Time-resolved Coulomb collision of single electrons J.D. Fletcher1W. Park2S. Ryu3P. See1J.P. Griﬃths4G.A.C. Jones4I. Farrer4D.A. Ritchie4H.-S. Sim2and M. Kataoka1

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Time-resolved Coulomb collision of single electrons

J.D. Fletcher,1, ∗W. Park,2S. Ryu,3P. See,1J.P. Griﬃths,4G.A.C.

Jones,4I. Farrer,4D.A. Ritchie,4H.-S. Sim,2and M. Kataoka1

1National Physical Laboratory, Hampton Road, Teddington TW11 0LW, United Kingdom

2Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea

3Instituto de F´ısica Interdisciplinary Sistemas Complejos IFISC (CSIC-UIB), E-07122 Palma de Mallorca, Spain

4Cavendish Laboratory, University of Cambridge,

J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom

(Dated: October 10, 2022)

Precise control over interactions between bal-

listic electrons will enable us to exploit Coulomb

interactions in novel ways, to develop high-speed

sensing,1to reach a non-linear regime in elec-

tron quantum optics and to realise schemes for

fundamental two-qubit operations2on ﬂying elec-

trons. Time-resolved collisions between electrons

have been used to probe the indistinguishability,3

Wigner function4,5and decoherence6of single

electron wavepackets. Due to the eﬀects of

screening, none of these experiments were per-

formed in a regime where Coulomb interactions

were particularly strong. Here we explore the

Coulomb collision of two high energy electrons

in counter-propagating ballistic edge states.7,8We

show that, in this kind of unscreened device,

the partitioning probabilities at diﬀerent electron

arrival times and barrier height are shaped by

Coulomb repulsion between the electrons. This

prevents the wavepacket overlap required for the

manifestation of fermionic exchange statistics but

suggests a new class of devices for studying and

manipulating interactions of ballistic single elec-

trons.

In principle, time-resolved electronic interactions can

be studied with a wavepacket collider like that sketched

in Fig. 1. Single electron sources S1 and S2 emit parti-

cles with relative delay t21 into an experimentally-deﬁned

collision region. Interactions determine how particles are

partitioned into detectors D1 and D2 for diﬀerent injec-

tion time.3,9Under some conditions, fermionic exchange

eﬀects can create antibunching of wavepackets, detected

via reduced current noise at the detectors.3This eﬀect

can be used as a measurement of the indistinguishabil-

ity of the wavepackets, an important ﬁgure of merit for

quantum coherent transport.10 However, in general, un-

derstanding the behaviour of the electrons in the inter-

action region is not straightforward if direct Coulomb

eﬀects are present in addition to exchange eﬀects.11

For sources injecting electrons near the Fermi en-

ergy, the impact of the Coulomb interaction on the elec-

tron trajectory is diminished by screening.3Changes

to the single electron trajectories or velocity from di-

rect Coulomb interactions between single electrons has

not been seen, although coupling to nearby conduct-

ing channels has been detected via decoherence of the

wavepackets.6Where electronic density is reduced, or

∗jonathan.ﬂetcher@npl.co.uk

FIG. 1: Idealised electronic wavepacket collider

Sources S1, S2 inject electrons into a collision region

with a variable time delay. They interact and scatter

into detectors D1 and D2. This can be used to study

interactions between electrons, interactions with the

environment, and behaviour of the electron sources.

for particles injected at higher energy, the trajectories

and velocity of electrons are expected to be signiﬁcantly

modiﬁed.12,13 Understanding this regime is important for

the controlled use of Coulomb interactions for quantum

logic gates.14

To learn how to harness the Coulomb repulsion in bal-

listic electron systems, we have studied an electron col-

lider based on electron pumps which emit electrons into

edge states at high energy (E>100 meV).7,8,15,16 In this

case, the time-resolved Coulomb interaction between bal-

listic single electrons can be directly detected.

The collision region of the electron collider is shown in

Fig. 2a and the overall device structure in Fig. 2b. Elec-

tron pump sources S1 (left) and S2 (right) each inject

an electron every τ=2 ns.17 In a perpendicular mag-

netic ﬁeld these are conﬁned to states on the mesa-edge

with energies E1and E2typically in excess of 100 meV.7

These can be individually tuned7such that E1≃E2. Tra-

jectories meet at a barrier with height Ebset by a gate

voltage (see methods). At the barrier, electrons from

source 1 and 2 are transmitted with probabilities T1and

T2into output arms D2 and D1, respectively. The emis-

sion times from the two sources can be ﬁne-tuned so that

the nominal diﬀerence between arrival times at the centre

barrier t21 =t2−t1can be chosen arbitrarily. For t21 =0

arrival is synchronised (see methods), down to a residual

uncertainty determined by the emission time variation of

the sources, typically of order picoseconds.16

arXiv:2210.03473v1 [cond-mat.mes-hall] 7 Oct 2022

FIG. 2: Electron collider implemented with high energy single electrons (a) Partitioning of electrons in a

high energy device. Three outcomes are characterised by charge distributions (2,0) when both electrons enter D1,

