††thanks: Nguyen Ha My Dang and Simone Zanotti contributed equally to this
work as first authors. Corresponding authors:

# Long-range ballistic propagation of 80$\%$-excitonic-fraction polaritons in
a perovskite metasurface at room temperature

Nguyen Ha My Dang1 Simone Zanotti2<EMAIL_ADDRESS>Emmanuel Drouard1 Céline Chevalier1 Gaëlle Trippé-Allard3 Emmanuelle
Deleporte3 Christian Seassal1 Dario Gerace2 Hai Son Nguyen1,4 hai-
<EMAIL_ADDRESS>1Université Lyon, Ecole Centrale de Lyon, CNRS, INSA
Lyon, Université Claude Bernard Lyon 1, CPE Lyon, CNRS, INL, UMR5270, Ecully
69130, France  2Dipartimento di Fisica, Università di Pavia, via Bassi 6,
I-27100 Pavia, Italy  3 Université Paris-Saclay, ENS Paris-Saclay,
CentraleSupélec, CNRS, Lumière, Matière et Interfaces (LuMIn) Laboratory,
91190 Gif-sur-Yvette, France  4 Institut Universitaire de France (IUF), 75231
Paris, France

###### Abstract

Exciton-polaritons, hybrid light-matter elementary excitations arising from
the strong coupling regime between excitons in semiconductors and photons in
photonic nanostructures, offer a fruitful playground to explore the physics of
quantum fluids of light as well as to develop all-optical devices. However,
achieving room temperature propagation of polaritons with a large excitonic
fraction, which would be crucial, e.g., for nonlinear light transport in
prospective devices, remains a significant challenge. Here we report on
experimental studies of exciton-polariton propagation at room temperature in
resonant metasurfaces made from a sub-wavelength lattice of perovskite
pillars. Thanks to the large Rabi splitting, an order of magnitude larger than
the optical phonon energy, the lower polariton band is completely decoupled
from the phonon bath of perovskite crystals. The long lifetime of these cooled
polaritons, in combination with the high group velocity achieved through the
metasurface design, enables long-range propagation regardless of the polariton
excitonic fraction. Remarkably, we observed propagation distances exceeding
hundreds of micrometers at room temperature, even when the polaritons possess
a very high excitonic component, approximately 80$\%$. Furthermore, the design
of the metasurface introduces an original mechanism for directing uni-
directional propagation through polarization control. This discovery of a
ballistic propagation mode, leveraging high-speed cooled polaritons, heralds a
promising avenue for the development of advanced polaritonic devices.

##  I Introduction

Exciton-polaritons are hybrid light-matter excitations emerging from the
strong coupling regime between photons and excitons in semiconductors [1, 2].
The manipulation of these elementary excitations in confining geometries,
e.g., quantum wells embedded in microcavities [3], has paved the way for
probing the fundamental physics of out-of-equilibrium Bose-Einstein
condensation, as well as the rich physics of quantum fluids of light [4].
Exciton-polaritons are currently at the forefront of developing advanced all-
optical devices [5]. Investigating polariton propagation is crucial for
designing devices that fully exploit high-speed and efficient polaritonic
signal transmission. Thanks to their hybrid nature, polariton propagation
features unique properties not found in purely photonic or excitonic
transport. From their photonic component, exciton-polaritons benefit from a
small effective mass and high group velocity, enabling ballistic propagation
over macroscopic distances, and allowing for the tailoring of these
propagation properties through the engineering of their potential landscape
[6, 7]. In addition, their excitonic component introduces highly nonlinear
behaviors, giving rise to propagating solitons [8], nonlinear tunneling
effects [9], and even superfluid regimes [10, 11, 12]. Exciton-polaritons can
also be directed by external fields that interact with the polariton flow
through their excitonic component [13].

A wide range of active materials capable of sustaining an excitonic response
has been utilized to study polariton propagation, including GaAs-based quantum
wells for foundational research at cryogenic temperatures [14, 15, 16, 17, 18,
19, 20, 21], along with more recent studies involving materials that operate
at room temperature such as ZnO [22], organic materials [12, 23, 24], GaN
[25], transition-metal dichalcogenides (TMDs) [26, 27], and perovskites [28,
29, 30, 31, 32]. Nevertheless, much of this research has focused on photonic-
like polaritons with a low excitonic fraction (less than 50$\%$). Overcoming
the challenges associated with achieving macroscopic propagation of polaritons
with a high excitonic fraction (i.e., larger than 50$\%$) is essential to
ultimately harness the full potential of these hybrid excitations in
innovative communication devices. The primary obstacle is that the microcavity
design, a commonly employed system for generating exciton-polaritons,
generally offers relatively low group velocities, which decrease significantly
as the excitonic fraction increases. Moreover, the thermal broadening of the
excitonic resonance at room temperature poses a significant challenge for the
macroscopic propagation of highly excitonic polaritons. For example, one of
the most used perovskite materials, (C6H5C2H4NH3)2PbI4 (PEPI), displays a
strong excitonic resonance with $30\text{\,}\mathrm{meV}$ homegeneous
linewidth at room temperature [33, 34], thus suggesting that PEPI-based
polaritons with an excitonic fraction in excess of 50$\%$ would have a minimum
linewidth of $15\text{\,}\mathrm{meV}$, even in the best case scenario of
negligible photonic losses. For a group velocity of
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{ps}$, which is typical
of polaritons in microcavity samples, this linewidth corresponds to an
extremely short propagation length, i.e., less than $100\text{\,}\mathrm{nm}$.

In this work, we overcome the previously mentioned limitations by engineering
a photonic potential landscape, tailored through a sub-wavelength lattice of
pillars resonant with the active excitonic transition of the PEPI perovskite.
This dispersion engineering allows for the decoupling of high-speed and
excitonic-like polaritons from the thermal bath, ultimately enabling polariton
propagation over macroscopic distances at room temperature. As a result,
ballistic propagation of polaritons with 80$\%$ excitonic fraction at a speed
of $25\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{ps}$ and a
linewidth as small as a few meV is experimentally demonstrated across
distances exceeding 100 µm. Our findings suggest an original mechanism for
long distance polariton propagation, and provide a unique platform to engineer
polaritonic transport at room temperature.

