\section{Introduction}
Phase transitions have been at the heart of statistical physics over the past 60 years, and a central issue for most areas of physics. Whereas a thorough understanding of critical behaviours has been acquired for equilibrium systems, the theoretical description of non-equilibrium phase transitions, in particular the ones involving non-equilibrium steady states, is still a major challenge and has been the subject of intense work in the last decades. Remarkably, self-organised criticality can emerge in non-equilibrium systems, leading to the onset of scale invariance without the need to tune any external parameter. This is realised in the celebrated Kardar-Parisi-Zhang (KPZ) equation \cite{KardarParisiZhang}. Whereas it was originally derived to describe kinetic roughening of interfaces undergoing stochastic growth \cite{HalpinHealyZhang}, the KPZ critical properties have been shown to arise in many non-equilibrium or disordered systems, ranging from turbulent liquid crystals \cite{TakeuchiSano1} to non-equilibrium hydrodynamics \cite{Spohn2016} to name a few.
 
{More recently, Bose-Einstein condensates of exciton-polaritons (EP) \cite{EPBEC}, a  quantum fluid with markedly different properties from equilibrium Bose-Einstein condensate of ultracold atoms, have proven to be a promising playground to observe KPZ universal properties. EP are bosonic quasi-particles arising from the strong coupling of photons to excitons (electron-hole bound states) realised in a  semiconductor microcavity. They are formed under intrinsically driven-dissipative conditions, since one has to introduce an optical pump to overcome the leakage of photons out of the cavity mirrors and maintain a steady state. Properties of EP  have been thoroughly investigated both experimentally and theoretically \cite{CarusottoCiuti}. Recently, a striking connection to KPZ universality has been brought out by several theoretical approaches \cite{Altman2015,Ji2015,GladilinJiWouters}. More precisely, the dynamics of the phase of the condensate wavefunction at long distances has been shown to obey the KPZ equation, and KPZ scaling has been reported in various conditions \cite{Zamora2017,He2017,Comaron2018}. In particular, the KPZ exponents were found in numerical simulations of the one-dimensional \cite{HeSiebererAltmanDiehl} and two-dimensional \cite{Mei2020} EP systems, as well as of photonic cavity arrays \cite{Amelio2019}.}

However, the KPZ universality class encompasses much more than mere scaling. In particular, the exact long time probability distribution of the fluctuations of the height has been determined for a number of systems with a one-dimensional growing interface \cite{SasamotoSpohn, Kriecherbauer, Corwin}. A remarkable feature is that, while these systems share the same critical exponents, their probability distribution depends on the initial conditions of the growth, thereby distinguishing three main geometry-dependent universality sub-classes. For an initially flat, respectively curved, interface, the probability distribution coincides with that of the largest eigenvalue of random matrices in the  Gaussian Orthogonal Ensemble (GOE) \cite{CalabreseLeDoussal, LeDoussal, BaikRains2, BaikRainsbook, Sasamoto, FerrariSpohn}, respectively Gaussian Unitary Ensemble (GUE) \cite{AmirCorwinQuastel, Calabrese, SasamotoSpohn2, BaikDeiftJohansson, Johansson}, unveiling a non-trivial connection with random matrix theory,  where these distributions, called Tracy-Widom (TW), originally emerged \cite{TracyWidom}. The third sub-class corresponds to Brownian, also called stationary, initial conditions, with fluctuations following a Baik-Rains (BR) distribution \cite{ImamuraSasamoto1, ImamuraSasamoto2}.
    
These sub-classes also differ at the level of two-point statistics. More specifically, it was shown that the spatial correlations of the height fluctuations of the one-dimensional growing interface are identical to those of stochastic processes called Airy$_1$ \cite{BorodinFerrariSasamoto, Sasamoto} and Airy$_2$ \cite{PrahoferSpohn_2pt1, ProlhacSpohn} for the flat and curved interface respectively. On the experimental side, the realisation of growing interfaces in turbulent liquid-crystal systems stands as the most advanced platform to study one-dimensional KPZ dynamics. In these experiments, both the one-point and two-point statistics have been measured for both the flat and curved geometries, and they confirm the theoretical results with impressive accuracy \cite{TakeuchiSano1, TakeuchiSano2, TakeuchiSanoSasamotoSpohn}. 
     
In this work, we show that EP condensates appear in many respects as a very versatile set-up to futher investigate KPZ dynamics. While some of the advanced KPZ features were already observed in numerical simulations of EP condensates \cite{Davide}, in particular the TW-GOE distribution for the flat geometry, as well as the BR distribution for the stationary (Brownian) case, in the present paper we demonstrate that the curved KPZ sub-class can also be realised by tailoring a confinement potential that effectively bends the phase profile. Using numerical simulations we find that in the presence of this confinement the phase fluctuations follow the expected TW-GUE distribution. Moreover, we provide the first study of the two-point statistics of the phase of the EP. We first determine the scaling function, which displays similar features for all sub-classes. We then compute the two-point spatial correlations of the fluctuations of the phase in both geometries, and show that they reproduce with great accuracy the expected theoretical ones related to the Airy$_1$ and Airy$_2$ processes, although only locally for the curved case since the condensate phase is bent only over a limited space region. Our study hence shows that all the geometrical KPZ sub-classes can be accessed in EP condensates.

\section{Model for the dynamics of the EP condensate}
Our starting point is the mean-field generalised Gross-Pitaevskii equation for the EP Bose-Einstein condensate under incoherent pumping formulated in \cite{WoutersCarusotto},
\begin{align}
i \hbar\partial_t \psi = \left[ \mathcal{F}^{-1}[E_{LP}(k)]+\frac{i\hbar}{2}\left( Rn_r - \mathcal{F}^{-1}[\gamma_l(k)] \right) \nonumber \right.\\ \left.
+ \hbar g_{\textrm{int}}\left\lvert \psi \right\rvert^2 \right] \psi \,,
\label{eq: model}
\end{align}
where $\psi$ is the condensate wavefunction, $\mathcal{F}^{-1}[E_{LP}(k)]$ denotes the inverse Fourier transform of the dispersion relation of the lower-polariton branch in momentum space, $Rn_r$ is the amplification term, $\gamma_l(k)$ is the loss rate of polaritons, $g_{\textrm{int}}$ is the polariton-polariton interaction strength and $n_r$ is the reservoir density, whose evolution obeys the phenomenological rate equation $\partial_t n_r=P-\gamma_r n_r -R n_r |\psi|^2$,  where $P$ is the pumping strength and $\gamma_r$ the reservoir loss rate. Under the assumption that the time scales of the reservoir and of the condensate are well separated \footnote{Note that one can derive the mapping to the KPZ equation directly from the two-equation system without assuming the separation of time scales of the condensate and reservoir dynamics \cite{DavideKonstantinosLeonieAnna}.}, one may integrate out the reservoir dynamics, which, for sufficiently small field amplitudes, leads to a stochastic equation analogous to a complex Ginzburg-Landau equation,
\begin{align}
i \hbar \partial_t \psi = & \left[\mathcal{F}^{-1}[E_{LP}(k)]  +V(x) + \hbar g_{\text{int}}\left\lvert \psi\right\rvert^2 \right] \psi \nonumber \\
& + \frac{i\hbar}{2}\left[\frac{PR}{\gamma_r}-\frac{PR^2}{\gamma_r^2} \left\lvert \psi \right\rvert^2 - \mathcal{F}^{-1}[{\gamma_l}(k)] \right] \psi + \hbar\xi \,,
\label{eq: gGPEplus}
\end{align}
where the noise $\xi$, which arises from both dissipation and pumping stochastic processes, is complex and has zero mean and covariance $\left\langle \xi(x,t) \xi^*(x', t') \right\rangle = 2\sigma\delta(x-x') \delta(t-t')$ with $\sigma = \gamma_{l,0}(p+1)/2$ \cite{SiebererBuchholdDiehl2016}. This equation further accounts for two effects which were shown to be important to relate the KPZ regime to actual experimental systems: the quartic correction to the dispersion relation $E_{LP}(k)=\hbar \omega_{0,LP}+\frac{\hbar^2}{2m}k^2 - \frac{1}{2 \hbar \Omega} \left(\frac{\hbar^2}{2m}\right)^2 k^4$  with $\Omega$ the Rabi frequency and a momentum-dependent loss rate of polaritons ${\gamma_l}(k)=\gamma_{l,0} + k^2 \gamma_{l,2}$. We have also introduced a confinement potential $V(x)$, which is the cornerstone of this work.

