# Kardar-Parisi-Zhang universality in the linewidth of non-equilibrium 1D
quasi-condensates

Ivan Amelio  Center for Nonlinear Phenomena and Complex Systems, Université
Libre de Bruxelles, CP 231, Campus Plaine, B-1050 Brussels, Belgium  Institute
of Quantum Electronics ETH Zurich, CH-8093 Zurich, Switzerland  Pitaevskii BEC
Center, INO-CNR and Dipartimento di Fisica, Università di Trento I-38123
Trento, Italy  Alessio Chiocchetta  Institute for Theoretical Physics,
University of Cologne, Zülpicher Strasse 77, 50937 Cologne, Germany  Iacopo
Carusotto  Pitaevskii BEC Center, INO-CNR and Dipartimento di Fisica,
Università di Trento I-38123 Trento, Italy

###### Abstract

We investigate the finite-size origin of the emission linewidth of a
spatially-extended, one-dimensional non-equilibrium condensate. We show that
the well-known Schawlow-Townes scaling of laser theory, possibly including the
Henry broadening factor, only holds for small system sizes, while in larger
systems the linewidth displays a novel scaling determined by Kardar-Parisi-
Zhang physics. This is shown to lead to an opposite dependence of the
linewidth on the optical nonlinearity in the two cases. We then study how sub-
universal properties of the phase dynamics such as the higher moments of the
phase-phase correlator are affected by the finite size and discuss the
relation between the field coherence and the exponential of the phase-phase
correlator. We finally identify a configuration with enhanced open boundary
conditions, which supports a spatially uniform steady-state and facilitates
experimental studies of the linewidth scaling.

##  I Introduction

The statistical theory of critical phenomena in systems at thermal equilibrium
is considered one of the most successful branches of theoretical physics Huang
(1963); Brezin (2010). The central result is that at criticality the low
energy correlation functions are characterized by universal exponents,
insensitive to the microscopic details of the system but only determined by
dimensionality and symmetries, in particular spontaneously broken ones. As a
result, seemingly distant phenomena such as the gas-liquid transition and
Ising ferromagnetism end up belonging to the same universality class. While
the general theory refers to a spatially infinite setting, real systems
necessarily have a finite spatial extent, so that the study of finite-size
effects is of great importance in this context. In particular, finite-size
scaling methods Fisher and Barber (1972); Cardy (2012) have been developed to
obtain precise estimates of the critical exponents from measurements on
systems of different sizes on the order of the correlation length. Such a tool
has turned out to be of tremendous utility in numerical simulations.

Even though criticality is omnipresent also in out-of-equilibrium systems –
power law correlators are present for instance in avalanches, percolation,
social networks, and many other natural phenomena –, here a general
classification scheme is still lacking. In contrast to equilibrium where scale
invariance only appears in the proximity of phase transitions and requires a
fine-tuning of one or more parameters, in non-equilibrium systems it can be
observed without a fine-tuning of the parameters, hence the name of self-
organized criticality Bak  _et al._ (1988). This provides a further motivation
to explore the interplay between finite-size effects and universality in non-
equilibrium systems.

In this work we study the linewidth of a driven-dissipative 1D
(quasi-)condensate. Experimentally relevant platforms to investigate this
physics include lasing in 1D spatially extended systems such as photonic Zhang
_et al._ (1995) or polariton Wertz  _et al._ (2010) wires, discrete arrays of
polariton micropillars Amo and Bloch (2016); Fontaine  _et al._ (2022) or
VCSELs Grabherr  _et al._ (1999); Cui  _et al._ (2009), or even the edge modes
of 2D topological lasers Harari  _et al._ (2018); Bahari  _et al._ (2017);
Loirette-Pelous  _et al._ (2021); Amelio and Carusotto (2020). In all these
systems, a natural and technologically very relevant observable is the
emission linewidth, namely the spectral width of the light emitted from a
given point of the device.

