cavitiesemittwoelectrons(twoholes)simultaneously,see
undertheadiabaticconditionofemissionwasanalyzedin
Appendix C2c, the energy carried by the pair of parti-
Refs. 27,28. Here we analyze it under the non-adiabatic
cles, ee = hh, is enhanced two times compared to the
condition of emission when each cavity is driven by the Ena Ena
condition of independent emission (when two separate
pulsed potential, see Eq. (9).
We calculate the DC heat flow, IQ , generated by the particles carry energy 2Ena):
DC
two-cavity source as a quantity able to differentiate vari-
ouscases: (i)Separateemissionofparticles,(ii)electron- ee =4 . (19)
holeemission,(iii)two-electron(two-hole)emission. The Ena Ena
details of calculations are given in Appendix C. Here we
discuss the results. The same two-time enhancement was found under the
Westartfromconditionswhenthecavitiesemitparti- adiabatic emission condition.28 The enhancement fac-
cles at different times, see Appendix C2a. The DC heat tor two can be understood using the Joule-Lenz law,
flow, IQ = 4 / , is due to four particles emitted by Eq. (18), since if the two particles are emitted simul-
DC Ena T
both cavities during each period. Each particle carries taneously, then the amplitude of a current pulse is dou-
an excess (over the Fermi sea level) energy bled. The reason that an energy enhancement can not
beavoidedfollowsfromthePauliblocking: ThecavityB
cannotemitaparticlewithenergy ifthereisaparticle
∆ E
= . (17) with the same energy (emitted by the cavity A). There-
na
E 2
fore,thecavityBhastoemitaparticlewithanenhanced
energy. Under the non-adiabatic condition the cavity B
The above result is clear, since the potential Uj(t), excitesanelectrontothenextavailablequantumlevelof
Eq. (9), moves a quantum level of the cavity j by ∆/2 the cavity and then an electron having energy ∆/2+∆
above(below)theFermienergywhenanelectron(ahole) leavesacavity. Thisscenarioagreeswithanon-adiabatic
is emitted. excitationofanelectroninadynamicalquantumdotob-
Thesingle-particleenergy canbealsounderstoodon served in Ref. 57. The direct spectroscopy of energies of
E
the base of the Joule-Lenz law, electrons emitted by the two-particle source can be done
in the same way as proposed in Ref. 58 for the single-
(cid:90) particle emitter.
=R dtI2(t). (18) The last operating condition we want to discuss is, see
E Appendix C2b, when one cavity emits an electron at
the time the other one emits a hole, see Fig. 2 (b). We
Here we integrate over a single-particle current pulse. find that the DC heat flow is not changed compared to
Under the adiabatic emission condition we use Eq. (4) thecasewhentheparticlesareemittedatdifferenttimes.
7
Thismeansthatnowoursourceemitselectron-holepairs through and heat released in the macroscopic conduc-
each carrying finite heat eh = 2 but zero charge. tor also holds for the single-particle excitation: The
Ena Ena
This is in striking contrast with the adiabatic emission square of the single electron (hole) current pulse inte-
casewhenaparticleemittedbyonesourceisreabsorbed grated over time gives the heat carried by this particle
by the other source thus nullifying both the charge cur- fromthesourceandreleasedinthemacroscopicreservoir.
rent and the DC heat flow.27,28 The nullification of the This law works under either adiabatic or non-adiabatic
DC heat flow can also be understood as a work transfer emission conditions though with different relevant resis-
between the external forces59,60 driving the two particle tances. However it is violated completely for the two-
sources. We remark that in the electron-hole emission particle source emitting an electron-hole pair under the
case the Joule-Lenz law, Eq. (18), holds under the adia- non-adiabatic emission condition.
baticcondition,whereasitseemstobeviolatedunderthe
non-adiabatic condition. Note that also the fluctuation-
dissipation relation is broken in the adiabatic operating
Acknowledgments
conditions when the two cavities generate electron-hole
pairs.28
G. H. acknowledges the support of the Alexander von
Humboldt Foundation in the framework of the Alexan-
der von Humboldt Professorship, endowed by the Fed-
IV. CONCLUSION
eral Ministry of Education and Research. This work was
supportedbytheSwissNationalScienceFoundationand
Here we developed an analytical Floquet scattering
Swiss National Center for Competence in Research on
matrix approach to describe the chiral single-electron
Quantum Science and Technology QSIT.
source driven by the pulsed potential and, therefore,
emittingparticles,electronsandholes,non-adiabatically.
