# Two electrons interacting at a mesoscopic beam splitter

Niels Ubbelohde<EMAIL_ADDRESS>Physikalisch-Technische Bundesanstalt,
38116 Braunschweig, Germany  Lars Freise  Physikalisch-Technische
Bundesanstalt, 38116 Braunschweig, Germany  Elina Pavlovska  Department of
Physics, University of Latvia, 3 Jelgavas street, LV-1004 Riga, Latvia  Peter
G. Silvestrov  Institut für Mathematische Physik, Technische Universität
Braunschweig, D-38106 Braunschweig, Germany  Patrik Recher  Institut für
Mathematische Physik, Technische Universität Braunschweig, D-38106
Braunschweig, Germany  Laboratory for Emerging Nanometrology Braunschweig,
D-38106 Braunschweig, Germany  Martins Kokainis  Department of Physics,
University of Latvia, 3 Jelgavas street, LV-1004 Riga, Latvia  Faculty of
Computing, University of Latvia, 19 Raina boulevard, LV-1586 Riga, Latvia
Girts Barinovs  Department of Physics, University of Latvia, 3 Jelgavas
street, LV-1004 Riga, Latvia  Frank Hohls  Physikalisch-Technische
Bundesanstalt, 38116 Braunschweig, Germany  Thomas Weimann  Physikalisch-
Technische Bundesanstalt, 38116 Braunschweig, Germany  Klaus Pierz
Physikalisch-Technische Bundesanstalt, 38116 Braunschweig, Germany
Vyacheslavs Kashcheyevs  Department of Physics, University of Latvia, 3
Jelgavas street, LV-1004 Riga, Latvia

###### Abstract

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 correlations 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 tomography 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, $U/(\hbar\omega)>10$, 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 propagating quanta of radiation and matter [1, 2, 3, 4, 5]. In linear
quantum optics these correlations are driven by the second-order coherence
manifested in the Hong-Ou-Mandel (HOM) effect [1], and thereby quantum
indistinguishability of the particles can be inferred and certified with
extensive applications in photonic quantum information technologies [6, 7].
More recently, a fermionic analogue of linear quantum optics has emerged [8,
9, 10, 11, 12, 13], leading to quantum tomography of electrical currents [14]
via fermionic HOM for well-screened excitations in chiral one-dimensional
conductors [5, 15, 16, 17, 18, 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 generation and high fidelity detection
and demonstrate the measurement of coincidence correlations between individual
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 quantitative
microscopic modelling of the interplay between dispersion of the beam splitter
and the Coulomb repulsion. 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.

Figure 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 $\pm 5\text{\,}\sigma$-bounds for identification of
electron number. (c) The matrix of all counting statistics elements of
$P_{nm}$ corresponding to the time trace shown in (b).

In a HOM-type experiment [1], high time resolution — not limited by detector
bandwidth — is achieved by precise 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 detectors 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 electron (2), with fixed average emission energy
$\langle\epsilon_{2}\rangle$, and a scan electron (1), the energy of which
($\langle\epsilon_{1}\rangle$) is varied. The emission energy and time are
well approximated by a correlated [23] bivariate normal distribution
$\rho_{i}(\epsilon_{i},t_{i})$, parameterized by the widths $\sigma_{E}$,
$\sigma_{t}$, and the correlation coefficient $r$. 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, 25, 26] and below the regime of LO-phonon emission [27, 28, 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 [27, 30, 26]. A notched gate electrode forms a barrier
serving as an electronic beam splitter with energy-dependent transmittance
$T$, which partitions an incident electron with a transmission probability
$T_{i}(\langle\epsilon_{i}\rangle)$. The energy broadening of the transmission
threshold is here assumed to be dominated by the energy width $\sigma_{E,i}$
of the energy-time distribution supplied by the source
$\rho_{i}(\epsilon_{i},t_{i})$, while the barrier is comparatively sharp [31]
(see also Methods). The parameters of $\rho_{i}(\epsilon_{i},t_{i})$ are
inferred for each individual source separately via maximum-likelihood
estimation 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, $T_{i}(0)=0.5$. The average interarrival time
$\langle\Delta t\rangle$ at the beam splitter of the electrons is controlled
by changing the emission time $t_{i}$ of one source with respect to the other,
$\langle\Delta t\rangle=\langle t_{1}\rangle-\langle t_{2}\rangle$. The
transmitted and reflected electrons are then trapped via controlled 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 [32, 26],
which then can be applied to the time-trace to determine the full counting
statistics of coincidence correlations $N_{nm}$. The joint detection
probability $P_{nm}$ is then estimated as $N_{nm}/\sum_{n,m}N_{nm}$, where $n$
and $m$ are 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 probability for coincident arrival
($\langle\epsilon_{1}\rangle=\langle\epsilon_{2}\rangle=\langle\Delta
t\rangle=0$).

First we focus on lossless partitioning of two colliding electrons,
$n=2-m=0,1,\text{ or }2$, and $P_{20}+P_{11}+P_{02}=1$. We characterize
coincidence correlations by the first and the second order correlation
signals, $s^{(1)}$ and $s^{(2)}$, defined as deviations from the uncorrelated
baseline of the intensity difference, $(\langle n\rangle-\langle
m\rangle)/2=P_{20}-P_{02}$ and the interbeam intensity correlation $\langle
n\,m\rangle=P_{11}$, 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,
$P_{11}=T_{1}\ T_{2}+(1-T_{1})(1-T_{2})$ and $P_{20}-P_{02}=T_{1}-T_{2}$. As
the probe electron is always kept at $\langle\epsilon_{2}\rangle\\!=\\!0$
regardless of the scan electron tuning, the correlation signals for our
special case $T_{2}=0.5$ are

$\displaystyle s^{(1)}$ $\displaystyle=P_{20}-P_{02}-T_{1}+T_{2}\,,$ (1)
$\displaystyle s^{(2)}$ $\displaystyle=P_{11}-0.5\,.$ (2)

In linear quantum optics, first order quantities are unaffected by coincident
arrival ($s^{(1)}=0$ for any $T_{1,2}$), but quantum mechanical interference
of two-particle amplitudes modifies the antibunching probability by the
overlap of wave-functions of the transmitted ($\ket{\psi_{T}}$) and the
reflected ($\ket{\psi_{R}}$) particles, $P_{11}\to P_{11}\pm
2\,|\langle\psi_{R}|\psi_{T}\rangle|^{2}$ with $-$ ($+$) sign corresponding to
boson (fermion) exchange statistics (the HOM effect). However, for the single-
electron sources in this experiment the estimated upper bound on the
visibility of this HOM effect in $s^{(2)}$,
$\mathrm{Tr}\left(\hat{\rho}_{1}\hat{\rho}_{2}\right)\simeq 0.07$, 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, interpretation 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.

Figure 2: First and second order correlation signals (a) for
$\langle\epsilon_{1}\rangle=\langle\epsilon_{2}\rangle=0$ and (b) for
$\langle\Delta t\rangle=0,\langle\epsilon_{2}\rangle=0$ 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, $\epsilon_{1}^{\ast}(\epsilon_{2})$ and
$\epsilon_{2}^{\ast}(\epsilon_{1})$, and the resulting partitioning (colored
domains) for $\Delta t=0$. The ellipses mark the joint probability
distribution of $\epsilon_{1}$ and $\epsilon_{2}$ for pairs emitted at $\Delta
t=0$ in case B ($\langle\epsilon_{i}\rangle=0$); contour levels are chosen at
intervals equivalent to the coverage factor of a normal distribution.

The clear definition of mechanism-independent signals allows us now to probe
the time-width of coincidence correlations for conditions, where the scan
source is tuned most similar to the probe source and the main difference is
set by a non-zero interarrival time $\langle\Delta t\rangle$. In Fig. 2a,
$s^{(2)}$ shows a clear correlation signature with a peak amplitude of 0.2 and
the second moment width of
$\sigma^{(B)}_{t}=$20\text{\,}\mathrm{p}\mathrm{s}$$, while $s^{(1)}$ does not
a show a discernible deviation from zero within the measurement uncertainty.
This result is seemingly consistent with second-order coherence between
sources of indistinguishable particles but on its own does not rule out
interaction-driven correlations. 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) generate indistinguishable
energy-time distributions, then $\langle\Delta t\rangle\to-\langle\Delta
t\rangle$ is equivalent to $1\leftrightarrow 2$, and $s^{(1)}$ must vanish at
$\langle\Delta t\rangle=0$. Whether $s^{(1)}\not=0$ is resolved as function of
$\langle\Delta t\rangle$ is determined by the competition between the
symmetry-breaking effect of arrival time difference and the reduction of the
interaction strength at large $|\langle\Delta t\rangle|$. The experimental
results for the $T_{1}=T_{2}=0.5$ setting 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 tuning the average emission energy
$\langle\epsilon_{1}\rangle$ of 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, $s^{(1)}\not=0$ as a function of
$\langle\epsilon_{1}\rangle$. In the following, specific key features of both
signals, labelled A–D in Fig. 2b, will be discussed to build a generic
qualitative picture of the correlation-generating mechanism.