Schawlow and Townes in their seminal work Schawlow and Townes (1958) predicted
that the ultimate linewidth of a single mode laser is set by the spontaneous
emission and scales inversely to the number of photons in the laser cavity.
This scaling is accurate for the simplest, textbook case of a zero-dimensional
device where the spatial dynamics of the light field is frozen and a single
cavity mode can be considered. Ramping up in geometrical complexity, two of us
recently remarked in Amelio and Carusotto (2020) how a finite value of the
linewidth of a spatially extended laser device can be viewed as a finite-size
effect. While in a zero-dimensional system the phase dynamics is diffusive and
the phase-phase correlator grows linearly in time, a different behaviour is
anticipated for infinite-size, low-dimensional systems, where the phase-phase
correlator is characterized by specific exponents. As proven in a series of
recent studies Gladilin  _et al._ (2014); Altman  _et al._ (2015); He  _et
al._ (2015); Squizzato  _et al._ (2018), non-equilibrium 1D quasi-condensates
belong in fact to the Kardar-Parisi-Zhang (KPZ) Kardar  _et al._ (1986)
universality class. Here we undertake the study of finite-size effects in this
context and, in particular, show that, for large enough systems, the
signatures of the KPZ universal exponents are visible in the scaling of the
linewidth with the spatial size of the device. This provides an exciting link
between crucial concepts of non-equilibrium statistical mechanics and an
observable quantity of central importance in the general theory of lasing as
well as for technological applications.

The structure of this work is the following. In Sec. II we start introducing
the model for the quasi-condensate and reviewing the general KPZ theory of the
phase dynamics. In Sec. III we report numerical simulations of the stochastic
Complex Ginzburg Landau Equation (CGLE) describing the field evolution and we
compare the result with the predictions of the Kuramoto-Sivashynskii equation
(KSE) describing the phase dynamics and of the low energy KPZ equation. For
both the full CGLE dynamics and the KSE, the numerical calculations clearly
show that for small system sizes $L$ the coherence time (or inverse linewidth)
scales linearly in $L$ according to the Schawlow-Townes prediction, while for
large systems the scaling is instead proportional to $\sqrt{L}$. For the low-
energy KPZ evolution, only the latter scaling is observed, as entailed by the
universal 1D KPZ exponents under a finite-size scaling hypothesis. The KPZ
scaling properties are then used to explore the effect of an optical
nonlinearity, namely a photon-photon interaction term, on the linewidth:
interestingly, this nonlinearity has opposite effects in the two regimes, so
the optimal linewidth is obtained for an intermediate value of the
interactions.

In Sec. IV we investigate the higher moments of the phase-phase correlator,
that are known to exhibit sub-universal features Takeuchi (2018); Squizzato
_et al._ (2018). In particular, we present numerical evidence that, as a
finite-size effect, the probability distribution for the phase transits from a
skewed distribution, as expected by KPZ, to an approximately Gaussian
distribution at large times. This entails that re-exponentiation of the phase-
phase correlator is allowed when computing the linewidth.

In view of facilitating experiments, in Sec. V we propose a lattice
configuration with enhanced open boundary conditions: the spatially uniform
profile of the steady-state is of great utility for the accurate extraction of
the scaling of the linewidth with $L$ in experiments. Conclusions are finally
drawn in Sec. VI

##  II KPZ universality in one-dimensional non-equilibrium condensates

We start by reviewing the theory of KPZ universality in one-dimensional
geometries Gladilin  _et al._ (2014); He  _et al._ (2015); Altman  _et al._
(2015); Squizzato  _et al._ (2018); Amelio and Carusotto (2020). Even though
we will focus on the case of a continuous wire, the main results also apply to
discrete lattices, e.g. the Lieb arrays of polariton micropillars considered
in Fontaine  _et al._ (2022) as well to the edge modes of 2D topological
lasers Amelio and Carusotto (2020); someth-pelous2021. Assuming that the
reservoir of carriers can be adiabatically eliminated, the field dynamics is
described by the stochastic complex Ginzburg-Landau equation (CGLE)