\section{KPZ mapping with the confinement}
To establish the mapping to the KPZ equation, one usually decomposes the wavefunction $\psi$ in the density-phase representation $\psi=\sqrt{\rho} e^{i \theta}$ and expands Eq.~(\ref{eq: gGPEplus}) for small variations around the mean-field solution and in powers of gradients. Following a similar strategy, one finds that in the presence of the static potential $V(x)$, the dynamics of the phase of the EP condensate at long distances obeys an inhomogeneous KPZ equation (see  Sec. \ref{S2} for the derivation),
\begin{equation}
\partial_t \theta = \nu(x) \partial_x^2 \theta + \dfrac{\lambda(x)}{2} \left( \partial_x \theta \right)^2 + \sqrt{D(x)}\eta \,,
\label{eq: KPZ}
\end{equation}
where $\eta$ is a white noise with zero mean and covariance $\left\langle \eta(x,t) \eta(x', t') \right\rangle = 2\delta(x-x') \delta(t-t')$, and 
\begin{align}
& \nu(x)= \frac{\gamma_{l,2}}{2} + \frac{\hbar \rho_0(x)}{m}\tilde{u}(x) \,,\nonumber\\
& \lambda(x)=-\frac{\hbar}{m}+2\gamma_{l,2}\rho_0(x)\tilde{u}(x)\,,\nonumber\\ 
& D(x)=\frac{\sigma}{2\rho_0(x)}(1+4\rho_0^2(x)\tilde{u}^2(x))\,,\nonumber\\
& \tilde{u}(x)=\dfrac{\frac{\hbar}{2m} \left[ \frac{(\partial_x \rho_0)^2}{2\rho_0^3} - \frac{\partial_x^2 \rho_0}{2\rho_0^2} \right]- \frac{\gamma_{l,2}}{2} \frac{(\partial_x \rho_0)( \partial_x \theta_0)}{\rho_0^2} - g_{\text{int}}}{\frac{\gamma_{l,2}}{2}\left[ \frac{(\partial_x \rho_0)^2}{2\rho_0^2} - 2(\partial_x \theta_0)^2 \right] - \frac{\hbar}{m}\partial_x^2 \theta_0 + \tilde{p}(x)}\,, \label{eq:param}
\end{align}
where $\theta_0(x,t)$ and $\rho_0(x)$ are the mean-field inhomogeneous solutions and $\tilde{p}(x)=\left(p-1-\frac{2pR}{\gamma_r}\rho_0(x)\right)\gamma_{l,0}$ with dimensionless pumping parameter $p=PR/(\gamma_{l,0} \gamma_r)=P/P_{\textrm{th}}$, $P_{\textrm{th}}$ being the threshold pumping strength for condensation.

When $V(x)=0$, one recovers the standard mapping to the homogeneous KPZ equation where the parameters $\nu$, $\lambda$ and $D$ are constant \cite{Davide}. With a non-vanishing potential, these parameters continuously vary with space. One can get an intuition of how this may affect the dynamics by noting that the average velocity of the phase is proportional to the KPZ non-linearity $\lambda$. Qualitatively, when $V\neq 0$, the non-linearity $\lambda(x)$, and thus the velocity, increases where the potential is larger. Therefore, if one implements a suitable enhancement of the potential at the boundaries, one may induce an effective drag at the boundaries prone to bend the phase profile. We found that this can indeed be realised.

In our work, we consider a harmonic potential 
\begin{equation}
V(x) = \frac{1}{2}m \omega_0^2 x^2,
\label{eq:pot}
\end{equation}
where the frequency trap $\omega_0$ can be adjusted. We found that the most favorable trap to study KPZ properties is a shallow parabola (we choose $\omega_0 \simeq 4\times10^{-4} \gamma_{l,0}$ in the following), since it allows one to keep the density fluctuations tame and the KPZ parameters slowly varying in the vicinity of $x=0$. Furthermore, we checked that the  curved KPZ universality sub-class is robust and can be observed for different confinement potentials, in particular Gaussian walls (see Sec. \ref{S4}).

\section{Numerical simulations}
We solve the generalised Gross-Pitaevskii equation (\ref{eq: gGPEplus}) for a  system size of $L/\hat{x}=2^{10}$, where we have chosen $\hat{x}=\sqrt{\hbar/(2m\gamma_{l,0})}\simeq \,2\mu$m, $\hat{t} = \gamma_{\ell,0}^{-1}$ and $\hat{\epsilon}=\hbar \gamma_{l,0}$ as units for length, time and energy. We take for the parameters of Eq.~(\ref{eq: gGPEplus}) the typical values for the CdTe experiments conducted in Grenoble \cite{RojanEtAl2017, Maxime}:  $m=4\times10^{-5}$m$_{\textit{e}}$, $\gamma_{l,0}=0.5$ps$^{-1}$, $\gamma_r = \gamma_{l,0}/25$, $g_{\textrm{int}}=7.59 \times 10^{2}$m$\times$s$^{-1}$, $R=4 \times 10^2$m$\times$s$^{-1}$, $P=4\times10^{19}$(m$\times$s)$^{-1}$, $\gamma_{l,2}=1.29$m$^2 \times$s$^{-1}$, $\Omega=100$ THz.
We record the wavefunction $\psi(x,t)$ during the time evolution and extract its phase $\theta(x,t)$ at suitable time intervals for half of the spatial grid, thanks to the symmetry of the potential. 

We work in the low-noise regime, which ensures that the density fluctuations are negligible and also that there are no topological defects, such as solitons or phase slips. This allows us to unwind the phase, $\theta(x,t) \in (-\pi, \pi] \rightarrow \theta(x,t) \in (-\infty, \infty)$. This is necessary since the KPZ universality class describes fluctuations continuously growing in space and time, which cannot be realised for a compact field. In order to achieve the unwinding in the numerical simulations, we constrain the phase such that the difference between neighouring space-time points is less than $0.8 \times 2\pi$, where the factor $0.8$ is chosen empirically in order to take into account unwinding errors due to space and time discretization. We checked that the unwinding protocol is robust, {\it i.e.} the results do not depend on the specific value of this factor as long as it is close to 1.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.31]{fig1.pdf}
\caption{Typical spatial phase profiles at different times during the evolution, with lighter colours corresponding to larger times, (i) in the absence of confinement potential, leading to a flat profile, and (ii) in the presence of the parabolic confinement potential, leading to a bent profile.
}
\label{fig: fig1}
\end{center}
\end{figure}

In Fig.~\ref{fig: fig1} we display typical phase profiles obtained in the homogeneous case with no external potential $V=0$, which leads to a flat profile, and in the inhomogeneous case with the parabolic  confinement potential (\ref{eq:pot}), which leads to a curved profile. One can observe that in this case the phase profile indeed propagates faster near the boundaries where the EPs feel the largest potential as anticipated. Around the central tip at $x=0$, the phase presents a local curvature, as we evidence in the following.

