# Non-thermal Tomonaga-Luttinger liquid eventually emerging from hot electrons
in the quantum Hall regime

Kotaro Suzuki  Tokuro Hata  Yuya Sato  Department of Physics, Tokyo Institute
of Technology, 2-12-1 Ookayama, Meguro, Tokyo, 152-8551, Japan.  Takafumi
Akiho  Koji Muraki  NTT Basic Research Laboratories, NTT Corporation, 3-1
Morinosato-Wakamiya, Atsugi 243-0198, Japan.  Toshimasa Fujisawa
<EMAIL_ADDRESS>Department of Physics, Tokyo Institute of
Technology, 2-12-1 Ookayama, Meguro, Tokyo, 152-8551, Japan.

Dynamics of integrable systems, such as Tomonaga-Luttinger (TL) liquids, is
deterministic, and the absence of stochastic thermalization processes provides
unique characteristics, such as long-lived non-thermal metastable states with
many conserved quantities. Here, we show such non-thermal states can emerge
even when the TL liquid is excited with extremely high-energy hot electrons in
chiral quantum-Hall edge channels. This demonstrates the robustness of the
integrable model against the excitation energy. Crossover from the single-
particle hot electrons to the many-body TL liquid is investigated by using on-
chip detectors with a quantum point contact and a quantum dot. The charge
dynamics can be understood with a single-particle picture only for hot
electrons. The resulting electron-hole plasma in the TL liquid shows a non-
thermal metastable state, in which warm and cold electrons coexist without
further thermalization. The multi-temperature constituents are attractive for
transporting information with conserved quantities along the channels.

Introduction

The Coulomb interaction in one-dimensional (1D) conductors plays an essential
role in non-equilibrium transport characteristicsBookGiamarchi ; ReviewElecQO
; TaruchaSSC1995 ; BockrathNature1999 ; LorenzNature2002 ;
AuslaenderScience2005 . Two distinct transport regimes, Tomonaga-Luttinger
(TL) liquid in the low-energy regime and single-particle hot-electron
transport in the high-energy regime, appear in chiral 1D channels of an
integer quantum Hall (QH) region at Landau-level filling factor $\nu$ = 2 in
AlGaAs/GaAs heterostructuresBookEzawa ; ChangRMB ; LevkivskyiBook2012 . In the
low-energy regime, the energy dispersion near the Fermi energy can be
approximated to be linear and the system is well described with collective
excitation modes in the TL liquid modelvonDelftAnnPhys1998 ; BergPRL2009 .
Arbitrary electronic excitation can be represented by spin and charge
collective modes. The collective excitations propagate at different velocities
for each mode, a phenomenon known as spin-charge separationFreulonNatComm-SC ;
Bocquillon-NatCom2013 ; Inoue-PRL2014 ; Kamata-NatNano2014 ; Hashisaka-
NatPhys2017 . Owing to the integrable nature of the TL model, the electronic
excitation will be equilibrated into an ensemble of spin and charge
excitations but not fully relaxed into a thermal equilibrium
stateGutmanPRB2010 ; GutmanPRL2008 ; Iucci2009 ; KovrizhinPRB2011 ;
LevkivskyiPRB2012 . This non-thermal metastable state (prethermalized state)
survives even after traveling over a distance ($>$ 20 $\mu$m) much longer than
required for the spin-charge separation ($<$ 0.1 $\mu$m)WashioPRB2016 ; Itoh-
PRL2018 ; RodriguezNatComm2020 . In these experiments, the initial state is
excited with low energies ($<$ 0.2 meV from the Fermi energy) where the TL
model holds well. The validity of the TL model, or the system’s integrability,
is no longer guaranteed in the high-energy regime, where due to the deviation
from the linear energy dispersion the Coulomb interaction induces weak
electron-electron (e-e) scattering between the hot and cold
electronsTaubertPRB2011 ; OtaPRB2019 .

In the high energy regime, the hot electron travels for a long distance ($>$
10 $\mu$m) without losing energy when the Coulomb interaction is suppressed,
for example, by screening the interaction with metal on the
surfaceFletcherPRL2013 ; KataokaPRL2016 ; JohnsonPRL2018 ; FletcherNatCom2019
; Ubbelohde-NatNano2014 . Such ballistic hot electron transport can be seen at
high energy ($>$ 50 meV) from the Fermi energy. In the intermediate energy
region, the e-e scattering becomes significant, where the crossover between
the TL liquid and single-particle hot electron transport is expected. However,
the crossover regime remained veiled as the cold- and hot-electron dynamics
have been studied separately by using different experimental schemes.

Here, we investigate the crossover dynamics from high-energy hot electrons to
a non-thermal TL liquid, where the hot electrons lose their energy by exciting
the TL liquid under the Coulomb interaction. The resulting state is unusual as
warm and cold electrons coexist in the same channel even in the quasi-steady
state. The emergence of the non-thermal many-body state is striking as the
hot-electron energy is well beyond the low energy regime for the TL liquid.
The obtained non-thermal state is stable for long transport even after full
relaxation of hot electrons, which represents the robustness of the integrable
nature. We find that the final state depends on the initial hot-electron
energy even after the equilibration, while energy-dependent dissipation is
observed at higher energy. The crossover includes intra-channel scattering
generating a hot spot in the spin-up channel and subsequent inter-channel
scattering generating another hot spot in the spin-down channel. The
characteristics are experimentally obtained by using a quantum point contact
(QPC) as a spin-dependent bolometer to extract the energy-space trajectory of
hot electrons and a quantum dot (QD) as a narrow-band spectrometer to evaluate
the non-thermal TL liquid. The QH channel is a promising platform to study the
interplay between single-particle and many-body physics.

Figure 1: Crossover dynamics of single-particle scattering processes and non-
thermal Tomonaga-Luttinger (TL) liquid. a Schematic spatial evolution of
energy distribution functions $f_{\uparrow}$ and $f_{\downarrow}$ for up- and
down-spins, respectively. Hot electrons are injected into the spin-up channel
at energy $eV_{\mathrm{inj}}$ (i) and relax with intra-channel electron-
electron scattering (e↑-e↑) and inter-channel scattering (e↑-e↓) (i)-(iv).
Electron-hole plasma is excited in the channels, and hot spots appear at
$x_{\mathrm{H\uparrow}}$ and $x_{\mathrm{H\downarrow}}$ in the spin-up and
-down channels, respectively. The system ends up in a non-thermal metastable
state (v) before reaching thermal equilibrium (vi). b Energy profiles for the
lowest Landau levels (LLLs) for up and down spins. Intra-channel electron-
electron scattering (e↑-e↑) is shown for a hot electron at energy
$E_{\mathrm{h}}$. c Intra-channel scattering (e↑-e↑) for hot and cold
electrons at energies $E_{\mathrm{h}}$ and $E_{\mathrm{c}}$, respectively, and
final states at $E_{\mathrm{h}}-\varepsilon$ and $E_{\mathrm{c}}+\varepsilon$
with energy exchange $\varepsilon$. d Inter-channel scattering (e↑-e↓) and
suppressed intra-channel scattering (e↑-e↑) with destructive interference for
$E_{\mathrm{h}}<$ $\Delta E_{\uparrow\uparrow}$. e Formation of a TL liquid
within the energy range $\Delta E_{\mathrm{TL}}$.