$i\partial_{t}\psi=\left[-\frac{1}{2m}\nabla^{2}+gn+\frac{i}{2}\left(\frac{P}{1+n/n_{S}}-\gamma\right)\right]\psi+\sqrt{2D}\xi,$
(1)

where $\psi$ is the semi-classical field, $m$ the photon mass, $g$ the
strength of the Kerr optical nonlinearity, namely the photon-photon
interactions, and $n=|\psi|^{2}$ the density. The non-equilibrium features
enter through the loss rate $\gamma$, the effective pumping rate $P$ and the
saturation density scale $n_{S}$. Finally, $\xi$ is a Gaussian-distributed
white noise term
$\langle\xi^{*}(x,t)\xi(x^{\prime},t^{\prime})\rangle=\delta(x-x^{\prime})\delta(t-t^{\prime})$
and $D$ is the noise strength coefficient.

At mean-field level, the steady-state above the condensation threshold
$P>P_{th}=\gamma$ is characterized by the density $n_{0}=n_{S}(P/P_{th}-1)$.
Because of the $U(1)$ symmetry of the CGLE, the global phase of the steady-
state is spontaneously selected and the Bogoliubov excitation spectrum
(reviewed in the Appendix) contains a gapless branch.

Upon inclusion of noise, provided density fluctuations are small, one can
focus on the phase dynamics, which occurs on much longer timescales compared
the density relaxation rate $\Gamma=\frac{\gamma(P-\gamma)}{P}$. By
adiabatically eliminating the density fluctuations, the CGLE (1) reduces to
the Kuramoto-Sivashinsky equation (KSE) 111Note how our Eq. (2) differs from
the equation considered in Gladilin  _et al._ (2014) in that we are not making
the near-threshold approximation $\alpha\ll 1$.

$\partial_{{t}}{\phi}=\frac{1}{2m}\left[-\frac{\Gamma^{-1}}{2m}\partial^{4}_{{x}}{\phi}+\alpha\partial^{2}_{{x}}{\phi}-(\partial_{{x}}{\phi})^{2}\right]+\sqrt{\frac{D(1+\alpha^{2})}{n_{0}}}{\xi_{1}},$
(2)

where we have introduced the blueshift of the unperturbed steady-state
$\mu=gn_{0}$ and the so-called Henry factor $\alpha=2\mu/\Gamma$: the
fluctuations of the density determine a local variation of the refractive
index of the optical medium, which results in an extra noise source in the
phase equation, the so-called Henry linewidth broadening effect Henry (1982).
Note that the phase $\phi$ indicates here an unwound phase variable that is
not restricted to the $[0,2\pi]$ interval. As such, the theory based on the
KSE (2) does not capture the physics of (spatio-temporal) vortices discussed
in He  _et al._ (2017); Fontaine  _et al._ (2022): this approximation is
legitimate as long as noise is sufficiently weak and density fluctuations are
small.

Measuring space, time, blueshift and (unwound) phase in terms of the
characteristic scales defined in terms of the microscopic parameters by

$l^{*}=\left[(2m)^{4}\Gamma^{3}Dn_{0}^{-1}\right]^{-1/7},\ \
t^{*}=\left[(2m)^{2}\Gamma^{5}(Dn_{0}^{-1})^{4}\right]^{-1/7},$
$\phi^{*}=\left[\frac{2m(Dn_{0}^{-1})^{2}}{\Gamma}\right]^{1/7},\ \
\mu^{*}=\frac{1}{2}\left[2m\Gamma^{6}(Dn_{0}^{-1})^{2}\right]^{1/7}.$ (3)

leads to the adimensional form of the KSE,

$\partial_{\tilde{t}}\tilde{\phi}=\tilde{\mu}\partial^{2}_{\tilde{x}}\tilde{\phi}-\partial^{4}_{\tilde{x}}\tilde{\phi}-(\partial_{\tilde{x}}\tilde{\phi})^{2}+\sqrt{1+\alpha^{2}}\
\tilde{\xi}$ (4)

which, at large distances and long times, renormalizes to a KPZ equation of
the form Ueno  _et al._ (2005)