##  II Polariton eigenmodes from a large nano-imprinted perovskite metasurface

Our sample consists of a high-quality and homogeneous PEPI metasurface,
covering a large area of $\sim$ 3 cm2. This metasurface was successfully
fabricated using the thermal imprinting method [35, 36] (further details on
the fabrication are reported in the Methods section and detailed in Ref.
[36]). A sketch of the fabricated metasurface is shown in Fig. 1(a). The
overall PEPI thickness is estimated as $110\text{\,}\mathrm{nm}$, while the
patterned thickness is $t_{\text{pattern}}=$70\text{\,}\mathrm{nm}$$ (i.e.,
unpatterned thickness $t_{\text{slab}}=$40\text{\,}\mathrm{nm}$$). The
metasurface consists of a square lattice of pillars, with both the period and
diameter being $300\text{\,}\mathrm{nm}$ each, as depicted in Fig. 1(b).

Figure 1: (a) Three-dimensional sketch of the active metasurface studied in
this work (not to scale). (b) Scanning electron microscope image of the
sample. (c) Reflectivity ($R$) and (d) photoluminescence ($PL$) spectra for
$E_{y}$-polarized light with wavevector pointing along the $k_{x}$ and $k_{y}$
directions, respectively, and obtained from the structure pictured in (a). In
this paper, our focus lies on the study of the propagation from the lower
polariton branch, referred to as “LP” in panels (c)-(d). The broadband signal
that is visible in (d) around 2.394 eV is attributed to bare PEPI exciton
emission.

Signatures of polariton excitations in the system are experimentally probed by
angle-resolved reflectance (ARR) and photoluminescence (ARPL), respectively
(we refer to the Methods section for futher details of the experimental
setup). For both the ARR and ARPL measurements, the signals can be analyzed in
two different polarizations: $E_{y}$ (i.e., S-polarization with respect to
$xz$ plane) and $E_{x}$ (i.e., P-polarization with respect to $xz$ plane). In
Fig. 1(c,d) we show results for the $E_{y}$ polarization only, along both
$k_{x}$ and $k_{y}$. Due to the C4 symmetry of the square lattice, the results
for the $E_{x}$ polarization along $k_{x}$ and $k_{y}$ (not shown here) are
identical to those in $E_{y}$ polarization along $k_{y}$ and $k_{x}$,
respectively. We now focus on the polariton mode denoted as LP (for Lower
Polariton) in Figs. 1(c,d). The strong coupling regime is clearly evidenced by
the anticrossing effect observed in both ARR and ARPL spectra. This is
highlighted by the bending of the LP mode as it approaches the exciton
resonant energy in PEPI ($E_{X}=2.394$ eV). We observe that the dispersion of
the LP mode is almost flat along $k_{y}$, but highly dispersive along $k_{x}$,
with a relatively large group velocity. Therefore the LP mode is a perfect
candidate for studying polariton propagation along the $x$ direction. In
addition, as previously discussed, the C4 symmetry imposes a similar
polaritonic mode for the $E_{x}$ polarization, which is the ideal candidate
for studying propagation along the $y$ direction.

Figure 2: The yellow dots in panels (a), (b) and (c) represent the energy peak
position, the linewidth (full width at half maximum), and the group velocity,
respectively, measured for the polaritonic branch (LP in Fig. 1), as a
function of the wave vector, $k_{x}$. The full lines in (a), (b), and (c) are
theoretical curves for the LP polaritonic mode obtained from the coupled
oscillators model, with the excitonic resonance indicated as a dashed line in
(a). From the same model, we plot in (d) the Hopfield coefficients, the dashed
(full) line representing the calculated photonic (excitonic) fraction of this
polaritonic mode as a function of the wave vector. In (a,b,c), the vertical
dashed lines correspond to the wave vector for which polaritons possess
excitonic fraction $W_{X}=63\%$, corresponding to the propagation measurements
in Fig. 3 and Fig.4 .

Figures 2(a,b,c) present the resonance energy, linewidth (measured as the full
width at half maximum from the PL spectra), and group velocity of LP,
extracted from ARPL measurements. The experimental results are nicely fitted
by the coupled oscillator model between PEPI excitons and photonic Bloch
resonances, given by:

$\tilde{E}_{pol}(k_{x})=\frac{\tilde{E}_{ph}(k_{x})+\tilde{E}_{X}}{2}-\frac{1}{2}\sqrt{4V^{2}+\left[\tilde{E}_{ph}(k_{x})-\tilde{E}_{X}\right]^{2}}\,.$
(1)

Here, we can explicitly separate the real part contribution (i.e., the
resonances dispersion) from the losses due to the imaginary part:
$\tilde{E}_{pol}(k_{x})=E_{pol}(k_{x})+i\Gamma_{pol}(k_{x})/2$. In the
expression given in Eq. (1), $V$ is the exciton-photon coupling strength,
$\tilde{E}_{X}=E_{X}+i\Gamma_{X}/2$ is the dispersionless excitonic band of
energy $E_{X}$ and linewidth $\Gamma_{X}$, and $\tilde{E}_{ph}(k_{x})$ is the
complex photonic mode dispersion, which can be approximated by an analytic
expression as Eq.(2) (the full expression of all photonic modes along both
$k_{x}$ and $k_{y}$ is reported in the Supplemental Information, for
completeness):

$\tilde{E}_{ph}(k_{x})=E_{0}+i\frac{\Gamma_{0}}{2}+\sqrt{\left(\hbar
vk_{x}\right)^{2}+\left(U+i\frac{\Gamma_{0}}{2}\right)^{2}}\,.$ (2)

In Eq. (2), $v$ is the group velocity of the guided modes, which are then
folded and coupled by the periodic metasurface. The coupling strength between
guided modes is given by the parameter $U$, while $\Gamma_{0}$ accounts for
the coupling between guided modes and the radiative continuum, due to the
periodic dielectric modulation. From Eq. (2), one may extract the photonic
dispersion, $E_{ph}(k_{x})$, as well as the photonic losses,
$\Gamma_{ph}(k_{x})$, by calculating the real and imaginary parts of
$\tilde{E}_{ph}(k_{x})$, respectively.

The group velocity along $x$ for the given polaritonic branch is
straightforwardly obtained from the derivative of real part of the polaritonic
dispersion. In the wavevector range shown in Fig. 2(c), the group velocity
rapidly increases from $v_{g}=$0\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{ps}}^{-1}$$ to
$v_{g}=$50\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{ps}}^{-1}$$
for a slight wavevector increase; then, its value remains above
$v_{g}=$25\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{ps}}^{-1}$$
at larger wavevector.