In regions A and A′ the energy $\langle\epsilon_{1}\rangle$ is so far detuned,
that the scan electron is always reflected (A: $P_{20}=0$) or always
transmitted (A′: $P_{02}=0$), and as a consequence, $s^{(2)}=+s^{(1)}$ or
$-s^{(1)}$, respectively. Equation (1) with $T_{1}=0$ or $1$ suggests
interpreting $\mp s^{(1)}=T_{2}^{\ast}-T_{2}<0$ as a non-linear modification
of the probe electron transmission function, quantifiable by the shift of the
corresponding half-transmission threshold from $0$ to $\epsilon_{2}^{\ast}>0$
where $T_{2}^{\ast}=T_{2}(\epsilon_{2}^{\ast})$. This simple heuristic can be
generalized to a quantitative statistical model of two-electron collision, if
the randomness in partitioning comes predominantly from the spread of the
incoming energy and arrival time distributions $\rho_{i}(\epsilon_{i},t_{i})$.
We introduce two functions, $\epsilon_{2}^{\ast}(\epsilon_{1},\Delta t)$ and
$\epsilon_{1}^{\ast}(\epsilon_{2},\Delta t)$, which, plotted jointly in the
$(\epsilon_{1},\epsilon_{2})$ plane for a fixed value of $\Delta t$, define
four domains corresponding to definite scattering outcomes: transmission
($\epsilon_{i}>\epsilon_{i}^{\ast}$) or reflection
($\epsilon_{i}<\epsilon_{i}^{\ast}$) of the electron incoming from the source
$i$ with a well-defined energy $\epsilon_{i}$ and interarrival time $\Delta
t$. A schematic of such four-fold partitioning of the initial conditions is
shown in Figure 2d for the time slice $\Delta t=0$. Subsequently, the counting
statistics is obtained by integrating over the corresponding domains,

$\displaystyle(P_{20},P_{02})=\int\prod_{i=1,2}d\epsilon_{i}\,dt_{i}\,\Theta[\pm\epsilon_{i}\mp\epsilon_{i}^{\ast}(\epsilon_{\bar{\imath}},t_{1}-t_{2})]\,\rho_{i}(\epsilon_{i},t_{i})\,,$
(3)

where the upper sign is for $P_{20}$, $\bar{\imath}=3-i$ is the electron index
opposite to $i$. Here Heaviside step function $\Theta$ approximates a sharp
transmission function of negligible quantum width in energy.

For an inversion-symmetric beam splitter and simultaneous arrival ($\Delta
t=0$), the shifted thresholds have to be symmetric with respect to exchange of
the sources, $1\leftrightarrow 2$, hence both can be described by one single-
argument function,
$\epsilon_{i}^{\ast}(\epsilon,0)=\epsilon^{\ast}(\epsilon)$. The symmetric
crossing point $\epsilon^{\ast}(3U/4)=3U/4$ defines a characteristic energy
$U$ of 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 signals in Figure
2b. At B, the joint distribution symmetrically contributes to $P_{20}$ and
$P_{02}$, and the difference in $s^{(1)}$ will therefore vanish, whereas at C,
$\epsilon_{1}$ is sufficiently large compared to $\epsilon^{\ast}$ for only
$50\text{\,}\%$ of the incident electron pairs to be jointly deflected,
indicated by the zero crossing of $s^{(2)}$. When the joint distribution
mostly probes the difference between the zero shifted and shifted threshold at
positive energies, $0<\langle\epsilon_{1}\rangle<\epsilon_{1}^{\ast}(0)$ (the
area between $\epsilon_{2}=\epsilon^{\ast}_{2}$ and $\epsilon_{2}=0$ in Fig.
2d), $s^{(1)}$ will assume its most negative value (D). The mutual gating
captured by the shifted-threshold function $\epsilon^{\ast}(\epsilon)$ thus
offers qualitatively a complete picture of both correlation signals at
$\langle\Delta t\rangle=0$, but does not quantify the scaling of the
thresholds with $\Delta t$, nor, without binding $\epsilon^{\ast}$ to a
specific model, predicts the dependence on $\sigma_{E},\sigma_{t}$ of the
sources.

Figure 3: (a),(b) Measured and (c),(d) simulated correlation signals
$s^{(2)}$, $s^{(1)}$. For $s^{(1)}$,
$T_{1}(\langle\epsilon_{1}\rangle)-T_{2}(\langle\epsilon_{2}\rangle)$ is
estimated at large $|\langle\Delta t\rangle|$ and subtracted from
$P_{20}-P_{02}$. Simulation parameters:
$U=$2.3\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$,
$\omega=$250\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$,
$d_{\text{cutoff}}=$75\text{\,}\mathrm{n}\mathrm{m}$$ and
$\omega_{x}/\omega_{y}=1$. The white, dashed, straight line indicates the
expected energy-time correlation of the source emission, offset to coincide
with the maximal joint detection probability $P_{11}$ at
$\langle\epsilon_{1}\rangle=0$, which also defines $\Delta t=0$ for the
simulation. The labels identify key features discussed in the main text.

To compute the threshold functions and interpret the decay of the correlation
signals, we introduce a microscopic model. The separation of scales between
the fast cyclotron motion localized into the lowest Landau state and
$\bm{E}\times\bm{B}$ drift [33] of the corresponding guiding centres along
smoothly varying equipotential lines [34] allows application of the classical
equations of motion for two interacting electrons [35], once the beam splitter
$V(x_{i},y_{i})$ and the interaction $V_{\text{int}}(r)$ potentials are
specified. A versatile analytical model is enabled [35] by approximating the
beam splitter potential with a quadratic saddle [36, 37, 35, 38], as defined
in Methods and summarized below. First, the symmetric threshold function
$\epsilon^{\ast}(\epsilon)$ is implicitly expressed via the interaction-
induced change of the relative coordinate traversal time $\Delta\tau(E)$

$\left(\epsilon^{\ast}/\epsilon\right)^{\pm
1}=\tanh[\omega\Delta\tau(\epsilon^{\ast}+\epsilon)/2]\,,$ (4)

where the ‘$+$’ sign is for $\epsilon^{\ast}<|\epsilon|$. Equation (4)
reflects the competition [38, 35] of the energy-dependent interaction effect
encoded in $\Delta\tau(E)$ and the linear dispersion of the beam splitter
characterized by $\omega=m\omega_{x}\,\omega_{y}/(eB)$. Second, the $\Delta t$
dependence is obtained from Eq. (4) by scaling relations [35]
$\epsilon_{i}^{\ast}(\epsilon_{\bar{\imath}},\Delta
t)=2\epsilon^{\ast}([1+e^{\pm\omega\,\Delta
t}]\epsilon_{\bar{\imath}}/2)/(1+e^{\mp\omega\,\Delta t})$ with the upper sign
for $i=1$. These relations obey the symmetry-dictated invariance under $\Delta
t\to-\Delta t$ and $1\leftrightarrow 2$. In particular, the crossing point of
the thresholds stays on a straight line (dashed line in Fig. 2d),
$\epsilon_{1c}^{\ast}+\epsilon_{2c}^{\ast}=1.5U$ with late-to-arrive electron
experiencing a stronger gating effect,
$\epsilon_{1c}^{\ast}>\epsilon_{2c}^{\ast}$ for $\Delta t>0$.