$\partial_{\tilde{t}}\tilde{\phi}=\nu\partial^{2}_{\tilde{x}}\tilde{\phi}+\frac{\lambda}{2}(\partial_{\tilde{x}}\tilde{\phi})^{2}+\sqrt{\mathcal{D}}{\xi_{1}}.$
(5)

In particular, the Galilean invariance of the KSE and KPZ Takeuchi (2018)
dictates that the coupling of the nonlinear term does not get renormalized and
remains fixed to $\lambda=-2$ along the renormalization flow.

Neglecting finite-size effects, it can be shown Kardar  _et al._ (1986) that
the phase-phase correlator (corresponding to the height-height correlator in
the KPZ literature)

$\Delta\tilde{\phi}_{\tilde{x},\tilde{t}}^{2}\equiv\langle[\tilde{\phi}(\tilde{x},\tilde{t})-\tilde{\phi}(0,0)]^{2}\rangle$
(6)

has the scalings $\Delta\tilde{\phi}_{\tilde{x},0}^{2}\sim\tilde{x}^{2\chi}$
and $\Delta\tilde{\phi}_{0,\tilde{t}}^{2}\sim\tilde{t}^{2\chi/z}$ in space and
time, respectively. In $1$D the correlator is known exactly Prähofer and Spohn
(2004), giving values $\chi=1/2$ and $z=3/2$ for the so-called roughness and
dynamical exponents.

From these results for the phase-phase correlation function, one may be
tempted to perform an exponentiation of the phase variance to directly extract
the field coherence

$g^{(1)}(t)=\frac{1}{n_{0}}\left|\langle\psi^{*}(x,t)\psi(x,0)\rangle\right|\sim
e^{-\frac{1}{2}\Delta\tilde{\phi}_{\tilde{x},\tilde{t}}^{2}}\,,$ (7)

which is the typical quantity that is measured in optical experiments Fontaine
_et al._ (2022). However, as it was remarked in Squizzato  _et al._ (2018),
this “cumulant approximation” procedure is not generally legitimate, since the
KPZ height profile at a given point is not a Gaussian random variable, but is
rather given by

$\phi(x,t)-\phi(x,0)=v_{\infty}t+\sigma t^{1/3}X+...$ (8)

where $X$ is a non-Gaussian random variable. For a system at the steady-state,
the distribution of $X$ is of the Baik-Rains type Baik and Rains (2001); under
different initial or boundary conditions, the distribution may fall in other
KPZ universality subclasses, see Takeuchi (2018) for a review.

As we are going to see in what follows, in spite of the non-Gaussianity of
$X$, the re-exponentiation encoded in Eq. (7) can still be used in the very
long time regime to extract the emission linewidth.

##  III Scaling of the linewidth

Figure 1:  Coherence time $\tau_{c}(L)$ of a one-dimensional condensate
plotted as a function of the system length $L/l^{*}$ measured in natural
units. The coherence time is here normalized to $\tau_{c}(l^{*})$ as computed
for $L=l^{*}$. The three numerical data series correspond to the full CGLE
(green squares) and the KSE Eq. (4) (blue circles) and KPZ Eq.(5) (red
triangles) approximations. The interaction constant is set to zero, $g=0$, and
the other CGLE parameters are $\gamma=0.1$, $m=10$, $P=2\gamma$, $D=\gamma$,
$n_{S}=1000$. In the present $g=0$ case, the KSE has $\tilde{\mu},\alpha=0$.
The parameters of the KPZ are chosen as $D/\nu=1.82,\lambda=2$, so to match
the same universal long-wavelength physics as the CGLE and KSE. For small
sizes Bogoliubov theory holds and the coherence time scales proportionally to
the system length as predicted by the Schawlow-Townes theory (black dashed
line). For longer systems, instead, the nonlinearity of the phase dynamics
determines a stronger broadening of the linewidth and a scaling
$\tau_{c}\propto L^{1/2}$ (red line).