\section{Results for the scaling}
The KPZ scaling properties can be studied directly from the first-order correlation function of the EP condensate wavefunction,
\begin{equation}
g_1(\Delta x,\Delta t) =  \dfrac{\left \lvert \left\langle \psi^*(x+\Delta x, t+\Delta t) \psi(x,t) \right \rangle \right \rvert}{\left\langle \sqrt{\rho(x+\Delta x, t+\Delta t) \rho(x,t)} \right\rangle},
\end{equation}
Throughout this work, $\left\langle ... \right\rangle$ denotes the average over noise realisations. Note that the first-order coherence $g_1$ is routinely measured in EP experiments, which renders the following analysis easily accessible. By performing a cumulant expansion and neglecting density-phase correlations, one can relate $g_1$ to the connected correlation function of the phase $C$, obtaining to first order
\begin{align}
-2 \ln \left[ g_1(\Delta x,\Delta t)  \right] & = \left\langle \left[ \theta(x+\Delta x, t+\Delta t) - \theta(x,t) \right]^2 \right\rangle\nonumber\\
 &- \Big\langle \theta(x+\Delta x, t+\Delta t) - \theta(x,t) \Big \rangle^2\nonumber\\
 &\equiv C(\Delta x, \Delta t) \,.
\label{eq: g1}
\end{align}
If the phase follows the 1D KPZ dynamics, it should endow the Family-Vicsek scaling form \cite{FamilyVicsek}
\begin{equation}
 C(\Delta x, \Delta t) = C_0 {\Delta t}^{2/3} g\left(y_0 \frac{\Delta x}{\Delta t^{2/3}}\right)\,,
\label{eq:defC}
\end{equation}
where $g(y)$ is a universal scaling function and $C_0$, $y_0$ are normalisation constants defined as
\begin{equation}
 y_0 = (2A \lambda^2)^{-1/3},\, C_0 =  {\Gamma}^{2/3}, \; A = \frac D \nu, \, \Gamma =\frac{\lambda}{2} A^2\,,
\label{eq:norm}
\end{equation}
where the numerical prefactors are conventional. The precise form of the scaling function $g(y)$ is known exactly only for the stationary interface \cite{Prahofer}. However, the scaling function satisfies the same asymptotics in all sub-classes
\begin{equation}
g(y) \stackrel{y\to 0}{\longrightarrow}  g_0 \; , \quad  g(y) \stackrel{y\to \infty}{\sim} 2 y\,,
\label{eq: gy}
\end{equation}
where $g_0$ is a universal constant depending on the geometrical sub-class, whose values are known exactly \cite{Spohn}.
Therefore, one expects a  similar behaviour for the scaling functions in the three sub-classes, apart from small vertical shifts reflecting the differences in $g_0$ and thus small changes in the intermediate crossover region between the two asymptotic limits. In our simulations, we determined $C(\Delta x,\Delta t)$ from the wavefunction correlation function $g_1$ using Eq.~(\ref{eq: g1}). We first estimated the KPZ scaling exponents with and without the  confinement potential by studying the equal-time and equal-space correlation functions, which, according to Eqs.~(\ref{eq:defC}) and (\ref{eq: gy}), should behave as
\begin{subequations}
\begin{align}
C(\Delta x, \Delta t=0) &\sim  \Delta x^{2\chi}, 
\label{eq: scalinglaw1} \\
C(\Delta x=0, \Delta t) &\sim  \Delta t^{2\beta} \,,
\label{eq: scalinglaw2}
\end{align}
\end{subequations} 
with $\chi=1/2$ and $\beta=1/3$ the 1D KPZ roughness and growth critical exponents. We found $\chi=0.49 \pm 0.01$ and $\beta =0.30\pm 0.01$ in both the flat and the curved cases for the purely spatial and purely temporal correlations, see Fig.~\ref{fig: fig6}. The value of the growth exponent $\beta$ slightly differs from the theoretical one but it is comparable with values reported in previous studies of EP condensate for this system size \cite{Davide} for the flat geometry. 

In order to construct the universal scaling function $g(y)$ defined in Eq.~(\ref{eq:defC}), we first selected all the data points lying in the correct scaling regime by filtering out the points differing by more than small cutoffs $\epsilon_x, \epsilon_t$ from the expected scaling laws in Eqs.~(\ref{eq: scalinglaw1}, \ref{eq: scalinglaw2}) for each value of spatial and temporal separation. We extracted the normalisation parameters $A$ and $\Gamma$ in (\ref{eq:norm}) from our numerical data. Note that in the curved case, these parameters are not homogeneous. However, in the vicinity of the central tip, where the confinement potential is nearly vanishing, they are effectively almost constant and coincide with the values for the homogeneous case (see Sec. \ref{S3} for details and the corresponding data).

The scaling function is obtained by plotting $C(\Delta x,\Delta t)/(C_0 {\Delta t}^{2/3})$ as a function of $y_0 \Delta x/\Delta t^{2/3}$. The results are displayed in Fig.~\ref{fig: gy} together with the theoretical curve $g(y)_{\rm stat}$ for the stationary case \cite{Prahofer}. For both the flat and the curved cases, we observe a reasonable collapse of all the data points onto a single function $g$, which demonstrates that $C$ indeed takes a scaling form. Additionally, we confirm that the scaling functions $g$ are quite similar for the three cases. However, it is not a perfectly one-dimensional curve, it has a finite (small) thickness, and the numerical values for $g_0=g(0)$ differ of about $\sim 40\%$  from the theoretical exact values in both the flat and curved cases  ($g^{\rm num}_{0,{\rm flat}}\simeq  0.95$ vs $g^{\rm th}_{0,{\rm flat}}=  0.63805...$, $g^{\rm num}_{0,{\rm curved}}\simeq  1.23$ vs $g^{\rm th}_{0,{\rm curved}}=  0.8132...$). These discrepancies  may originate from the fact that the actual growth exponent is slightly smaller than the theoretical one, and also from the fact that the $g_1$ function includes other contributions beside the phase correlations, even if they are assumed to be small (higher-order cumulants of the phase or density-phase correlations). However, let us emphasise that the values for $g_0$ are clearly distinct in the two cases, and their ratio (or relative difference) turns out to be within 3\% accuracy with the theoretical ratio (or relative difference). This already indicates that the mapping from the Gross-Pitaevskii to the KPZ equation is well-grounded, and that both the flat and the curved universality sub-classes can be probed in EP systems.
\begin{figure}
\begin{center}
\includegraphics[scale=0.31]{fig2.pdf}
\caption{Universal scaling function $g(y)$ for the flat (blue dots) and curved (red triangles) phase profiles. The theoretical result for the stationary interface $g_{\textrm{stat}}(y)$ is shown for comparison (solid line). The theoretical values $g^{\rm th}_{0,{\rm flat}}$, $g^{\rm th}_{0,{\rm curved}}$, $g^{\rm th}_{0,{\rm stat}}$ are indicated in the inset, together with the two numerical curves for $y_0 \Delta x/ \Delta t^{2/3} \rightarrow 0$.}
\label{fig: gy}
\end{center}
\end{figure}

\section{Results for the phase fluctuations}
\subsection{One-point statistics -- Tracy-Widom distributions}
The precise geometry of the phase profile affects the distribution of the fluctuations of the phase. More precisely, as the phase profile propagates linearly in time with fluctuations growing as $t^{1/3}$, one introduces the rescaled fluctuation field $\chi$, defined from the long time behaviour of the phase in 1D as 
\begin{equation}
\theta(x,t) \stackrel{t\to \infty}{\sim} \omega_{\infty}t + (\Gamma t)^{1/3} \chi(\zeta, t)\,,
\label{eq: ansatz}
\end{equation}
where $\omega_{\infty}$ is the asymptotic velocity of the phase, which has a non-trivial dependence on the KPZ parameters \cite{DavideKonstantinosLeonieAnna}, and $\zeta$ is the spatial coordinate rescaled by the correlation length of fluctuations $\zeta \equiv x/\xi(t)$ with $\xi(t)=(\Gamma t)^{2/3}\frac{2}{A}$ \cite{FukaiTakeuchi}. We focus on the universal statistical properties of the centered unwound phase $\Delta \theta(x_0,t) = \theta(x_0,t) - \left\langle\theta(x_0,t) \right\rangle$. 
This allows one to subtract the drift term in Eq.~(\ref{eq: ansatz}), and we henceforth omit the arguments, since there is translational invariance in time all through the KPZ regime (which occurs for stationary condensates), and it is sufficient to consider only the central point $x_0 = 0$. Let us emphasise that Eq.~(\ref{eq: ansatz}) merely stands as an ansatz for the long-time behaviour of the phase. We estimated the numerical value of $\Gamma$ from the time dependence of $\Delta\theta^2$ (see Sec. \ref{S5} for details).

We compute the rescaled fluctuation field $\chi$ from $\Delta\theta$ following  Eq.~(\ref{eq: ansatz}). Note that for the flat case, we conform to the standard definition of the TW-GOE random variable found in the literature, and further rescale $\chi$ as $\chi\to 2^{-2/3} \chi$. We construct the histograms of $\chi$ both for the flat case without confinement and for the curved case with the parabolic confinement $V$. The resulting distributions are displayed in Fig.~\ref{fig: fig3}, where they are compared with the theoretical distributions (more precisely to the mirror ones $P_{\text{TW-GOE}}(-\chi)$ and $P_{\text{TW-GUE}}(-\chi)$ since the KPZ non-linearity $\lambda$ is negative for our choice of experimental parameters).
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.31]{fig3.pdf}
\caption{Centered distribution of the rescaled phase fluctuations $\chi$ sampled at $x=0$, for flat (blue dots) and curved (red triangles) phase profiles, together with the theoretical TW-GOE and TW-GUE distributions.}
\label{fig: fig3}
\end{center}
\end{figure}
One observes a clear distinction between the two cases, and moreover the distribution for the curved case is in excellent agreement with the TW-GUE distribution, thus demonstrating that one can indeed tune the KPZ geometrical sub-class realised in the EP system. We emphasise that this result is very robust with respect to the choice of confinement potential, and we found that the agreement remains excellent for a Gaussian walls potential (see Sec. \ref{S4}).

