Coulomb-mediated antibunching of an electron pair surng on sound Junliang Wang1Hermann Edlbauer1Aymeric Richard1Shunsuke Ota2 3Wanki Park4Jeongmin Shim4Arne Ludwig5Andreas D. Wieck5Heung-Sun Sim4Matias Urdampilleta1Tristan Meunier1Tetsuo Kodera2

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Coulomb-mediated antibunching of an electron pair surﬁng on sound

Junliang Wang,1Hermann Edlbauer,1Aymeric Richard,1Shunsuke Ota,2, 3 Wanki Park,4Jeongmin Shim,4Arne

Ludwig,5Andreas D. Wieck,5Heung-Sun Sim,4Matias Urdampilleta,1Tristan Meunier,1Tetsuo Kodera,2

Nobu-Hisa Kaneko,3Hermann Sellier,1Xavier Waintal,6Shintaro Takada,3and Christopher B¨auerle1, ∗

1Universit´e Grenoble Alpes, CNRS, Grenoble INP, Institut N´eel, F-38000 Grenoble, France

2Department of Electrical and Electronic Engineering,

Tokyo Institute of Technology, Tokyo 152-8550, Japan

3National Institute of Advanced Industrial Science and Technology (AIST),

National Metrology Institute of Japan (NMIJ), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8563, Japan

4Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, South Korea

5Lehrstuhl f¨ur Angewandte Festk¨orperphysik, Ruhr-Universit¨at Bochum,

Universit¨atsstraße 150, D-44780 Bochum, Germany

6Universit´e Grenoble Alpes, CEA, INAC-Pheliqs, F-38000 Grenoble, France

(Dated: October 10, 2022)

Electron ﬂying qubits are envisioned as potential information link within a quantum computer [1],

but also promise – alike photonic approaches [2] – a self-standing quantum processing unit [3, 4].

In contrast to its photonic counterpart, electron-quantum-optics implementations are subject to

Coulomb interaction, which provide a direct route to entangle the orbital [5, 6] or spin [7–10] degree

of freedom. However, the controlled interaction of ﬂying electrons at the single particle level has

not yet been established experimentally. Here we report antibunching of a pair of single electrons

that is synchronously shuttled through a circuit of coupled quantum rails by means of a surface

acoustic wave. The in-ﬂight partitioning process exhibits a reciprocal gating eﬀect which allows us

to ascribe the observed repulsion predominantly to Coulomb interaction. Our single-shot experiment

marks an important milestone on the route to realise a controlled-phase gate for in-ﬂight quantum

manipulations.

Collision experiments provide fundamental insights

into the quantum statistics of elementary particles.

A prime example is the well-known Hong-Ou-Mandel

(HOM) interferometer [11] where two incident particles

are simultaneously scattered at a beam splitter. For

the case of indistinguishable photons, they bunch due

to Bose-Einstein statistics leading to an increased prob-

ability of the two particles arriving at the same detector.

For colliding electrons, on the other hand, antibunch-

ing occurs because of two coexisting mechanisms – the

Pauli exclusion principle and Coulomb repulsion – caus-

ing coincidental counts at the two detectors. In collision

experiments performed within the two-dimensional elec-

tron gas (2DEG) in a solid-state device, it is typically

assumed that Coulomb interaction is negligible due to

screening by the surrounding Fermi sea and, therefore,

Pauli exclusion is the dominant repulsion mechanism [12–

14]. Coulomb interaction provides however a direct route

for orbital entanglement [15], enabling experiments on

quantum nonlocality [16, 17] and the implementation of

a two-qubit gate for single ﬂying electrons [3, 4, 7, 18].

Whether such a controlled Coulomb interaction is exper-

imentally feasible and suﬃcient for orbital entanglement,

however, has not yet been demonstrated.