Results and discussion

Crossover dynamics. We investigate the dynamics of hot and cold electrons in
the two chiral edge channels at $\nu$ = 2 in a high magnetic field $B$. Figure
1a illustrates our finding on how the energy distribution functions
$f_{\uparrow}$ and $f_{\downarrow}$ of up and down spins, respectively, change
with traveling along the $x$-axis. With the spin-up and down channels
equilibrated with cold electrons prepared at the base temperature
$T_{\mathrm{B}}$ at $x<0$, we inject hot electrons with energy
$eV_{\mathrm{inj}}$ selectively to the spin-up channel at $x=0$ [panel (i)].
For moderate $eV_{\mathrm{inj}}$, the Coulomb interaction induces intra-
channel e-e scattering between the hot and cold spin-up electrons (e↑-e↑), and
electron-hole plasma (the light magenta regions) is generated in the spin-up
channel [panel (ii)] OtaPRB2019 . We neglect three-body e-e scattering
throughout this paper for simplicityKhodasPRB2007 . The hot-electron
relaxation can be characterized by the energy decay rate per unit length,
$\gamma_{\uparrow\uparrow}=-dE_{\mathrm{h}}/dx$, for hot-electron’s energy
$E_{\mathrm{h}}$ and the maximum energy exchange in a single scattering event,
$\Delta E_{\uparrow\uparrow}$LundePRB2010-EE ; LundePRB-EE .
$\gamma_{\uparrow\uparrow}$ is obtained by measuring the
energy($E_{\mathrm{h}}$)-space($x$) trajectory of the hot electrons, and
$\Delta E_{\uparrow\uparrow}$ is roughly estimated from the spread of the
distribution function, $W_{\mathrm{d}}$, near the chemical potential $\mu$. As
we show later, both $\gamma_{\uparrow\uparrow}$ and $\Delta
E_{\uparrow\uparrow}$ increase rapidly as the hot electrons lose energy. The
resultant sharp drop in the hot electron’s energy creates a hot spot at
$x=x_{\mathrm{H\uparrow}}$, where the spin-up electrons are maximally excited
[panel (iii)]. In addition, the inter-channel e-e scattering between up and
down spins (e↑-e↓) excites plasma (the dark magenta regions) and creates
another hot spot at $x=x_{\mathrm{H\downarrow}}$ in the spin-down channel
[panel (iv)]. The e↑-e↓ scattering equilibrates the two channels further
during transport. Importantly, once the electron-hole plasma relaxes to fit in
the energy range where the TL physics governs, the plasma cannot equilibrate
further and remains in a non-thermal metastable state in which warm and cold
electrons coexist in the same channel [panel (v)]. Such crossover from hot
electrons into a non-thermal TL liquid is the subject of this paper. Due to
the coupling to the environment, the system finally reaches thermal
equilibrium at $T_{\mathrm{B}}$ after long transport [panel (vi)].

The crossover dynamics can be understood by considering microscopic processes
in the edge states, where the Landau levels are bent upwards by the edge
potential, as schematically shown in Fig. 1b. This energy($E$) - space($y$)
diagram can be understood with the energy($E$) - wavenumber($k_{x}$)
dispersion, as the relation $k_{x}=-eBy/\hbar$ holds in the QH regime under
the Landau gaugeBookEzawa . Due to the nonlinearity of the dispersion, e-e
scattering is suppressed as it cannot conserve both energy and momentum,
particularly when the energy $E_{\mathrm{h}}$ and the energy exchange
$\varepsilon$ are large (Fig. 1c). In reality, the scattering is allowed by
the presence of disorder that breaks the translational invariance and lifts
the momentum conservation to some extent. This should determine the cutoff
$\Delta E_{\uparrow\uparrow}$ in the e-e scattering. For the concave
dispersion relation, $\Delta E_{\uparrow\uparrow}$ should increase with
decreasing $E_{\mathrm{h}}$. The Coulomb interaction increases with decreasing
the distance at lower $E_{\mathrm{h}}$. Both effects induce the rapid
reduction of $E_{\mathrm{h}}$ with large $\gamma_{\uparrow\uparrow}$ and
$\Delta E_{\uparrow\uparrow}$ near $x=x_{\mathrm{H\uparrow}}$.

When $E_{\mathrm{h}}<\Delta E_{\uparrow\uparrow}$ is reached, the system
enters a new regime, where the scattering involves two competing processes
with the swapped final states [the dashed lines with an arrow labeled e↑-e↑ in
Fig. 1d]. They destructively interfere with each other, and thus the intra-
channel scattering of the same spins should be suppressed at low
$E_{\mathrm{h}}$ ($<\Delta E_{\uparrow\uparrow}$)LundePRB-EE . When the intra-
channel scattering is inefficient, the inter-channel scattering between
different spins should dominate the relaxation. This region can be referred to
as the inter-channel scattering regime. The hot electrons in the spin-up
channel lose their energies by exciting an electron-hole plasma in the spin-
down channel, and no competing process is present for scattering between
different spins. The inter-channel scattering subsequently excites the spin-up
channel and thus equilibrates the two channels in the end. As the spin-down
channel is located on the inner side, where the edge potential is more gentle,
the maximum energy exchange $\Delta E_{\uparrow\downarrow}$ and the energy
loss rate $\gamma_{\uparrow\downarrow}$ for different spins should be smaller
than $\Delta E_{\uparrow\uparrow}$ and $\gamma_{\uparrow\uparrow}$,
respectively.

The above single-particle picture fails when the hot electron is further
relaxed within the energy range $\Delta E_{\mathrm{TL}}$ of the TL regime
($E_{\mathrm{h}}\lesssim\Delta E_{\mathrm{TL}}$). The dispersion can be
approximated to be linear, as shown in Fig. 1e. All electrons within the
energy range $\Delta E_{\mathrm{TL}}$ near the Fermi level interact with each
other. Such interacting electrons can be understood with non-interacting
bosonic excitations in the TL model vonDelftAnnPhys1998 ; KovrizhinPRB2011 .
The inter-channel interaction induces spin-charge separation, and the intra-
channel interaction enhances the charge velocity.Hashisaka-NatPhys2017  Once
all electron-hole plasma enters the TL regime, no further transformation is
expected in the plasma. Therefore, a non-thermal metastable state eventually
shows up after stochastic processes. In the following experiment, the
coexistence of warm and cold electrons remains even after long-distance
transport. While the TL model is based on low-energy physics, such non-thermal
states can emerge from high-energy hot electrons. This demonstrates the
robustness of the prethermalized state against the excitation energy.

Electronic distribution functions of non-thermal states have been studied
previously. When an initial state with a double-step distribution function is
prepared by a QPC with bias voltage $V_{\mathrm{b}}$ and low transmission
probability, the non-thermal state shows an arctangent distribution function
in the low-energy limit at $\left|E-\mu\right|\ll eV_{\mathrm{b}}$
LevkivskyiPRB2012 ; Itoh-PRL2018 . Phenomenologically, the non-thermal states
in the high-energy region can be approximated by a binary distribution
function

$f\left(E\right)=\left(1-p\right)f_{\mathrm{FD}}\left(E;k_{\mathrm{B}}T_{0}\right)+pf_{\mathrm{FD}}\left(E;k_{\mathrm{B}}T_{1}\right)$
(1)

consisting of high- ($T_{1}$) and low-temperature ($T_{0}$, close to the base
temperature $T_{\mathrm{B}}$) components with Fermi distribution
$f_{\mathrm{FD}}\left(E;k_{\mathrm{B}}T\right)=1/\left[1+\exp\left(E/k_{\mathrm{B}}T\right)\right]$,
which is valid for small fraction $p$ ($\ll 1$) WashioPRB2016 . The binary
form captures the exponential current profiles $\sim
e^{-E/k_{\mathrm{B}}T_{1}}$ observed in the low-energy regime WashioPRB2016 ;
Itoh-PRL2018 ; RodriguezNatComm2020 . As we do not know the theoretical form
for hot-electron injection, we use this binary form to analyze the non-thermal
states in this paper.

Experimental setup. We investigated the crossover dynamics with the device
fabricated in a standard AlGaAs/GaAs heterostructure with an electron density
of 2.5$\times$1011 cm-2 and low-temperature mobility of about 106 cm2/Vs, as
shown in Fig. 2a. The electron system is divided into three conductive regions
that serve as the source (S), base (B), and drain (D) by applying large
negative gate voltages $V_{\mathrm{G}i}$ on gate $i=1,2,3,\cdots$. A
perpendicular magnetic field $B=$ 5.0 T is applied to set the electron system
in the $\nu=2$ QH regime and form two parallel edge channels (the blue lines)
in each region.

Hot electrons with up spin and energy $eV_{\mathrm{inj}}$ ($=$ 1 $-$ 100 meV)
are injected from S to B, by applying voltage $-V_{\mathrm{inj}}$ to the ohmic
contact of S and tuning the transmission of the injector point contact (IPC).
The electronic distribution function of the up-spin channel near the chemical
potential is investigated with a QD, and electron-hole plasma in up- and down-
spin channel are studied by activating a QPC as shown in Fig. 2b. The QD or
QPC is located at a distance $L$ from the IPC, where several injection gates
at different distances $L=$ 1, 2, 5, 10, 20, and 30 $\mu$m were selectively
activated [see the scanning electron micrograph of Fig. 2c]. As shown in the
schematic energy profile of Fig. 2d, the hot spot ($x_{\mathrm{H\uparrow}}$)
can be changed by tuning $eV_{\mathrm{inj}}$. The non-thermal state in the TL
regime can be investigated at $x=L>x_{\mathrm{H\uparrow}}$, and e-e scattering
processes can be studied at $x=L<x_{\mathrm{H\uparrow}}$. The current
$I_{\mathrm{det}}$ through the QD or QPC is measured with a bias voltage
$V_{\mathrm{D}}$ on the detector (D) ohmic contact. The base current
$I_{\mathrm{B}}$ is monitored to ensure $I_{\mathrm{inj}}\simeq
I_{\mathrm{det}}+I_{\mathrm{B}}$ [see Fig 10].