(0,2) when both enter D2, and (1,1) one enters each. Scalebar length is 500 nm. Proximity of electrons is

exaggerated for clarity (see methods for timing information). (b) False colour scanning electron microscopy image of

overall device structure and measurement system. See methods for details. Electrons emitted from sources S1 (left,

red) and S2 (right, blue) eject electrons follow trajectories indicated by arrows, colliding at the central barrier

(green). Dashed region indicates the collision region shown in (a). Inset: Energy level drawing to show adjustable

ejection energy.

When only one source is active, individual Tn(Eb)can

be measured from the time-averaged current.7,16 For our

two sources we ﬁnd that T1(Eb)≃T2(Eb). When both

sources are active, the current at detector D2 is IDC

D2 =

ef(T1+1−T2)i.e. collecting any electrons from source S1

transmitted through the barrier and those from source S2

reﬂected by the barrier. Similarly IDC

D1 =ef(T2+1−T1).

For deliberately mismatched arrival times IDC

D1 =IDC

D2 ≃

ef, a nearly constant background. When arrival times

are synchronised to within a few picoseconds (Supple-

mentary Section 1) an imbalance in transmission is seen.

We show this, calculated using T1−T2=(IDC

D2 −IDC

D1 )/2ef,

in Fig. 3a. For values near t21 =0 we see a pronounced

current imbalance with T1−T2≃+0.2 for t21 >0 and

T1−T2≃−0.2 for t21 <0. The polarity of this signal cor-

responds to preferential transmission of the earlier elec-

tron.

Measuring the time-dependent noise current, IAC

D1 , IAC

using the additional connections shown in Fig. 2b enables

us to measure correlations3,8in partitioning outcomes

from individual scattering cycles that are not accessible

in a time-averaged measurement. The shot noise cur-

rent S12, simultaneously measured in contacts D1 and

D2 (see methods), is mapped as a function of Eband

t21 in Fig. 3b. In the case ∣t21∣≫0, electrons arrive at

very diﬀerent times and do not interact. The noise sig-

nal in this regime is explained by independent stochastic

transmission of the electrons into either D1 or D2 given

by S12 =2e2f∑n=1,2[Tn(1−Tn)].8The noise S12 ≃0

when the barrier is much higher or lower than the in-

jection energy, but reaches a peak S12∣max =e2fwhen

Eb≃E1≃E2and T1, T2≃0.5 (i.e. the noise peak can

pick out the electron energy).

The noise is modiﬁed in several ways by Coulomb in-

teractions. Firstly, for values of t21 ≃0 (i.e. electrons

collide), the position of the noise peak moves to a lower

value of the programmed barrier height Eb. This appar-

ent shift in detected energy, E1→E′

1and E2→E′

2is

a manifestation of the Coulomb repulsion between the

electrons which gives a transient increase in the eﬀective

barrier height of ≃0.5−1 meV. Conservation of energy

is preserved by a reduction in the kinetic energy corre-

sponding to the increased Coulomb repulsion.13 Secondly,

at delay times near t21 =0 the maximum partition noise

S12∣max is reduced from the independent scattering limit

by up to ≃30%, as shown in Fig.3(c). This eﬀect is driven

by an increase in the number of events where both elec-

trons are reﬂected due to a repulsive interaction. While

the noise reduction is in superﬁcial resemblance to a par-

tial Pauli dip,3we show below that the full partitioning

statistics, calculated from the time-averaged current and

noise data,8reveal a Coulomb-dominated system.

The partition probabilities for the three possible detec-

tor charge states Pij are P20, P11, P02 where i, j are the

number of electrons in detector D1, D2 and ∑Pij =1.