To characterize the extent of the correlation signatures in relative energy
and time, in Fig. 3 both $\langle\epsilon_{1}\rangle$ and $\langle\Delta
t\rangle$ are scanned for a modified operating point of the sources with
reduced driving amplitude. In addition to reproducing the key features
identified and labelled in Fig. 2, these results illustrate how the
correlation signatures both in $s^{(1)}$ and $s^{(2)}$ persist over a wide
range of energies beyond the energy width of the source distributions,
covering both zero and full transmission of the scan electron. The width of
the correlation signal as function of $\Delta t$ remains fairly constant; in
region A at
$\langle\epsilon_{1}\rangle=-$2.8\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ we
measure $\sigma^{(A)}_{t}=$14\text{\,}\mathrm{p}\mathrm{s}$$. Subtracting the
broadening due to source distributions and using the perturbative limit of Eq.
(4) yields an estimate of
$\omega=$270\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$ (see Methods). Taking
the maximal joint detection probability $P_{11}=s^{(2)}+0.5$ to be occurring
close to coincident arrival, the apparent shift of this point with
$\langle\epsilon_{1}\rangle$ should follow the energy-time correlation
$r\sigma_{t}/\sigma_{E}$ of the source, since for tunable barrier pumps the
emission energy is linked to the emission time via the drive signal modulating
the dynamic quantum dot (the correlation coefficient $r$ is a fingerprint of
the same mechanism [23]). The predicted time shift (white dashed line) agrees
well with observed correlation signal.

In Fig. 3c and d, the experimental results are compared to a simulation using
a Coulomb potential with adjustable parameters $\omega$,
$\omega_{x}/\omega_{y}$ and $U$ (which leaves the effective charge as a free
parameter). To account for both the limitations of the quadratic saddle
approximation and screening, we neglect correlations beyond a distance
$d_{\text{cutoff}}$ between the electrons, see Methods. Magnitude and extent
of both positive and negative correlations are well reproduced for both first
and second order signals, including the positive excess energy required to
overcome gating by the second electron (C in $s^{(2)}$), as well the large
area of the interaction-driven correlation signal in region A. Quantitatively,
the estimated parameters are in reasonable agreement with unscreened Coulomb
interactions ($U=$2.3\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ versus
$U=$2.6\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$, see Supplementary Note VI)
near the centre of the beam splitter. The large ratio $U/(\hbar\omega)=14$
justifies the semiclassical approach to interactions [35] and confirms that
the experiment is in the interaction-dominated regime [39]. The simulations
also explain the unresolved sign of the interarrival time for statistically
equal energy collisions, $s^{(1)}(\langle\Delta t\rangle)\approx 0$ for
$\langle\epsilon_{i}\rangle=0$ in Figures 1a, 3b and 3d, as the dispersive
delay $\sim\omega^{-1}\ln(d_{\text{cutoff}}/d_{\text{min}})$, cut off by the
finite size of the beam splitter, limits the dynamical range required to cover
the full emission time distribution. The discrepancies in the comparison with
the experimental data, such as the excessive shift of $s^{(1)}$ minimum (point
D) with respect to the inferred coincidence line and the faster decay of the
$s^{(1)}\approx s^{(2)}$ signal for
$\langle\epsilon_{1}\rangle\lesssim-$3\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$,
are likely to stem from microscopic details beyond the idealizations of the
model.

Figure 4: Detection probabilities, (a) $P_{10}$ and (b) $P_{01}$,
corresponding to the acquisition cycles, where one electron was lost before
reaching the detector. (c), (d) Simulation of $P_{10}$ and $P_{01}$ taking
into account the elastic energy exchange with parameters as in Fig. 3 and an
energy-dependent loss rate during propagation after the collision, as detailed
in Supplementary Note VIII. The white dashed line again indicates estimated
coincident arrival. The inset shows example trajectories with
$\epsilon_{1}=$0.1\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ and $\Delta
t=$17\text{\,}\mathrm{p}\mathrm{s}$$ leading to large energy exchange using
the same style as in Fig. 2c.

In addition to coincidence correlations, access to the full counting
statistics also allows to analyze cases where less than two electrons reach
the detectors (see Fig. 1c). For the injected electrons, any energy loss will
drastically increase the rate of scattering with electrons from the Fermi sea
during propagation [25, 26] and thus make it unlikely for the electron to pass
over the barrier at the entrance towards the detection node (see also Fig.
1a). Rare cases where one electron is lost are counted in $P_{10}$ and
$P_{01}$ in Fig. 4. Both signals show a $\langle\Delta t\rangle$-independent
background, which reflects the energy dependence of loss rate and transmission
probability, as it is more likely to find at least one electron at detector 1
(enhanced $P_{10}$) instead of detector 2 (reduced $P_{01}$) for positive
$\langle\epsilon_{1}\rangle$. The most striking feature in $P_{10}$ is the
appearance of enhanced losses near the coincidence line, which cannot be
explained by the concept of shifted transmission thresholds alone, yet will be
confirmed by the microscopic model. The analytic computation of the energy
exchange between the two electrons (see Supplementary Note VIII), which
together with an approximation of the energy-dependent loss rate after the
collision yields a prediction for $P_{10}$ and $P_{01}$ shown in Fig. 4c and d
using the simulation parameter set from Fig. 3. Based on the good agreement,
we identify the enhanced signal in $P_{10}$ as explicit evidence for
interaction-mediated energy exchange at the beam splitter. In the saddle
approximation, the energy exchange is maximized, when
$\langle\epsilon_{1}\rangle$ and $\langle\Delta t\rangle$ are just large
enough to allow electron 1 to be transmitted and then trail behind electron 2
(see inset in Fig. 4c). A complimentary signal of energy exchange in $P_{01}$
is more difficult to discern above the noise. Due to the large
$\langle\epsilon_{1}\rangle$ and $\langle\Delta t\rangle$ offsets in the
latter case, interactions in the experiment are likely attenuated by
screening. An additional fingerprint of energy exchange at the beam splitter
can be observed in $P_{00}$ (see Supplementary Fig. II), where a small but
clearly discernible signal traces the coincidence line. Loss of both electrons
might be induced by sufficiently large energy transfer in the elastic
collision such that subsequent LO phonon emission becomes relevant, yet other,
unexplored, inelastic loss mechanisms may also be present.

The results above demonstrate how precise coincidence counting measurements of
correlation signatures can be used to explore interactions between individual
electrons colliding at a beam splitter and to quantify the non-linear response
by the shift of the transmission threshold — a measure of parametric non-
linearity at a single-quantum level. The capability for the energy
spectroscopy of the scattering outcomes promotes further experimental studies.
While in linear optics the HOM effect enables metrology and tomography by
being time-sensitive to the overlap of two impinging wave packets, in the non-
linear regime $U/\hbar\gg\omega$, interactions at the beam splitter provide a
classical counterpart in measuring the statistical indistinguishability
between independent sources with $\omega\gg\sigma_{t}^{-1}$. Here the
dispersion of the beam splitter $\omega$ emerges as a crucial scale for
metrology: large $\omega$ boosts the sensitivity to interarrival times, while
small $\hbar\omega$ is essential for the selectivity of energy-time filters
[40, 31]. The width of the emission-time distribution measured here exhibits
consistency between the two tomography techniques.

Furthermore, the demonstrated interaction strength could be exploited in
quantum information technology as follows. An electron travelling
ballistically in a superposition of two short wave-packets spaced at distances
of multiple $\omega^{-1}$ forms a time-bin encoded qubit which can experience
a logic-state dependent phase rotation by $\Delta\varphi$ from the other
electron counterpropagating on the opposite side of the beam splitter. To
estimate feasibility of quantum logical operations, $\Delta\varphi\sim\pi$,
consider coincident arrival with $\epsilon_{1}=\epsilon_{2}=\epsilon/2<0$,
such that both electrons are reflected. The phase shift accumulated along the
classical trajectory in the perturbative limit of Eq. (4) evaluates exactly to

$\displaystyle\Delta\varphi=\hbar^{-1}\int
V_{\text{int}}[r(t)]dt=\frac{\epsilon^{\ast}(\epsilon)}{\hbar\omega}$ (5)

without violating the Heisenberg uncertainty for the wave-packet time-energy
localization if $U\gg\hbar\omega$ (see Supplementary Note VII).
$\Delta\varphi$ of order one is possible as
$\hbar\omega\sim\epsilon^{\ast}(\epsilon)\ll U$ is the classical perturbative
limit condition. Using the parameter range from Fig. 3 where the analytic
model has been validated with
$\epsilon^{\ast}($-2.8\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$)=$0.88\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$,
we estimate $\Delta\varphi=1.7\pi$, supporting the potential for quantum
applications. Equation (5) is also an important figure of merit for sensing,
e.g. single-shot readout of the flying qubit described above: shift of the
threshold of a probe electron by $\epsilon^{\ast}$ can be efficiently detected
by scattering if it exceeds the quantum width $\sim\hbar\omega$ of the
transmission function. The above estimates show that efficient exploitation of
interactions requires strong non-linearity, $U/(\hbar\omega)>1$, as
$\epsilon^{\ast}(\epsilon)$ is at most $\sim U$.