In addition, we report in Fig. 2(d) the calculated Hopfield coefficients
(i.e., excitonic and photonic fractions, respectively), given by:

$W_{X},W_{ph}\approx\frac{1}{2}\left(1\pm\frac{E_{ph}-E_{X}}{\sqrt{(E_{ph}-E_{X})^{2}+4V^{2}}}\right).$
(3)

Our results demonstrate that perovskite metasurfaces can exhibit highly
excitonic polaritons with high group velocity (tens of
$\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{ps}}^{-1}$) and low losses
(few $\mathrm{meV}$).

In terms of model parameters, the exciton energy is extracted from the
absorption measurement on an unpatterned PEPI thin film, resulting in
$E_{X}=$2.394\text{\,}\mathrm{eV}$$. From a fitting procedure, we estimated
$E_{0}=$2.294\text{\,}\mathrm{eV}$$,
$v=$77.5\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{ps}}^{-1}$$,
$U=$10\text{\,}\mathrm{meV}$$, $\Gamma_{0}=$2.2\text{\,}\mathrm{meV}$$ and
$\Gamma_{X}=$0.3\text{\,}\mathrm{meV}$$. The exciton-photon coupling strength
is estimated as $V=$127\text{\,}\mathrm{meV}$$, leading to a Rabi-splitting
energy of $\sim$254\text{\,}\mathrm{meV}$$ at zero detuning, which is in good
agreement with previous reports [37, 38]. An inhomogeneous broadening
quantified as $\Gamma_{inh}=$3\text{\,}\mathrm{meV}$$ is added to the total
polariton linewidth:
$\Gamma_{pol}^{\text{(total)}}=\Gamma_{pol}+\Gamma_{inh}$. Notice that these
parameters well reproduce both the real and imaginary parts of the complex LP
mode dispersion, as measured from the data shown, e.g., in Figs. 2(a,b,c), as
well as from the iso-frequency measurements and propagation experiments in
both real and momentum space, that we are going to present in Sec. III.

##  III Suppression of thermal broadening of perovskite polaritons

Intriguingly, the excitonic losses estimated from the fit in the previous
Section, $\Gamma_{X}=0.3$ meV, are orders of magnitude smaller than the value
($\sim$$30\text{\,}\mathrm{meV}$) that is either reported in the literature
for the homogeneous linewidth of PEPI excitons at room temperature [33, 34],
or obtained from a rough estimate of the full width at half maximum of
uncoupled excitons in the PL spectra of Fig. 1(d). Moreover, as shown in Fig.
2(b), the polariton linewidth evidently narrows down as the excitonic fraction
increases. As a direct consequence, at $63\%$ excitonic fraction (i.e.,
$k_{x}\approx$2.8\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$), the
measured polariton linewidth is about $4\text{\,}\mathrm{meV}$. Additionally,
from the propagation experiment described in the next section, we extract a
polariton lifetime of 0.7 ps, corresponding to a homogeneous linewidth of 0.94
meV for polaritonic states with $63\%$ excitonic fraction. This unusual
narrowing of the polariton linewidth is explained by the suppression of the
thermal broadening for our polaritonic states, as thoroughly described in the
following.

In general, exciton-photon interaction is orders of magnitude stronger than
exciton-phonon coupling, as explained by Savona and Piermarocchi [39]. Thus,
polaritons are primarily formed from photons and “isolated” excitons.
Subsequently, the thermal broadening is due to polariton-phonon interaction
and not to exciton-phonon interaction. As a consequence, the polariton
homogeneous linewidth is given by:

$\Gamma_{pol}\left(T\right)=\underbrace{W_{C}\cdot\Gamma_{ph}+W_{X}\cdot\Gamma_{X}}_{\text{coupled-
oscillator linewidth}}+\Gamma_{\text{th}}\left(T\right)\,,$ (4)

in which $\Gamma_{X}$ is the homogeneous exciton linewidth at
$0\text{\,}\mathrm{K}$ (notice, not at the temperature $T$!), and
$\Gamma_{\text{th}}\left(T\right)$ is the thermal broadening induced from the
polariton-phonon interaction.

The main contribution to polaritonic dephasing at high-temperature is the
optical phonon-polariton interaction, which scatters polaritons into the
excitonic states [40, 41, 42]. This process is particularly favourable given
the high density of states available in the excitonic reservoir at $E_{X}$.
But the energy conservation allows for such scattering in the energy range
$E_{pol}\in\left[E_{X}-E_{\text{LO}},E_{X}\right]$, where $E_{\text{LO}}\sim
20-30$ meV is the optical phonons energy in PEPI [43, 33, 34, 44]. This
essentially divides the spectrum into two regions, as represented in Fig.
2(a): the first region is defined by $E_{pol}>E_{X}-E_{\text{LO}}$, where
polaritons can efficiently scatter with phonons at room temperature; the
second region is defined by $E_{pol}<E_{X}-E_{\text{LO}}$, where the phonon-
polariton scattering is suppressed and polaritons effectively behave like in
an environment at $T_{\text{eff}}=$0\text{\,}\mathrm{K}$$. Strikingly, all the
experimental data taken for the LP band are compatible with these polaritons
being located in the $T_{\text{eff}}=$0\text{\,}\mathrm{K}$$ region,
regardless of the excitonic fraction. Therefore, these polaritons are “cooled”
and do not exhibit any thermal broadening: $\Gamma_{th}\approx 0$. This
argument gives a remarkably simple but effective account for the reduced
polariton linewidth observed in the experimental data reported in Fig. 2(b).

We further notice that the suppression of the polariton-phonon scattering
channel was theoretically suggested and experimentally demonstrated by Trichet
et al. [40, 41] for ZnO-based polaritons. A similar mechanism was recently
predicted for TMD-based polaritons. [42]. In the hypothesis of low losses
($\Gamma_{pol}\ll E_{pol}$, $\Gamma_{X}\ll E_{X}$), the condition
$E_{pol}<E_{X}-E_{\text{LO}}$ can be rewritten from Eq.(1) and Eq.(3) as:

$V>E_{\text{LO}}\sqrt{\frac{W_{X}}{1-W_{X}}}\,.$ (5)

In particular, for the case of $W_{X}=50\%$, this condition corresponds to
$V>E_{\text{LO}}$ [40, 41] that is difficult to be satisfied for exciton-
polaritons in III-V semiconductors or TMD monolayers, but it can be easily met
in materials with stronger oscillator strengths, such as the PEPI under
investigation in the present work. Thus, for PEPI polaritons we can safely
assume that $\Gamma_{th}\approx 0$, even for polariton eigenmodes with very
large excitonic fraction. Specifically, for our PEPI metasurface, using
$V=$127\text{\,}\mathrm{meV}$$ and $E_{\text{LO}}=$30\text{\,}\mathrm{meV}$$,
the condition of cooled polaritons Eq.(5) is satisfied for $W_{X}<95\%$.