\subsection{Two-point statistics -- correlations of Airy processes}
Besides the probability distribution, the geometry also influences the two-point statistics of the rescaled fluctuations, differing in the three universality sub-classes. In particular, it was shown that the connected correlation function of rescaled fluctuations of the height of the interface at equal time, defined as
\begin{equation} 
C_\chi(\Delta \zeta) \equiv \left\langle \chi(\zeta+\Delta \zeta,t) \chi(\zeta,t) \right\rangle - \left\langle \chi(\zeta+\Delta \zeta,t\right\rangle \left\langle \chi(\zeta,t) \right\rangle\,,
\label{eq:Cchi}
\end{equation}
with $\Delta\zeta=\Delta x / \xi(t)$, is given by the time correlation function of the Airy$_1$, respectively Airy$_2$ process in the asymptotic limit, in the flat, respectively curved geometry 
\begin{equation}
C_\chi(\Delta \zeta) = {\cal G}_i(\Delta\zeta) = \left \langle \mathcal{A}_i(t^{\prime}+\Delta \zeta) \mathcal{A}_i(t^{\prime}) \right\rangle\,,
\end{equation}
where $i=1,2$ stands for Airy$_1$, respectively Airy$_2$, processes. Let us mention one subtle issue here. While the Airy$_2$ process was found to coincide with the dynamics of the largest eigenvalue of GUE matrices \cite{Johansson2}, the Airy$_1$ process differs from the largest-eigenvalue dynamics of GOE matrices \cite{Bornemann2}. This indicates that while one-point statistics of the fluctuations of the growing interface are connected to random matrix theory, this connection is flawed at the two-point level in the flat case. 

In our simulations, we computed the correlation function Eq.~(\ref{eq:Cchi}) of the rescaled phase fluctuations  as
\begin{equation}
C_\chi(\Delta \zeta) = \dfrac{\left\langle \Delta \theta(x,t) \Delta \theta(x+\Delta x, t) \right\rangle}{\left( \Gamma t \right)^{2/3}}\,.
\label{eq: Cchiplot}
\end{equation}
The results we obtained for the two geometries are presented in Fig.~\ref{fig: fig4}. We note that in line with the rescaling of $\chi$ in the flat case mentioned previously, we also perform in this case the following rescaling $t \rightarrow t/2^{-2/3}$ and ${\cal G}_1 \rightarrow {\cal G}_1 / 2^{-2/3}$ to conform to the standard definition of the Airy$_1$ process \cite{TakeuchiSano1} and compare with the theoretical results from \cite{Bornemann1}. Furthermore, in the curved geometry, the limit stochastic process is shown to be $\chi_{\textrm{curved}}(\zeta,t) \stackrel{d}{\rightarrow} \mathcal{A}_2 - \zeta^2$, where $-\zeta^2$ reflects the influence of the mean profile \cite{PrahoferSpohn_2pt1}, which is automatically subtracted in our case since we consider the connected function.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.31]{fig4.pdf}
\caption{Correlation function $C_\chi$ of the rescaled phase fluctuations as a function of the rescaled length $\Delta \zeta$ for different times for the flat case (blue dots) and for the curved case (green and orange triangles), together with the theoretical results for the correlation of the Airy$_1$ ${\cal G}_1(\Delta \zeta)$ (dashed line) and Airy$_2$ ${\cal G}_2(\Delta \zeta)$ (solid line) processes presented in \cite{Bornemann1, Bornemann2, Bornemann3}. For the flat phase profile, data corresponding to  times $t/\hat{t}=7.5\times10^3, 8.125\times10^3, 1\times10^4$ is shown, and for the curved phase profile, data corresponding to $t/\hat{t}=4\times10^2, 1.6 \times 10^3$ is shown (lighter to darker colors correspond to increasing times). For the curved case, the red shades correspond to the small $\Delta \zeta$ regime where the phase profile is locally curved, while green correspond to large $\Delta \zeta$, where the profile is shaped by the effective drag -- see inset.}
\label{fig: fig4}
\end{center}
\end{figure}

For the flat case, we observe that the correlation function is stable in time, from $t/\hat{t}=7\times10^3$ to, approximately, $t/\hat{t}=1\times10^4$ and we find a good agreement with the theoretical Airy$_1$ correlations ${\cal G}_1(\Delta \zeta)$, even for large $\zeta$. The small shift visible in the figure can be traced back to the fact that the parameter $\Gamma$ in Eq.~(\ref{eq: Cchiplot}) is extracted from the numerical simulations and is found to be a bit larger than the theoretical value (see Sec. \ref{S5}).

For the curved case, the correlation function is stable over a time window from $t/\hat{t}=4 \times 10^2$ until $t/\hat{t} = 1.6 \times 10^3$ where the KPZ scaling regime is observed. We focus on the small$-\Delta \zeta$ limit, corresponding to small spatial separation around $x=0$. Indeed, we expect a local curvature only around the central tip, since the profile near the boundaries is affected by the effective drag ensuing from the confinement potential. The phase correlations are found to behave as an Airy$_2$ process only over this limited range of space, but the agreement with the theoretical Airy$_2$ correlations ${\cal G}_2(\Delta \zeta)$ on this range is extremely satisfactory. 

We emphasise that we have provided in this section the first analysis for both the flat and the curved phase profiles of a very fine statistical quantity: We have here probed reduced correlations, which are  the sub-leading behaviour emerging once the dominant scaling one (studied in the scaling part) has been subtracted. In this respect, the agreement found with the theoretical results for the Airy processes is remarkable. All the results presented here very deeply root in the relevance of KPZ dynamics for the EP system.

\section{Summary and Outlook}
In this work, we have shown that by engineering the confinement potential of 1D exciton polaritons one can tune the geometry of the phase of the condensate and thus access both the flat and curved KPZ universality sub-classes. In particular, we have found excellent agreement with the theoretical exact results, not only for the scaling properties, but also  at the level of one-point statistics (probability distributions), as well as two-point statistics of rescaled fluctuations, though locally for the curved case since the condensate phase is only curved on a limited range by the confinement potential. Our results show the remarkable emergence of KPZ universal properties from the microscopic Gross-Pitaevskii equation for exciton polaritons  at all levels explored so far: not only in the scaling function, but also in the probability distributions and even in the sub-leading behaviour for the two-point correlations, and all this for both the universality sub-classes. We believe that our findings pave the way for stimulating new protocols for investigating KPZ universality in experiments, since the KPZ sub-classes can be accessed through simple engineering of the EP system. In particular, harmonic confinement can be implemented by a suitable engineering of the pumping mechanism \cite{TosiEtAl2012}.

Whereas probing the scaling properties is readily accessible from the measurement of the first-order correlation function, the experimental determination of the probability distributions may be more challenging as it involves the measurement of the time and space resolved phase of the condensate. This requires the development of specific interferometric techniques capable of resolving very small times. On the other hand, higher-order correlations have been measured e.g in \cite{last1,last2} for the case of ultracold atoms. Similar techniques could be implemented in the EP system and lead to the possibility of accessing universal ratios of cumulants, thus enabling the demonstration of the typical non-gaussian shape of the TW-GOE and TW-GUE probability distributions and the characterization of universality sub-classes. Last but not least, the investigation of KPZ universality in 2D is an exciting perspective, both from a theoretical viewpoint, where few indications of KPZ scaling in EP systems have been reported \cite{Zamora2017,Mei2020}, and from an experimental viewpoint, where a high-precision platform for exploring KPZ in 2D is still missing.

\acknowledgments
We acknowledge stimulating discussions with Alberto Amo, Jacqueline Bloch, and  Maxime Richard.  We wish to thank Prof. F. Bornemann for kindly providing us with the theoretical data for the correlation of the Airy processes used in Fig.~\ref{fig: fig4}. K.D. acknowledges the European Union Horizon 2020 research and innovation programme under the Marie Sk\l{}odowska-Curie grant agreement No 754303. L.C. acknowledges support from the French ANR through the project NeqFluids (grant ANR-18-CE92-0019) and support from Institut Universitaire de France (IUF).