## Methods

### Experimental realization

The circuit has been realized [41] using a GaAs/AlGaAs heterostructure with
$97\text{\,}\mathrm{nm}$ nominal 2DEG depth, charge carrier density
$1.9\text{\times}{10}^{11}\text{\,}{\mathrm{cm}}^{-2}$, mobility
$1.15\text{\times}{10}^{6}\text{\,}{\mathrm{cm}}^{2}\text{\,}{\mathrm{V}}^{-1}\text{\,}{\mathrm{s}}^{-1}$,
and quantum life time $2.9\text{\,}\mathrm{ps}$. The circuit elements are
formed by an shallow-etched channel and Cr/Au gate electrodes. Measurements
were performed in a dilution refrigerator with base temperature below
$50\text{\,}\mathrm{mK}$ and at $B=$10\text{\,}\mathrm{T}$$ perpendicular
magnetic field (filling factor $\nu\approx 1$). Electrons are generated on-
demand by driving the single electron pumps with a single period of a
$300\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ sine signal. The interarrival
time is tuned by adjusting the channel skew between the signal sources. A
depletion gate [33, 28] over the mesa edge of the wave guide is set to
$-200\text{\,}\mathrm{m}\mathrm{V}$ resulting in a transverse confinement
energy $\hbar\omega_{y}$ at the mesa edgee of
$2.7\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$, estimated from the energy
dependence of LO phonon emission [29]. For the electron propagating along the
path towards the beam splitter, the dispersion in the transverse confinement
potential adds an additional contribution to the energy-time correlation of
the source shown and discussed in Fig. 3 and Fig. 4, but which, with a coarse
estimate of $\sim$4\text{\,}\mathrm{p}\mathrm{s}$$, can be considered
negligible relative to the source contribution. The dispersive beam splitter
potential induced by a notched gate finger (see also Supplementary Fig. I) can
be characterized in time-of-flight measurements [33] yielding an order of
magnitude estimate of $\omega=$300\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$
[35] for comparable gate geometries. The parameter estimates for two source
distributions
$\rho(\epsilon,t)=\rho(\sigma_{E}\tilde{\epsilon},\sigma_{t}\tilde{t})=$
$1/\left(2\pi\sqrt{1-r^{2}}\right)\exp\left[-\left(\tilde{\epsilon}^{2}-2r\tilde{\epsilon}\tilde{t}+\tilde{t}^{2}\right)/\left(2(1-r^{2})\right)\right]$
are $\\{\sigma_{E},\sigma_{t},r\\}$ =
$\\{$1.08\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$,$8.3\text{\,}\mathrm{p}\mathrm{s}$,0.50\\}$
for the source electron and
$\\{$1.07\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$,$6.9\text{\,}\mathrm{p}\mathrm{s}$,0.52\\}$
for the probe electron. Here $\epsilon$ and $t$ are statistical deviations
from the central averages, $\langle\epsilon_{i}\rangle$ and $\langle
t_{i}\rangle$.

### Analytical model

For analytical calculations we use quadratic saddle approximation [36] of the
single-particle beam splitter potential
$V(x_{i},y_{i})=m(\omega_{y}^{2}y_{i}^{2}-\omega_{x}^{2}x_{i}^{2})/2$, where
$m$ is the effective mass. Recent work [35] has shown that in this case,
besides the total energy $\epsilon_{1}+\epsilon_{2}$, an additional quantity
$\mathcal{H}=\epsilon_{1}^{\prime}+\epsilon_{2}^{\prime}$ with
$\epsilon_{i}^{\prime}=(1+e^{\mp\omega\Delta t})\epsilon_{i}/2$ (upper sign
for $i=1$) is conserved in collisions, for any interaction potential
$V_{\text{int}}$ decaying sufficiently fast within the range of validity of
the quadratic saddle approximation. This extra conservation law corresponds to
equipotential drift of the relative coordinates $x=x_{2}-x_{1}$,
$y=y_{2}-y_{1}$ generated by the Hamiltonian flow,
$\mathcal{H}(x,y)=V(x,y)/2+V_{\text{int}}\left(\sqrt{x^{2}+y^{2}}\right)$ such
that $p=eBy$ is the canonical momentum, conjugate to $x$ [35]. The effects of
interactions relevant for the computation of shifted thresholds and the energy
exchange are conveniently encoded in terms of
$\Delta\tau(E)=\tau(E)-\tau_{0}(E)$ where
$\tau=\int|dx/(\partial\mathcal{H}/\partial p)|$ is total traversal time for
the relative coordinate, computed along the trajectory $\mathcal{H}(x,y)=E$
implicitly defined by the conservation law, and $\tau_{0}$ denotes $\tau$ in
the absence of interactions ($V_{\text{int}}=0$ identically). $\Delta\tau(E)$
diverges at $E=3U/2$ which is the minimal $E$ required for the two electrons
to pass each other (i.e., reach $x=0$), thus setting the scale for the
magnitude of the mutual gating effect. For the unscreened Coulomb potential,
$V_{\text{int}}(r)=U\,d_{0}/r$, we derive a closed form expression for
$\Delta\tau(E)$ which is then used to determine the thresholds via Eq. (4) and
compute the final energies after the collision. In simulations, the finite
size of the beam splitter region is accounted for by setting $\Delta\tau(E)=0$
for $|E|>U_{c}^{\pm}$ where the thresholds $U^{\pm}_{c}$ are chosen by fixing
an interaction cut-off distance $d_{\text{cutoff}}$. Derivation details are
given in the Supplementary Notes III, IV, V, and VII.

### Perturbative limit for the non-linear shifts and $\omega$ estimation

Under the condition $\epsilon^{\ast}\ll|\epsilon|$ Eq. (4) simplifies to
$\epsilon^{\ast}=\epsilon\,\omega\,\Delta\tau(\epsilon)/2=c_{\pm}V_{\text{int}}(d_{\text{min}})$
where $d_{\text{min}}(\epsilon)$ is the closest approach distance between
particles and $c_{\pm}$ is a constant of order $1$ which depends on the exact
shape of the potentials. We derive the exact prefactors for the case of
$V_{\text{int}}(r)\propto r^{-\alpha}$ to prove Eq. (5) and relate the tail of
the interaction potential to that of the shifted threshold,
$\epsilon^{\ast}(\epsilon)\propto(U/|\epsilon|)^{\alpha/2}U$, see
Supplementary Notes VI and VII. The latter relation captures the contribution
of the increasing distance $d_{\text{min}}$ ($\propto|\epsilon|^{-1/2}$ in the
quadratic saddle) and the decay of the interaction potential. Further applying
the scaling relations, we get $\epsilon_{2}^{\ast}\propto(1+e^{-\omega\,\Delta
t})^{-\alpha/2}(1+e^{+\omega\,\Delta t})^{-1}$ for constant
$|\epsilon_{1}|\gg\epsilon^{\ast}_{2}$. The second moment of the latter
$\Delta t$-dependence is
$(\sigma_{t}^{\text{min}})^{2}=[\pi^{2}/6+\Psi^{\prime}(\alpha/2)]\,\omega^{-2}$
where $\Psi$ is the digamma function. This gives
$\omega=(2/3)^{1/2}\pi/\sigma_{t}^{\text{min}}$ for the Coulomb ($\alpha=1$)
potential.

This is used to assess sensitivity of the probe electron for small gating
effect, $\epsilon^{\ast}_{2}\lesssim\sigma_{E2}$, and large energy detuning of
the scan electron, $|\langle\epsilon_{1}\rangle|\ll\sigma_{E1}$, as
$s^{(1)}=s^{(2)}\propto\langle\epsilon^{\ast}_{2}\rangle$ in region A.
Neglecting the $\epsilon_{1}$-dependence of the threshold
$\epsilon_{2}^{\ast}$ (which is weak on the scale of $\sigma_{E}$ around
$\langle\epsilon_{1}\rangle$), the convolution with the source parameter
distributions given by Eq. (3) gives
$\sigma_{t}^{(A)}=(\sigma^{\min}_{t})^{2}+\sigma^{2}_{\Delta t}$ where
$\sigma^{2}_{\Delta t}=\sigma_{t1}^{2}+(1-r_{2}^{2})\sigma_{t2}^{2}$, form
which the beam splitter contribution $\sigma^{\min}_{t}$, and hence $\omega$,
can be estimated bypassing the need for a simulation.