Figure 2: Schematic measurement setup. a Device structure. By applying gate
voltages on the gates (yellow), an injector point contact (IPC) and a detector
quantum dot (QD) are formed between the source (S), base (B), and drain (D)
regions. The blue parallel lines show edge channels at $\nu=2$. b Device
configuration for a quantum point contact (QPC) detector. c Scanning electron
micrograph with the false color of a control device with several IPC gates. d
Schematic energy diagram of the measurement. Electrons are filled with the
Fermi energy $\varepsilon_{\mathrm{F}}$ for up- and down-spin channels. The
energy-space trajectory of hot electrons is shown by the red arrows. Electron-
hole plasma near the chemical potential in B is analyzed by measuring
$I_{\mathrm{det}}$ through the QD or QPC.

Energy-space trajectory of hot electrons. First, we investigate the hot-spot
positions $x_{\mathrm{H\uparrow}}$ and $x_{\mathrm{H\downarrow}}$ by using QPC
detectionOtaPRB2019 ; AkiyamaAPL2019 . As shown in Fig. 3a, the QPC
conductance $G$ obtained without injecting hot electrons (by opening the IPC
with $V_{\mathrm{inj}}=$ 0 and $V_{\mathrm{Ginj}}=$ 0) shows standard
quantized conductances as a function of the gate voltage $V_{\mathrm{G4}}$.
The onsets for spin-up and -down transport are seen at
$V_{\mathrm{G4}}\sim-0.4$ V and $-0.2$ V, respectively. Additional peaks at
$V_{\mathrm{G4}}\sim-0.5$ V are associated with parasitic impurity states,
which play minor roles in the following analysis. The following QPC detection
was performed at $V_{\mathrm{D}}=0$. When hot spin-up electrons at energy
$eV_{\mathrm{inj}}$ are injected from the IPC for $L=$ 1 $\mu$m, distinct
features appear in the detector current profile $I_{\mathrm{det}}$, as shown
in Fig. 3b. For the bottom trace at small $eV_{\mathrm{inj}}=$ 2 meV, a single
current step with a height comparable to $I_{\mathrm{inj}}=$ 1 nA is seen at
the opening of the spin-up transport ($V_{\mathrm{G4}}\sim-0.4$ V). This
ensures no tunneling between spin-up and -down channels. With increasing
$eV_{\mathrm{inj}}$, current peaks (rather than steps) show up for both spin-
up and -down transports (in the pink stripes). As shown in the insets (i) and
(ii) to Fig. 3a, each peak can be understood by considering the bolometric
detection for each channel, where higher-energy electrons are transmitted but
lower-energy holes are reflectedOtaPRB2019 . Therefore, the peak current is
proportional to the density of the electron-hole plasma in the respective
channel, while the sensitivity for up spins may be greater than that for down
spins. Since the hot spot is the point where the detected current is maximal,
the strategy to determine the hot spot position (for a given $L$) is to look
for the voltage $V_{\mathrm{inj}}$ that maximizes the detected current. The
peak currents for up spins ($V_{\mathrm{G4}}=-0.42$ V) and down spins
($V_{\mathrm{G4}}=-0.18$ V) are plotted as a function of $eV_{\mathrm{inj}}$
in Fig. 3c. The spin-up current is maximized at $eV_{\mathrm{inj}}$ = 22 meV,
which indicates that the position of the hot spot in the spin-up channel
coincides with that of the QPC detector ($x_{\mathrm{H\uparrow}}=$ 1 $\mu$m
for $eV_{\mathrm{inj}}$ = 22 meV). Similarly, the spin-down current is
maximized at $eV_{\mathrm{inj}}$ = 14 meV, where the hot spot in the spin-down
channel is considered to be located at the position of the QPC detector
($x_{\mathrm{H\downarrow}}=$ 1 $\mu$m for $eV_{\mathrm{inj}}$ = 14 meV).

Figure 3: Quantum point contact (QPC) detection of electron-hole plasma.  a
Conductance $G=I_{\mathrm{det}}/V_{\mathrm{D}}$ of the detector QPC measured
with $V_{\mathrm{D}}=$ 0.1 mV. Transmission and reflection of hot electrons
(the blue arrow) and cold electrons and holes (the magenta arrows) are
illustrated in the insets (i), (ii), and (iii). b $I_{\mathrm{det}}$ profiles
obtained at various $eV_{\mathrm{inj}}$. $V_{\mathrm{G4}}$ in the horizontal
axis was simultaneously swept with $V_{\mathrm{G5}}$ ($=V_{\mathrm{G4}}$). c
The peak current of $I_{\mathrm{det}}$ for spin-up (red) and -down (blue) are
plotted as a function of $eV_{\mathrm{inj}}$. The maximum peak current appears
when the hot spot is located close to the detector.

We repeated similar measurements for various $L$, and the obtained hot-spot
positions are summarized in Fig. 4a. In addition to the data obtained with the
QPC (the circles), data obtained with the QD only for spin-up (described
below) are also plotted by the squares. Both data for up spins agree well with
each other within the experimental reproducibility. The smooth connection of
the data points can be used to extract the energy-space trajectory of hot
electrons. $\gamma_{\uparrow\uparrow}$ can be obtained from
$\gamma_{\uparrow\uparrow}=d\left(eV_{\mathrm{inj}}\right)/dL$, as the hot
spot $x_{\mathrm{H\uparrow}}$ ($=L$) increases with $eV_{\mathrm{inj}}$ at
rate $\gamma_{\uparrow\uparrow}^{-1}$. Empirically, the power-law dependence
in a form of $\gamma_{\uparrow\uparrow}=aE_{\mathrm{h}}^{-\lambda}$ for hot-
electron energy $E_{\mathrm{h}}$ explains the data with parameters $a$ and
$\lambda$OtaPRB2019 . This suggests an energy($E_{\mathrm{h}}$)-space($x$)
trajectory

$E_{\mathrm{h}}\left(x\right)=\left[\left(eV_{\mathrm{inj}}\right)^{\lambda+1}-a\left(\lambda+1\right)x\right]^{1/\left(\lambda+1\right)},$
(2)

and the hot-spot condition

$eV_{\mathrm{inj}}=\left[a\left(\lambda+1\right)x_{\mathrm{H\uparrow}}\right]^{1/\left(\lambda+1\right)}.$
(3)

The latter reproduces our data very well with $\lambda=1.8$, as shown by the
red curve in Fig. 4a. Corresponding energy loss rate
$\gamma_{\uparrow\uparrow}\left(E_{\mathrm{h}}\right)$ increases with
decreasing $E_{\mathrm{h}}$, as shown in Fig. 4b. The energy-space trajectory
$E_{\mathrm{h}}\left(x\right)$ is shown for several $eV_{\mathrm{inj}}$ values
in Fig. 4c. As the energies of incident electrons should be distributed at
around $eV_{\mathrm{inj}}$ mostly due to the energy-dependent tunneling
probability of the IPC, possible dispersion of the trajectories for $\pm$1 meV
distribution is shown by pink stripes. Notice that hot-electron relaxation
proceeds rapidly just before the hot spot $x_{\mathrm{H\uparrow}}$, which can
be tuned with $V_{\mathrm{inj}}$. This is useful for investigating the decay
length of non-thermal states, as one can change the distance $D$ between the
hot spot and the detector.

Figure 4: Energy-space trajectory of hot electrons. a The hot-spot conditions
for up (red) and down (blue) spins measured with  quantum point contact (QPC)
(circles) and quantum dot (QD) (squares) detectors. The error bar shows the
variations due to the injector and detector conditions. The solid lines are
fitted to the data with $\lambda=1.8$ (the uncertainty between 1.6 and 2.2)
and $\ell_{\uparrow\downarrow}=0.9$ $\mu$m. The inset shows the data in the
double logarithmic plot. (b) The estimated energy loss rate
$\gamma_{\uparrow\uparrow}$ by assuming the power dependence of hot-electron
energy $E_{\mathrm{h}}$. (c) The estimated energy($E_{\mathrm{h}}$)-space($x$)
trajectory of hot electrons for $eV_{\mathrm{inj}}=$ 40, 60, and 80 meV. Each
stripe represents the spread associated with variation of $eV_{\mathrm{inj}}$
by $\pm$1 meV.

