Two electrons interacting at a mesoscopic beam splitter Niels Ubbelohde1Lars Freise1Elina Pavlovska2Peter G. Silvestrov3Patrik Recher3 4Martins Kokainis2 5Girts Barinovs2Frank Hohls1Thomas Weimann1Klaus Pierz1and Vyacheslavs Kashcheyevs2

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Two electrons interacting at a mesoscopic beam splitter

Niels Ubbelohde,1, ∗Lars Freise,1Elina Pavlovska,2Peter G. Silvestrov,3Patrik Recher,3, 4 Martins

Kokainis,2, 5 Girts Barinovs,2Frank Hohls,1Thomas Weimann,1Klaus Pierz,1and Vyacheslavs Kashcheyevs2

1Physikalisch-Technische Bundesanstalt, 38116 Braunschweig, Germany

2Department of Physics, University of Latvia, 3 Jelgavas street, LV-1004 Riga, Latvia

3Institut f¨ur Mathematische Physik, Technische Universit¨at Braunschweig, D-38106 Braunschweig, Germany

4Laboratory for Emerging Nanometrology Braunschweig, D-38106 Braunschweig, Germany

5Faculty of Computing, University of Latvia, 19 Raina boulevard, LV-1586 Riga, Latvia

The non-linear response of a beam splitter to the coincident arrival of interacting particles enables

numerous applications in quantum engineering and metrology yet poses considerable challenge to

achieve focused interactions on the individual particle level. Here we probe the coincidence correla-

tions at a mesoscopic constriction between individual ballistic electrons in a system with unscreened

Coulomb interactions and introduce concepts to quantify the associated parametric non-linearity.

The full counting statistics of joint detection allows us to explore the interaction-mediated energy

exchange. We observe an increase from 50% up to 70% in coincidence counts between statistically

indistinguishable on demand sources, and a correlation signature consistent with independent tomog-

raphy of the electron emission. Analytical modeling and numerical simulations underpin consistency

of the experimental results with Coulomb interactions between two electrons counterpropagating in

a dispersive quadratic saddle, and demonstrate interactions suﬃciently strong, U/(~ω)>10, to

enable single-shot in-ﬂight detection and quantum logic gates.

Coincidence correlations generated by the arrival at

separate input ports of a beam splitter have been proven

essential for metrology and utilization of single propa-

gating quanta of radiation and matter [1–5]. In lin-

ear quantum optics these correlations are driven by

the second-order coherence manifested in the Hong-Ou-

Mandel (HOM) eﬀect [1], and thereby quantum indistin-

guishability of the particles can be inferred and certiﬁed

with extensive applications in photonic quantum infor-

mation technologies [6, 7]. More recently, a fermionic

analogue of linear quantum optics has emerged [8–13],

leading to quantum tomography of electrical currents [14]

via fermionic HOM for well-screened excitations in chiral

one-dimensional conductors [5, 15–19]. An alternative

source of controlled two-particle correlations, pursued

for photonic technologies, is direct interaction in the so

called quantum non-linear regime [20] when the presence

of a single quantum suﬃciently changes the transparency

of the substrate for the other one. While the non-linear

response is naturally weak for photons, thus making the

engineering of suﬃciently strong interactions particularly

challenging, the opposite applies to electrically charged

fermions, where the natural Coulomb interaction is key

to many envisioned applications such as the solid-state

ﬂying qubit platform [21]. Here we present a single-

electron quantum optics platform with on-demand gen-

eration and high ﬁdelity detection and demonstrate the

measurement of coincidence correlations between individ-

ual ballistic electrons at a mesoscopic beam splitter with

a picosecond-scale time resolution. We identify generic

signatures of the quantum non-linear regime in the ﬁrst-

and second-order correlation signals, supported by quan-

titative microscopic modelling of the interplay between

dispersion of the beam splitter and the Coulomb repul-

sion. Further we utilize the full counting statistics of joint

detection to investigate the interaction-mediated energy

exchange between the propagating electrons. Our results

demonstrate a mesoscopic beam splitter with Coulomb

interactions suﬃciently strong for in-ﬂight single-electron

detection and potentially a quantum logic gate between

time-bin encoded qubits.

In a HOM-type experiment [1], high time resolution —

not limited by detector bandwidth — is achieved by pre-

cise control of the interarrival time at the beam splitter.

In its essential elements the setup is an exact analogue

of a two-port scattering experiment and therefore ideally

suited to probe interactions utilizing single-particle de-

tectors in this electron quantum optics experiment (for a

schematic see Fig. 1a and Supplementary Fig. I). Two

electrons are generated on-demand by tunable barrier

non-adiabatic single electron pumps [22]: a probe elec-

tron (2), with ﬁxed average emission energy h2i, and

a scan electron (1), the energy of which (h1i) is varied.