## Acknowledgements

We acknowledge discussions with Piet W. Brouwer, Jonathan Fletcher, and Masaya
Kataoka, and thank Pascal Degiovanni for suggesting the phase shift
considerations. E.P., M.K., G.B., and V.K. are supported by grant no.
lzp-2021/1-0232 from the Latvian Council of Science. P.S. and P.R. acknowledge
financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) within the framework of Germany’s Excellence Strategy-EXC-2123
QuantumFrontiers-390837967. This work was supported in part by the Joint
Research Project SEQUOIA (17FUN04) which received funding from the European
Metrology Programme for Innovation and Research (EMPIR) cofinanced by the
Participating States and from the European Unions Horizon 2020 research and
innovation programme.

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## Supplementary information

###### Contents

  1. I Micrograph of the sample
  2. II Additional data for energy exchange
  3. III Analytic expression for $\Delta\tau$ for the unscreened Coulomb interaction
  4. IV Relation between the shifted transmission threshold $\epsilon^{\ast}$ and the modified relative traversal time $\Delta\tau$
  5. V Cut-off implementation
  6. VI Power-law interaction potential and first-principles estimates
  7. VII Perturbation theory in the interaction strength
  8. VIII Energy exchange and electron loss calculation

##  I Micrograph of the sample

Figure S1: SEM image showing the layout of the CrAu top gates over the etched
mesa channel forming the pumps (red), sidewall depletion (green), quantum dot
for charge read out and coupling floating gate (blue, red), and entrance/exit
to the detector nodes (yellow). The blue arrows indicate the path of the
injected electrons.

##  II Additional data for energy exchange

Figure S2: The probability $P_{00}$ that both detectors recorded no electron
for the dataset shown in Fig. 3 and Fig. 4.

##  III Analytic expression for $\Delta\tau$ for the unscreened Coulomb
interaction

Here we show that for the Coulomb potential parametrized as
$V_{\text{int}}(r)=Ud_{0}/r$ the interaction-induced delay time $\Delta\tau$
can be expressed as

$\exp[\omega\,\Delta\tau(E)]=|z|\frac{1+\kappa^{2}}{2\,\kappa^{2}\,\xi_{m}^{2}(z)}\times\\\
\exp\left\\{2\\!\\!\int_{\xi_{m}(z)}^{\infty}\\!\left[\frac{\kappa\,\xi^{2}}{(\xi^{3}\,\kappa^{2}-2+2\,\xi\,z)^{1/2}(\xi^{3}\,+2-2\,\xi\,z)^{1/2}}-\frac{1}{\xi}\right]d\xi\right\\}$
(6)

where $z=E/U$, $\kappa=\omega_{x}/\omega_{y}$, and

$\xi_{m}(z)=\begin{cases}\kappa^{-1}\left[(\sqrt{(2z/3)^{3}+\kappa^{2}}+\kappa)^{1/3}+(\kappa-\sqrt{(2z/3)^{3}+\kappa^{2}})^{1/3}\right],&z<3/2,\\\
(8z/3)^{1/2}\cos\left(\arccos\left[\left(2z/3\right)^{-3/2}\right]/3-\pi/3\right),&z>3/2,\end{cases}$
(7)

with $(\cdot)^{1/3}$ denoting the arithmetical cube root, and the exact
minimal distance reached in Coulomb scattering [35] is

$\displaystyle d_{\text{min}}^{(\alpha=1)}=\xi_{m}(E/U)\,d_{0}$ (8)

for $E=\epsilon_{1}^{\prime}+\epsilon_{2}^{\prime}$.

To formally define $\Delta\tau$, we first introduce the traversal time
$\tau[x_{0}]=\int dx/(\partial\mathcal{H}/\partial p)$, with integral computed
along the trajectory $\mathcal{H}(x,y)=E$, for the relative coordinate to
travel from $-x_{0}<0$ to either $x_{0}>0$ (in case of transmission of the
relative coordinate) or back to $x_{0}$ (in case of reflection of the relative
coordinate), as per [35, Eq. (15)]. Due to the symmetry,
$\tau[x_{0}]=2\int_{-x_{0}}^{0}dx/(\partial\mathcal{H}/\partial p)$ or
$\tau[x_{0}]=2\int_{-x_{0}}^{-d_{\text{min}}}dx/(\partial\mathcal{H}/\partial
p)$ (in case of transmission or reflection of the relative coordinate,
respectively). With $\tau_{0}[x_{0}]$ standing for $\tau[x_{0}]$ in the
absence of interactions, the interaction-induced delay time
$\Delta\tau[x_{0}]$ on the described segment of the trajectory is defined as
$\tau[x_{0}]-\tau_{0}[x_{0}]$. Now the total delay time $\Delta\tau$ is
defined as the limit of $\Delta\tau[x_{0}]$ as $x_{0}\to+\infty$.

We proceed to prove (6) in three steps: by a change of variables we obtain an
expression for $\tau[x_{0}]$; then we characterize the minimal distance
reached between the two electrons via (7); finally, we similarly describe
$\tau_{0}[x_{0}]$ and its asymptotics that allows to find the required limit
of $\Delta\tau[x_{0}]$.

#### Characterization of the traversal time $\tau[x_{0}]$

Explicitly, the trajectory $\mathcal{H}(x,y)=E$ in the relative coordinates is
described by

${y^{2}-\kappa^{2}x^{2}}{}+\frac{2d_{0}^{3}}{\sqrt{x^{2}+y^{2}}}=2d_{0}^{2}z,$
(9)

Introduce new variables $r,\phi$ by $x=d_{0}r\cos\phi$, $y=d_{0}r\sin\phi$,
then (9) becomes

$r^{3}\left(\sin^{2}\phi-\kappa^{2}\cos^{2}\phi\right)-2rz+2=0.$ (10)

As $x$ changes along the trajectory from $-x_{0}$ to either $0$ or
$-d_{\text{min}}$, the variable $r$ changes from $r_{0}$ to
$d_{\text{min}}/d_{0}$. We express the derivative of $r$ w.r.t. $\tau$ as
$\frac{dr}{\omega d\tau}=\frac{\partial r}{\partial x}\cdot\frac{dx}{\omega
d\tau}+\frac{\partial r}{\partial y}\cdot\frac{dy}{\omega d\tau}$, where the
velocity vector $(\frac{dx}{\omega d\tau},\frac{dy}{\omega d\tau})$ is found
as a tangent to the trajectory, arriving at

$\frac{dr}{\omega d\tau}=-\frac{(\kappa^{2}+1)r\sin(2\phi)}{2\kappa}.$

To determine $\sin(2\phi)$, rewrite (10) as
$r^{3}\left(\sin^{2}\phi-\kappa^{2}\cos^{2}\phi\right)=2(rz-1)(\sin^{2}\phi+\cos^{2}\phi)$
and express $\tan\phi$,

$\tan\phi=\pm\sqrt{\frac{r^{3}\kappa^{2}+rz-2}{r^{3}-2rz+2}}.$

Here $\pm$ equals the sign of $x\cdot y$ (which is positive before $x$ reaches
0 or $-d_{\text{min}}$, since $x<0$ and $y<0$ until then). The identity
$\sin(2\phi)=\frac{2\tan\phi}{\tan^{2}\phi+1}$ then leads to

$\frac{\omega d\tau}{dr}=-\frac{\kappa
r^{2}}{\sqrt{(r^{3}-2rz+2)(\kappa^{2}r^{3}+2rz-2)}}.$

Consequently,

$\omega\tau[x_{0}]/2=\int_{d_{\text{min}}/d_{0}}^{r_{0}}\frac{\kappa
r^{2}\,dr}{\sqrt{(r^{3}-2rz+2)(\kappa^{2}r^{3}+2rz-2)}}.$

#### Characterization of the minimal distance $d_{\text{min}}$

Furthermore, the value $d_{\text{min}}/d_{0}$ itself can be found from (10),
since the minimal distance is achieved either when $(x,y)=(-d_{\text{min}},0)$
or when $(x,y)=(0,-d_{\text{min}})$. In the former case (reflection of the
relative coordinate), $z<3/2$ and $\phi=-\pi$; in the latter (transmission of
the relative coordinate), $z>3/2$ and $\phi=-\pi/2$. Thus
$d_{\text{min}}/d_{0}=r$ is the solution of $-\kappa^{2}r^{3}=2rz-2$ or
$r^{3}=2rz-2$, respectively.