##  IV Polariton propagation: measurements in real space

In this section we present the results of a propagation experiment involving
polaritonic eigenmodes with $\sim 63\%$ excitonic fraction. From the PL signal
under non-resonant pumping, this fraction is selected by using a spectral
band-pass filter centered at $E_{f}=$2.283\text{\,}\mathrm{eV}$$. This
spectral filtering corresponds to an average wave vector of
$2.8\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$, with an excitonic fraction
of $63\%$.

Figure 3: (a) Schematic representation of the in-plane propagation of
polaritonic modes excited with $E_{x}$ and $E_{y}$ polarised light,
respectively. (b)-(c) Iso-frequency wavevector dispersion of exciton-
polaritons, filtered at $E_{0}=$2.283\text{\,}\mathrm{eV}$$, as obtained from
mapping the far-field emission in (b) $E_{y}$ and (c) $E_{x}$ polarisations,
respectively. The arrows indicate the group velocity patterns. The dashed
circles indicate the numerical aperture of the objective lens, while the solid
lines are the iso-frequency cuts of the bands calculated with Eq. (1), using
the same parameters as in Fig. 2. Details on the theoretical model for the
description of polaritonic modes, both as a function of $k_{x}$ and $k_{y}$,
are given in the Supplemental Material.

Since the propagation direction is dictated by the group velocity vectors, it
is crucial to investigate the group velocity pattern of the polaritons at the
selected energy before studying the propagation itself. This pattern can be
directly visualized from the iso-frequency mapping of the polaritonic modes.
Indeed, the group velocity follows the normal direction of the iso-frequency
curves, since it is defined from the gradient of the energy surface in
momentum space: $\mathbf{v}_{g}=1/\hbar\,\nabla_{\mathbf{k}}E_{pol}$. Figure
3(b)-(c) presents the measured iso-frequency map for two different
polarizations, obtained by projecting the far-field PL emission onto a camera
sensor. Two polaritonic modes at this energy are revealed: one is
$E_{y}$-polarized, with a dispersion only along $k_{x}$, and corresponding to
the LP mode shown in Fig. 1(c)-(d); the other is $E_{x}$-polarized, with
dispersion only along $k_{y}$. The results of the iso-frequency curves
calculated from the theory are also plotted, showing very good agreement with
experimental results. The iso-frequency cuts reveal that the
$E_{y}$-($E_{x}$-)polarized mode contains two approximately parallel bands
along $k_{y}$ ($k_{x}$), with a group velocity pattern that is almost uniform
along the $x$ ($y$) direction. This is further confirmed by the measurement of
the group velocity of the LP mode for different values of $k_{y}$, as shown in
Fig. S2 of the Supplemental Material.

Figure 4: (a) Spatially resolved photoluminescence image of the polaritonic
states filtered at $2.283\text{\,}\mathrm{eV}$, with $E_{y}$ polarization
selection. The blue-shaded area in the centre corresponds to the pump spot
region. (b) Intensity profile selected at $y=0$. The scattered points
represent the experimental data, while the dashed line is the fitting results
obtained from Eq. (6), and the white line with blue contour is calculated with
the theoretical model for the polariton propagation presented in the
Supplementary Material.

From the results shown above, we expect that the polariton propagation in the
2D metasurface consists of two one-dimensional flows of uniform propagating
fronts with constant group velocity, respectively along the $x$ and $y$
directions, with orthogonal polarizations, as pictorially depicted in Fig.
3(a). Such a propagation scenario is experimentally demonstrated by
visualizing the farfield image in real space. In particular, Fig. 4(a)
presents the spectrally filtered PL image in real space, measured in $E_{y}$
polarization. In this experiment, the non-resonant pumping of $\sim 2-3$
$\mathrm{\SIUnitSymbolMicro m}$ diameter is focused at $x=y=0$. These results
clearly show that polaritons are locally injected at $x=y=0$ under the pump
spot, and then propagate along the $x$ direction. Strikingly, the polariton
flow remains tightly focused along $y$ direction, even about
$x=$60\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ away from the pumping spot, as
it is evident from Fig. 4(a). This observation confirms that the velocity
vector is aligned along the $x$ direction. Evidently, by switching
polarization from $E_{y}$ to $E_{x}$, also the propagation direction switches
from $x$ to $y$.

To quantitatively evaluate the propagation properties, the intensity profile
at $y=0$ is extracted from the PL image and reported in Fig. 4(b). In the
real-space PL, we still have part of the PEPI uncoupled excitonic signal
around the excitation spot. Still, these excitons possess high effective mass
and zero group velocity, which prevents their propagation over long distances
to a large extent. Hence, it is safe to confirm that the PL signal detected
far away from the pumping spot comes from propagating polaritons. The PL
intensity is then fitted using a phenomenological law of two components
function: a Gaussian centered in the origin to account for the excitonic
reservoir, an exponential decay curve outside the spot, where the dynamics is
dominated by polariton propagation:

$I(x)=I_{0}e^{-\frac{x^{2}}{2\sigma^{2}}}+I_{1}e^{-\frac{|x|}{l}}\,.$ (6)

Here, $\sigma$ is the size of the pumping spot and $l$ is the decay length. By
applying this function to fit the experimental data, we estimated a
propagation constant $l=$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and a
pumping spot size $\sigma\sim$3.5\text{\,}\mathrm{\SIUnitSymbolMicro m}$$.
This value of decay length corresponds to a polariton lifetime of
$0.7\text{\,}\mathrm{ps}$, if ballistic propagation is assumed. This lifetime
corresponds to a homogeneous linewidth of $0.94\text{\,}\mathrm{meV}$. We
notice that this linewidth is much smaller than the value
$4\text{\,}\mathrm{meV}$ obtained from the fit in the previous section. This
indicates that the inhomogeneous broadening effects are the dominant
contributions to the total polariton linewidth in the emission spectra. Such
an observation is in good agreement with the value $\Gamma_{inh}=3$ meV, which
is used as fitting parameter for the inhomogeneous broadening in the results
of Fig. 2. Finally, the same parameters for the polaritonic modes are
implemented in a 1D propagation model (see the Supplemental Material for
details). The calculated spatial decay given by this model, whose results are
shown in Fig. 4, perfectly reproduces the experimental results and the
phenomenological law.