\label{Bibliography}

\section{Introduction}
Phase transitions have been at the heart of statistical physics over the past 60 years, and a central issue for most areas of physics. Whereas a thorough understanding of critical behaviours has been acquired for equilibrium systems, the theoretical description of non-equilibrium phase transitions, in particular the ones involving non-equilibrium steady states, is still a major challenge and has been the subject of intense work in the last decades. Remarkably, self-organised criticality can emerge in non-equilibrium systems, leading to the onset of scale invariance without the need to tune any external parameter. This is realised in the celebrated Kardar-Parisi-Zhang (KPZ) equation \cite{KardarParisiZhang}. Whereas it was originally derived to describe kinetic roughening of interfaces undergoing stochastic growth \cite{HalpinHealyZhang}, the KPZ critical properties have been shown to arise in many non-equilibrium or disordered systems, ranging from turbulent liquid crystals \cite{TakeuchiSano1} to non-equilibrium hydrodynamics \cite{Spohn2016} to name a few.
 
{More recently, Bose-Einstein condensates of exciton-polaritons (EP) \cite{EPBEC}, a  quantum fluid with markedly different properties from equilibrium Bose-Einstein condensate of ultracold atoms, have proven to be a promising playground to observe KPZ universal properties. EP are bosonic quasi-particles arising from the strong coupling of photons to excitons (electron-hole bound states) realised in a  semiconductor microcavity. They are formed under intrinsically driven-dissipative conditions, since one has to introduce an optical pump to overcome the leakage of photons out of the cavity mirrors and maintain a steady state. Properties of EP  have been thoroughly investigated both experimentally and theoretically \cite{CarusottoCiuti}. Recently, a striking connection to KPZ universality has been brought out by several theoretical approaches \cite{Altman2015,Ji2015,GladilinJiWouters}. More precisely, the dynamics of the phase of the condensate wavefunction at long distances has been shown to obey the KPZ equation, and KPZ scaling has been reported in various conditions \cite{Zamora2017,He2017,Comaron2018}. In particular, the KPZ exponents were found in numerical simulations of the one-dimensional \cite{HeSiebererAltmanDiehl} and two-dimensional \cite{Mei2020} EP systems, as well as of photonic cavity arrays \cite{Amelio2019}.}

However, the KPZ universality class encompasses much more than mere scaling. In particular, the exact long time probability distribution of the fluctuations of the height has been determined for a number of systems with a one-dimensional growing interface \cite{SasamotoSpohn, Kriecherbauer, Corwin}. A remarkable feature is that, while these systems share the same critical exponents, their probability distribution depends on the initial conditions of the growth, thereby distinguishing three main geometry-dependent universality sub-classes. For an initially flat, respectively curved, interface, the probability distribution coincides with that of the largest eigenvalue of random matrices in the  Gaussian Orthogonal Ensemble (GOE) \cite{CalabreseLeDoussal, LeDoussal, BaikRains2, BaikRainsbook, Sasamoto, FerrariSpohn}, respectively Gaussian Unitary Ensemble (GUE) \cite{AmirCorwinQuastel, Calabrese, SasamotoSpohn2, BaikDeiftJohansson, Johansson}, unveiling a non-trivial connection with random matrix theory,  where these distributions, called Tracy-Widom (TW), originally emerged \cite{TracyWidom}. The third sub-class corresponds to Brownian, also called stationary, initial conditions, with fluctuations following a Baik-Rains (BR) distribution \cite{ImamuraSasamoto1, ImamuraSasamoto2}.
    
These sub-classes also differ at the level of two-point statistics. More specifically, it was shown that the spatial correlations of the height fluctuations of the one-dimensional growing interface are identical to those of stochastic processes called Airy$_1$ \cite{BorodinFerrariSasamoto, Sasamoto} and Airy$_2$ \cite{PrahoferSpohn_2pt1, ProlhacSpohn} for the flat and curved interface respectively. On the experimental side, the realisation of growing interfaces in turbulent liquid-crystal systems stands as the most advanced platform to study one-dimensional KPZ dynamics. In these experiments, both the one-point and two-point statistics have been measured for both the flat and curved geometries, and they confirm the theoretical results with impressive accuracy \cite{TakeuchiSano1, TakeuchiSano2, TakeuchiSanoSasamotoSpohn}. 
     
In this work, we show that EP condensates appear in many respects as a very versatile set-up to futher investigate KPZ dynamics. While some of the advanced KPZ features were already observed in numerical simulations of EP condensates \cite{Davide}, in particular the TW-GOE distribution for the flat geometry, as well as the BR distribution for the stationary (Brownian) case, in the present paper we demonstrate that the curved KPZ sub-class can also be realised by tailoring a confinement potential that effectively bends the phase profile. Using numerical simulations we find that in the presence of this confinement the phase fluctuations follow the expected TW-GUE distribution. Moreover, we provide the first study of the two-point statistics of the phase of the EP. We first determine the scaling function, which displays similar features for all sub-classes. We then compute the two-point spatial correlations of the fluctuations of the phase in both geometries, and show that they reproduce with great accuracy the expected theoretical ones related to the Airy$_1$ and Airy$_2$ processes, although only locally for the curved case since the condensate phase is bent only over a limited space region. Our study hence shows that all the geometrical KPZ sub-classes can be accessed in EP condensates.

\section{Model for the dynamics of the EP condensate}
Our starting point is the mean-field generalised Gross-Pitaevskii equation for the EP Bose-Einstein condensate under incoherent pumping formulated in \cite{WoutersCarusotto},
\begin{align}
i \hbar\partial_t \psi = \left[ \mathcal{F}^{-1}[E_{LP}(k)]+\frac{i\hbar}{2}\left( Rn_r - \mathcal{F}^{-1}[\gamma_l(k)] \right) \nonumber \right.\\ \left.
+ \hbar g_{\textrm{int}}\left\lvert \psi \right\rvert^2 \right] \psi \,,
\label{eq: model}
\end{align}
where $\psi$ is the condensate wavefunction, $\mathcal{F}^{-1}[E_{LP}(k)]$ denotes the inverse Fourier transform of the dispersion relation of the lower-polariton branch in momentum space, $Rn_r$ is the amplification term, $\gamma_l(k)$ is the loss rate of polaritons, $g_{\textrm{int}}$ is the polariton-polariton interaction strength and $n_r$ is the reservoir density, whose evolution obeys the phenomenological rate equation $\partial_t n_r=P-\gamma_r n_r -R n_r |\psi|^2$,  where $P$ is the pumping strength and $\gamma_r$ the reservoir loss rate. Under the assumption that the time scales of the reservoir and of the condensate are well separated \footnote{Note that one can derive the mapping to the KPZ equation directly from the two-equation system without assuming the separation of time scales of the condensate and reservoir dynamics \cite{DavideKonstantinosLeonieAnna}.}, one may integrate out the reservoir dynamics, which, for sufficiently small field amplitudes, leads to a stochastic equation analogous to a complex Ginzburg-Landau equation,
\begin{align}
i \hbar \partial_t \psi = & \left[\mathcal{F}^{-1}[E_{LP}(k)]  +V(x) + \hbar g_{\text{int}}\left\lvert \psi\right\rvert^2 \right] \psi \nonumber \\
& + \frac{i\hbar}{2}\left[\frac{PR}{\gamma_r}-\frac{PR^2}{\gamma_r^2} \left\lvert \psi \right\rvert^2 - \mathcal{F}^{-1}[{\gamma_l}(k)] \right] \psi + \hbar\xi \,,
\label{eq: gGPEplus}
\end{align}
where the noise $\xi$, which arises from both dissipation and pumping stochastic processes, is complex and has zero mean and covariance $\left\langle \xi(x,t) \xi^*(x', t') \right\rangle = 2\sigma\delta(x-x') \delta(t-t')$ with $\sigma = \gamma_{l,0}(p+1)/2$ \cite{SiebererBuchholdDiehl2016}. This equation further accounts for two effects which were shown to be important to relate the KPZ regime to actual experimental systems: the quartic correction to the dispersion relation $E_{LP}(k)=\hbar \omega_{0,LP}+\frac{\hbar^2}{2m}k^2 - \frac{1}{2 \hbar \Omega} \left(\frac{\hbar^2}{2m}\right)^2 k^4$  with $\Omega$ the Rabi frequency and a momentum-dependent loss rate of polaritons ${\gamma_l}(k)=\gamma_{l,0} + k^2 \gamma_{l,2}$. We have also introduced a confinement potential $V(x)$, which is the cornerstone of this work.