Consider the case $z>3/2$ and the respective equation $r^{3}-2rz+2=0$. The
discriminant of this polynomial is $4\left(8z^{3}-27\right)>0$, therefore the
equation has three real roots. Vieta’s formulas imply that exactly one of them
is negative. It can be deduced from Vieta’s formulas that one of the two
positive roots is less than 1 and the other root is larger than one; since
$d_{\text{min}}\geq d_{0}$ in the case of transmission of the relative
coordinate, the relevant root is the largest one. By applying Vieta’s
trigonometric expression for the cubic equation roots, the three real roots
are

$r^{[k]}=2\sqrt{\frac{2z}{3}}\cos\left(\frac{1}{3}\arccos\left(-(2z/3)^{-3/2}\right)-k\frac{2\pi}{3}\right),\quad
k\in\\{0,1,2\\},$

which can be simplified to

$r^{[k]}=2\sqrt{\frac{2z}{3}}\cos\left(\frac{1}{3}\arccos\left((2z/3)^{-3/2}\right)+\frac{(2k-1)\pi}{3}\right),\quad
k\in\\{0,1,2\\},$

with $r^{[0]}$ being the largest of the three roots. Therefore, for $z>3/2$
the value of $d_{\text{min}}/d_{0}$ is equal to

$\sqrt{8z/3}\cos\left(\arccos\left[\left(2z/3\right)^{-3/2}\right]/3-\pi/3\right).$

Consider the case $z<3/2$ and the respective equation
$\kappa^{2}r^{3}+2rz-2=0$. The discriminant of the left-hand side is
$-4\kappa^{2}\left(27\kappa^{2}+8z^{3}\right)$, which is negative for
$z>z_{*}:=-\frac{3}{2}\kappa^{2/3}$ and positive for $z<z_{*}$. Hence, $r$ is
the unique real (and positive) solution for $z\in(z_{*},3/2)$. When $z<z_{*}$,
there are three distinct real roots; Vieta’s formulas imply that exactly one
of them is positive, thus $r$ is the unique positive solution. In both cases,
it follows from Cardano’s formula that the positive solution is

$\frac{(\,\sqrt{(2\,z/3)^{3}+\kappa^{2}}+\kappa\,)^{1/3}+(\kappa-\sqrt{(2\,z/3)^{3}+\kappa^{2}}\,)^{1/3}}{\kappa}.$

This completes the proof that $d_{\text{min}}/d_{0}$ is given by the function
$\xi_{m}(z)$, defined in (7).

#### Characterization of the traversal time $\tau_{0}[x_{0}]$

In the absence of interactions, the trajectory in the relative coordinates is
described by

${y^{2}-\kappa^{2}x^{2}}{}=2d_{0}^{2}z.$ (11)

Similarly as we did for $\omega\tau[x_{0}]$, one can derive

$\displaystyle\omega\tau_{0}[x_{0}]$
$\displaystyle=2\int_{\rho_{m}}^{\rho_{0}}\frac{\kappa
r\,dr}{\sqrt{r^{2}-2z}\sqrt{\kappa^{2}r^{2}+2z}}=\left.-2\ln\left(\sqrt{\kappa^{2}r^{2}+2z}-\kappa\sqrt{r^{2}-2z}\right)\right|_{r=\rho_{m}}^{r=\rho_{0}},$

where $\rho_{0}=\sqrt{x_{0}^{2}+y_{0}^{2}}/d_{0}$, with $y_{0}$ determined by
(11), and the minimal distance reached between the two electrons now is

$\rho_{m}=\begin{cases}\sqrt{2z},&z>0,\\\ \sqrt{-2z}/\kappa,&z<0.\end{cases}$

Both for $z>0$ and $z<0$ one obtains

$\omega\tau_{0}[x_{0}]=\ln\left(2\left(\kappa^{2}+1\right)|z|\right)-2\ln\left|\sqrt{\kappa^{2}\rho_{0}^{2}+2z}-\kappa\sqrt{\rho_{0}^{2}-2z}\right|.$

Furthermore, since

$\lim_{\rho\to\infty}\rho\left(\sqrt{2z+\rho^{2}\kappa^{2}}-\kappa\sqrt{\rho^{2}-2z}\right)=\frac{z(\kappa^{2}+1)}{\kappa},$

we have

$-2\ln\left|\sqrt{\kappa^{2}\rho_{0}^{2}+2z}-\kappa\sqrt{\rho_{0}^{2}-2z}\right|=\ln\frac{\kappa^{2}}{z^{2}(\kappa^{2}+1)^{2}}+2\ln\rho_{0}+o(\rho_{0})$

as $\rho_{0}\to\infty$. This gives the following asymptotics for large
$\rho_{0}$ (equivalently, large $x_{0}$):

$\omega\tau_{0}[x_{0}]=\ln\frac{2\kappa^{2}\rho_{0}^{2}}{|z|(\kappa^{2}+1)}+o(x_{0}).$

By expressing $\ln r_{0}=\ln\xi_{m}+\int_{\xi_{m}}^{r_{0}}\frac{d\xi}{\xi}$ we
now see that

$\displaystyle\omega\Delta\tau[x_{0}]$
$\displaystyle=\omega\tau[x_{0}]-\omega\tau_{0}[x_{0}]$
$\displaystyle=2\int_{\xi_{m}}^{r_{0}}\frac{\kappa\xi^{2}\,d\xi}{\sqrt{(\xi^{3}-2\xi
z+2)(\kappa^{2}\xi^{3}+2\xi
z-2)}}-\ln\frac{2\kappa^{2}\rho_{0}^{2}}{|z|(\kappa^{2}+1)}+o(x_{0})$
$\displaystyle=2\int_{\xi_{m}}^{r_{0}}\frac{\kappa\xi^{2}\,d\xi}{\sqrt{(\xi^{3}-2\xi
z+2)(\kappa^{2}\xi^{3}+2\xi
z-2)}}-2\int_{\xi_{m}}^{r_{0}}\frac{d\xi}{\xi}-\ln\frac{2\kappa^{2}\rho_{0}^{2}\xi_{m}^{2}}{|z|(\kappa^{2}+1)r_{0}^{2}}+o(x_{0})$
$\displaystyle=\ln\frac{|z|(\kappa^{2}+1)}{2\kappa^{2}\xi_{m}^{2}}+2\int_{\xi_{m}}^{r_{0}}\left[\frac{\kappa\xi^{2}}{\sqrt{(\xi^{3}-2\xi
z+2)(\kappa^{2}\xi^{3}+2\xi z-2)}}-\frac{1}{\xi}\right]d\xi+o(x_{0}).$

The last equality is justified by the fact that from (9) and (11) it follows
that

$\displaystyle\frac{r_{0}^{2}}{\rho_{0}^{2}}$
$\displaystyle=\frac{x_{0}^{2}(\kappa^{2}+1)-\frac{2d_{0}^{3}}{r_{0}^{2}}+2d_{0}^{2}z}{x_{0}^{2}(\kappa^{2}+1)+2d_{0}^{2}z}=1-\frac{2d_{0}^{3}}{r_{0}^{2}\rho_{0}^{2}}=1-o(x_{0}).$

Taking the limit $x_{0}\to\infty$, we arrive at (6). Finally, we note that by
the limit comparison test the improper integral
$\lim\limits_{r_{0}\to+\infty}\int_{\xi_{m}}^{r_{0}}\left[\frac{\kappa\xi^{2}}{\sqrt{(\xi^{3}-2\xi
z+2)(\kappa^{2}\xi^{3}+2\xi z-2)}}-\frac{1}{\xi}\right]d\xi$ converges, since

$\frac{\kappa\xi^{2}}{\sqrt{(\xi^{3}-2\xi z+2)(\kappa^{2}\xi^{3}+2\xi
z-2)}}-\frac{1}{\xi}\sim\begin{cases}z(\kappa^{2}-1)/\kappa^{2}\xi^{-3},&\kappa\neq
1,\,z\neq 0,\\\ -(\kappa^{2}-1)/\kappa^{2}\xi^{-4},&\kappa\neq 1,\,z=0,\\\
2z^{2}\xi^{-5},&\kappa=1,\,z\neq 0,\\\ 2\xi^{-7},&\kappa=1,\,z=0.\end{cases}$

##  IV Relation between the shifted transmission threshold $\epsilon^{\ast}$
and the modified relative traversal time $\Delta\tau$

Condition for either electron being on the threshold is derived in Ref. [35]
to be