As for the spin-down hot spot $x_{\mathrm{H\downarrow}}$ shown by the blue
circles in Fig. 4a, the $L$ dependence can be reproduced by the blue curve
which is obtained just by shifting the red curve horizontally by
$\ell_{\mathrm{\uparrow\downarrow}}$ = 0.9 $\mu$m. This suggests that finite
length $\ell_{\mathrm{\uparrow\downarrow}}$, independent of
$eV_{\mathrm{inj}}$, is required for exciting spin-down electrons through the
inter-channel interaction. This $\ell_{\mathrm{\uparrow\downarrow}}$ is much
longer than the length $l_{\mathrm{SC}}$ required for spin-charge separation
in the TL regime. Similar devices show $l_{\mathrm{SC}}\simeq$ 0.18 $\mu$m at
$eV_{\mathrm{inj}}=$ 0.6 meV, and $l_{\mathrm{SC}}$ decreases with increasing
$eV_{\mathrm{inj}}$ ($l_{\mathrm{SC}}\propto 1/V_{\mathrm{inj}}$)Itoh-PRL2018
. The difference ($\ell_{\mathrm{\uparrow\downarrow}}>l_{\mathrm{SC}}$)
indicates the existence of the stochastic inter-channel e-e scattering regime
between $x_{\mathrm{H\uparrow}}$ and $x_{\mathrm{H\downarrow}}$.

The transition between intra- and inter-channel scattering regimes can be
confirmed from the data in Fig. 3c. In the range of $eV_{\mathrm{inj}}$
between 30 and 70 meV for $L<x_{\mathrm{H\uparrow}}$, spin-up electrons are
highly excited ($I_{\mathrm{det}}>I_{\mathrm{inj}}$) but no spin-down
electrons are excited ($I_{\mathrm{det}}\simeq I_{\mathrm{inj}}$) because the
intra-channel scattering is dominant. In contrast, both signals become large
at $eV_{\mathrm{inj}}<$ 25 meV and change in a similar manner at
$eV_{\mathrm{inj}}<$ 15 meV ($L>x_{\mathrm{H\downarrow}}$), where the inter-
channel interaction equilibrates the two channels. The sequence of the intra-
channel scattering regime [from (i) to (iii) in Fig. 1a] followed by the
inter-channel scattering regime [(iii)-(iv)] is justified with the result.

QD spectroscopy. We investigate electronic excitation in the spin-up channel
by using the QD detectorAltimiras-NatPhys10 ; Itoh-PRL2018 . Coulomb diamond
characteristics of the QD are shown in Fig. 11. Figure 5a shows the Coulomb
blockade (CB) oscillations obtained with $V_{\mathrm{D}}=$ 0.2 mV under hot-
electron injection at $L$ = 10 $\mu$m. The horizontal axis $\Delta
V_{\mathrm{G5}}$ is the relative voltage of $V_{\mathrm{G5}}$ from the left
onset of the right CB peak. The width of the CB peaks in the bottom trace
measures the transport window of $eV_{\mathrm{D}}$. When hot electrons are
injected at various $eV_{\mathrm{inj}}$ and fixed $I_{\mathrm{inj}}=$ 1.8 pA,
excess current appears in the entire CB region, as highlighted by the pink
color. This current measures the electrons excited above the QD level. For
instance, the current at $\Delta V_{\mathrm{G5}}=$ $-$19 mV (the dashed line)
becomes maximum at $eV_{\mathrm{inj}}=$ 62 meV, as shown in Fig. 5b. This
measures the spin-up hot-spot condition, where the hot electrons injected at
$eV_{\mathrm{inj}}=$ 62 meV just relaxed to the Fermi energy at $L$ = 10
$\mu$m. This coincides with the QPC data, as shown by the squares in Fig. 4a.
In the following analysis, we use the hot-electron trajectory given by Eq. (2)
at $\lambda=1.8$, while the parameter $a$ for each geometry ($L$) is
determined from $eV_{\mathrm{inj}}$ that maximizes $I_{\mathrm{det}}$ by using
Eq. (3).

Figure 5: Quantum dot (QD) detection of electron-hole plasma. a Coulomb
oscillations of the QD observed in detector current $I_{\mathrm{det}}$ as a
function of relative gate voltage $\Delta V_{\mathrm{G5}}$ under hot-electron
injection at various injection voltage $V_{\mathrm{inj}}$. The excess current
is highlighted by the pink color. The inset shows the energy diagram of the
QD. b $I_{\mathrm{det}}$ measured in the deep Coulomb blockade (CB) region at
$\Delta V_{\mathrm{G5}}=$ -19 mV [the dashed line in a]. The left inset shows
the energy diagram for $eV_{\mathrm{inj}}\lesssim$ 60 meV, where the hot
electrons relax before reaching the QD. The right inset shows the diagram for
$eV_{\mathrm{inj}}\gtrsim$ 60 meV, where the hot electrons pass over the QD.

The broad current profile in Fig. 5a suggests that the electrons are excited
well above the charging energy of the QD ($U\simeq$ 1.5 meV) as well as the
single-particle energy spacing of the QD ($\bar{\Delta}\simeq$ 0.2 meV on
average). The current decreases with decreasing $\Delta V_{\mathrm{G5}}$
reflecting the electron energy distribution function $f\left(E\right)$ of the
channel. To evaluate $f\left(E\right)$, the current profiles are plotted in
the logarithmic scale in Fig. 6a for $eV_{\mathrm{inj}}\leq$ 60 meV, where the
hot electrons are fully relaxed before reaching the QD detector
($x_{\mathrm{H\uparrow}}<L$). The current profile in the CB region shows
exponential dependence $I_{\mathrm{det}}\propto\exp\left(\alpha\Delta
V_{\mathrm{G5}}/W_{\mathrm{c}}\right)$ as shown by the dashed lines. Here,
$\alpha\simeq 0.044e$ is the lever arm factor to convert $\Delta
V_{\mathrm{G5}}$ into the electrochemical potential $\mu_{N}=-\alpha\Delta
V_{\mathrm{G5}}$ of the $N$-electron QD, and $W_{\mathrm{c}}$ is the
characteristic energy of the current profile. For $eV_{\mathrm{inj}}>$ 62 meV,
the hot electrons pass over the QD potential ($x_{\mathrm{H\uparrow}}>L$), as
shown in the inset to Fig. 6b. Noting that all the hot electrons are
eventually absorbed by the ammeter, we plot in Fig. 6b
$I_{\mathrm{det}}-I_{\mathrm{inj}}$ as the net current in the logarithmic
scale. The exponential profile, $\propto\exp\left(\alpha\Delta
V_{\mathrm{G5}}/W_{\mathrm{c}}^{\prime}\right)$, in the CB region is
characterized by $W_{\mathrm{c}}^{\prime}$. We confirmed that $W_{\mathrm{c}}$
and $W_{\mathrm{c}}^{\prime}$ do not change significantly with
$I_{\mathrm{inj}}$, as shown in Figs. 6c and 6d, as well as $N$ [partially
seen at $\Delta V_{\mathrm{G5}}<-30$ mV in Figs. 6a and 6b]. Some other data
obtained at different $L$ and $I_{\mathrm{inj}}$ are shown in Fig. 12.

To analyze the exponential tail in the CB region, we rely on the orthodox CB
theory with a continuous density of QD statesBookNazarov , as all features
associated with the excited states of the QD are smeared out by the broad
profile. We also assume the QD is relaxed to the ground state before accepting
a single hot-electron. Under these assumptions, the detector current yields

$I_{\mathrm{det}}=\int_{\mu_{N}}^{\infty}\frac{G_{\mathrm{QD}}}{e}f\left(E\right)dE$
(4)

with $G_{\mathrm{QD}}$ the tunneling conductance (see Methods). If
$G_{\mathrm{QD}}\propto e^{E/W_{\mathrm{t}}}$ increases exponentially with
energy $E$ at characteristic energy $W_{\mathrm{t}}$ of the tunneling, the
model suggests exponential energy dependence of the distribution function
$f\left(E\right)\propto\exp\left(-E/W_{\mathrm{d}}\right)$ with
$W_{\mathrm{d}}^{-1}=$ $W_{\mathrm{t}}^{-1}+W_{\mathrm{c}}^{-1}$. While we do
not know $W_{\mathrm{t}}$ of our device, the spread of the current tail
($W_{\mathrm{c}}$) should reflect the spread of the distribution function
($W_{\mathrm{d}}$) when $W_{\mathrm{c}}$ is not too large.