We analyzed an electronic collider and the two-particle
emitter circuits with such sources and compared them Appendix A: The Floquet scattering matrix
to the analogous circuits with single-electron emitters
driven by the smooth potential, i.e., emitting particles In Ref. 37 the scattering amplitude S (t,E) for an
in
adiabatically. electronwithalineardispersionbeingscatteredoffaone-
We found that the collision of electrons approaching a dimensionalcircularedgestate(acavity)wascalculated.
quantum point contact from different sides, see Fig. 3, Thecavityisdrivenbytheuniforminspaceandperiodic
suppresses the shot noise. This effect is similar to the in time potential, U(t) = U(t+ ). This amplitude can
one found under the adiabatic emission condition18 and be presented as the sum of partTial amplitudes classified
it is due to the Pauli repulsion between the overlapping bythenumberofturnsqmadebyanelectronwithenergy
electrons, which forces them to go to the different out- E entering the cavity before leaving it at time t:
puts thus regularizing the outgoing particle flows. The
sharper suppressing factor, Fig. 4 (black solid line), is
due to the spatial asymmetry of traveling wave-packets ∞
(cid:88)
generated non-adiabatically. Sin(t,E)=r+t˜2 rq−1ei{qϕ(E)−Φin,q(t)}. (A1)
Amorestrikingdifferencewasfoundforacircuitcom- q=1
prising two cavities connected to the same edge state,
Fig. 2. If both cavities emit particles at close times such Here r/t˜is the reflection/transmission amplitude of the
a circuit serves as a two-particle emitter. The differ- quantumpointcontactconnectingthecavityandthechi-
ence between adiabatic and non-adiabatic emission con- ral one-dimensional conductor (an electron waveguide),
ditions appears when cavities emit particles of different ϕ(E) = ϕ(µ)+(τ/¯h)(E µ) is the kinematic phase ac-
kind,i.e.,onecavityemitsaholeatthesametimeasthe quiredbyanelectrondur−ingoneturninthecavity,Φ is
q
other cavity emits an electron. If particles are emitted the phase due to the time-dependent potential acquired
adiabatically, then the cavity B reabsorbs27,28 what was by an electron during q turns,
emitted by the cavity A, whereas in the non-adiabatic
emission mode a neutral electron-hole pair having a fi-
nite energy is emitted. This can be verified by looking t
(cid:90)
e
at the DC heat flowing out of the system: It is zero un- Φ (t)= dt(cid:48)U(t(cid:48)), (A2)
in,q
der the adiabatic emission condition but finite under the ¯h
non-adiabatic one. If both cavities emit particles of the t−qτ
same kind (two electrons or two holes) then under either
adiabatic or non-adiabatic emission conditions we found where τ is a duration of one turn. Details of the deriva-
doubling of heat compared to the case when all particles tion can be found in Ref. 1
are emitted at different times. The Fourier coefficients of S define the elements of
in
Surprisingly the Joule-Lenz law relating a current the Floquet scattering matrix (in the energy space),
8
It is instructive to look at the scattering amplitude
S(t,E) as a function of its arguments, Fig. 5. The nar-
T
(cid:90) dt rowchinevisualizesaquantumlevelinthecavitymoving
S (E+n¯hΩ,E)=S (E) einΩtS (t,E),
F in,n in under the action of the potential U(t). At zero tem-
≡
0 T perature the scattering amplitude at the Fermi energy,
(A3a) E = µ, is sufficient to calculate the emitted current:
I (t) = ie/(2π)S∂S∗/∂t.56,62,63 Thus the cross-section
a
−
onFig.5atE =0showsuswhenaquantumlevelcrosses
which are amplitudes corresponding to photon-assisted
theFermienergyand,hence,whenthecurrentpulsesap-
scatteringwithexchangeofnenergyquanta¯hΩbetween
pear. Importantly, the shape of peaks of the aforemen-
an electron and the driving field. Here Ω = 2π/ is the
tioned cross-section (at T 1) is similar to the shape of
T
frequency of the drive. For n > 0 the electron absorbs (cid:28)
a current pulse (up to a normalization factor). This can
energy whereas for n<0 it emits energy.
beeasilyshownifoneconsidersthescatteringamplitude
For some calculations the dual amplitude S (E,t),
out close to, say, the time of an electron emission, t−. It
which fixes the time when an electron enters the dot61, reads:18
is more natural to use. Its Fourier coefficients relate to
the Floquet scattering amplitudes in the following way:
S(t,µ)=(t t +iΓ )/(t t iΓ ). (A6)
− τ − τ
T − − −
(cid:90)
dt
S (E,E n¯hΩ)=S (E) einΩtS (E,t).