\section{KPZ mapping with the confinement}
To establish the mapping to the KPZ equation, one usually decomposes the wavefunction $\psi$ in the density-phase representation $\psi=\sqrt{\rho} e^{i \theta}$ and expands Eq.~(\ref{eq: gGPEplus}) for small variations around the mean-field solution and in powers of gradients. Following a similar strategy, one finds that in the presence of the static potential $V(x)$, the dynamics of the phase of the EP condensate at long distances obeys an inhomogeneous KPZ equation (see  Sec. \ref{S2} for the derivation),
\begin{equation}
\partial_t \theta = \nu(x) \partial_x^2 \theta + \dfrac{\lambda(x)}{2} \left( \partial_x \theta \right)^2 + \sqrt{D(x)}\eta \,,
\label{eq: KPZ}
\end{equation}
where $\eta$ is a white noise with zero mean and covariance $\left\langle \eta(x,t) \eta(x', t') \right\rangle = 2\delta(x-x') \delta(t-t')$, and 
\begin{align}
& \nu(x)= \frac{\gamma_{l,2}}{2} + \frac{\hbar \rho_0(x)}{m}\tilde{u}(x) \,,\nonumber\\
& \lambda(x)=-\frac{\hbar}{m}+2\gamma_{l,2}\rho_0(x)\tilde{u}(x)\,,\nonumber\\ 
& D(x)=\frac{\sigma}{2\rho_0(x)}(1+4\rho_0^2(x)\tilde{u}^2(x))\,,\nonumber\\
& \tilde{u}(x)=\dfrac{\frac{\hbar}{2m} \left[ \frac{(\partial_x \rho_0)^2}{2\rho_0^3} - \frac{\partial_x^2 \rho_0}{2\rho_0^2} \right]- \frac{\gamma_{l,2}}{2} \frac{(\partial_x \rho_0)( \partial_x \theta_0)}{\rho_0^2} - g_{\text{int}}}{\frac{\gamma_{l,2}}{2}\left[ \frac{(\partial_x \rho_0)^2}{2\rho_0^2} - 2(\partial_x \theta_0)^2 \right] - \frac{\hbar}{m}\partial_x^2 \theta_0 + \tilde{p}(x)}\,, \label{eq:param}
\end{align}
where $\theta_0(x,t)$ and $\rho_0(x)$ are the mean-field inhomogeneous solutions and $\tilde{p}(x)=\left(p-1-\frac{2pR}{\gamma_r}\rho_0(x)\right)\gamma_{l,0}$ with dimensionless pumping parameter $p=PR/(\gamma_{l,0} \gamma_r)=P/P_{\textrm{th}}$, $P_{\textrm{th}}$ being the threshold pumping strength for condensation.

When $V(x)=0$, one recovers the standard mapping to the homogeneous KPZ equation where the parameters $\nu$, $\lambda$ and $D$ are constant \cite{Davide}. With a non-vanishing potential, these parameters continuously vary with space. One can get an intuition of how this may affect the dynamics by noting that the average velocity of the phase is proportional to the KPZ non-linearity $\lambda$. Qualitatively, when $V\neq 0$, the non-linearity $\lambda(x)$, and thus the velocity, increases where the potential is larger. Therefore, if one implements a suitable enhancement of the potential at the boundaries, one may induce an effective drag at the boundaries prone to bend the phase profile. We found that this can indeed be realised.

In our work, we consider a harmonic potential 
\begin{equation}
V(x) = \frac{1}{2}m \omega_0^2 x^2,
\label{eq:pot}
\end{equation}
where the frequency trap $\omega_0$ can be adjusted. We found that the most favorable trap to study KPZ properties is a shallow parabola (we choose $\omega_0 \simeq 4\times10^{-4} \gamma_{l,0}$ in the following), since it allows one to keep the density fluctuations tame and the KPZ parameters slowly varying in the vicinity of $x=0$. Furthermore, we checked that the  curved KPZ universality sub-class is robust and can be observed for different confinement potentials, in particular Gaussian walls (see Sec. \ref{S4}).

\section{Numerical simulations}
We solve the generalised Gross-Pitaevskii equation (\ref{eq: gGPEplus}) for a  system size of $L/\hat{x}=2^{10}$, where we have chosen $\hat{x}=\sqrt{\hbar/(2m\gamma_{l,0})}\simeq \,2\mu$m, $\hat{t} = \gamma_{\ell,0}^{-1}$ and $\hat{\epsilon}=\hbar \gamma_{l,0}$ as units for length, time and energy. We take for the parameters of Eq.~(\ref{eq: gGPEplus}) the typical values for the CdTe experiments conducted in Grenoble \cite{RojanEtAl2017, Maxime}:  $m=4\times10^{-5}$m$_{\textit{e}}$, $\gamma_{l,0}=0.5$ps$^{-1}$, $\gamma_r = \gamma_{l,0}/25$, $g_{\textrm{int}}=7.59 \times 10^{2}$m$\times$s$^{-1}$, $R=4 \times 10^2$m$\times$s$^{-1}$, $P=4\times10^{19}$(m$\times$s)$^{-1}$, $\gamma_{l,2}=1.29$m$^2 \times$s$^{-1}$, $\Omega=100$ THz.
We record the wavefunction $\psi(x,t)$ during the time evolution and extract its phase $\theta(x,t)$ at suitable time intervals for half of the spatial grid, thanks to the symmetry of the potential. 

We work in the low-noise regime, which ensures that the density fluctuations are negligible and also that there are no topological defects, such as solitons or phase slips. This allows us to unwind the phase, $\theta(x,t) \in (-\pi, \pi] \rightarrow \theta(x,t) \in (-\infty, \infty)$. This is necessary since the KPZ universality class describes fluctuations continuously growing in space and time, which cannot be realised for a compact field. In order to achieve the unwinding in the numerical simulations, we constrain the phase such that the difference between neighouring space-time points is less than $0.8 \times 2\pi$, where the factor $0.8$ is chosen empirically in order to take into account unwinding errors due to space and time discretization. We checked that the unwinding protocol is robust, {\it i.e.} the results do not depend on the specific value of this factor as long as it is close to 1.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.31]{fig1.pdf}
\caption{Typical spatial phase profiles at different times during the evolution, with lighter colours corresponding to larger times, (i) in the absence of confinement potential, leading to a flat profile, and (ii) in the presence of the parabolic confinement potential, leading to a bent profile.
}
\label{fig: fig1}
\end{center}
\end{figure}

In Fig.~\ref{fig: fig1} we display typical phase profiles obtained in the homogeneous case with no external potential $V=0$, which leads to a flat profile, and in the inhomogeneous case with the parabolic  confinement potential (\ref{eq:pot}), which leads to a curved profile. One can observe that in this case the phase profile indeed propagates faster near the boundaries where the EPs feel the largest potential as anticipated. Around the central tip at $x=0$, the phase presents a local curvature, as we evidence in the following.