In this work, we address this question by implement-

ing the HOM interferometer in a depleted single-electron

circuit with coupled quantum rails. In the absence of the

Fermi sea along the transport paths, the screening eﬀect

∗corresponding authors: christopher.bauerle@neel.cnrs.fr

is expected to be signiﬁcantly reduced. We move a pair of

isolated electrons from two diﬀerent input ports towards

a tunnel-coupled region employing the conﬁnement po-

tential accompanying a surface acoustic wave (SAW) [19–

21]. In order to make the co-propagating electron pair

collide, we tune the transmission in this coupling region

such that the individual electrons are equally partitioned

towards the two outputs. We synchronise the transport

via triggered-sending processes [22] that we apply inde-

pendently on each electron source. This control of the

time delay between the two electrons allows us to con-

trast the full-counting statistics of the single-shot scatter-

ing events with and without interaction. Comparing our

experimental results to numerical simulations, we iden-

tify the major cause of in-ﬂight interaction and assess its

applicability for orbital entanglement.

The experimental setup consists of a surface-gate-

deﬁned circuit hosting a pair of coupled quantum rails

(see Fig. 1a). The SAW is emitted from a regular inter-

digital transducer (IDT) that is located around 1.6 mm

to the left of the single-electron circuit. It travels with

a speed of 2.86 µm/ns and has a wavelength of 1 µm.

When propagating along the nanoscale device, the SAW

allows shuttling of a single electron between distant quan-

tum dots (QD) [19, 20] that are located at the respective

ends of the coupled transport paths (see Fig. 1b). The

presence of the electron in a QD is traced via the cur-

rent ﬂowing through a nearby quantum point contact

(QPC) as a non-invasive electrometer. Enhancing the

SAW potential modulation up to a peak-to-peak ampli-

tude of 42±13 meV (see Appendix A), we ensure that the

arXiv:2210.03452v1 [cond-mat.mes-hall] 7 Oct 2022

transported electrons are strongly conﬁned during ﬂight

[23]. The two injection paths of our single-electron cir-

cuit converge to a tunnel-coupled wire (TCW). Over a

length of 40 µm, the two quantum rails in this region are

only separated by a narrow barrier that is deﬁned via a

30-nm-wide surface gate. Before being projected to the

upper (U) or lower (L) output channels, a transported

electron experiences thus a ﬂight-time of ≈14 ns in this

double-well potential.

A key requirement to realise the HOM interferometer

is to control the delay between individual electrons from

the two source QDs. To synchronise the sending process,

we apply a 90-ps voltage pulse on the plunger gate of

each QD to trigger SAW-driven electron transport on

demand [22]. In order to characterise the eﬃciency of

this triggering approach, we ﬁrst tune the voltages on

the surface gates into a condition where the two quantum

rails are decoupled. Sweeping the delay of the sending-

trigger pulse with respect to the arrival time of the SAW,

we observe distinct peaks in the transfer probability as

shown in Fig. 2a. The spacing of the peaks coincides

with the SAW period TSAW which indicates that we are

able to address a speciﬁc minimum of the SAW train

to transport the electron. The increase of the transfer

probability from 0.35 ±0.24% to 99.77 ±0.25% for both

source QDs demonstrates our ability to synchronise the

electrons with high accuracy.

To implement the analog of an optical beam splitter for

SAW-driven electrons [24], we investigate the partition-

ing of a single ﬂying electron through the coupled quan-

tum rails. For this purpose, we lower the barrier potential

of the TCW such that the electron sent from the up-

per (lower) source QD can transit into the lower (upper)

quantum rail with probability PU→L(PL→U). To control

the in-ﬂight partitioning, we use the side-gate voltages

VUand VLto induce a detuning ∆ = VU−VLbetween

the two channels. Figure 2b shows transfer probabilities

for VB=−1.10 V as we detune the double-well potential

within the TCW. We observe a gradual transition which

follows a Fermi function:

Pi→L(∆) = P10(∆) = 1

exp ∆−∆S

σ+ 1,(1)

with i∈[U,L]. Here, ∆Sindicates the detuning for 50%

transmission – that is ideally zero for a symmetric device

–, and σis the characteristic transition width which is

related to the energy distribution of the electron. Com-

pared to previous work [22], we observe a reduced σdue

to mitigated excitation, which we attribute to the in-

creased SAW conﬁnement [23] and the improved surface-

gate design at the transition region to the TCW employ-

ing realistic electrostatic potential simulations [25]. To

maximise the probability of interaction, it is necessary

to prepare an electron pair with similar energy, and thus

equal in-ﬂight partitioning in the coupling region. We

ﬁnd (see Appendix B) that such condition is satisﬁed for

VB&−1.15 V.