While these universal features have been derived for the case of spatially
infinite systems, they provide an accurate description also for finite systems
up to a saturation time scaling as $L^{z}$. For longer times/shorter systems,
the physics is instead dominated by finite-size effects, which are also
expected Amelio and Carusotto (2020) to display remarkable features as we are
now going to see.

If spatial fluctuations are neglected in a sort of single-mode approximation,
the long time behavior of the correlation function is dominated by the
diffusion of the phase. Since any restoring force is forbidden by the
microscopic $U(1)$ symmetry of the model, the phase performs a random walk in
time. In this case, re-exponentiation (7) is exact and yields an exponential
decay of the coherence function

$g^{(1)}(t)=e^{-\frac{\gamma_{ST}}{2}|t|},$ (9)

at a rate

$\gamma_{ST}=\frac{D}{n_{0}L}(1+\alpha^{2})\,.$ (10)

As it was first pointed out by Schawlow and Townes Schawlow and Townes (1958),
the coherence time $\tau_{c}\equiv 2/\gamma_{ST}$ is proportional to the total
number $N_{ph}=n_{0}L$ of photons present in the lasing mode and this scaling
directly extends to generic dimensions. In all those cases when this scaling
holds, we will refer to $\gamma_{ST}$ as the Schawlow-Townes (ST) linewidth.
The factor $(1+\alpha^{2})$ was later introduced by Henry to account for the
additional broadening due to refraction index fluctuations with no change in
the overall scaling Henry (1982).

As it is discussed in detail in Gladilin  _et al._ (2014); He  _et al._
(2015); Amelio and Carusotto (2020), for spatially extended yet sufficiently
small systems, the spatial fluctuations of the phase remain weak, so one can
neglect the nonlinear term of the KSE/KPZ dynamics. As a consequence, the
Bogoliubov theory of non-equilibrium condensates Wouters and Carusotto (2007)
can be used and a ST scaling holds Amelio and Carusotto (2022). Except at very
short times when density fluctuations are still important, the equal-space
correlators mostly probe the ST physics of the single condensate mode.

For larger systems, instead, a crossover between KPZ physics and a Schawlow-
Townes-like exponential decay of coherence is visible, the latter behaviour
taking over at times longer than a characteristic saturation time proportional
to $L^{z}$. Nonetheless, as anticipated in Amelio and Carusotto (2020), the
nonlinearity of the phase equation keeps having a key impact and results in a
stronger broadening of the linewidth compared to the standard ST prediction.
We therefore coin the expression generalized Schawlow-Townes (gST) regime to
indicate the long-time behaviour where $g^{(1)}(t)$ decays exponentially but
the standard ST scaling $\tau_{c}\propto N_{ph}$ no longer holds. In the
following of this Section, we specifically investigate the dependence of the
coherence time $\tau_{c}$ on the system size $L$ in one dimension and we
highlight the possibility of different scalings.

###  III.1 Non-interacting $g=0$ case

We start from the the non-interacting case $g=0$ with parameters that are
chosen in a way to have relatively small density fluctuations of the order of
10% but sizable spatial fluctuations of the phase.

####  III.1.1 CGLE simulations

The coherence time extracted from an exponential fitting $g^{(1)}(t)\sim
e^{-t/\tau_{c}}$ of a numerical simulation of the CGLE is reported in Fig. 1
as green squares. For small systems of length up to $L\sim 5l^{*}$, Bogoliubov
theory holds and the effect of the KPZ nonlinearity in the phase equation is
negligible: correspondingly, the coherence time scales as the system size, as
predicted by the ST formula (10). For longer systems with $L>10l^{*}$, the
behavior changes and the scaling is well captured by a different scaling law,
$\tau_{c}\propto L^{1/2}$. At the crossover between the two regimes, a
peculiar non-monotonic feature is visible whose understanding remains an open
question.