##  V Polariton propagation: measurements in momentum space

Figure 5: (a)-(b)-(c) Normalized PL spectra obtained by spatially filtering
the signal at $x=$-100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$,
$x=$0\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$x=$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, respectively. The pump spot
is focused at $x=$0\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. (d)-(e)-(f)
Simulated PL spectra calculated for the same positions as (a)-(b)-(c). Details
on the theoretical model for the description of polariton propagation, as well
as on the signal denoising, are given in the Supplemental Material.

To investigate the ballistic nature of the polariton propagation in the nano-
imprinted 2D metasurfaces, we show the monitoring of this propagation in
momentum space, with the aim of probing possible backscattering. To do so, a
slit of width $S=$17\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, playing the role
of a spatial filter, is put at the intermediate image position in the optical
path of the ARPL experiment setup [45]. By moving the slit position along the
$x-$axis, we can probe the polariton dispersion either immediately below or
far from the excitation spot. In this experiment, the spectral filter from the
previous experiment is removed. The experimental results corresponding to
three different slit positions are shown in Fig. 5(a-c). The corresponding
theoretical calculations, obtained from the 1D propagation model, are shown in
Fig 5(d-e).

Under the pumping spot at $x=0$, the ARPL results show all of the polaritonic
branches that were previously visible in the measured reflectivity and PL
spectra of Fig. 1(c)-(d). We notice that all the branches reported in Fig. 5
are broadened as compared to the ones measured without spatial filter in Fig.
1(c)-(d). This is due to an uncertainty on the emission wavevector,
$\Delta_{k}\approx 1/S$, introduced by the finite extent of the spatial
filter. Moreover, the intensity distribution is very different from the one
measured without the spatial filter: the strongest signals are observed from
the uncoupled excitons and the zero-group velocity point in the polariton
branches. Indeed, polaritons/excitons injected in these states cannot
propagate, and would radiatively emit photoluminescence under the pump spot,
while states of non-zero group velocity propagate out of the pump spot region.

At $x=$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
($x=-$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) away from the excitation,
only polaritonic states at positive (negative) group velocity are observed.
This demonstrates that polaritons propagate ballistically along the $x$
direction without any backscattering. Moreover, thanks to the absence of the
signal from uncoupled excitons far from the pump spot, we can appreciate the
linewidth narrowing of the LP branch when it approaches the excitonic energy.
This appears as a further confirmation of our interpretation about the
quenching of the thermal broadening. Most impressively, this propagation
across $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is observed even with
polaritons at $k_{x}=$4\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$,
corresponding to an excitonic fraction of 80$\%$. Since the extracted lifetime
for polaritonic eigenmodes with $55\%$ excitonic fraction is
$0.7\text{\,}\mathrm{ps}$, which is only limited by the photonic losses, we
estimate that the lifetime of these 80$\%$ excitonic fraction polaritons is
even longer, about $1.26\text{\,}\mathrm{ps}$.

##  VI Conclusion and Perspectives

We have successfully demonstrated ballistic propagation, in the hundred-
micrometer-range, of high-excitonic-fraction polaritons at room temperature.
This significant achievement is enabled by the suppression of thermal-induced
broadening in perovskite polaritons, formed within a large-scale and
homogeneous perovskite metasurface fabricated through direct nano-imprint
technology. The metasurface design introduces an innovative mechanism for
directing unidirectional and high-speed propagation across macroscopic
distances through polarization control. Given the strong nonlinearity
exhibited by PEPI polaritons at high excitonic fractions [46], our platform
opens new avenues for exploring nonlinear transport of quantum fluids of light
beyond traditional microcavity architectures. By leveraging methods such as
the precise tailoring photonic dispersion [47], or employing novel concepts of
loss engineering such as bound states in the continuum [48, 49, 50], we
anticipate unlocking new regimes of superfluidity and non-equilibrium
hydrodynamics. Finally, from the potential applications perspective, the
ability to route and confine low-loss, highly nonlinear polaritons at room
temperature using a scalable and cost-effective fabrication method paves the
way for integrating polariton physics into all-optical and integrated devices,
marking a significant advancement for the field where polariton-based devices
could revolutionize signal processing and transmission.

Acknowledgement: The authors would like to thank the staff from the Nanolyon
Technical Platform for helping and supporting all nanofabrication processes.
We also thank Le Si Dang and Lucio Claudio Andreani for fruitful discussions.
H.S.N, C.S, E.D acknowledge financial support from by the French National
Research Agency (ANR) under the project POPEYE (ANR-17-CE24-0020), project
EMIPERO (ANR-18-CE24-0016). S.Z. and D.G. acknowledge financial support from
PNRR MUR project PE0000023-NQSTI.

##  VII METHODS

###  VII.1 Sample fabrication

The fabrication process initiates by depositing a thin layer of PEPI onto
substrates composed of $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ thick SiO2 on
silicon (Si). This is achieved by spin-coating a 20% wet solution of PEPI in
Dimethylformamide (DMF) at 5000 rpm for 30 seconds. Following this step, the
PEPI film undergoes annealing at 95°C for 90 seconds to induce crystallization
before proceeding to the imprinting step. Subsequently, a Si mold with the
desired structure is pressed onto the PEPI film using a thermal press. This
imprinting process occurs at 100°C under a pressure of 100 bar for
approximately 10 minutes. As a result, the structure from the mold is directly
transferred onto the PEPI film. Details of the fabrication method were already
reported in Ref. [36].

###  VII.2 Experimental setup

The sample is illuminated via a microscope objective (20x magnification,
NA=0.42) using a white-light beam (halogen lamp) for ARR experiments or a non-
resonant laser ($400\text{\,}\mathrm{nm}$, $80\text{\,}\mathrm{MHz}$,
$50\text{\,}\mathrm{ps}$) for ARPL experiments . The signal that is scattered
or emitted from the sample is then collected through the same objective, and
analyzed by imaging the back-focal plane of the microscope objective for
measurements in momentum-space, or the sample plane for measurements in real-
space. For dispersion measurements, the signal is first projected onto the
entrance of a spectrometer, whose output is coupled to the sensor of a CCD
camera. For the propagation experiment in real-space, the signal is spectrally
filtered by a band pass filter at 543 nm, and then projected onto the sensor
of a SCMOS camera. For the propagation experiment in momentum space, the
signal is spatially filtered by a slit that is positioned in correspondence of
the plane of the intermediate image [45].