Before carrying out the collision experiment, we tune

the partitioning of each individual ﬂying electron to be

50% in the coupling region via the voltages VU=VL=

−1.00 V and VB=−1.15 V. Employing the delays, tU

and tL, of the sending triggers of the upper and lower

source QDs, we control the relative timing between the

two transported electrons as sketched in Fig. 3a. In par-

ticular, we ﬁx the delay of the upper electron (tU= 0)

and step the delay for the lower triggering pulse in mul-

tiples of the SAW period (tL=k·TSAW where k∈Z) in

order to address diﬀerent SAW minima for transport. If

the electrons tunnel without experiencing the presence of

the other, the probabilities at the detectors would follow

a Poissonian distribution. In this case, we expect 50%

for the probability P11 ≡PUL→UL to ﬁnd one electron

in both the upper and the lower detector, and, accord-

ingly, P20 ≡PUL→LL and P02 ≡PUL→UU to be 25%.

Figure 3b shows such a measurement of the antibunch-

ing probability P11 as a function of the trigger delay tL

of the electron sent from the lower source QD. We ﬁnd

P11 ≈50% as expected when the two electrons are trans-

ported in diﬀerent SAW minima (tL6= 0). As the sending

triggers are synchronised (tL= 0) and the electron pair

is thus sent within the same SAW minimum, we observe

in contrast a signiﬁcant increase of P11 up to 80% re-

sulting from the interaction between the two electrons.

The distinct P11 peak underpins our expectation that

the ﬂying electrons remain within the initially addressed

SAW minimum during transport. Our observation fur-

ther indicates that beyond a distance of one SAW period

(≈1µm) the interaction of the electron pair gets negli-

gible.

In order to investigate the nature of the antibunch-

ing eﬀect – Pauli exclusion or Coulomb repulsion –, we

perform the partitioning experiment by varying the de-

tuning of the TCW, from the previous symmetric case

to the fully detuned situation with the electron pair

forced in the same channel. As reference, we ﬁrst con-

sider the non-interacting case shown in Fig. 4a by the

semi-transparent data obtained with the two electrons

travelling in diﬀerent SAW minima (∆t= 5 ·TSAW).

The observed probabilities are a direct consequence of

the partitioning distribution of the individual electrons

shown in Fig. 2b. Since the electrons do not interact, the

probability to ﬁnd both electrons in the lower channel

is simply the product of the single-electron cases, P20 =

PU→L·PL→L. Similarly, we have P02 =PU→U·PL→U, and

P11 =PU→L·PL→U+PU→U·PL→L= 1 −P20 −P02 due

to charge conservation. The semi-transparent lines indi-

cate the course resulting from this non-interacting model

that shows good agreement with the experimental data.

As we send the two electrons synchronously within the

same SAW minimum (non-transparent data), we observe

a change in the functional course of P20 and P02 leading

to a signiﬁcant increase and broadening of P11 compared

to the non-interacting case.

To ﬁnd out the physical eﬀect that causes the observed

in-ﬂight partitioning of the two interacting electrons,

we focus on the Coulomb potential that is experienced

by one electron due to the presence of the other. We

perform three-dimensional electrostatic simulations [25]

taking into account the geometry and electronic prop-

erties of the presently investigated nanoscale device –

see Methods. For the sake of simplicity, we consider

a symmetric conﬁguration of the surface gate voltages

(VU=VL=−1.00 V and VB=−1.15 V). Figure 4b

shows the result of an electrostatic simulation (dotted

line) by adding the density of an electron-charge in the

lower or upper rail. We observe that the double-well

potential is tilted by the presence of the electron with

an induced asymmetry of 3.7 meV, which can be repro-

duced by considering an eﬀective gate-voltage detuning

δ≈18.5±0.4 mV (solid line). Therefore, these numeri-

cal results indicate that the electron in the lower rail (L)

experiences a potential landscape that is eﬀectively de-

tuned due to the presence of the electron in the upper

rail (U), and vice versa.