Figure 6: Quantum dot (QD) spectroscopy for energy distribution functions. a
Detector current $I_{\mathrm{det}}$ as a function of relative gate voltage
$\Delta V_{\mathrm{G5}}$ at various injection energy $eV_{\mathrm{inj}}\leq$
60 meV for studying non-thermal metastable states after long transport. b
Excess current  $I_{\mathrm{det}}-I_{\mathrm{inj}}$ for
$eV_{\mathrm{inj}}\geq$ 62 meV for studying non-equilibrium states in the
course of electron-electron scattering. The dashed lines,
$\propto\exp\left(\alpha V_{\mathrm{G5}}/W_{\mathrm{c}}\right)$ in a and
$\propto\exp\left(\alpha V_{\mathrm{G5}}/W_{\mathrm{c}}^{\prime}\right)$ in b,
are fitted to the data. The current tails in a and b are characterized by cut-
off energies $W_{\mathrm{c}}$ and $W_{\mathrm{c}}^{\prime}$, respectively. The
insets show the energy diagrams around the QD. c $I_{\mathrm{det}}$ for
several injection current $I_{\mathrm{inj}}$ at $eV_{\mathrm{inj}}=$ 20 meV.
No significant change in the slope (the dashed lines) is seen. d Injection
current $I_{\mathrm{inj}}$ dependence of  cut-off energies $W_{\mathrm{c}}$
(red) and $W_{\mathrm{c}}^{\prime}$ (blue).

Non-thermal metastable state. The state after equilibration at
$x>x_{\mathrm{H\downarrow}}$ can be studied with the data in Fig. 6a. The
coexistence of the small exponential current tail with large $W_{\mathrm{c}}$
and the large CB current peak with a sharp onset associated with cold
electrons manifests the non-thermal metastable state of the TL
liquidWashioPRB2016 ; Itoh-PRL2018 ; RodriguezNatComm2020 . In the present
case, the non-thermal states are analyzed with the binary form of Eq. (1). The
low temperature ($T_{0}$) is always close to the base temperature
($T_{\mathrm{B}}$), as the CB peak remains sharp at any $V_{\mathrm{inj}}$ in
Fig. 6a. The high temperature ($T_{1}$) is related to the spread of the
current tail ($k_{\mathrm{B}}T_{1}\sim W_{\mathrm{c}}$). Significantly
different temperatures ($T_{1}\gg T_{0}$) are seen at higher
$V_{\mathrm{inj}}$. These current profiles are qualitatively the same as those
reported for small excitation energies (10 - 100 $\mu$eV) where the low-energy
TL physics governs. Surprisingly, low-energy physics emerges from the high-
energy source (1 - 60 meV). Note that the non-thermal metastable state is
distinct from the ordinary non-equilibrium state often seen in higher
dimensionsBookAltshulerAronovEE ; PothierThermalize .

As several QD levels are involved in the transport for the highly non-
equilibrium state, $p$ in Eq. (1) cannot be estimated from the current
profile. Instead, $p$ can be obtained from the electrical power
$P_{\mathrm{inj}}=I_{\mathrm{inj}}V_{\mathrm{inj}}$ injected from the IPC. It
is known that a single edge channel carries heat current
$J=\frac{\pi^{2}}{6h}\left(k_{\mathrm{B}}T\right)^{2}$ if the electrons are in
thermal equilibrium at temperature $T$JezouinScience2013 . Therefore, if the
spin-up and -down channels were thermally equilibrated with
$P_{\mathrm{inj}}$, the thermal energy would increase to
$k_{\mathrm{B}}T_{\mathrm{th}}=\sqrt{\frac{3h}{\pi^{2}}P_{\mathrm{inj}}}$ by
neglecting the thermal energy of the base temperature ($T_{\mathrm{B}}$). This
yields $k_{\mathrm{B}}T_{\mathrm{th}}=$ 10 $\mu$eV for $P_{\mathrm{inj}}=$ 13
fW ($I_{\mathrm{inj}}=$ 1.8 pA and $V_{\mathrm{inj}}=$ 7 mV), which is much
smaller than $W_{\mathrm{c}}=$ 0.3 meV at $V_{\mathrm{inj}}=$ 7 mV for the
third trace from the bottom in Fig. 6a. The heat conservation suggests that
only a small fraction, about
$p=\left(k_{\mathrm{B}}T_{\mathrm{th}}/W_{\mathrm{c}}\right)^{2}\simeq$ 10-3
of electrons, is highly excited to the high-temperature $T_{1}$. Such an
unusual energy distribution is a consequence of the crossover from hot
electrons.

The exponential current tail in Fig. 6a starts to develop from
$eV_{\mathrm{inj}}\simeq$ 2 meV, and the corresponding $W_{\mathrm{c}}$ value
rapidly increases with $eV_{\mathrm{inj}}$. The $W_{\mathrm{c}}$ values
obtained at various $eV_{\mathrm{inj}}$ and $L$ are summarized in Fig. 7.
First, we focus on the low $eV_{\mathrm{inj}}$ ($\lesssim$ 8 meV) regime for
the shortest $L=$ 1 $\mu$m, where $W_{\mathrm{c}}$ increases linearly with
$eV_{\mathrm{inj}}$ (the dashed line). As the hot electron relaxes within a
short propagation length ($x_{\mathrm{H\uparrow}}<0.1$ $\mu$m as shown in the
auxiliary scale) in this case, the distribution function should measure the
non-thermal metastable state after traveling of $\sim$ 1 $\mu$m. Actually, the
slope $W_{\mathrm{c}}/eV_{\mathrm{inj}}=0.2$ is comparable to the values
obtained in the low-energy limit of the TL regime
($W_{\mathrm{c}}/eV_{\mathrm{inj}}=0.17$ for $W_{\mathrm{c}}=$ 34 $\mu$eV at
$eV_{\mathrm{inj}}=$ 200 $\mu$eV with QPC injection in Ref. Itoh-PRL2018 ,
$W_{\mathrm{c}}/eV_{\mathrm{inj}}=0.22-0.25$ for $W_{\mathrm{c}}=$ 14 - 17
$\mu$eV at $eV_{\mathrm{inj}}=$ 63 $\mu$eV with QD injection in Ref.
RodriguezNatComm2020 , and $W_{\mathrm{c}}/eV_{\mathrm{inj}}=0.17$ for
$W_{\mathrm{c}}=$ 20 - 35 $\mu$eV at $eV_{\mathrm{inj}}=$ 100 - 200 $\mu$eV
with indirect QPC injection in Ref. WashioPRB2016 ). Our experiment confirms
that the same ratio ($W_{\mathrm{c}}/eV_{\mathrm{inj}}$) holds up to
$W_{\mathrm{c}}\simeq$ 1 meV, which is more than 10 times greater than the
values in the previous works. The linearity implies that the injection energy
fits in the energy range of the TL liquid without experiencing e-e scattering.
All electrons within the range interact with each other to form the collective
plasmon mode, as discussed in Fig. 1e. The memory of the initial state at
$eV_{\mathrm{inj}}$ remains in the broad electronic distribution function with
$W_{\mathrm{d}}$ ($\simeq W_{\mathrm{c}}\simeq 0.2eV_{\mathrm{inj}}$). This
can be understood with the integrable nature of holding many conserved
quantities.

Figure 7: The cutoff energy $W_{\mathrm{c}}$ for the non-thermal states. The
dashed line shows the  linear relation, $W_{\mathrm{c}}\simeq
0.2eV_{\mathrm{inj}}$, obtained in the low-energy limit of Tomonaga-Luttinger
(TL) liquids. The inset shows the energy-space trajectory of hot electrons.
The hot-spot position, $x_{\mathrm{H\uparrow}}$, obtained with Eq. (3) at
$\lambda=$ 1.8 is shown in the upper scale.

However, $W_{\mathrm{c}}$ deviates from the linear dependence to lower values
at $eV_{\mathrm{inj}}\gtrsim 8$ meV for $L=$ 1 $\mu$m, as seen in Fig. 7.
$W_{\mathrm{c}}$ obtained with longer $L$ may follow the linear dependence at
small $eV_{\mathrm{inj}}\lesssim 1$ meV and becomes substantially smaller at
larger $eV_{\mathrm{inj}}$. The reduction in $W_{\mathrm{c}}$ might originate
from the dissipationleSueurPRL2010 ; ProkudinaPRL2014  or the departure from
the TL regime. To characterize the decay of non-thermal states, the
$W_{\mathrm{c}}$ data are plotted as a function of the distance
$D=L-x_{\mathrm{H\uparrow}}$ from the hot spot to the detector, as shown in
Fig. 8. The corresponding $eV_{\mathrm{inj}}$ values are shown in the
auxiliary scale for each $L$. Each trace features a gradual reduction of
$W_{\mathrm{c}}$ (except for $L=$ 1 $\mu$m) at $eV_{\mathrm{inj}}>$ 50 meV
followed by a sudden reduction at $eV_{\mathrm{inj}}<$ 10 meV ($D\simeq L$).
Both features can be explained consistently with the energy-dependent
dissipation during traveling.