$\displaystyle E_{-}\,\Phi_{\kappa}=\pm E_{+}\,$ (12)

where in our notation $E_{\pm}=\epsilon_{2}^{\prime}\pm\epsilon_{1}^{\prime}$
and $\Phi_{\kappa}=\operatorname{sgn}(E_{+})\exp[\omega\,\Delta\tau(E_{+})]$.
The equation is valid with both signs and encodes both
$\epsilon_{2}^{\ast}(\epsilon_{1})$ and $\epsilon_{1}^{\ast}(\epsilon_{2})$.
For the function $\epsilon^{\ast}(\epsilon)$ defining the symmetric ($\Delta
t=0$) shifted transmission threshold, we consider
$\epsilon_{2}^{\prime}=\epsilon_{2}=\epsilon^{\ast}$ and
$\epsilon_{1}^{\prime}=\epsilon_{1}=\epsilon$, hence
$E_{\pm}=\epsilon^{\ast}\pm\epsilon$. $\Phi_{\kappa}$ as function of $E_{+}$
changes sign at $E_{+}=0$ ($\Delta\tau\to-\infty$) and diverges at
$E_{+}=3U/2$ ($\Delta\tau\to+\infty$). For the solution branch defining
$\epsilon^{\ast}>0$ as function of $\epsilon$ one has to pick the following
signs in the corresponding three ranges of $E_{+}$:

$\displaystyle\begin{cases}(\epsilon^{\ast}-\epsilon)\,\exp[\omega\,\Delta\tau]=-(\epsilon^{\ast}+\epsilon)\,,&E_{+}<0\,,\\\
(\epsilon^{\ast}-\epsilon)\,\exp[\omega\,\Delta\tau]=(\epsilon^{\ast}+\epsilon)\,,&0<E_{+}<1.5\,U\,,\\\
(\epsilon^{\ast}-\epsilon)\,\exp[\omega\,\Delta\tau]=-(\epsilon^{\ast}+\epsilon)\,,&E_{+}>1.5\,U\,.\end{cases}$
(13)

The solutions are $\epsilon^{\ast}/\epsilon=\tanh[\omega\,\Delta\tau/2]$
($E_{+}<0$ and $E_{+}>1.5\,U$) and
$\epsilon^{\ast}/\epsilon=\coth[\omega\,\Delta\tau/2]$ ($0<E_{+}<1.5\,U$). As
$\epsilon^{\ast}=-\epsilon$ at $E_{+}=0$ and $\epsilon^{\ast}=\epsilon=3U/4$
at $E_{+}=3U/2$, the results can be expressed as equation (4) in the main
text.

##  V Cut-off implementation

Figure S3: Illustration of the cut-off procedure used in the simulation in
terms of the shifted transmission thresholds at $\Delta t=0$. Here $\kappa=1$,
$U=$2.3\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$,
$U_{c}=$7.0\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$, corresponding to the
simulation parameters used for Fig. 3 in the main text.

Two physical effects limit applicability of the exactly solvable analytical
model for which Eqs. (6) and Eq. (7) apply: finite extent of the beam splitter
where the quadratic saddle potential is applicable and the screening of two-
electron interaction at large distances. We account for these limitations by a
single condition: if the two electrons are predicted within the full model to
reach a minimal distance $d_{\text{min}}$ that is less than a fixed cut-off
distance $d_{\text{cutoff}}$, the full model is applied. If, however,
$d_{\text{min}}>d_{\text{cutoff}}$ then the electrons are treated as non-
interacting and the baseline transmission thresholds,
$\epsilon^{\ast}_{1}=\epsilon^{\ast}_{2}=0$ are applied. This approach
underestimates the effect of interactions outside of the beam splitter region,
but has the advantage of being compatible with the $\Delta t$ scaling
relations which also prevents unphysical extrapolation of exponential
acceleration due the quadratic roll-off of the saddle potential for large
$|\Delta t|$.

We apply the above cut-off procedure by defining
$U_{c}=m\omega_{x}^{2}d_{\text{cutoff}}^{2}/4$ and expressing the cutoff
values of the conserved integral of motion for the relative coordinate,

$\displaystyle U_{c}^{-}$
$\displaystyle=\mathcal{H}(-d_{\text{cutoff}},0)=-U_{c}+\frac{U^{3/2}\kappa}{\sqrt{2U_{c}}}$
(14) $\displaystyle U_{c}^{+}$
$\displaystyle=\mathcal{H}(0,-d_{\text{cutoff}})=\frac{U_{c}}{\kappa^{2}}+\frac{U^{3/2}\kappa}{\sqrt{2U_{c}}}$
(15)

Note that $d_{\text{cutoff}}$ as a function of $U_{c}^{\pm}$ in the Eq. (14)
and (15) equals $d_{\min}$ as a function of $E$ in the equation (8) and the
cut-off condition $d_{\text{min}}>d_{\text{cutoff}}$ on $E$ is implemented as
as: $E>U_{c}^{+}(d_{\text{cutoff}})$ or $E<U_{c}^{-}(d_{\text{cutoff}})$.

Figure S3 illustrates the shifted transmission thresholds that correspond to
this procedure.

##  VI Power-law interaction potential and first-principles estimates

Parametrizing a power-law interaction potential as

$\displaystyle
V_{\text{int}}(r)=\frac{3\,U}{2+\alpha}\,\left(\frac{d_{0}}{r}\right)^{\alpha}$
(16)

ensures that the minimum of $\mathcal{H}(0,y)$ as function of $y$ is reached
at $y=\pm d_{0}$ and $\mathcal{H}=3U/2$ where $d_{0}$ satisfies
$m\omega_{y}^{2}d_{0}^{2}/2=\alpha\,V_{\text{int}}(d_{0})$, which in turn
implies that $\Delta\tau(E)$ diverges as $E\to 3U/2$ (setting the crossing
point of the shifted thresholds to $\epsilon_{1c}+\epsilon_{2c}=3U/2$) and
makes $d_{0}$ the closest possible distance between the electrons passing each
other at the beam splitter, in consistency with the definitions of $U$ and
$d_{0}$ introduced for the special case of $\alpha=1$ in Ref. [35].

Explicitly,

$\displaystyle
d_{0}=\sqrt{\frac{6\alpha}{2+\alpha}}\sqrt{\frac{U}{m\omega_{y}^{2}}}$ (17)

Unscreened Coulomb potential in GaAs [42] can be quantified in terms of
effective hartree
$\hbar\omega^{\star}=$12\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ and
effective Bohr radius $a^{\star}=$9.8\text{\,}\mathrm{n}\mathrm{m}$$ as
$V_{\text{int}}(r)=\hbar\omega^{\star}\,a^{\star}/r$, which gives, taking into
account that $\hbar/m=\omega^{\star}(a^{\star})^{2}$,

$\displaystyle U$
$\displaystyle=2^{-1/3}\hbar\left(\omega^{\star}\,\omega_{y}^{2}\right)^{1/3}$
(18) $\displaystyle d_{0}$
$\displaystyle=2^{1/3}\,a^{\star}\,(\omega^{\star}/\omega_{y})^{2/3}$ (19)

From $\omega_{x}/\omega_{y}=1$,
$\omega=\omega_{x}\,\omega_{y}/\omega_{c}=$250\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$$
inferred from simulations and $\hbar\omega_{c}=\hbar
eB/m=$17.5\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$, we get
$\hbar\omega_{y}=$1.7\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ and, using
equation (18) above, $U=$2.6\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ which is
compared to $U=$2.3\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ obtained by
adjusting $U$ in the simulations. From Eq. (19) we get
$d_{0}=$45\text{\,}\mathrm{n}\mathrm{m}$$.

From $U_{c}=$7\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$$ we get
$d_{\text{cutoff}}=$75\text{\,}\mathrm{n}\mathrm{m}$$.