\section{Results for the scaling}
The KPZ scaling properties can be studied directly from the first-order correlation function of the EP condensate wavefunction,
\begin{equation}
g_1(\Delta x,\Delta t) =  \dfrac{\left \lvert \left\langle \psi^*(x+\Delta x, t+\Delta t) \psi(x,t) \right \rangle \right \rvert}{\left\langle \sqrt{\rho(x+\Delta x, t+\Delta t) \rho(x,t)} \right\rangle},
\end{equation}
Throughout this work, $\left\langle ... \right\rangle$ denotes the average over noise realisations. Note that the first-order coherence $g_1$ is routinely measured in EP experiments, which renders the following analysis easily accessible. By performing a cumulant expansion and neglecting density-phase correlations, one can relate $g_1$ to the connected correlation function of the phase $C$, obtaining to first order
\begin{align}
-2 \ln \left[ g_1(\Delta x,\Delta t)  \right] & = \left\langle \left[ \theta(x+\Delta x, t+\Delta t) - \theta(x,t) \right]^2 \right\rangle\nonumber\\
 &- \Big\langle \theta(x+\Delta x, t+\Delta t) - \theta(x,t) \Big \rangle^2\nonumber\\
 &\equiv C(\Delta x, \Delta t) \,.
\label{eq: g1}
\end{align}
If the phase follows the 1D KPZ dynamics, it should endow the Family-Vicsek scaling form \cite{FamilyVicsek}
\begin{equation}
 C(\Delta x, \Delta t) = C_0 {\Delta t}^{2/3} g\left(y_0 \frac{\Delta x}{\Delta t^{2/3}}\right)\,,
\label{eq:defC}
\end{equation}
where $g(y)$ is a universal scaling function and $C_0$, $y_0$ are normalisation constants defined as
\begin{equation}
 y_0 = (2A \lambda^2)^{-1/3},\, C_0 =  {\Gamma}^{2/3}, \; A = \frac D \nu, \, \Gamma =\frac{\lambda}{2} A^2\,,
\label{eq:norm}
\end{equation}
where the numerical prefactors are conventional. The precise form of the scaling function $g(y)$ is known exactly only for the stationary interface \cite{Prahofer}. However, the scaling function satisfies the same asymptotics in all sub-classes
\begin{equation}
g(y) \stackrel{y\to 0}{\longrightarrow}  g_0 \; , \quad  g(y) \stackrel{y\to \infty}{\sim} 2 y\,,
\label{eq: gy}
\end{equation}
where $g_0$ is a universal constant depending on the geometrical sub-class, whose values are known exactly \cite{Spohn}.
Therefore, one expects a  similar behaviour for the scaling functions in the three sub-classes, apart from small vertical shifts reflecting the differences in $g_0$ and thus small changes in the intermediate crossover region between the two asymptotic limits. In our simulations, we determined $C(\Delta x,\Delta t)$ from the wavefunction correlation function $g_1$ using Eq.~(\ref{eq: g1}). We first estimated the KPZ scaling exponents with and without the  confinement potential by studying the equal-time and equal-space correlation functions, which, according to Eqs.~(\ref{eq:defC}) and (\ref{eq: gy}), should behave as
\begin{subequations}
\begin{align}
C(\Delta x, \Delta t=0) &\sim  \Delta x^{2\chi}, 
\label{eq: scalinglaw1} \\
C(\Delta x=0, \Delta t) &\sim  \Delta t^{2\beta} \,,
\label{eq: scalinglaw2}
\end{align}
\end{subequations} 
with $\chi=1/2$ and $\beta=1/3$ the 1D KPZ roughness and growth critical exponents. We found $\chi=0.49 \pm 0.01$ and $\beta =0.30\pm 0.01$ in both the flat and the curved cases for the purely spatial and purely temporal correlations, see Fig.~\ref{fig: fig6}. The value of the growth exponent $\beta$ slightly differs from the theoretical one but it is comparable with values reported in previous studies of EP condensate for this system size \cite{Davide} for the flat geometry. 

In order to construct the universal scaling function $g(y)$ defined in Eq.~(\ref{eq:defC}), we first selected all the data points lying in the correct scaling regime by filtering out the points differing by more than small cutoffs $\epsilon_x, \epsilon_t$ from the expected scaling laws in Eqs.~(\ref{eq: scalinglaw1}, \ref{eq: scalinglaw2}) for each value of spatial and temporal separation. We extracted the normalisation parameters $A$ and $\Gamma$ in (\ref{eq:norm}) from our numerical data. Note that in the curved case, these parameters are not homogeneous. However, in the vicinity of the central tip, where the confinement potential is nearly vanishing, they are effectively almost constant and coincide with the values for the homogeneous case (see Sec. \ref{S3} for details and the corresponding data).

The scaling function is obtained by plotting $C(\Delta x,\Delta t)/(C_0 {\Delta t}^{2/3})$ as a function of $y_0 \Delta x/\Delta t^{2/3}$. The results are displayed in Fig.~\ref{fig: gy} together with the theoretical curve $g(y)_{\rm stat}$ for the stationary case \cite{Prahofer}. For both the flat and the curved cases, we observe a reasonable collapse of all the data points onto a single function $g$, which demonstrates that $C$ indeed takes a scaling form. Additionally, we confirm that the scaling functions $g$ are quite similar for the three cases. However, it is not a perfectly one-dimensional curve, it has a finite (small) thickness, and the numerical values for $g_0=g(0)$ differ of about $\sim 40\%$  from the theoretical exact values in both the flat and curved cases  ($g^{\rm num}_{0,{\rm flat}}\simeq  0.95$ vs $g^{\rm th}_{0,{\rm flat}}=  0.63805...$, $g^{\rm num}_{0,{\rm curved}}\simeq  1.23$ vs $g^{\rm th}_{0,{\rm curved}}=  0.8132...$). These discrepancies  may originate from the fact that the actual growth exponent is slightly smaller than the theoretical one, and also from the fact that the $g_1$ function includes other contributions beside the phase correlations, even if they are assumed to be small (higher-order cumulants of the phase or density-phase correlations). However, let us emphasise that the values for $g_0$ are clearly distinct in the two cases, and their ratio (or relative difference) turns out to be within 3\% accuracy with the theoretical ratio (or relative difference). This already indicates that the mapping from the Gross-Pitaevskii to the KPZ equation is well-grounded, and that both the flat and the curved universality sub-classes can be probed in EP systems.
\begin{figure}
\begin{center}
\includegraphics[scale=0.31]{fig2.pdf}
\caption{Universal scaling function $g(y)$ for the flat (blue dots) and curved (red triangles) phase profiles. The theoretical result for the stationary interface $g_{\textrm{stat}}(y)$ is shown for comparison (solid line). The theoretical values $g^{\rm th}_{0,{\rm flat}}$, $g^{\rm th}_{0,{\rm curved}}$, $g^{\rm th}_{0,{\rm stat}}$ are indicated in the inset, together with the two numerical curves for $y_0 \Delta x/ \Delta t^{2/3} \rightarrow 0$.}
\label{fig: gy}
\end{center}
\end{figure}

\section{Results for the phase fluctuations}
\subsection{One-point statistics -- Tracy-Widom distributions}
The precise geometry of the phase profile affects the distribution of the fluctuations of the phase. More precisely, as the phase profile propagates linearly in time with fluctuations growing as $t^{1/3}$, one introduces the rescaled fluctuation field $\chi$, defined from the long time behaviour of the phase in 1D as 
\begin{equation}
\theta(x,t) \stackrel{t\to \infty}{\sim} \omega_{\infty}t + (\Gamma t)^{1/3} \chi(\zeta, t)\,,
\label{eq: ansatz}
\end{equation}
where $\omega_{\infty}$ is the asymptotic velocity of the phase, which has a non-trivial dependence on the KPZ parameters \cite{DavideKonstantinosLeonieAnna}, and $\zeta$ is the spatial coordinate rescaled by the correlation length of fluctuations $\zeta \equiv x/\xi(t)$ with $\xi(t)=(\Gamma t)^{2/3}\frac{2}{A}$ \cite{FukaiTakeuchi}. We focus on the universal statistical properties of the centered unwound phase $\Delta \theta(x_0,t) = \theta(x_0,t) - \left\langle\theta(x_0,t) \right\rangle$. 
This allows one to subtract the drift term in Eq.~(\ref{eq: ansatz}), and we henceforth omit the arguments, since there is translational invariance in time all through the KPZ regime (which occurs for stationary condensates), and it is sufficient to consider only the central point $x_0 = 0$. Let us emphasise that Eq.~(\ref{eq: ansatz}) merely stands as an ansatz for the long-time behaviour of the phase. We estimated the numerical value of $\Gamma$ from the time dependence of $\Delta\theta^2$ (see Sec. \ref{S5} for details).

We compute the rescaled fluctuation field $\chi$ from $\Delta\theta$ following  Eq.~(\ref{eq: ansatz}). Note that for the flat case, we conform to the standard definition of the TW-GOE random variable found in the literature, and further rescale $\chi$ as $\chi\to 2^{-2/3} \chi$. We construct the histograms of $\chi$ both for the flat case without confinement and for the curved case with the parabolic confinement $V$. The resulting distributions are displayed in Fig.~\ref{fig: fig3}, where they are compared with the theoretical distributions (more precisely to the mirror ones $P_{\text{TW-GOE}}(-\chi)$ and $P_{\text{TW-GUE}}(-\chi)$ since the KPZ non-linearity $\lambda$ is negative for our choice of experimental parameters).
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.31]{fig3.pdf}
\caption{Centered distribution of the rescaled phase fluctuations $\chi$ sampled at $x=0$, for flat (blue dots) and curved (red triangles) phase profiles, together with the theoretical TW-GOE and TW-GUE distributions.}
\label{fig: fig3}
\end{center}
\end{figure}
One observes a clear distinction between the two cases, and moreover the distribution for the curved case is in excellent agreement with the TW-GUE distribution, thus demonstrating that one can indeed tune the KPZ geometrical sub-class realised in the EP system. We emphasise that this result is very robust with respect to the choice of confinement potential, and we found that the agreement remains excellent for a Gaussian walls potential (see Sec. \ref{S4}).