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— SUPPLEMENTAL MATERIAL —

##  I Suppression of thermal broadening

In order to investigate the impact of exciton losses on the polariton, we
performed the fit of Eq. (1) reported in the main text in two different
configurations. In the first case, we fixed the excitonic linewidth at
$\Gamma_{X}=$0.3\text{\,}\mathrm{meV}$$, representing excitonic losses without
any thermal broadening effect. Secondly, we set the exciton linewidth
corresponding to the bare exciton as $\Gamma_{X}=$30\text{\,}\mathrm{meV}$$,
i.e., including thermal broadening effects. The resulting polaritonic
linewidths are compared in Fig. S1 with experimental data extracted from the
photoluminescence (PL) spectrum of Fig. 1(d) in the main text. Remarkably, we
observed a significant agreement between the fitted model and experimental
results for the choice $\Gamma_{X}=$0.3\text{\,}\mathrm{meV}$$, meaning that
polariton losses are not affected by thermal broadening of the bare excitons.

Figure S1: The lines depict the polaritonic linewidths for various excitonic
fractions $W_{X}$, as fitted from Eq. (1) in the main text. The solid line
represents the fit with an excitonic linewidth
$\Gamma_{X}=$0.3\text{\,}\mathrm{meV}$$, while the dashed line corresponds to
$\Gamma_{X}=$30\text{\,}\mathrm{meV}$$. Scattered data points are derived from
experimental PL measurements.

##  II Measurement of the group velocity pattern for 63$\%$ excitonic-fraction
polaritons

With the aim of highlighting the uniform group velocity along $x$-direction
for $y$-polarized polaritons in the 63$\%$ excitonic fraction case, we
experimentally measured the group velocity, $v_{g}$, corresponding to
different values of $k_{y}$. Here, the velocity is extracted from the slope of
the dispersion along $k_{x}$, at various values of $k_{y}$. Experimentally,
the scanning of $k_{y}$ is achieved by moving the image of the back-focal
plane of the microscope objective with respect to the entrance slit of the
spectrometer [45]. The results, shown in Fig. S2(a), confirm that the group
velocity (for both forward and backward propagating polaritons) does not
depend on $k_{y}$.

Moreover, we can also extract the polariton linewidth, $\Gamma_{pol}$, at
different values of $k_{y}$ from the very same measurements. The results,
shown in Fig. S2(b), confirm that the polariton linewidth does not depend on
$k_{y}$, as predicted by the analytical model.

Figure S2: Experimental results for (a) the group velocity along $x$
direction, and (b) the linewidth of $y$-polarized polaritons having 63$\%$
excitonic fraction. The colored areas represent 1 standard deviation around
the average values.

##  III Theoretical model for photonic and polaritonic dispersions

###  III.1 Photonic modes

As discussed in detail in Ref. [36], the two photonic modes that give rise to
the two polaritonic branches observed in Figs. 1(c,d) can be described by the
following analytic expression:

$\tilde{E}_{ph(\pm)}=E_{0}+i\frac{\Gamma_{0}}{2}\pm\sqrt{\left(\hbar
vk_{x}\right)^{2}+\left(U+i\frac{\Gamma_{0}}{2}\right)^{2}}+\frac{\hbar
va}{4\pi}k_{y}^{2}\,,$ (S1)

in which $v$ is the group velocity of the photonic guided modes that are
folded and coupled by the periodic metasurface with lattice pitch $a$. The
coupling strength between guided modes is quantified by the parameter $U$,
while $\Gamma_{0}$ accounts for the coupling between guided modes and the
radiative continuum, due to the periodic dielectric modulation. From this
expression, we separate the real part contribution (i.e., the resonances
dispersion) from the the imaginary part (i.e., the contribution to the
losses):

$\tilde{E}_{ph(\pm)}=E_{ph(\pm)}+i\frac{\Gamma_{ph(\pm)}}{2}\,.$ (S2)

Notice that the $\tilde{E}_{ph+}$ photonic mode in the latter expression
corresponds to the mode indicated as $\tilde{E}_{ph}$ in Eq. (2) of the main
text.

###  III.2 Strong coupling regime

The coupling between these photonic modes and perovskite excitons gives rise
to four polartonic branches [36], analytically described as:

$\displaystyle\tilde{E}_{LP(\pm)}$
$\displaystyle=\frac{\tilde{E}_{ph(\pm)}+\tilde{E}_{X}}{2}-\sqrt{V^{2}+\left[\frac{\tilde{E}_{ph(\pm)}-\tilde{E}_{X}}{2}\right]^{2}}\,,$
(S3) $\displaystyle\tilde{E}_{UP(\pm)}$
$\displaystyle=\frac{\tilde{E}_{ph(\pm)}+\tilde{E}_{X}}{2}+\sqrt{V^{2}+\left[\frac{\tilde{E}_{ph(\pm)}-\tilde{E}_{X}}{2}\right]^{2}}\,,$
(S4)

from which the decomposition into real and imaginary parts can be explicitly
written as:

$\displaystyle\tilde{E}_{LP(\pm)}$
$\displaystyle=E_{LP(\pm)}+i\frac{\Gamma_{LP(\pm)}}{2}\,,$ (S5)
$\displaystyle\tilde{E}_{UP(\pm)}$
$\displaystyle=E_{UP(\pm)}+i\frac{\Gamma_{UP(\pm)}}{2}\,.$ (S6)

Also in this case, notice that the polaritonic mode here indicated as
$\tilde{E}_{LP+}$ corresponds to the one denoted as $\tilde{E}_{pol}$ in Eq.
(1) of the main text. The calculated dispersion and linewidth of photonic
modes and lower polaritonic modes, obtained from Eq. (S1) and Eq. (S3), are
presented in Fig. S3. The model parameters used to obtain the results for
these plots read as follows: $a=$300\text{\,}\mathrm{nm}$$,
$v=$77.5\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{ps}$$,
$U=$10\text{\,}\mathrm{meV}$$, $\Gamma_{0}=$2.2\text{\,}\mathrm{meV}$$,
$E_{X}=$2.394\text{\,}\mathrm{eV}$$, $E_{0}=$2.294\text{\,}\mathrm{eV}$$ and
$\Gamma_{X}=$0.3\text{\,}\mathrm{meV}$$. Finally, to take into account the
effect of disorder, an inhomogeneous broadening of $\Gamma_{inh}=3$ meV is
added to the polariton linewidths, such that:

$\displaystyle\Gamma_{LP(\pm)}^{\text{(total)}}$
$\displaystyle=\Gamma_{LP(\pm)}+\Gamma_{inh}\,,$ (S7)
$\displaystyle\Gamma_{UP(\pm)}^{\text{(total)}}$
$\displaystyle=\Gamma_{UP(\pm)}+\Gamma_{inh}\,.$ (S8)

As shown in the main text, we obtain an almost perfect matching between
experimental data and the analytic curves obtained from this model, both for
the real and imaginary parts of polaritonic modes, respectively.