To model the two-electron partitioning process with

interaction, we include such a reciprocal electron-gating

eﬀect (parameterized by δ) in the single-electron parti-

tioning distribution (see Eq. 1) as Pi→j(∆ ±δ) where

i, j ∈[U,L]. In combination with the Bayes’ theorem,

we derive – see Appendix C – the following expression:

P20(∆) = PL→L(∆ + δ)·PL→L(∆ −δ)

PL→L(∆+δ)

PU→L(∆+δ)+PL→L(∆ −δ)−PL→L(∆ + δ),

(2)

which allows us to construct P02(∆) and P11(∆). The

solid lines shown in Fig. 4a indicate the courses of P20,

P11 and P02 resulting from Eq. 2 with δ= 18.5 mV,

and PL→Land PU→Lextracted from the individual non-

interacting partitioning data. Since the Bayesian model

is solely based on electrostatics, the excellent agreement

with the experimental data without adjustable parame-

ters indicates that the Coulomb interaction is the ma-

jor source of the increased antibunching probability. We

further verify this conclusion by performing exact diago-

nalization calculations – see Appendix D – in which the

long-range Coulomb repulsion is taken into account, and

ﬁnd a good quantitative agreement both on the increased

antibunching probability and on the increased transition

width.

Having identiﬁed Coulomb interaction as the main

cause of antibunching for a speciﬁc conﬁguration, we now

check whether this assertion also holds when the barrier

potential is changed. For this purpose, we investigate

the antibunching probability P11 at a symmetric detun-

ing (∆ = 0) as a function of the barrier gate voltage

VB(see Fig. 5a). Focusing on the non-interacting case

(semi-transparent data), increasing the barrier height

(VB<−1.15 V) reduces the transmission of each elec-

tron to the opposite channel, leading to a gradual in-

crease of P11 above 50% and up to 100% when both

rails are fully separated. This regime of barrier volt-

ages with progressively decoupled rails is therefore not

suitable to investigate the inﬂuence of the electron-pair

interaction solely. When the electron pair is transported

synchronously (black data), we observe a similar increase

of P11 in this regime starting from the optimal value

of 80% discussed previously. For lower barrier heights

(VB>−1.15 V; grey area), the antibunching probabil-

ity P11 decreases gradually below 80% while the non-

interacting data is saturated at 50%. To model this de-

pendence on the barrier height, we extract the Coulomb-

equivalent detuning δfrom two-electron partitioning ex-

periments performed at three diﬀerent barrier voltages

VB∈ {−1.150,−1.125,−1.100}V (see Appendix E). Us-

ing a linear course of δ(VB), the simulation from the

Bayesian model (red) shows excellent agreement with the

experimental data. The quantitative comparison indi-

cates that Coulomb interaction is dominant for a wide

range of barrier voltages.

Next, we address the question of what limits the max-

imum observed antibunching probability at P11 ≈80%.

A possible explanation could be the occupation of ex-

cited states by the ﬂying electrons [22]. If their energy

overcomes the Coulomb repulsion, P11 is expected to be

reduced. To check this possibility, we numerically inves-

tigate the eﬀect of excitation in the antibunching process

using the Bayesian model – see Appendix F. We ﬁnd that

P11 is expected to exceed 99% if the eﬀective thermal ex-

citation of the electron is reduced from the present 3 meV

to below 1 meV.

For the implementation of the two-qubit gate with ﬂy-

ing electrons [7, 8, 26], let us estimate the extent of the

reciprocal phase shift, ϕ=UC·t/~, induced on the wave-

functions of the electron pair after an interaction time t.