For $eV_{\mathrm{inj}}<$ 10 meV, the electron-hole excitation at the hot spot
is developing with increasing $eV_{\mathrm{inj}}$, while $D$ ($\simeq L$) is
almost constant for each $L$. Therefore, the dissipation can be studied by
varying $L$ for a given $eV_{\mathrm{inj}}$, as shown by the two dotted lines
connecting the data points for $eV_{\mathrm{inj}}$ = 5 and 10 meV. The steeper
slope at higher $W_{\mathrm{c}}$ suggests increased dissipation at higher
excitation energy.

In contrast, the excitation at the hot spot does not increase further for
$eV_{\mathrm{inj}}>$ 30 meV. As seen in the energy-space trajectory of Fig.
4c, the excess energy beyond 20 - 30 meV is lost during the long-distance
transport with the remaining 20 - 30 meV used to build the hot spot with the
highest excitation energy. Therefore, the $eV_{\mathrm{inj}}$ dependence at
$eV_{\mathrm{inj}}>$ 30 meV in Fig. 8 should be understood as representing the
$D$ dependence of $W_{\mathrm{c}}$ . The overall feature at
$eV_{\mathrm{inj}}>$ 50 meV follows a monotonic reduction of $W_{\mathrm{c}}$
with $D$, as shown by the long dashed line. This slope is comparable to the
slope of the short-dashed lines if the data in the same range of
$W_{\mathrm{c}}$ are compared. The data shows how the electronic distribution
function changes with transport. The characteristic energy $W_{\mathrm{d}}$
($\simeq W_{\mathrm{c}}$) decays fast if $W_{\mathrm{d}}$ is large (the decay
length $\ell_{\mathrm{d}}\sim 5$ $\mu$m at $W_{\mathrm{c}}\simeq$ 1 meV), but
the decay length becomes longer for smaller $W_{\mathrm{d}}$ ($\sim$ 30 $\mu$m
at 0.2 meV). This is comparable to $\ell_{\mathrm{d}}\sim$ 20 $\mu$m obtained
from the low-energy experiment at $W_{\mathrm{c}}=$ 34 $\mu$eVItoh-PRL2018 .
The energy dependent $\ell_{\mathrm{d}}$ may be explained by energy-dependent
dissipation associated with the characteristics of the environment. If the
reduction signifies the departure of the TL regime, the energy range for the
TL physics could be $\Delta E_{\mathrm{TL}}\sim$ 1 meV in our device.

Figure 8: Decay of non-thermal metastable states.  The characteristic energy
$W_{\mathrm{c}}$ is plotted as a function of the distance $D$ from the hot
spot to the quantum dot (QD). $D$ is obtained from Eq. (3) at $\lambda=$ 1.8.
The inset shows the energy-space trajectory. The $eV_{\mathrm{inj}}$ value for
each $L$ is shown in the respective auxiliary scale. The decay of
$W_{\mathrm{c}}$ is shown by the short dashed lines connecting the data points
obtained with the same $eV_{\mathrm{inj}}$ ($\leq$ 10 meV) and the long dashed
line as a guide to the eye for the data at $eV_{\mathrm{inj}}\gtrsim$ 50 meV.

e-e scattering. The electron-hole plasma generated in the course of the e-e
scattering was analyzed with the current profile,
$I_{\mathrm{det}}-I_{\mathrm{inj}}$, at $x_{\mathrm{H\uparrow}}>L$, in Fig.
6b. The data indicates that $W_{\mathrm{c}}^{\prime}$ (the inverse of the
slope) decreases with increasing $eV_{\mathrm{inj}}$. Considering that the
energy loss rate $\gamma_{\uparrow\uparrow}$ increases as $E_{\mathrm{h}}$
decreases during the transport from the IPC to the detector [see Fig. 4b], it
is likely that the obtained $W_{\mathrm{c}}^{\prime}$ value reflects most
sensitively the e-e scattering near the detector QD. From this point of view,
we plot in Fig. 9 $W_{\mathrm{c}}^{\prime}$ as a function of
$E_{\mathrm{h}}\left(L\right)$, the hot-electron energy at the detector
position. The two data sets for $L=$ 1 and 10 $\mu$m show similar dependency,
which justifies this approach. The data show that $W_{\mathrm{c}}^{\prime}$
decreases with increasing $E_{\mathrm{h}}$, as shown by the dashed line.
Namely, the electron-hole plasma is highly excited in the vicinity of the hot
spot, as illustrated in the inset.

Figure 9: The characteristic energy $W_{\mathrm{c}}^{\prime}$ for the
electron-electron scattering. The hot-electron energy
$E_{\mathrm{h}}\left(L\right)$ at the quantum dot (QD) detector is obtained
from Eq. (2) at $\lambda=$ 1.8. The $eV_{\mathrm{inj}}$ value for each $L$ is
shown in the respective auxiliary scale. The dashed line is a guide to the eye
by using a function proportional to $E_{\mathrm{h}}^{-1.5}$. The inset shows
the energy-space trajectory and buildup of electron-hole plasma.

The physical meaning of $W_{\mathrm{c}}^{\prime}$ can be understood by
considering the electron-hole generation process. Suppose that an intra-
channel e-e scattering between a hot electron with energy $E_{\mathrm{h}}$ and
a cold electron with $E_{\mathrm{c}}$ ($<0$) generates an electron-hole pair
with a hole at $E_{\mathrm{c}}$ and an electron near the Fermi energy [at
$E_{\mathrm{c}}+\varepsilon$ ($>0$)] by the energy exchange $\varepsilon$, as
shown in Fig. 1c. Scattering with various combinations of $E_{\mathrm{c}}$ and
$\varepsilon$ induces the plasma. Here, $\varepsilon$ is bounded by the
momentum conservation. For a given $E_{\mathrm{h}}$, we assume that the
scattering probability
$P\left(E_{\mathrm{c}},\varepsilon;E_{\mathrm{h}}\right)\varpropto
e^{-\varepsilon/\Delta E_{\uparrow\uparrow}}$ for $E_{\mathrm{c}}$ and
$\varepsilon$ depends only on $\varepsilon$ with the upper bound $\Delta
E_{\uparrow\uparrow}$ that depends on $E_{\mathrm{h}}$. For the present case,
we are interested in the electronic energy distribution function
$f\left(E\right)$ above the Fermi energy ($E>0$). We further assume that
$f\left(E\right)$ is determined by the e-e scattering in the vicinity of the
detector by neglecting the scattering that occurs in the upstream region.
Then, $f\left(E\right)$ is given by the instantaneous e-e scattering with
$E_{\mathrm{h}}$ at $x=L$ as

$f\left(E\right)\propto\int_{-\infty}^{0}P\left(E_{\mathrm{c}},\varepsilon;E_{\mathrm{h}}\right)dE_{\mathrm{c}}\propto
e^{-E/\Delta E_{\uparrow\uparrow}},$ (5)

where $E=E_{\mathrm{c}}+\varepsilon$ was used in the integration. Therefore,
provided that $W_{\mathrm{c}}^{\prime}$ measures the spread $W_{\mathrm{d}}$
of the distribution function, the obtained $W_{\mathrm{c}}^{\prime}$ can be
understood as the energy cutoff $\Delta E_{\uparrow\uparrow}$ of the intra-
channel e-e scattering.

The monotonic reduction of $W_{\mathrm{c}}^{\prime}$ with $E_{\mathrm{h}}(L)$
in Fig. 9 is consistent with the concave dispersion in Fig. 1b, as explained
above. In this way, the energy loss of hot electrons ($E_{\mathrm{h}}>$ 20
meV) and excitation of cold electrons ($E_{\mathrm{h}}<$ 5 meV) are
consistently explained with the intra-channel e-e scattering. It should be
noted that the energy dependency (larger $W_{\mathrm{c}}^{\prime}$ for smaller
$E_{\mathrm{h}}$) of e-e scattering in Fig. 9 is opposite to that (larger
$W_{\mathrm{c}}$ for larger $eV_{\mathrm{inj}}$) for TL states in Fig. 7.