Particular values of P02, P20 map directly to current

noise and time-averaged current distribution.8,18 Com-

puted values of P02, P20 at diﬀerent barrier height are

shown in Fig. 3d-i. The non-interacting case in Fig. 3d

(∣t21∣=20 ps) has P02, P20 ≃0, P11 ≃1 for high and

FIG. 3: Current imbalance, shot noise suppression and partition statistics (a) Current imbalance T1−T2

for diﬀerent barrier height Eband time delay t21. Favoured events associated with the negative and positive current

imbalances are indicated schematically. (b) Map of shot noise S12 for barrier height Eband time delay t21. Dashed

line is a guide to peak noise position. (c) Shot noise S12 in the non-interacting (t21 =−20 ps) and strongly

interacting regime (t21 =+2 ps) where the eﬀect is most pronounced. Reduction in noise is associated with repulsion

events as indicated. For estimation of errors bars, see Supplementary Section 1. Each panel (d-i) shows partition

probabilities (P20,P02) at diﬀerent barrier heights for a chosen t21. Underlying colour map shows noise S12 (contour

lines are at 0.1e2fintervals). (d) The non-interacting case (electrons miss each other) at t21 =−20 ps (dark grey)

+20 ps (light grey). Arrows show the sweep from high to low barrier height, reaching the maximum noise at

independent scattering limit (thick solid line). At this point (P20, P02)=(0.25,0.25)(circled) there are an equal

statistical mixture of each outcome event. (e-i) as (d) but for selected interacting cases t21 =−9,−2,0,+2,+9 ps

corresponding to arrows in panel b. These cases show a strongly modiﬁed behaviour as described in the text.

low barriers and the expected statistical mixture of out-

comes when Eb≃E1, E2namely P02∣max, P20∣max ≃0.25

and P11∣min ≃0.5. The results in the interacting regime

(∣t21∣<10 ps) are shown in Fig. 3e-i. Preferential trans-

mission into D1 or D2 causes an imbalance in P20 and

P02 (a tilt toward the P02 or P20 axis). P20∣max and

P02∣max are also suppressed below the independent scat-

tering value √P20 +√P02 =1 (thick solid line in Fig. 3d-

i). This corresponds to a reduction in noise or an excess

antibunching ∆P11 =P11∣min −0.5≃0.15.

We show below that the full partitioning dataset is

fully captured by a model of the microscopic trajectories

which follow the E×Bdrift motion. The shape of these

trajectories is set by the partitioning barrier potential

and the (weakly screened) Coulomb interaction. We also

show that the small timing and energy ﬂuctuations of

the source16 are important in determining the statistical

outcomes seen experimentally.

We compute the trajectories for an assumed form of

the potential19 near the barrier U2D(x, y)and an interac-

tion potential Uee(⃗

r1,⃗

r2)∝1/rat short range, but which

weakens at longer distances (r12 ≫90 nm, see methods)

due to screening by metallic surface gates.20 Interactions

are limited to a region within the screening length near

the eﬀective tunneling point, making a saddle point bar-

rier potential an appropriate approximation.19 Calcula-

tions based on the E×Btrajectories are valid in the

regime of our experiments where the energy window of

quantum scattering (over which the barrier transmission

probability changes from 0 to 1) is much smaller than

the energy uncertainty of the electrons and/or the purity

of the electrons is suﬃciently low16 (see Supplementary

Section 2).

Example trajectories are shown in Fig. 4for injection

energy exceeding the barrier height En−Eb=0.5 meV.

This illustrates three important cases; completely mis-

matched arrival Fig. 4a, near-synchronised Fig. 4b,c and

closely synchronised Fig. 4d. In the non-interacting case

Fig. 4a, trajectories follow the equipotential contours of

U2Dand the transmitted charge (QT1, QT2)=(1e, 1e).

In contrast, close to synchronisation as in Fig. 4d, both

electrons are deﬂected such that (QT1, QT2)=(0e, 0e).

For the near-synchronised cases in Fig. 4b,c interactions

modify the trajectory of the electrons in a more compli-

cated way. The late-arriving electron (from S1 in these

examples) tends to be deﬂected by the earlier-arriving

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Time-resolvedCoulombcollisionofsingleelectronsJ.D.Fletcher,1,W.Park,2S.Ryu,3P.See,1J.P.Griths,4G.A.C.Jones,4I.Farrer,4D.A.Ritchie,4H.-S.Sim,2andM.Kataoka11NationalPhysicalLaboratory,HamptonRoad,TeddingtonTW110LW,UnitedKingdom2DepartmentofPhysics,KoreaAdvancedInstituteofScienceandTechnology,Daejeon...

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Time-resolved Coulomb collision of single electrons J.D. Fletcher1W. Park2S. Ryu3P. See1J.P. Griﬃths4G.A.C. Jones4I. Farrer4D.A. Ritchie4H.-S. Sim2and M. Kataoka1.pdf

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Time-resolved Coulomb collision of single electrons J.D. Fletcher1W. Park2S. Ryu3P. See1J.P. Griﬃths4G.A.C. Jones4I. Farrer4D.A. Ritchie4H.-S. Sim2and M. Kataoka1

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