The energy due to the Coulomb interaction is represented

here as UC(r) = e2

4πε0εr

rwhere ris the distance between

the two electrons, ε0is the vacuum permittivity and

εr= 12.88 is the dielectric constant of GaAs. From po-

tential simulations, we extract a distance of r≈230 nm,

which gives a Coulomb energy UC≈0.5 meV. Consid-

ering the SAW velocity vSAW ≈2.86 µm/ns, we expect

a phase rotation ϕ=π(Bell state formation) over a

propagation distance l=π~vSAW/UC≈12 nm. This es-

timation shows that in-ﬂight Coulomb interaction within

a TCW introduces a signiﬁcant reciprocal phase shift ca-

pable of entangling the orbits in a SAW-driven single-

electron circuit.

In conclusion, we have demonstrated the controlled in-

teraction between two single ﬂying electrons transported

by sound. This has been achieved through the imple-

mentation of the HOM interferometer with a circuit of

coupled quantum rails. Synchronising the transport of

a pair of individual electrons, we witnessed single-shot

events of fermionic antibunching. To address the under-

lying mechanism, we performed quantitative electrostatic

simulations, and observed a reciprocal electron-gating ef-

fect. Developing a Bayesian model, which contains no

adjustable parameter, we showed quantitative agreement

with the entire set of two-electron collision data. This

provides strong evidence that the observed antibunch-

ing is mediated by Coulomb repulsion. Further estimat-

ing the strength of this Coulomb interaction, we high-

light that it is more than suﬃcient for the formation of

a fully entangled Bell state. Combining this controlled

interaction with novel, scalable single-electron-transport

techniques [27], our results set an important milestone

towards the implementation of the controlled-phase gate

for SAW-driven ﬂying electron qubits.

METHODS

SAW transducer. The employed IDT consists of 111

cells of period λ0= 1 µm. The resonance frequency is

f0=vSAW/λ0≈2.86 GHz at cryogenic temperatures.

To reduce internal reﬂections at resonance, we employ a

double-electrode pattern for the transducers. The sur-

face electrodes of the IDTs are fabricated using stan-

dard electron-beam lithography with successive thin-ﬁlm

evaporation (Ti 3 nm, Al 27 nm) on GaAs/AlGaAs het-

erostructure. The transducer has an aperture of 30 µm

with the SAW propagation direction along [1¯

10]. For

single-electron transport, we employ an input signal at

the resonance frequency with a duration of 50 ns. To

achieve strong SAW conﬁnement, the input signal for

SAW formation is enhanced by a high-power ampliﬁer

(ZHL-4W-422+; +25 dB) prior injection.

Electron-transport experiments. We use a Si-

modulation-doped GaAs/AlGaAs heterostructure grown

by molecular beam epitaxy (MBE). The two-dimensional

electron gas (2DEG) is located 110 nm below the surface,

with an electron density of n≈2.8×1011 cm−2and a mo-

bility of µ≈9×105cm2V−1s−1. Metallic surface gates

(Ti 3 nm, Au 14 nm) deﬁne the nanostructures. The ex-

periment is performed at a temperature of about 20 mK

in a 3He/4He dilution refrigerator. At low temperatures,

the 2DEG below the transport channels and the QDs are

completely depleted via a set of negative voltages applied

on the surface gates. To enable triggering of the sending

process, the plunger gate of each source QD is connected

to a broadband bias tee (SHF AG; 20 kHz to 40 GHz).

Synchronisation between SAW emission and

triggered sending process. We employ two dual-

channel arbitrary waveform generators (AWG, Keysight

M8195A) synchronised via an synchronisation unit

(Keysight M8197A) for the antibunching experiments. A

small jitter of ≈1 ps between the AWG channels allows

to control precisely the timing between SAW emission

and the triggered sending process at each QD.

Potential simulations. The simulations are per-

formed with the commercial Poisson solver nextnano [28].

We deﬁne a three-dimensional structure with realistic

heterostructure layers and gate geometries where the cor-

responding materials’ properties are taken into account.