###  III.3 iso-frequency curves in high momentum limit

In the limit of a large momentum, $\hbar vk_{x}\gg U,\Gamma_{0}$, the
expressions for the photonic modes given in Eq. Eq. (S1) simplify as:

$\tilde{E}_{ph(\pm)}\approx E_{0}+i\frac{\Gamma_{0}}{2}\pm\hbar
vk_{x}+\frac{\hbar va}{4\pi}k_{y}^{2}\,.$ (S9)

Moreover, considering that the losses $\Gamma_{0}$,$\Gamma_{X}$ are
negligeable with respect to $\hbar vk_{x}$ and $V$, the real part dispersion
of the lower polariton modes, given by Eq. (S3), can be rewritten as:

$E_{LP(\pm)}\approx\frac{E_{0}+E_{X}}{2}\pm\frac{\hbar vk_{x}}{2}+\frac{\hbar
va}{8\pi}k_{y}^{2}-\sqrt{V^{2}+\left[\frac{E_{0}-E_{X}}{2}\pm\frac{\hbar
vk_{x}}{2}+\frac{\hbar va}{8\pi}k_{y}^{2}\right]^{2}}\,.\\\ $ (S10)

We notice that the expressions above are for polaritonic modes coming from
$E_{y}$-polarized photonic branches. For polaritonic modes arising from
$E_{x}$-polarized photons, we will have similar expression but switching
between $k_{x}$ and $k_{y}$. Therefore, the iso-frequency curves at $E_{iso}$
for $LP(+)$ modes are given by:

$\displaystyle\frac{\hbar vk_{x}}{2}+\frac{\hbar
va}{8\pi}k_{y}^{2}-\sqrt{V^{2}+\left[\frac{E_{0}-E_{X}}{2}+\frac{\hbar
vk_{x}}{2}+\frac{\hbar
va}{8\pi}k_{y}^{2}\right]^{2}}+\frac{E_{0}+E_{X}}{2}-E_{iso}$
$\displaystyle=0\,,$ (S11) $\displaystyle\frac{\hbar vk_{y}}{2}+\frac{\hbar
va}{8\pi}k_{x}^{2}-\sqrt{V^{2}+\left[\frac{E_{0}-E_{X}}{2}+\frac{\hbar
vk_{y}}{2}+\frac{\hbar
va}{8\pi}k_{x}^{2}\right]^{2}}+\frac{E_{0}+E_{X}}{2}-E_{iso}$
$\displaystyle=0\,$ (S12)

for $E_{y}$ and $E_{x}$ polarization respectively.

Figure S3: Theoretical calculation of the (a) energy dispersion and (b)
linewidth along $k_{x}$ and $k_{y}$ of photonic modes $\tilde{E}_{ph(\pm)}$
and lower polaritonic modes $\tilde{E}_{LP(\pm)}$ that are given by Eq. (S1)
and Eq. (S3) , respectively. The parameters are:
$a=$300\text{\,}\mathrm{nm}$$, $v=$77.5\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}\mathrm{ps}$$, $U=$10\text{\,}\mathrm{meV}$$,
$\Gamma_{0}=$2.2\text{\,}\mathrm{meV}$$, $E_{X}=$2.394\text{\,}\mathrm{eV}$$,
$E_{0}=$2.294\text{\,}\mathrm{eV}$$, $\Gamma_{X}=$0.3\text{\,}\mathrm{meV}$$
and $\Gamma_{inh}=3$ meV

.

##  IV Theoretical model for polariton propagation

In this section, we present the theoretical model corresponding to the results
shown in Figs. 5(d,e,f) of the main text.

###  IV.1 Polariton density

In order to study the propagation of polaritonic excitations within the
metasurface plane, we developed an approximated method to calculate the steady
state spatial density distribution of polaritons with complex energy
dispersion $\tilde{E}_{pol}=E_{pol}+i\Gamma_{pol}/2$, defined as
$n(x,\tilde{E}_{pol})$. The input parameters of the model are given in the
previous Sec. III. We discuss only the case of $y-$polarised polaritons here,
while the case of $x-$polarised polaritons can be easily derived from the
former. In the case of $y-$polarised polaritons, the propagation can be well
approximated by a group velocity pointing in the $x$ direction,
$\mathbf{v}_{g}=v_{g}\hat{x}$, where $\hat{x}$ is the unit vector of the $x$
coordinate axis. This assumption is justified from the energy cut shown in
Fig. 3(a). We start by considering a generation rate $G(x,E_{pol})$ for
polaritons with energy $E_{pol}$ at position $x$, as obtained from a non-
resonant Gaussian excitation spot centered at $x=0$, i.e.,

$G(x,E_{pol})=G_{0}(E_{pol})e^{-x^{2}/2\sigma^{2}}\,,$ (S13)

in which the amplitude $G_{0}(E_{pol})$ represents the spectral distribution
of polariton injection, and $\sigma$ determines the spot size. For a given
polariton energy $E_{pol}$, a polaritonic mode has a well defined group
velocity $v_{g}$. On a first approximation, we can neglect any nonlinear
effect, i.e., we assume that each energy component of the polaritonic
dispersion is independent from the others. In steady state, the total net flux
of polaritons is null, and we get the condition:

$\frac{\partial n(x,\tilde{E}_{pol})}{\partial
x}v_{g}+\Gamma_{pol}n(x,\tilde{E}_{pol})=G(x,E_{pol})\,.$ (S14)

Using the generation rate obtained from Eq. (S13), we have the differential
equation:

$\frac{\partial n(x,\tilde{E}_{pol})}{\partial
x}v_{g}+\Gamma_{pol}n(x,\tilde{E}_{pol})=G_{0}(E_{pol})e^{-x^{2}/2\sigma^{2}}.$
(S15)

At $x=0$, the polariton density is maximized for any eigenmode of energy
$E_{pol}$, thus $\frac{\partial n(0,\tilde{E}_{pol})}{\partial x}=0$, which
gives the boundary condition:

$n(0,\tilde{E}_{pol})=\frac{G_{0}(E_{pol})}{\Gamma_{pol}}\,.$ (S16)

Moreover, for each polariton eigenmode with complex energy $\tilde{E}_{pol}$,
using Eq. (S16) as initial value at $x=0$, Eq. (S15) can be easily integrated
numerically, which gives the solution $n(x,\tilde{E}_{\text{pol}})$ of the
polariton density.