##  VII Perturbation theory in the interaction strength

Applying classical perturbation theory to the definition of $\tau(E)$ to the
first order in $V_{\text{int}}(r)$ gives

$\displaystyle\omega\Delta\tau(E)=E^{-1}\kappa_{\pm}\int_{1}^{\infty}\left[\frac{(1+\kappa_{\pm}^{2})\,\xi\,V_{\text{int}}(\xi
d_{\text{min}})}{\sqrt{\xi^{2}-1}(\kappa_{\pm}^{2}\xi^{2}+1)^{3/2}}-\frac{d_{\text{min}}\,V_{\text{int}}^{\prime}(\xi
d_{\text{min}})}{\sqrt{\xi^{2}-1}\sqrt{\kappa_{\pm}^{2}\xi^{2}+1}}\right]d\xi$
(20)

where $\kappa_{+}=\omega_{x}/\omega_{y}=1/\kappa_{-}$,
$\pm=\operatorname{sgn}E$, $V_{\text{int}}^{\prime}(r)=dV_{\text{int}}(r)/dr$,
and

$\displaystyle
d_{\text{min}}^{(U=0)}=\sqrt{4|E|\kappa_{\pm}/(m\omega_{x}\omega_{y})}$ (21)

(which is consistent with $U\to 0$ limit of Eq. (8)).

The two terms in the integrand in Eq. (20) account for the interaction-induced
change of velocity and the change of trajectory, respectively.

The quantum phase shift $\Delta\varphi(E)$ along the non-perturbed trajectory
is expressed similarly

$\displaystyle\Delta\varphi(E)=(\hbar\omega)^{-1}\int_{-\infty}^{+\infty}V_{\text{int}}[r(t)]dt=\frac{2\,\kappa_{\pm}}{\hbar\omega}\int_{1}^{\infty}\frac{\xi\,V_{\text{int}}(\xi
d_{\text{min}})\,d\xi}{\sqrt{\xi^{2}-1}\sqrt{\kappa_{\pm}^{2}\xi^{2}+1}}$ (22)

For the scale-free power law potential (16), the integrals in the linear
expansion (20) can be computed exactly (regardless of the prefactor notation)

$\displaystyle\omega\,\Delta\tau(E)\,E=c(\kappa_{\pm},\alpha)\,V_{\text{int}}(d_{\text{min}})$
(23)

where the dimensionless coefficient is

$\displaystyle c(\kappa,\alpha)$
$\displaystyle=\kappa\int_{1}^{\infty}\frac{\xi^{-1-\alpha}\left[\alpha+\left(1+\kappa^{2}+\alpha\,\kappa^{2}\right)\xi^{2}\right]}{{\sqrt{\xi^{2}-1}(\kappa^{2}\xi^{2}+1)^{3/2}}}d\xi=\int_{1}^{\infty}\frac{\alpha\,\kappa\,\xi^{1-\alpha}\,d\xi}{\sqrt{\xi^{2}-1}\sqrt{\kappa^{2}\xi^{2}+1}}$
(24)
$\displaystyle=\frac{\sqrt{\pi}\,\Gamma\left(1+\frac{\alpha}{2}\right)}{\Gamma\left(\frac{1+\alpha}{2}\right)}{}_{2}F_{1}\left(\frac{1}{2},\frac{\alpha}{2};\frac{1+\alpha}{2};-\frac{1}{\kappa^{2}}\right)$
(25) $\displaystyle c(1,\alpha)$
$\displaystyle=\sqrt{\pi}\,\frac{\Gamma\left(1+\frac{\alpha}{4}\right)}{\Gamma\left(\frac{1}{2}+\frac{\alpha}{4}\right)}=\begin{cases}1+\frac{(\ln
2)\alpha}{2},&\alpha\ll 1\\\ \frac{\sqrt{\alpha\pi}}{2},&\alpha\gg
1\end{cases}$ (26) $\displaystyle c(\kappa,1)$
$\displaystyle=K(-1/\kappa^{2})=\begin{cases}\kappa\ln\frac{4}{\kappa},&\kappa\ll
1\\\ \frac{\pi}{2}-\frac{\pi}{8\kappa^{2}},&\kappa\gg 1\end{cases}$ (27)

where $\Gamma$ is the Gamma function, ${}_{2}F_{1}$ is the hypergeometric
function, and $K$ is the complete elliptic integral of the first kind.

As explained in the main text, in the small $\Delta\tau$ limit, Eq. (23) with
$E=\epsilon$ gives directly the threshold function

$\displaystyle\epsilon^{\ast}(\epsilon)=2c(\kappa_{\pm},\alpha)V_{\text{int}}[d_{\text{min}}(\epsilon)]\ll|\epsilon|$
(28)

Using the prefactor defined in Eq. (16) with Eq. (21) gives the
large-$|\epsilon|$ asymptotics of the shifted thresholds:

$\displaystyle\epsilon^{\ast}(\epsilon)=U\,(U/|\epsilon|)^{-\alpha/2}\,6\,\left(\frac{3\alpha}{2}\right)^{\alpha/2}(2+\alpha)^{-1-\alpha/2}\times\begin{cases}c(\kappa,\alpha)\,,&\epsilon>0\\\
k^{\alpha}c(\kappa^{-1},\alpha)&\epsilon<0\end{cases}$ (29)

For the quantum phase estimate (22) and the power-law interaction (16), we
find with the help of Eq. (24)

$\displaystyle\Delta\varphi(E)=\frac{2\,c(\kappa_{\pm},\alpha)}{\alpha}\frac{V_{\text{int}}(d_{\text{min}})}{\hbar\omega}$
(30)

With $E=\epsilon/2+\epsilon/2$ and Eq. (28) this proves the relation
$\Delta\varphi(\epsilon)=\alpha^{-1}\epsilon^{\ast}(\epsilon)/(\hbar\omega)$
quoted in the main text for the experimentally relevant case of the Coulomb
potential $\alpha=1$.

Consistency of using Eq. (22), and, subsequently Eq (5) of the main text,
requires localization of the quantum wave-packet in time domain within the
beam splitter time-scale, $\sigma_{t}<\omega^{-1}$, and the no-reflection
condition $|\epsilon/2|\gg\sigma_{E}$. Due to energy-time uncertainty
$\sigma_{E}\sigma_{t}>\hbar/2$ this requires $|\epsilon|\gg\hbar\omega$ which
is compatible with $\Delta\varphi$ of order one as
$\hbar\omega\sim\epsilon^{\ast}(\epsilon)\ll|\epsilon|$ is possible in the
perturbative regime for large $U\gg\hbar\omega$.

##  VIII Energy exchange and electron loss calculation

The motion of the two electrons in the quadratic saddle potential can be fully
separated into relative and center-of-mass coordinates that implies not only
the conservation of the true asymptotic energy $\epsilon_{1}+\epsilon_{2}$ but
also of the integral of motion $E=\epsilon_{1}^{\prime}+\epsilon_{2}^{\prime}$
associated with the relative coordinate [35]. These two conservation laws
allow one to express the relation between the incoming initial
($\epsilon_{1}$, $\epsilon_{2}$, $\Delta t$) and the outgoing final
($\epsilon_{1}^{f}$, $\epsilon_{2}^{f}$, $\Delta t^{f}$) non-interacting
parameters via a single function,
$\Phi_{\kappa}=\operatorname{sgn}(E)\exp[\omega\,\Delta\tau(E)]$, for which we
provide a closed form expression, Eq. (6). The relation for $\Delta t^{f}$ is
derived in Ref. [35]. Similarly, we derive the expression for the energy
transfer,

$\epsilon_{2}^{f}-\epsilon_{2}=-(\epsilon_{1}^{f}-\epsilon_{1})=\frac{\pm
1+\Phi_{\kappa}}{2\,\Phi_{\kappa}}\left[\mp(\epsilon_{2}^{\prime}-\epsilon_{1}^{\prime})\,\Phi_{\kappa}+E\tanh\frac{\omega\Delta
t}{2}\right]\,,$ (31)

where the upper sign is for $E<3U/2$. Note that the energy exchange is
singular at $E=3U/2$ as at this combination of initial conditions the
electrons stay together ($\Delta t^{f}=0$) indefinitely within the classical
solution. The real amount of the energy exchange is limited by the finite
extent of the beam splitter but may exceed both $|\epsilon_{1}|$ and
$|\epsilon_{2}|$ significantly, leading to the distinct loss signal.

The losses in each channel are calculated using a simple model for loss rate
energy dependence
$P^{\text{loss}}_{1,2}=1-\exp\left[-C_{1,2}\exp(-\beta_{0}\,\epsilon_{1,2}^{f})\right]$
at detector D1 or D2, respectively. The parameters $C_{1}=0.014$,
$C_{2}=0.004$ and
$\beta_{0}=$0.25\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}^{-1}$$ are adjusted to
reproduce the absolute contrast of the the loss signals $P_{10}$ and $P_{01}$
while all the other parameters of collision kinematics and statistical
broadening are fixed by the simulation of the correlation signatures in the
lossless collision outcomes, $P_{20}$ and $P_{02}$.