The emission energy and time are well approximated by a

correlated [23] bivariate normal distribution ρi(i, ti), pa-

rameterized by the widths σE,σt, and the correlation co-

eﬃcient r. The average emission energy of both electrons

is chosen to be some ten meV above the Fermi energy and

thereby above the threshold of strong energy relaxation

by scattering with the Fermi sea [24–26] and below the

regime of LO-phonon emission [27–29] to ensure low-loss

transport. The injected electrons are conducted along

the gated mesa edge towards the beam splitter following

the chirality imposed by the magnetic ﬁeld [26, 27, 30].

A notched gate electrode forms a barrier serving as an

electronic beam splitter with energy-dependent trans-

mittance T, which partitions an incident electron with

a transmission probability Ti(hii). The energy broad-

arXiv:2210.03632v1 [cond-mat.mes-hall] 7 Oct 2022

〈∆t〉

〈ε1〉 (VExit,1)

〈ε2〉 (VExit,2)

V,B

2.5 µm

a b

Pnm

0.0

0.2

0.4

0.6

0.8

1.0

0 5 10 15 20 25

Cycle #

100

n = 0

m = 0

n = 1

m = 1

n = 2

m = 2

D1 (pA)

D2 (pA)

FIG. 1. (a) Schematic of the single electron circuit showing the paths (blue, red line) of the two electrons in the transverse

conﬁnement potential from the sources (S1, S2) through the beam splitter potential over the entrance barrier towards the

single electron detector (D1, D2). Scattered electrons lost to the Fermi sea (blue plane) are removed via ground contacts. (b)

Example detector signal for coincident arrival at equal energies. The red/blue bars indicate ±5σ-bounds for identiﬁcation of

electron number. (c) The matrix of all counting statistics elements of Pnm corresponding to the time trace shown in (b).

ening of the transmission threshold is here assumed to

be dominated by the energy width σE,i of the energy-

time distribution supplied by the source ρi(i, ti), while

the barrier is comparatively sharp [31] (see also Meth-

ods). The parameters of ρi(i, ti) are inferred for each

individual source separately via maximum-likelihood es-

timation applied to a tomographic protocol utilizing the

beam splitter as a tunable energy-ﬁltering barrier [31]

(with the respective opposite source switched oﬀ). The

point of half transmission deﬁnes the reference levels for

the emission energies, Ti(0) = 0.5. The average interar-

rival time h∆tiat the beam splitter of the electrons is

controlled by changing the emission time tiof one source

with respect to the other, h∆ti=ht1i−ht2i. The trans-

mitted and reﬂected electrons are then trapped via con-

trolled energy relaxation in a detection node and ﬁnally

read out using a capacitively coupled quantum dot with

near unity detection eﬃciency. A detailed description of

this circuit building block is found in Ref. [26]. Repeating

the sequence of electron injection, detection, and circuit-

level reset builds the counting statistics for the occupancy

of each detector node as a result of the partitioning at the

beam splitter. A histogram of each detector signal allows

to infer a mapping to the corresponding charge transition

[26, 32], which then can be applied to the time-trace to

determine the full counting statistics of coincidence corre-

lations Nnm. The joint detection probability Pnm is then

estimated as Nnm/Pn,m Nnm, where nand mare the

number of injected electrons reaching detectors 1 and 2

respectively. Fig. 1b and c show an example of measured

detector signal time trace and inferred detection proba-

bility for coincident arrival (h1i=h2i=h∆ti= 0).

First we focus on lossless partitioning of two colliding

electrons, n= 2−m= 0,1,or 2, and P20 +P11 +P02 = 1.

We characterize coincidence correlations by the ﬁrst and

the second order correlation signals, s(1) and s(2), de-

ﬁned as deviations from the uncorrelated baseline of the

intensity diﬀerence, (hni−hmi)/2 = P20 −P02 and the in-

terbeam intensity correlation hn mi=P11, respectively.