\subsection{Two-point statistics -- correlations of Airy processes}
Besides the probability distribution, the geometry also influences the two-point statistics of the rescaled fluctuations, differing in the three universality sub-classes. In particular, it was shown that the connected correlation function of rescaled fluctuations of the height of the interface at equal time, defined as
\begin{equation} 
C_\chi(\Delta \zeta) \equiv \left\langle \chi(\zeta+\Delta \zeta,t) \chi(\zeta,t) \right\rangle - \left\langle \chi(\zeta+\Delta \zeta,t\right\rangle \left\langle \chi(\zeta,t) \right\rangle\,,
\label{eq:Cchi}
\end{equation}
with $\Delta\zeta=\Delta x / \xi(t)$, is given by the time correlation function of the Airy$_1$, respectively Airy$_2$ process in the asymptotic limit, in the flat, respectively curved geometry 
\begin{equation}
C_\chi(\Delta \zeta) = {\cal G}_i(\Delta\zeta) = \left \langle \mathcal{A}_i(t^{\prime}+\Delta \zeta) \mathcal{A}_i(t^{\prime}) \right\rangle\,,
\end{equation}
where $i=1,2$ stands for Airy$_1$, respectively Airy$_2$, processes. Let us mention one subtle issue here. While the Airy$_2$ process was found to coincide with the dynamics of the largest eigenvalue of GUE matrices \cite{Johansson2}, the Airy$_1$ process differs from the largest-eigenvalue dynamics of GOE matrices \cite{Bornemann2}. This indicates that while one-point statistics of the fluctuations of the growing interface are connected to random matrix theory, this connection is flawed at the two-point level in the flat case. 

In our simulations, we computed the correlation function Eq.~(\ref{eq:Cchi}) of the rescaled phase fluctuations  as
\begin{equation}
C_\chi(\Delta \zeta) = \dfrac{\left\langle \Delta \theta(x,t) \Delta \theta(x+\Delta x, t) \right\rangle}{\left( \Gamma t \right)^{2/3}}\,.
\label{eq: Cchiplot}
\end{equation}
The results we obtained for the two geometries are presented in Fig.~\ref{fig: fig4}. We note that in line with the rescaling of $\chi$ in the flat case mentioned previously, we also perform in this case the following rescaling $t \rightarrow t/2^{-2/3}$ and ${\cal G}_1 \rightarrow {\cal G}_1 / 2^{-2/3}$ to conform to the standard definition of the Airy$_1$ process \cite{TakeuchiSano1} and compare with the theoretical results from \cite{Bornemann1}. Furthermore, in the curved geometry, the limit stochastic process is shown to be $\chi_{\textrm{curved}}(\zeta,t) \stackrel{d}{\rightarrow} \mathcal{A}_2 - \zeta^2$, where $-\zeta^2$ reflects the influence of the mean profile \cite{PrahoferSpohn_2pt1}, which is automatically subtracted in our case since we consider the connected function.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.31]{fig4.pdf}
\caption{Correlation function $C_\chi$ of the rescaled phase fluctuations as a function of the rescaled length $\Delta \zeta$ for different times for the flat case (blue dots) and for the curved case (green and orange triangles), together with the theoretical results for the correlation of the Airy$_1$ ${\cal G}_1(\Delta \zeta)$ (dashed line) and Airy$_2$ ${\cal G}_2(\Delta \zeta)$ (solid line) processes presented in \cite{Bornemann1, Bornemann2, Bornemann3}. For the flat phase profile, data corresponding to  times $t/\hat{t}=7.5\times10^3, 8.125\times10^3, 1\times10^4$ is shown, and for the curved phase profile, data corresponding to $t/\hat{t}=4\times10^2, 1.6 \times 10^3$ is shown (lighter to darker colors correspond to increasing times). For the curved case, the red shades correspond to the small $\Delta \zeta$ regime where the phase profile is locally curved, while green correspond to large $\Delta \zeta$, where the profile is shaped by the effective drag -- see inset.}
\label{fig: fig4}
\end{center}
\end{figure}

For the flat case, we observe that the correlation function is stable in time, from $t/\hat{t}=7\times10^3$ to, approximately, $t/\hat{t}=1\times10^4$ and we find a good agreement with the theoretical Airy$_1$ correlations ${\cal G}_1(\Delta \zeta)$, even for large $\zeta$. The small shift visible in the figure can be traced back to the fact that the parameter $\Gamma$ in Eq.~(\ref{eq: Cchiplot}) is extracted from the numerical simulations and is found to be a bit larger than the theoretical value (see Sec. \ref{S5}).

For the curved case, the correlation function is stable over a time window from $t/\hat{t}=4 \times 10^2$ until $t/\hat{t} = 1.6 \times 10^3$ where the KPZ scaling regime is observed. We focus on the small$-\Delta \zeta$ limit, corresponding to small spatial separation around $x=0$. Indeed, we expect a local curvature only around the central tip, since the profile near the boundaries is affected by the effective drag ensuing from the confinement potential. The phase correlations are found to behave as an Airy$_2$ process only over this limited range of space, but the agreement with the theoretical Airy$_2$ correlations ${\cal G}_2(\Delta \zeta)$ on this range is extremely satisfactory. 

We emphasise that we have provided in this section the first analysis for both the flat and the curved phase profiles of a very fine statistical quantity: We have here probed reduced correlations, which are  the sub-leading behaviour emerging once the dominant scaling one (studied in the scaling part) has been subtracted. In this respect, the agreement found with the theoretical results for the Airy processes is remarkable. All the results presented here very deeply root in the relevance of KPZ dynamics for the EP system.

\section{Summary and Outlook}
In this work, we have shown that by engineering the confinement potential of 1D exciton polaritons one can tune the geometry of the phase of the condensate and thus access both the flat and curved KPZ universality sub-classes. In particular, we have found excellent agreement with the theoretical exact results, not only for the scaling properties, but also  at the level of one-point statistics (probability distributions), as well as two-point statistics of rescaled fluctuations, though locally for the curved case since the condensate phase is only curved on a limited range by the confinement potential. Our results show the remarkable emergence of KPZ universal properties from the microscopic Gross-Pitaevskii equation for exciton polaritons  at all levels explored so far: not only in the scaling function, but also in the probability distributions and even in the sub-leading behaviour for the two-point correlations, and all this for both the universality sub-classes. We believe that our findings pave the way for stimulating new protocols for investigating KPZ universality in experiments, since the KPZ sub-classes can be accessed through simple engineering of the EP system. In particular, harmonic confinement can be implemented by a suitable engineering of the pumping mechanism \cite{TosiEtAl2012}.

Whereas probing the scaling properties is readily accessible from the measurement of the first-order correlation function, the experimental determination of the probability distributions may be more challenging as it involves the measurement of the time and space resolved phase of the condensate. This requires the development of specific interferometric techniques capable of resolving very small times. On the other hand, higher-order correlations have been measured e.g in \cite{last1,last2} for the case of ultracold atoms. Similar techniques could be implemented in the EP system and lead to the possibility of accessing universal ratios of cumulants, thus enabling the demonstration of the typical non-gaussian shape of the TW-GOE and TW-GUE probability distributions and the characterization of universality sub-classes. Last but not least, the investigation of KPZ universality in 2D is an exciting perspective, both from a theoretical viewpoint, where few indications of KPZ scaling in EP systems have been reported \cite{Zamora2017,Mei2020}, and from an experimental viewpoint, where a high-precision platform for exploring KPZ in 2D is still missing.

\acknowledgments
We acknowledge stimulating discussions with Alberto Amo, Jacqueline Bloch, and  Maxime Richard.  We wish to thank Prof. F. Bornemann for kindly providing us with the theoretical data for the correlation of the Airy processes used in Fig.~\ref{fig: fig4}. K.D. acknowledges the European Union Horizon 2020 research and innovation programme under the Marie Sk\l{}odowska-Curie grant agreement No 754303. L.C. acknowledges support from the French ANR through the project NeqFluids (grant ANR-18-CE92-0019) and support from Institut Universitaire de France (IUF).

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