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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 [1]. Whereas it was originally derived to describe kinetic roughening of interfaces undergoing stochastic growth [2], the KPZ critical properties have been shown to arise in many non-equilibrium or disordered systems, ranging from turbulent liquid crystals [3] to non-equilibrium hydrodynamics [4] to name a few. More recently, Bose-Einstein condensates of exciton-polaritons (EP) [5], 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 [6]. Recently, a striking connection to KPZ universality has been brought out by several theoretical approaches [7–9]. More precisely, the dynamics of the phase of the condensate wave function at long distances has been shown to obey the KPZ equation, and KPZ scaling has been reported in various conditions [10–12]. In particular, the KPZ exponents were found in numerical simulations of the one-dimensional [13] and two-dimensional [14] EP systems, as well as of photonic cavity arrays [15]. 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 [16–18]. 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) [19–24], respectively, Gaussian Unitary Ensemble (GUE) [25–29], unveiling a non-trivial connection with random matrix theory, where these distributions, called Tracy-Widom (TW), originally emerged [30]. The third sub-class corresponds to Brownian, also called stationary, initial conditions, with fluctuations following a Baik-Rains (BR) distribution [31,32]. 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 Airy1 [23,33] and Airy2 [34,35] 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 [3,36,37]. 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 [38], 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 Airy1 and Airy2 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. Model for the dynamics of the EP condensateOur starting point is the mean-field generalised Gross-Pitaevskii equation for the EP Bose-Einstein condensate under incoherent pumping formulated in [39], ![]() where ψ is the condensate wave function, ![]() where the noise ξ, which arises from both dissipation and pumping stochastic processes, is complex and has zero mean and covariance To establish the mapping to the KPZ equation, one usually decomposes the wave function ψ in the density-phase representation ![]() where η is a white noise with zero mean and covariance ![]() where When In our work, we consider a harmonic potential ![]() where the frequency trap We solve the generalised Gross-Pitaevskii equation (2) for a system size 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, In fig. 1 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 (5), which leads to a curved profile. One can observe that in this case the phase profile indeed propagates faster near the boundaries where the EP 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. ![]() Fig. 1: 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. Download figure: Standard image Results for the scalingThe KPZ scaling properties can be studied directly from the first-order correlation function of the EP condensate wave function, ![]() Throughout this work, ![]() If the phase follows the 1D KPZ dynamics, it should endow the Family-Vicsek scaling form [44] ![]() where g(y) is a universal scaling function and C0, y0 are normalisation constants defined as ![]() where the numerical prefactors are conventional. The precise form of the scaling function g(y) is known exactly only for the stationary interface [45]. However, the scaling function satisfies the same asymptotics in all sub-classes, ![]() where g0 is a universal constant depending on the geometrical sub-class, whose values are known exactly [46]. Therefore, one expects a similar behaviour for the scaling functions in the three sub-classes, apart from small vertical shifts reflecting the differences in g0 and thus small changes in the intermediate crossover region between the two asymptotic limits. In our simulations, we determined ![]()
![]() with In order to construct the universal scaling function g(y) defined in eq. (8), we first selected all the data points lying in the correct scaling regime by filtering out the points differing by more than small cutoffs The scaling function is obtained by plotting ![]() Fig. 2: Universal scaling function g(y) for the flat (blue dots) and curved (red triangles) phase profiles. The theoretical result for the stationary interface Download figure: Standard image Results for the phase fluctuationsOne-point statistics —Tracy-Widom distributionsThe 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 ![]() where ![]() Fig. 3: Centered distribution of the rescaled phase fluctuations χ sampled at x = 0, for flat (blue dots) and curved (red triangles) phase profiles, together with the theoretical TW-GOE and TW-GUE distributions. Download figure: Standard imageWe compute the rescaled fluctuation field χ from Δθ following eq. (12). 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 χ as 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 ![]() with ![]() where In our simulations, we computed the correlation function, eq. (13), of the rescaled phase fluctuations as ![]() The results we obtained for the two geometries are presented in fig. 4. We note that in line with the rescaling of χ in the flat case mentioned previously, we also perform in this case the following rescaling ![]() Fig. 4: Correlation function Download figure: Standard imageFor the flat case, we observe that the correlation function is stable in time, from For the curved case, the correlation function is stable over a time window from 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. Summary and outlookIn 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 [52]. 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 [53,54] 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 the theoretical viewpoint, where few indications of KPZ scaling in EP systems have been reported [10,14], and from the experimental viewpoint, where a high-precision platform for exploring KPZ in 2D is still missing. AcknowledgmentsWe 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. 4. KD acknowledges the European Union Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 754303. LC 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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