The uncorrelated baseline expectation for partitioning

probabilities at suﬃciently large arrival time mismatch is

set by addition of statistically independent transmission-

or-reﬂection events, P11 =T1T2+ (1 −T1)(1 −T2) and

P20 −P02 =T1−T2. As the probe electron is always

kept at h2i= 0 regardless of the scan electron tuning,

the correlation signals for our special case T2= 0.5 are

s(1) =P20 −P02 −T1+T2,(1)

s(2) =P11 −0.5.(2)

In linear quantum optics, ﬁrst order quantities are un-

aﬀected by coincident arrival (s(1) = 0 for any T1,2), but

quantum mechanical interference of two-particle ampli-

tudes modiﬁes the antibunching probability by the over-

lap of wave-functions of the transmitted (|ψTi) and the

reﬂected (|ψRi) particles, P11 →P11 ±2|hψR|ψTi|2with

−(+) sign corresponding to boson (fermion) exchange

statistics (the HOM eﬀect). However, for the single-

electron sources in this experiment the estimated up-

per bound on the visibility of this HOM eﬀect in s(2),

Tr (ˆρ1ˆρ2)'0.07, is small. Conversely, the essence of

non-linear correlations can be understood qualitatively

as mutual gating, if the presence of one electron at the

beam splitter changes the transmission probability for

the other. Yet without a microscopic model, interpreta-

tion of such conditional probabilities may be problematic,

since partitioning events are not statistically independent

in general, and a coarse mean-ﬁeld approximation would

average out possible dependence of the gating strengths

on the relative arrival time.

The clear deﬁnition of mechanism-independent signals

allows us now to probe the time-width of coincidence cor-

relations for conditions, where the scan source is tuned

most similar to the probe source and the main diﬀer-

ence is set by a non-zero interarrival time h∆ti. In

Fig. 2a, s(2) shows a clear correlation signature with a

peak amplitude of 0.2 and the second moment width of

σ(B)

t= 20 ps, while s(1) does not a show a discernible

deviation from zero within the measurement uncertainty.

This result is seemingly consistent with second-order co-

herence between sources of indistinguishable particles but

on its own does not rule out interaction-driven correla-

tions. Indeed, as deﬁned by Eqs. (1)-(2), s(2) is even and

s(1) is odd w.r.t. the exchange of the two electrons. If

the two sources (which are independent by design) gen-

erate indistinguishable energy-time distributions, then

h∆ti → −h∆tiis equivalent to 1 ↔2, and s(1) must van-

ish at h∆ti= 0. Whether s(1) 6= 0 is resolved as function

of h∆tiis determined by the competition between the

symmetry-breaking eﬀect of arrival time diﬀerence and

the reduction of the interaction strength at large |h∆ti|.

The experimental results for the T1=T2= 0.5 set-

ting show that electrons are insuﬃciently distinguished

in their transmission properties by the average diﬀerence

in emission time alone.

To reveal the unambiguous signature of interaction-

induced correlations, we break the symmetry by tun-

ing the average emission energy h1iof the scan electron

while keeping the probe electron at half transmission. In

Fig. 2b, also the ﬁrst-order signal now clearly presents

a correlation signature, s(1) 6= 0 as a function of h1i.

In the following, speciﬁc key features of both signals, la-

belled A–D in Fig. 2b, will be discussed to build a generic

qualitative picture of the correlation-generating mecha-

nism.

In regions A and A0the energy h1iis so far detuned,

that the scan electron is always reﬂected (A: P20 = 0) or

always transmitted (A0:P02 = 0), and as a consequence,

s(2) = +s(1) or −s(1), respectively. Equation (1) with

T1= 0 or 1 suggests interpreting ∓s(1) =T∗

2−T2<0 as

a non-linear modiﬁcation of the probe electron transmis-

sion function, quantiﬁable by the shift of the correspond-

ing half-transmission threshold from 0 to ∗

2>0 where

T∗

2=T2(∗

2). This simple heuristic can be generalized

to a quantitative statistical model of two-electron col-

lision, if the randomness in partitioning comes predomi-

nantly from the spread of the incoming energy and arrival

time distributions ρi(i, ti). We introduce two functions,

∗

2(1,∆t) and ∗

1(2,∆t), which, plotted jointly in the

(1, 2) plane for a ﬁxed value of ∆t, deﬁne four domains

corresponding to deﬁnite scattering outcomes: transmis-

sion (i> ∗

i) or reﬂection (i< ∗

i) of the electron incom-

ing from the source iwith a well-deﬁned energy iand

interarrival time ∆t. A schematic of such four-fold par-

titioning of the initial conditions is shown in Figure 2d

for the time slice ∆t= 0. Subsequently, the counting

statistics is obtained by integrating over the correspond-

ing domains,

(P20, P02) = ZY

i=1,2

didtiΘ[±i∓∗

i(¯ı, t1−t2)] ρi(i, ti),

(3)

where the upper sign is for P20, ¯ı= 3 −iis the elec-

tron index opposite to i. Here Heaviside step function Θ

approximates a sharp transmission function of negligible

quantum width in energy.