In order to understand the $\tau_{c}\propto L^{1/2}$ scaling, we can put
forward the following argument based on a finite-size scaling assumption.
Since the KPZ equation is by itself scale invariant, the only available length
scale is provided by the system size. On this basis, we can expect that the
equal-space correlator has the universal form

$\Delta\phi_{0,t}\sim t^{2\chi/z}f\left(\frac{t}{L^{z}}\right).$ (11)

where the function $f$ must asymptotically recover the KPZ result at small
times, $f(x\to 0)\sim 1$, and the ST behavior at long times,
$f(x\to\infty)\sim x^{1-2\chi/z}$. In particular, in this latter regime we
have $\Delta\phi_{0,t}\sim t/L^{z-2\chi}$, which implies that

$\tau_{c}\propto L^{z-2\chi}.$ (12)

and thus recovers the numerically observed scaling $\tau_{c}\propto L^{1/2}$.

In three (or higher) dimensions, long range order of the condensate is robust
Sieberer  _et al._ (2013), so we can expect that spatial fluctuations of the
phase do not affect the long time coherence. As a result, the simple Schawlow-
Townes scaling formula Eq. 10 should hold. The physics is more subtle in the
intermediate 2D case. Here, we can make use of the general KPZ result
$\chi+z=2$ and the known exponent $\chi\simeq 0.373$ Kloss  _et al._ (2012) to
predict $\tau_{c}\sim L^{0.881}$. In view of a numerical verification of this
prediction, a challenging issue will be to properly isolate the KPZ physics
from competing effects related to the proliferation of vortices Zamora  _et
al._ (2017); He  _et al._ (2017); Ferrier  _et al._ (2020). This will be the
topic of future numerical work.

####  III.1.2 KSE and KPZ simulations

Coming back to the one-dimensional case, it is important to verify that the
behavior observed in Fig. 1 does not arise from spurious physics related to
the UV sector and/or to density fluctuations nor from a violation of the
cumulant approximation We have then directly simulated the KSE Eq. (4) with
$\tilde{\mu}=0$. At long times, we observe that the phase-phase correlator (6)
grows linearly in time with a coefficient defined as twice the inverse
coherence time, $\Delta\tilde{\phi}_{\tilde{x},\tilde{t}}^{2}\sim
2t/\tau_{c}$.

In Fig. 1 we show as blue circles the coherence time predicted by the KSE for
different system sizes. These points show an excellent agreement with the
predictions of the full CGLE. This means that in the small density fluctuation
regime studied here, the KSE description can be considered an accurate
approximation for all system sizes and the cumulant approximation is a
legitimate approximation, at least for what concerns the emission linewidth.
Quite interestingly, also the crossover behavior is very well reproduced by
the KSE: this suggests that the kink is determined by the renormalization of
the KSE (2) and occurs at the emergent lengthscale at which the KSE flows into
the KPZ.

Further insight on this is provided by numerical simulations of the KPZ
equation (5), whose predictions for the coherence time is shown as red
triangles in Fig. 1. The parameters $D/\nu=1.92,\lambda=2$ in the KPZ equation
were chosen to match the scaling function yielded by the CGLE and KSE for
large system sizes (see Fig 6.b of Amelio and Carusotto (2020)). In other
words, the KSE renormalizes to a KPZ, whose parameters can be obtained by
extracting the scaling function from the numerical data. Contrary to the KSE,
the KPZ equation does not capture the phase dynamics for small sizes, but the
matching is excellent for systems that are long enough for the dynamics to be
renormalized into the KPZ equation.