Conclusions

We have investigated how high-energy hot electrons lose their energy by
exciting cold electrons. A QPC was used as a broad-band spin-dependent
bolometer to extract the energy-space trajectory of hot electrons. A QD was
employed as a narrow-band spectrometer to detect non-equilibrium electronic
excitation near the Fermi energy. The hot electrons lose their energy with the
e-e scattering and eventually create spin-dependent hot spots. The cold
electrons are maximally excited at the hot spot and equilibrate while
traveling the channels. Interestingly, the system always ends up with a non-
thermal metastable state without fully relaxing into thermal equilibrium, even
when the initial hot-electron energy is well beyond the TL regime. This
demonstrates the robustness of non-thermal states against the excitation
energy. The non-thermal state depends on the excitation energy even after
traveling 20 $\mu$m. This suggests that the final state remembers the initial
state even after the equilibration, in consistency with the integrable nature
with many conserved quantities. In addition, the non-equilibrium state
generated with the e-e scattering is in-situ investigated to evaluate the
maximum energy exchange in the scattering. The analysis implies the
significance of inter-channel scattering with different spins as well as
intra-channel scattering with the same spins. The experiment provides a novel
crossover from single particles to a non-equilibrium many-body state. The
scheme can be used for studying non-linear TL liquids in the presence of non-
linear dispersionImambekovRMP2012 . The robustness of non-thermal states is
attractive for transporting information and energy for long
distancesCalzonaPRB2017 ; KaminishiPRA2018 ; UedaNRP2020 .

Methods

Hot-electron injection. We use custom-made voltage source and ammeter to
inject hot electrons. A large input impedance of 100 k$\Omega$ in the ammeter
prevents unwanted large current during adjustments. $\ V_{\mathrm{Ginj}}$ is
precisely adjusted by a software to obtain the desired $I_{\mathrm{inj}}$
value (1 pA - 1 nA) within a few % error before measuring $I_{\mathrm{det}}$
at each condition ($V_{\mathrm{inj}}$, $V_{\mathrm{G4}}$, etc.).
$V_{\mathrm{Ginj}}$ depends strongly on $V_{\mathrm{inj}}$ (typically, from
$V_{\mathrm{Ginj}}\simeq$ -1.1 V at $V_{\mathrm{inj}}=$ 5 mV to
$V_{\mathrm{Ginj}}\simeq$ -1.9 V at $V_{\mathrm{inj}}=$ 80 mV with an almost
linear relation in between) for $I_{\mathrm{inj}}=$ 1 nA. Slightly more
negative $V_{\mathrm{Ginj}}$ by $\simeq$ -0.01 V was needed for
$I_{\mathrm{inj}}=$ 10 pA. While the location of the injector should be pushed
toward the other gate G1 with more negative $V_{\mathrm{Ginj}}$, we neglected
possible change in transport length $L$. $I_{\mathrm{inj}}$ is kept
sufficiently low, 1 $-$ 10 pA for the QD detection to minimize the heating
inside the QD and $\sim$ 1 nA for the QPC detection for better visibility.

As the e-e scattering is sensitive to the distance from the channel to the
gate metal, we fixed the gate voltages defining the channel
($V_{\mathrm{G1}}=V_{\mathrm{G3}}=$ -0.6 V and $V_{\mathrm{G2}}=$ -0.45 V)
throughout this paper. These gate voltages were chosen in such a way that the
edge potential becomes gentle enough to suppress unwanted LO phonon emission
while high enough to confine hot electronsOtaPRB2019 ; EmaryPRB2019 ;
AkiyamaAPL2019 . While $V_{\mathrm{inj}}$ (1 - 100 meV) is much larger than
the cyclotron energy of about 9 meV at 5 T, we did not observe any tunneling
transition to higher Landau levels in the present conditions.

QPC detection. We focused on the current peaks appearing at the onsets of the
conductance steps of the QPC, as described in the main text. Additional data
that support our observation are shown in Supplementary Note SN1. However, one
can investigate the hot-electron energy from the current step appearing at
$V_{\mathrm{G4}}<-0.8$ V and $eV_{\mathrm{inj}}>$ 30 meV in Fig. 3b. This
current step appears when the potential barrier height coincides with the hot-
electron energy, as illustrated in the inset (iii) to Fig. 3a, and disappears
at $eV_{\mathrm{inj}}\lesssim$ 25 meV in consistency with the hot spot
condition obtained above. In our previous report, the hot-electron energy is
obtained from $V_{\mathrm{G4}}$ at the current step by using a conversion
factor determined from the LO phonon replicas of the current stepOtaPRB2019 ;
AkiyamaAPL2019 . The obtained $\lambda$ in the range between 1 and 2 is
consistent with the present value ($\lambda=1.8$), while $\lambda$ may depend
on the device and the side gate voltage. However, the hot-electron energy
cannot be obtained for the present case, as we suppressed LO phonon emission.
Instead, the whole hot-electron energy is used for generating the electron-
hole plasma, which is used to estimate $p$ in this study.

QD detection. The QD characteristics ($U\simeq$ 1.5 meV, $\bar{\Delta}\simeq$
0.2 meV, and $\alpha\simeq 0.044e$) were obtained from standard Coulomb
diamond measurements at $V_{\mathrm{inj}}=0$ and $V_{\mathrm{Ginj}}=0$. The
base electron temperature $T_{\mathrm{B}}\simeq$ 110 mK was also obtained from
the onset of the CB peak. The backaction from the change in
$V_{\mathrm{Ginj}}$ and $V_{\mathrm{inj}}$ to the detector conditions is
present. While this is negligible for the QPC detection, the backaction to the
QD is significant, as the CB peaks were shifted by $\sim$ 30 mV in
$V_{\mathrm{G5}}$ for $L=$ 1 $\mu$m and $\sim$ 8 mV for $L=$ 10 $\mu$m when
$V_{\mathrm{inj}}$ was changed from 1 to 90 mV under the adjustment of
$V_{\mathrm{Ginj}}$ for $I_{\mathrm{inj}}=$ 5 pA. The relative value $\Delta
V_{\mathrm{G5}}$ from the CB peak is used in the main text. Additional data
that support our observation are shown in Supplementary Note SN2.

Analysis of the current profiles. We use the orthodox CB theory to analyze the
broad current profile in the CB region at $\mu_{N}>\mu>\mu_{\mathrm{D}}$ (see
inset to Fig. 5a). As the current is sufficiently low, the dot should remain
close to the base temperature. In this case, the tunneling rate from the edge
channel of interest to the dot is given by
$\Gamma_{\mathrm{dc}}\simeq\int_{\mu_{N}}^{\infty}\frac{1}{e^{2}}G_{\mathrm{L}}f\left(E\right)dE$
for the distribution function $f\left(E\right)$, where the tunneling
conductance $G_{\mathrm{L}}$ may depends on energy. By assuming
$T_{\mathrm{B}}=0$ for simplicity, the tunneling rates from the dot to the
channel and drain read
$\Gamma_{\mathrm{cd}}\simeq\int_{\mu}^{\mu_{N}}\frac{1}{e^{2}}G_{\mathrm{L}}dE$
and
$\Gamma_{\mathrm{Dd}}\simeq\int_{\mu_{\mathrm{D}}}^{\mu_{N}}\frac{1}{e^{2}}G_{\mathrm{R}}dE$,
where $G_{\mathrm{L}}$ and $G_{\mathrm{R}}$ are tunneling conductances of the
respective barriers. By solving the master equation, we obtain the steady
current
$I=e\Gamma_{\mathrm{dc}}\Gamma_{\mathrm{Dd}}/\left(\Gamma_{\mathrm{dc}}+\Gamma_{\mathrm{Dd}}+\Gamma_{\mathrm{cd}}\right)$.
By assuming energy-dependent conductances $G_{\mathrm{L}}\propto
e^{E/W_{\mathrm{t}}}$ and $G_{\mathrm{R}}\propto e^{E/W_{\mathrm{t}}}$ with
the same characteristic energy $W_{\mathrm{t}}$ for the tunneling, we obtain
$I$ $\simeq\int_{\mu_{N}}^{\infty}\frac{1}{e}G_{\mathrm{QD}}f\left(E\right)dE$
with $G_{\mathrm{QD}}\propto e^{E/W_{\mathrm{t}}}$ by neglecting factors that
weakly depend on energy. This form was used in the analysis of distribution
functions.