For an inversion-symmetric beam splitter and simul-

taneous arrival (∆t= 0), the shifted thresholds have to

be symmetric with respect to exchange of the sources,

1↔2, hence both can be described by one single-

argument function, ∗

i(, 0) = ∗(). The symmetric

crossing point ∗(3U/4) = 3U/4 deﬁnes a characteris-

tic energy Uof mutual gating. Having introduced the

generic transmission threshold shifts as quantiﬁers of

non-linearity in Figure 2d and Eq. (3), we can explain

the remaining marked features of the experimental sig-

nals in Figure 2b. At B, the joint distribution symmet-

rically contributes to P20 and P02, and the diﬀerence in

s(1) will therefore vanish, whereas at C, 1is suﬃciently

large compared to ∗for only 50 % of the incident electron

pairs to be jointly deﬂected, indicated by the zero cross-

ing of s(2). When the joint distribution mostly probes the

diﬀerence between the zero shifted and shifted threshold

−40 −30 −20 −10 0 10 20 30

−0.2

−0.1

0.0

0.1

0.2

−〈∆t〉 (ps)

s(1), s(2)

s(1)

s(2)

B C

−4 −2 0 2 4 6 8 10

〈ε1〉 (meV)

−0.2

−0.1

0.0

0.1

0.2

A A'D 1.0

1.2

1.4

1.6

1.8

2.0

s(1), s(2)

s(1)

∑Pn,m

s(2)

∑Pn,m

a b

−6 −4 −2 0 2 4 6

ε1 (meV)

ε2 (meV)

−6

−4

−2

02 11

11 20

A A'

1.5U

B CD

-120

-60

120

-120

-60

120

-20

-10

x (nm)

y (nm)

E (meV)

FIG. 2. First and second order correlation signals (a) for h1i=h2i= 0 and (b) for h∆ti= 0,h2i= 0 together with the

average number of electrons jointly detected. (c) The colored lines show individual electron trajectories in the horizontal plane

of two spatial dimensions with the height indicating the potential experienced by the scan (blue) and the probe (red) electrons

for three example initial conditions, corresponding to points A, B and A’ in (b). The surface and the associated level lines

show the external guiding potential with the dark gray shadows highlighting the external potential contribution to the total

potential guiding the motion of the interacting electrons. (d) Schematic showing the transmission thresholds, ∗

1(2) and ∗

2(1),

and the resulting partitioning (colored domains) for ∆t= 0. The ellipses mark the joint probability distribution of 1and 2

for pairs emitted at ∆t= 0 in case B (hii= 0); contour levels are chosen at intervals equivalent to the coverage factor of a

normal distribution.

at positive energies, 0 <h1i< ∗

1(0) (the area between

2=∗

2and 2= 0 in Fig. 2d), s(1) will assume its most

negative value (D). The mutual gating captured by the

shifted-threshold function ∗() thus oﬀers qualitatively a

complete picture of both correlation signals at h∆ti= 0,

but does not quantify the scaling of the thresholds with

∆t, nor, without binding ∗to a speciﬁc model, predicts

the dependence on σE, σtof the sources.

To compute the threshold functions and interpret the

decay of the correlation signals, we introduce a micro-

scopic model. The separation of scales between the fast

cyclotron motion localized into the lowest Landau state

and E×Bdrift [33] of the corresponding guiding centres

along smoothly varying equipotential lines [34] allows ap-

plication of the classical equations of motion for two in-

teracting electrons [35], once the beam splitter V(xi, yi)

and the interaction Vint(r) potentials are speciﬁed. A ver-

satile analytical model is enabled [35] by approximating

the beam splitter potential with a quadratic saddle [35–

38], as deﬁned in Methods and summarized below. First,

the symmetric threshold function ∗() is implicitly ex-

pressed via the interaction-induced change of the relative

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摘要：

TwoelectronsinteractingatamesoscopicbeamsplitterNielsUbbelohde,1,LarsFreise,1ElinaPavlovska,2PeterG.Silvestrov,3PatrikRecher,3,4MartinsKokainis,2,5GirtsBarinovs,2FrankHohls,1ThomasWeimann,1KlausPierz,1andVyacheslavsKashcheyevs21Physikalisch-TechnischeBundesanstalt,38116Braunschweig,Germany2Departme...

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Two electrons interacting at a mesoscopic beam splitter Niels Ubbelohde1Lars Freise1Elina Pavlovska2Peter G. Silvestrov3Patrik Recher3 4Martins Kokainis2 5Girts Barinovs2Frank Hohls1Thomas Weimann1Klaus Pierz1and Vyacheslavs Kashcheyevs2.pdf

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Two electrons interacting at a mesoscopic beam splitter Niels Ubbelohde1Lars Freise1Elina Pavlovska2Peter G. Silvestrov3Patrik Recher3 4Martins Kokainis2 5Girts Barinovs2Frank Hohls1Thomas Weimann1Klaus Pierz1and Vyacheslavs Kashcheyevs2

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