For the following, it is useful to obtain a precise expression of the
linewidth $\gamma_{KPZ}$ of the pure KPZ equation in terms of the parameters
$\nu,\lambda,D$. To this purpose, let us recall that the KPZ equation

$\partial_{t}\phi=\nu\partial^{2}_{x}\phi-\frac{\lambda}{2}(\partial_{x}\phi)^{2}+\sqrt{D}\xi,$
(13)

with $\langle\xi(x,t)\xi(0,0)\rangle=\frac{1}{2}\delta(x)\delta(t)$ is scale-
invariant and can be rescaled through

$x=\frac{\nu^{3}}{\lambda^{2}D}y,\ \ \ t=\frac{\nu^{5}}{\lambda^{4}D^{2}}s,\ \
\ \phi=\frac{\nu^{5}}{\lambda^{4}D^{2}}\varphi,$ (14)

to an adimensional form

$\partial_{s}\varphi=\partial^{2}_{y}\phi-\frac{1}{2}(\partial_{y}\varphi)^{2}+\eta,$
(15)

with $\langle\eta(y,s)\eta(0,0)\rangle=\frac{1}{2}\delta(y)\delta(s)$ and no
free parameters.

From the previous argument on the scaling function, we infer that the
linewidth predicted by this equation for a system of extension
$y\in[0,\mathcal{L}]$ has the functional form

$\gamma_{KPZ}(\nu=1,\lambda=1,D=1,\mathcal{L})=\lim_{s\to\infty}\frac{1}{s}\langle\Delta\varphi^{2}\rangle=\frac{\gamma_{1}}{\sqrt{\mathcal{L}}}\,.$
(16)

This scaling is confimed by the red triangles in Fig. 1, from which we can fit
the constant $\gamma_{1}=0.172\pm 0.005$.

In terms of the physical variables (in particular the system extension is
$x\in[0,L]$) this reads

$\gamma_{KPZ}(\nu,\lambda,D,L)=\lim_{t\to\infty}\frac{1}{t}\langle\Delta\phi^{2}\rangle=\frac{\lambda
D^{3/2}}{\nu^{3/2}}\frac{\gamma_{1}}{\sqrt{{L}}},$ (17)

where $\gamma_{KPZ}$ only depends on the ratio $D/\nu$. This form is
consistent with the fact that in the RG flow of the 1D KPZ the two parameters
$D$ and $\nu$ separately diverge but the fixed point is determined by their
ratio $D/\nu$ and by the $\lambda$ parameter, which is also not renormalized.
A most interesting feature of Eq. (17) is that it allows to predict the
linewidth of an arbitrary KPZ system, having just fitted once and for all the
constant $\gamma_{1}$.

Figure 2:  Coherence time in natural units
$\tilde{\tau}_{c}=(\phi^{*})^{2}\tau_{c}/t^{*}$ as a function of the Henry
parameter $\alpha=2gn_{0}/\Gamma$ for three different values of the system
size. The numerical points are obtained from simulations of the CGLE (1) using
the same parameters as in Fig. 1 except for $n_{S}=400$. The dotted lines are
the theoretical predictions: for large $\alpha$, we used the Henry-Schawlow-
Townes formula Eq. (10); for smaller but non-zero $\alpha$, we used the KPZ
scaling Eq. (18).

###  III.2 Effect of finite interactions $g\neq 0$

Let us know investigate the effect of a finite interaction constant, $g\neq
0$. On the one hand, the effective noise on the phase is enhanced via the same
mechanism underlying the Henry broadening of the linewidth in the single-mode
laser, as expressed by $\alpha$ in Eq. (10). On the other hand, for $g\neq 0$
the Laplacian term in the microscopic phase equation (4) is nonzero, which
tends to stabilize the fluid phase by reducing long wavelength fluctuations.
As a consequence, the Bogoliubov-Gaussian theory holds up to larger system
sizes and longer systems are needed to observe a clean KPZ scaling Gladilin
_et al._ (2014).