Supplementary Notes

SN1: QPC detection

We prepare the channel configuration by applying appropriate gate voltages, as
shown in Fig. 10a. $V_{\mathrm{G1}}=V_{\mathrm{G3}}=$ -0.6 V and
$V_{\mathrm{G2}}=$ -0.45 V are fixed throughout this study. We inject hot
electrons from the source (S) through the injection point contact (IPC) to the
base (B) by applying voltage $-V_{\mathrm{inj}}$ (= 1 - 100 mV) to the ohmic
contact. The injection current $I_{\mathrm{inj}}$ (= 1 nA) is adjusted by
tuning $V_{\mathrm{Ginj}}$. The resulting excitation in the edge channel of B
is analyzed with the quantum point contact (QPC) located at a distance $L$
from the IPC. The conductance $G$ of the QPC is varied in the range from $G=$
0 to $2e^{2}/h$ by changing $V_{\mathrm{G4}}$ and $V_{\mathrm{G5}}$
simultaneously ($V_{\mathrm{G4}}=V_{\mathrm{G5}}$). For each $V_{\mathrm{G4}}$
($=V_{\mathrm{G5}}$) setting, $V_{\mathrm{Ginj}}$ is readjusted to obtain the
desired $I_{\mathrm{inj}}$ by using a software feed-back control, and then
$I_{\mathrm{inj}}$, $I_{\mathrm{det}}$, and $I_{\mathrm{B}}$ are recorded at
nominally $V_{\mathrm{D}}=0$. Figure 10b-d show representative results
($I_{\mathrm{inj}}$, $I_{\mathrm{det}}$, and $I_{\mathrm{B}}$) obtained for
various $V_{\mathrm{G4}}$ values at $V_{\mathrm{inj}}$ = 40 mV and $L$ = 1
$\mu$m. One can confirm the relation $I_{\mathrm{inj}}$ $\simeq$
$I_{\mathrm{det}}+I_{\mathrm{B}}$ from the data. The remainder
($I_{\mathrm{inj}}$ $-$ $I_{\mathrm{det}}-I_{\mathrm{B}}$) should be drained
to the other ohmic contact to B, as the input impedance of the ammeter for
$I_{\mathrm{B}}$ is finite ($\sim$ 1 k$\Omega$).

Figure 10: a Schematic setup for the QPC detection under hot electron
injection from IPC. b $I_{\mathrm{inj}}$, c $I_{\mathrm{det}}$, and d
$I_{\mathrm{B}}$ as a function of $V_{\mathrm{G4}}$.

The current step at $V_{\mathrm{G4}}\simeq$ -0.9 V measures the energy of hot
electrons. The hot electrons are fully reflected at $V_{\mathrm{G4}}<$ -0.9 V
and fully transmitted through the QPC at $V_{\mathrm{G4}}>$ -0.9 V. As the
step height is identical to $I_{\mathrm{inj}}$, all hot electrons injected
from IPC are not fully relaxed to the Fermi energy. The current peak at
$V_{\mathrm{G4}}\simeq$ -0.5 V measures the electron-hole plasma in the spin-
up channel, which are generated by the intra-channel electron-electron
scattering. The other current peak at $V_{\mathrm{G4}}\simeq$ -0.2 V measures
the electron-hole plasma in the spin-down channel, which are generated by the
inter-channel electron-electron scattering. These scattering processes are
discussed in the main paper. At $V_{\mathrm{G4}}=$ 0 V, $I_{\mathrm{det}}$ is
slightly smaller than $I_{\mathrm{inj}}$, and $I_{\mathrm{B}}$ becomes
negative. This is an experimental artifact, in which small but finite
$V_{\mathrm{D}}$ (about 1 $\mu$V) was present in our circuit.

Figure 11: a and b Coulomb diamond characteristics in $I_{\mathrm{det}}$ as a
function of $V_{\mathrm{G5}}$ for various $V_{\mathrm{D}}$ voltages from -1 mV
to 1 mV in the step of 40 $\mu$V for the QD conditions at $L=$ 1 $\mu$m in a
and $L=$ 10 and 20 $\mu$m in b. Figure 12: QD detection current
$I_{\mathrm{det}}$ for studying electron-hole plasma near the Fermi energy.
Panels a, b, and c for $L=$ 1 $\mu$m, panels d, e, and f for $L=$ 10 $\mu$m,
and panels g, h, and i for $L=$ 20 $\mu$m are shown. Panels a, d, and g show
color plots of $I_{\mathrm{det}}$ as a function of $V_{\mathrm{G5}}$ and
$V_{\mathrm{inj}}$. The dashed line shows the condition
($V_{\mathrm{inj}}^{\left(\mathrm{H\uparrow}\right)}$) for maximized plasma
excitation. Panels b, e, and h show $I_{\mathrm{det}}$ as a function of
$\Delta V_{\mathrm{G5}}$ for various
$V_{\mathrm{inj}}<V_{\mathrm{inj}}^{\left(\mathrm{H\uparrow}\right)}$. Panels
c, f, and i show $I_{\mathrm{det}}-I_{\mathrm{inj}}$ as a function of $\Delta
V_{\mathrm{G5}}$ for various
$V_{\mathrm{inj}}>V_{\mathrm{inj}}^{\left(\mathrm{H\uparrow}\right)}$.

SN2: QD detection

Coulomb diamond characteristics of the QD in the absence of hot-electron
injection are shown in Figure 11a for the measurement at $L$ = 1 $\mu$m and
Fig. 11b for $L$ = 10 and 20 $\mu$m. The charging energy $U\simeq$ 1.5 meV,
typical energy spacing $\Delta\simeq$ 0.2 meV, and the lever arm factor
$\alpha\simeq 0.044e$ to obtain a change in the electrochemical potential
$\Delta\mu_{N}=\alpha\Delta V_{\mathrm{G5}}$ for a change $\Delta
V_{\mathrm{G5}}$ in $V_{\mathrm{G5}}$ were obtained from the data.

The electron-hole plasma under hot-electron injection is analyzed with the QD,
as shown in Figure 12. As shown in panels a, d, and g for $L=$ 1, 10, and 20
$\mu$m, respectively, the positions of the CB peaks shift to higher
$V_{\mathrm{G5}}$ side with increasing $V_{\mathrm{inj}}$, which can be
understood with the electrostatic effects from $V_{\mathrm{inj}}$ and
$V_{\mathrm{Ginj}}$. While the effect is largest in Figure 12a for the
shortest $L=$ 1 $\mu$m, the width and height of the CB peaks do not change so
much, indicating that the QD is well isolated from the IPC. Finite current in
the CB regions appears with increasing $V_{\mathrm{inj}}$ for a given
$I_{\mathrm{inj}}$. This current is maximized at
$V_{\mathrm{inj}}=V_{\mathrm{inj}}^{\left(\mathrm{H\uparrow}\right)}$, where
the hot spot $x_{\mathrm{H\uparrow}}$ of the spin-up channel coincides with
the QD position. The condition
$V_{\mathrm{inj}}^{\left(\mathrm{H\uparrow}\right)}$ increases with increasing
$L$ ($V_{\mathrm{inj}}^{\left(\mathrm{H\uparrow}\right)}=$ 28, 64, and 75 mV
for $L=$ 1, 10, and 20 $\mu$m, respectively), from which the hot-electron
trajectory is investigated in the main paper.

The non-thermal state after equilibration at $x=L>x_{\mathrm{H\downarrow}}$ is
studied with the data at
$V_{\mathrm{inj}}<V_{\mathrm{inj}}^{\left(\mathrm{H\uparrow}\right)}$ in
Figure 12b, e, and h. In these semi-logarithmic plots, the broad current
profiles in the CB regions and the sharp CB peaks coexist in the whole range.
The data suggests the coexistence of warm electrons ($T_{1}$) and cold
electrons ($T_{0}$ close to the base temperature), which justifies the binary
Fermi-Dirac distribution function as a non-thermal metastable state in the
Tomonaga-Luttinger (TL) liquid.

The non-equilibrium state in the course of e-e scattering at
$x=L<x_{\mathrm{H\uparrow}}$ was analyzed with the data at
$V_{\mathrm{inj}}>V_{\mathrm{inj}}^{\left(\mathrm{H\uparrow}\right)}$ in
Figure 12c, f and i. As the injected hot electrons pass over the QD and are
absorbed by the ammeter, $I_{\mathrm{det}}-I_{\mathrm{inj}}$ should be
evaluated. The excess current in the CB region decreases with increasing
$V_{\mathrm{inj}}$, which suggests suppression of e-e scattering for hot
electrons at higher energies. This suppression is discussed with momentum
conservation in the main paper.

Data availability

The data and analysis used in this work are available from the corresponding
author upon reasonable request.

Acknowledgements

We thank Taichi Hirasawa for preliminary measurements in the early stage of
the study. This study was supported by the Grants-in-Aid for Scientific
Research (KAKENHI JP19H05603) and the Nanotechnology Platform Program of the
Ministry of Education, Culture, Sports, Science and Technology, Japan.

Author contributions

T.F. designed and supervised this study. T.A. and K.M. grew the wafer. Y.S.
fabricated the device. K.S. performed the measurement and analysis with help
from T.H. and T.F. K.S. and T.F. wrote the manuscript. All authors discussed
the results and commented on the manuscript.

Competing interests

The authors declare no competing interests.

Correspondence and requests for materials should be addressed to T.F.

The final version of this paper was published in Communications Physics. 6,
103 (2023).

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