F out,n out
− ≡
0 T Then the current Ia(t), Eq. (4), is expressed in terms of
(A3b) the real part of the scattering amplitude as follows,
The amplitudes S (t,E) and S (E,t) are generally
in out
interrelated in a simple manner.61 In particular, for the e
I (t)= [1 ReS(t,µ)] . (A7)
model of interest here, the amplitude S (E,t) is given a 2πΓ −
out τ
by Eq. (A1) with Φ (t) replaced by
in,q
t+qτ
(cid:90)
e
Φ (t)= dt(cid:48)U(t(cid:48)). (A4)
out,q
¯h
t
Depending on the ratio between the time of a single
turnτ andacharacteristictimeduringwhichthedriving Re S(t,E)
{ }
potentialU(t)changeswedistinguishadiabaticandnon-
adiabatic operating conditions.
1. Adiabatic emission
IfthepotentialU(t)changesslowly,i.e.,themaximum
relevant frequency is much smaller than τ−1. Thus we
can keep U(t(cid:48)) constant while integrating over time in (E µ)/�
Eq. (A2) or Eq. (A4). We arrive at the frozen1,56 scat- �
t/
tering amplitude, S(U[t];E) S (t,E)=S (E,t):
≡ in out T
1 √Re−iφ(U[t];E)
S(U[t];E)= ei(φ(U[t];E)+θr) − ,(A5a)
− 1 √Reiφ(U[t];E)
−
FIG. 5: (Color online) Adiabatic emission: The real part of
where S(t,E),Eq.(A5a),isshown. Thetimetismeasuredinunits
of the period of the drive T = 2π/Ω. The energy E is mea-
suredfromtheFermienergyµinunitsofthelevelspacing∆.
E µ eU(t)
φ(U[t];E)=θ +ϕ(µ)+2π − 2π ,(A5b) OnlyoneperiodforbothtandE isshown. Thetransmission
r
h/τ − h/τ probability of a quantum point contact connecting the cav-
ity is T = 0.5. Other parameters correspond to the optimal
√R and θ are the absolute value and the phase of the operating conditions.
r
reflection amplitude, r =√Rexp(iθ ).
r
9
2. Non-adiabatic emission
with T = 1 R a transmission coefficient and φ (E) =
k
−
The periodic pulsed potential [U(t)=U(t+ )], φ(E;Uk), k =0, 1.
T ThequantityδS (t,E)characterizeshowS (t,E)de-
in in
 viates from the stationary scattering amplitude S(U;E)
U , /2<t<t ,
U0, −t T<t<t , − Uco(rtr)e.spTohnidsindgevitaotiotnheexiisntsstaonntlyanaefotuers thpeotpenotteianltiaUl wa=s
U(t)= 1 − + (A8) changed, t>0, and it decreases,
U0, t+ <t<T/2, − t
δSin e 2τD , t τ, (A12)
∼ (cid:29)
isanexample,relevanttoexperiment,2 leadingtoanon-
adiabaticemission. Thenon-adiabaticbehavioriscaused with a characteristic time
by the potential jumps, which formally have to be sharp
onthescaleofτ. Beforecalculatingthescatteringampli-
τ =τ/ln(1/R). (A13)
tudeforthepulsedpotentialU(t), Eq.(A8), weconsider D
the following auxiliary problem.
An analogous calculation gives
a. Single-step potential S (E,t)=S(U[t];E)+θ( t)δS (E,t). (A14a)
out out
−
Let us find scattering amplitudes for a cavity driven
where within each interval (N +1)τ <t< Nτ,
by the single-step potential,
− −

U0, t<0, δSout(E,t)=eiθrT RN2
U(t)= (A9)
(A14b)
U , t>0.
1 (cid:26) ei(N+1)φ0(E) e−i2πτtehδ/Uτ ei(N+1)φ1(E) (cid:27).
With this potential the time-dependent phase, say, × 1 √Reiφ0(E) − 1 √Reiφ1(E)
− −
Φ (t), Eq. (A2), can be easily calculated:
in,q
IncontrasttoS thescatteringamplitudeS (E,t)de-
in out
2πqheU/τ1 , t>qτ, vathtiaettiedmseevfsriaoptmiroenctheδdeSinfrgoztehdneeccsachyaastntegexerpinoogfnaeanmptiopatlleiltynutdwiaeil,t.hSAa(tEc|;th|Ua(cid:29)r[ta]c)τ-,
Φ (t)= 2πt eδU +2πqeU0 , 0<t<qτ, (A10) out
in,q τ h/τ h/τ teristic time τ .