We finally notice that the same excitation spot is also responsible for the
injection of non-propagating uncoupled excitons with density:

$n_{X}(x)=n_{0}e^{-x^{2}/2\sigma^{2}}\,.$ (S17)

###  IV.2 The PL signal

The photoluminesce (PL) spectrum measured at energy $E$ and due to a polariton
eigenmode with complex energy $\tilde{E}_{pol}=E_{pol}+i\Gamma_{pol}/2$ can be
modeled by the following Lorentzian profile:

$S_{pol}(E,\tilde{E}_{pol})=\frac{(\Gamma_{pol}+\Gamma_{inh})}{\pi\left[(E-E_{pol})^{2}+(\Gamma_{pol}+\Gamma_{inh})^{2}/4\right]}\,.$
(S18)

The PL intensity of the polariton mode at energy $E_{pol}$, when spatially
selected at position $x\in[x_{1},x_{2}]$, is given by:

$I_{pol}(E,\tilde{E}_{pol})\propto\underbrace{S_{pol}(E,\tilde{E}_{pol})}_{\text{spectrum}}.\underbrace{(\Gamma_{pol}-W_{X}.\Gamma_{X})}_{\text{radiative
losses}}.\underbrace{\int_{x_{1}}^{x_{2}}n(x,\tilde{E}_{pol})dx}_{\text{polariton
population}}.$ (S19)

Similarly, the PL spectrum and the PL intensity of uncoupled excitons, when
spatially selected in position $x\in[x_{1},x_{2}]$, are respectively given by:

$\displaystyle S_{X}(E)$
$\displaystyle=\frac{\Gamma_{X}^{\text{(uncoupled)}}}{\pi\left[(E-E_{X})^{2}+\left(\Gamma_{X}^{\text{(uncoupled)}}\right)^{2}/4\right]}$
(S20) $\displaystyle I_{X}(E)$ $\displaystyle\propto
S_{X}(E)\int_{x_{1}}^{x_{2}}n_{X}(x)dx\,.$ (S21)

Here, the linewidth $\Gamma_{X}^{\text{(uncoupled)}}$ of uncoupled excitons is
the room temperature one, i.e., including thermal broadening effects, and thus
$\Gamma_{X}^{\text{(uncoupled)}}=30$ meV.

Taking into account the two lower polaritonic branches, $LP(\pm)$, and the
uncoupled excitons, the total PL intensity at energy $E$ and wavevector
$k_{x}$, when spatially selected at position $x\in[x_{1},x_{2}]$, is given by:

$I_{total}(E,k_{x})=\sum_{pol=LP(\pm)}I_{pol}\left(E,\tilde{E}_{pol}(k_{x})\right)+I_{X}(E)\,,$
(S22)

in which $E_{LP(\pm)}(k_{x})$ represent the dispersions of the lower polariton
modes, respectively, as given in Eq. (S3). Finally, a selection in real space
(i.e., spatial selection) induces a broadening in momentum space, due to the
reciprocal relation $\Delta_{x}\cdot\Delta_{k}\sim 1$. This broadening is
taken into account by a convolution of $I_{total}(E,k_{x})$ with a Gaussian
function whose standard deviation is $\Delta_{k}$.

###  IV.3 Numerical parameters

The dependence of the polariton energy on the wave-vector along the
propagation direction, $E_{pol}(k_{x})$, and its corresponding linewidth,
$\Gamma_{pol}(k_{x})$, is calculated as detailed in Sec. III, for both
polaritonic eigenmodes. The only supplemental parameters to calculate the
spatial distribution of polariton density [see Eq. (S14)] and the PL in
momentum space [see Eq. (S22)] are the spectral distribution $G_{0}(E_{pol})$
for the polariton injection, and the injection density $n_{0}$ for uncoupled
excitons. For the former, we apply a simple phenomenological law, i.e.,
assuming $G_{0}(E_{pol})\propto\exp\left(-|E_{pol}-E_{c}|/W\right)$, which
exhibits a polariton concentration at energy $E_{c}$. Numerically, we assume
$W=$30\text{\,}\mathrm{meV}$$, $E_{c}=$2.209\text{\,}\mathrm{eV}$$ for LP(+)
mode and $E_{bn}=$2.17\text{\,}\mathrm{eV}$$ for LP(-) mode. For the amplitude
of injection, we assume that the LP(+) mode is pumped 80 times more
efficiently than the LP(-) one. Using these parameters, the theoretical
calculation for the decay of 63$\%$ excitonic-fraction polaritons in real-
space, as well as the polariton density in momentum-space after spatial
selection, are presented in Fig. 4 in the main text. We highlight the
remarkably good agreement between these calculation and the experimental data
presented in Figs. 4 and 5 of the main text.

##  V Denoising of PL signal

Figure S4: $S_{1}$, $S_{2}$, and $S_{3}$ are three PL signals coming from
three different positions on the sample. These signals are then convoluted
using the kernel $K$ defined in Eq. (S23), and reported in the plot as
$S_{1}\ast K$, $S_{2}\ast K$ and $S_{3}\ast K$. These convoluted signals are
the ones also reported in Fig. 5(a-c) of the main text.

The signal-to-noise ratio of propagated polaritons in Fig. 5(a-c) is
particularly low. This is due to the signal collection being performed at
$|x|=$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ from the excitation spot,
while the propagation length fitted from Fig. 4 is estimated to be of the
order of $l\approx$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. To “denoise”
the experimental data of Fig. 5(a-c), we performed a convolution $S\ast K$ of
the signal $S$ using a $5\times 5$ Gaussian kernel $K$ defined as:

$K=\frac{1}{2^{8}}\begin{bmatrix}1&4&6&4&1\\\ 4&16&24&16&4\\\ 6&24&36&24&6\\\
4&16&24&16&4\\\ 1&4&6&4&1\end{bmatrix}\,.$ (S23)

The convolution kernel in Eq. (S23) is normalized such that the sum of all its
elements is equal to 1. Since the Fourier transform of a Gaussian is still a
Gaussian, the convolution between the actual signal and $K$ quenches the high-
frequency components. The low-pass filtering effect of such a convolution
effectively increases the signal-to-noise ratio. As an example, this is
explicitly demonstrated in Fig. S4, in which we show three the signals
obtained from PL measurements before (upper panels) and after (lower panels)
the convolution. The same convolution technique has been used to denoise the
experimental data shown in Fig. 3 in the main text.