In our electrostatic model, a metallic gate is expressed

as a Schottky barrier [25, 29]. On the free surface, a

layer of surface charges simulates the Fermi-level-pinning

eﬀect that is well-known in GaAs substrates [30]. Us-

ing one-dimensional simulations, we ﬁrst calibrate the

dopant concentration with a surface gate such that it re-

produces the 2DEG density at the interface of GaAs and

AlGaAs. We then adjust similarly the surface charges in

the absence of the gate. The presence of an electron in

one side of the rail is emulated by inserting the charge of

an electron in a volume of ∆x= 150 nm, ∆y= 17 nm

and ∆z= 1 nm, where x(y) is parallel (perpendicular)

to the SAW propagation direction, and zis the growth

direction of the heterostructure.

ACKNOWLEDGMENTS

We acknowledge fruitful discussions with Vyacheslavs

Kashcheyevs and Elina Pavlovska. J.W. acknowledges

the European Union’s Horizon 2020 research and inno-

vation program under the Marie Sk lodowska-Curie grant

agreement No 754303. A.R. acknowledges ﬁnancial sup-

port from ANR-21-CMAQ-0003, France 2030, project

QuantForm-UGA. T.K. and S.T. acknowledge ﬁnancial

support from JSPS KAKENHI Grant Number 20H02559.

W.P., J.S., and H.-S.S. acknowledge support from Korea

NRF via the SRC Center for Quantum Coherence in Con-

densed Matter (Grant No. 2016R1A5A1008184). C.B.

acknowledges ﬁnancial support from the French Agence

Nationale de la Recherche (ANR), project QUABS ANR-

21-CE47-0013-01. This project has received funding from

the European Union’s H2020 research and innovation

program under grant agreement No 862683 ”UltraFast-

Nano”.

Appendix A: Estimation of SAW amplitude

In order to investigate how the input RF power Pin

applied on the IDT relates to the SAW peak-to-peak am-

plitude ASAW, we measure the SAW-induced modulation

of Coulomb-blockade resonances of a QD. For this pur-

pose, we polarise the surface gates such that the QD is

not depleted, and apply a bias voltage VSD across the two

leads. Varying VSD as a function of the plunger gate volt-

age VP, we measure the conductance across the QD and

obtain Coulomb diamonds as shown in Fig. S1a. This

data allows us to extract the quantum dot’s charging en-

ergy ECand the voltage spacing VCbetween Coulomb-

blockade peaks. The voltage-to-energy conversion factor

is thus η=EC/VC≈0.05 ±0.01 eV/V. Knowing η, we

can now deduce the SAW amplitude ASAW from a given

input power Pin via the relation:

ASAW [eV] = 2 ·η·10(Pin [dBm]−P0)/20, (A1)

where P0is a ﬁt parameter accounting for power losses.

P0is determined by comparison of Eq. A1 to the

SAW-induced broadening of the Coulomb-blockade res-

onances. Figure S1b shows a conductance measure-

ment as function of VPand Pin for VSD ≈20 µV. The

data shows Coulomb-blockade peaks that broaden ac-

cording to Eq. A1 with P0≈36.8±0.3 dBm as in-

dicated by the solid lines (the dashed lines represent

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Coulomb-mediatedantibunchingofanelectronpairsurngonsoundJunliangWang,1HermannEdlbauer,1AymericRichard,1ShunsukeOta,2,3WankiPark,4JeongminShim,4ArneLudwig,5AndreasD.Wieck,5Heung-SunSim,4MatiasUrdampilleta,1TristanMeunier,1TetsuoKodera,2Nobu-HisaKaneko,3HermannSellier,1XavierWaintal,6ShintaroTakada,3...

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Coulomb-mediated antibunching of an electron pair surng on sound Junliang Wang1Hermann Edlbauer1Aymeric Richard1Shunsuke Ota2 3Wanki Park4Jeongmin Shim4Arne Ludwig5Andreas D. Wieck5Heung-Sun Sim4Matias Urdampilleta1Tristan Meunier1Tetsuo Kodera2.pdf

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Coulomb-mediated antibunching of an electron pair surng on sound Junliang Wang1Hermann Edlbauer1Aymeric Richard1Shunsuke Ota2 3Wanki Park4Jeongmin Shim4Arne Ludwig5Andreas D. Wieck5Heung-Sun Sim4Matias Urdampilleta1Tristan Meunier1Tetsuo Kodera2

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