2πqeU0 , t<0. D
h/τ
b. Optimal operating conditions
Here δU = U U . To sum up over q in Eq. (A1) we
1 0
−
note that for a given t < 0 we have to use U q. In
0 The calculations are simplified greatly for the optimal
∀
contrast, as far as Nτ <t<(N +1)τ we have to use U1 operating conditions2,35 which lead to the emission of a
for q N and a more complicated phase for q >N. We
single electron and hole during each period. One con-
≤
can represent a time-dependent scattering amplitude as
dition is that the potential changes by exactly one level
follows:
spacing ∆=h/τ,
S (t,E)=S(U[t];E)+θ(t)δS (t,E). (A11a)
in in eδU = χ∆, (A15a)
−
Here θ(t) is the Heaviside theta-function, θ(t) = 0 for
t < 0 and θ(t) = 1 for t > 0. The frozen amplitude where χ = . In addition the Fermi energy should lie
∓
S(U[t];E) is given by Eq. (A5), and δS is given within exactly in the middle of two neighboring quantum levels
in
each interval Nτ <t<(N +1)τ (N =0,1,2,...) as of the cavity,
δSin(t,E)=eiθrT RN2 θr+ϕ(µ)=π. (A15b)
(A11b)
(cid:26) ei(N+1)φ1(E) e−i2πτtehδ/Uτ ei(N+1)φ0(E) (cid:27), With these conditions the frozen amplitude becomes in-
× 1−√Reiφ1(E) − 1−√Reiφ0(E) dependent of time, S(E) ≡ S(U0;E) = S(U1;E). In
10
other words, mere shaking of a potential would not c. Pulsed potential
disturb an electron system. What causes a dynami-
cal (transient) response is electrons entering and leaving Now we come back to the periodic pulsed potential,
the cavity at different potentials. That is described by Eq.(A8). FordefinitenessweuseeU >eU . Wesuppose
1 0
δSχ(E,t) δSin(t,E)=δSout(E,t), a drive with a delay between the potential steps that
≡
is long compared to the difference between absorption
and emission times which in turn are taken to be long
(cid:16) (cid:17)
RN2ei(N+1)φ(E) 1 eχi2πτt compared to the dwell time,
δS (E,t)=eiθrT − ,
χ
1 √Reiφ(E)
−
(A16a) >t+ t− τD. (A17)
T − (cid:29)
Thereforethetransientbehaviorcausedbyonepotential
with
step vanishes completely before the next step appears.
Thispermitsustousetheresultsforasingle-steppoten-
tial and get (t [ /2, /2])
E µ ∈ −T T
φ(E)=π+2π − . (A16b)
∆
S (t,E) = S(E)+θ(t t )δS (E,t t )
in − − −
For simplicity all calculations from here on are done for − −
+θ(t t )δS (E,t t ),
optimal operating conditions. However the formalism − + + − +
used in itself is not restricted to the optimal conditions, (A18)
Eqs. (A15a) and (A15b). S (E,t) = S(E)+θ(t t)δS (E,t t )
out − − −
− −
The real part of the scattering amplitude δSχ, +θ(t+ t)δS+(E,t t+).
Eq. (A16a), is shown in Fig. 6. Its overall behavior in − −
time reflects the asymmetry of the emitted state, in par-
ticular, of the current pulse Ina(t), Eq. (1). In addition Note that at the time t− an electron is emitted by the
it illustrates that the largest variations of the scattering driven cavity whereas at the time t+ a hole is emitted.
amplitude occur at the energy of the quantum state in
the cavity.
d. Fourier coefficients
To calculate the Floquet scattering amplitudes, see
Eqs. (A3), we need the Fourier transformation of
Re �S (t,E) Eq. (A18). To integrate over time we use the following
in
{ }
trick: Since δS is constant over an interval of duration
χ
τ, we integrate over this interval the factor exp(inΩt)
only and then sum up over N. Under the condition of
Eq.(A17,)thesumoverN runsfrom0to . Asaresult
∞
we find:
(cid:40) (cid:41)
einΩt− einΩt+
S (E) = S(E)δ A (E) + ,
in,n n,0− n 1 nh¯Ω 1+ nh¯Ω
(E µ)/� − ∆ ∆
�
(A19a)
t/⌧D Sout,n(E) = Sin,n(E n¯hΩ).
−
FIG. 6: (Color online) Non-adiabatic emission: The real Here
part of δS (t,E), Eq. (A16a) with χ = −, is shown close
in
to the time of emission of an electron, t ∼ t− . The time sin(cid:0)πnh¯Ω(cid:1)
t is measured in units of the dwell time τD = τ/T. The An(E)=S(E)eiπn∆h¯Ω ∆
energy E is measured from the Fermi energy µ in units of πn
the level spacing ∆. Only one period for E is shown. The (A19b)
visibleripplesreflectoscillationsintimewiththeperiodofτ. T
The transmission probability of the quantum point contact (cid:16) (cid:17)(cid:16) (cid:17),
connectingacavityisT =0.5. Otherparameterscorrespond × 1 √Re−iφ(E) 1 √Rei(φ(E)+nΩτ)
− −
to the optimal operating conditions.
and δ is the Kronecker symbol.
n,0