The latter observation entails that the scaling of the linewidth with $g$ (or,
more conveniently, with the adimensional $\alpha=2gn_{0}/\Gamma$ parameter) in
a long system may be different from what the standard Henry broadening Eq.
(10) of short systems. In the regime where the Laplacian term in KSE equation
(4) dominates over the quartic derivative term, one can in fact approximate
the KSE with a KPZ with $\nu\propto\alpha$ and $D\propto 1+\alpha^{2}$. The
scaling with $\alpha$ can then be straightforwardly obtained using formula
(17), which yields

$\tilde{\tau}_{c}=2/\gamma_{KPZ}(\tilde{\mu},2,1+\alpha^{2},L/l^{*})\propto\frac{\alpha^{3/2}}{(1+\alpha^{2})^{3/2}}\sqrt{{L}}.$
(18)

This scaling of the coherence time with $\alpha$ is reproduced by numerical
simulations of the CGLE for three different sizes, as illustrated in Fig. 2.
In these simulations, a smaller value of $n_{S}$ is used to increase
fluctuations and observe the physics of interest within a feasible integration
box.

In order to highlight the general trends, the coherence time is plotted in
natural units $\tilde{\tau}_{c}={(\phi^{*})^{2}}\tau_{c}/{t^{*}}$. At very
small $\alpha$, the same behavior studied in Fig. 1 for the $g=0$ case is
recovered: in this case, the phase dynamics is described by a KSE with
negligible Laplacian term and Eq. (18) does not apply. As a result, the
coherence time tends to a finite value in the $\alpha\to 0$ limit.

At small but finite values of $\alpha$, the Laplacian term starts dominating
over the quartic term and the points follow the trend predicted by Eq. (18).
Finally, at larger interactions, the spatial fluctuations are dominated by the
strong Laplacian term, so the physics turns out to be well described again by
Bogoliubov theory and the linewidth recovers the Henry scaling Eq. (10). The
dotted lines represent the minimum between the coherence time predicted by Eq.
(18) and Eq. (10) and the cusp that separating the two regimes is smoothened
out by the competition between the different effects.

It is interesting to note how the window in which KPZ physics is observed gets
wider in larger systems. In Fig. 2, this is visible as a shift of the cusp
towards larger $\alpha$ for growing $L/\ell^{*}$. Another interesting feature
visible in this plot is that for a given finite size, the optimal coherence
time is achieved for intermediate values of the interaction constant $g$.

##  IV Skewness and cumulant approximation

Figure 3:  Temporal evolution of the skewness $skew(t)$ of the phase for
different system sizes, from the CGLE with the same parameters as in Fig. 1.
The dotted line is the value expected from the Baik-Rains distribution.

The analysis reported in the previous Sections heavily relies on the scaling
behavior of the phase-phase correlator and the results have been translated to
the $g^{(1)}$ under the so called cumulant approximation mentioned in Eq. (7).
More explicitly, this approximation can be formulated as

$\left|\langle e^{-i\phi(x,t)}e^{i\phi(x,0)}\rangle\right|\simeq
e^{-\frac{1}{2}\Delta{\phi}_{{x},{t}}^{2}},$ (19)

which is actually exact if $\phi(x,t)$ is a Gaussian random variable and is a
good approximation if the field $\phi(x,t)$ is small: in this regime, one can
in fact expand the exponential to second order in $\phi$, compute the phase-
phase correlator (also called the second cumulant), and then re-exponentiate
the result.

However, in the KPZ regime the phase is not a Gaussian random variable and has
a more complex statistics given in Eq. (8). For a limited temporal window, the
fluctuations of $\phi$ are small and one can recover the KPZ phase-phase
correlator by taking the logarithm of $g^{(1)}(x,t)$ Amelio and Carusotto
(2020). At longer times, however, this procedure is no longer legitimate
Fontaine  _et al._ (2022).

It is then even more remarkable to notice the excellent agreement between the
CGLE and KSE predictions for the coherence time that is visible in Fig. 1. For
this, we recall that the fitted quantity in the first case is the logarithm of
$g^{(1)}(t)$, while it is directly the phase-phase correlator in the second
case.

This result suggests that the finite system version of Eq. (8) should have the
form

$\phi(x,t)-\phi(x,0)=v_{\infty}t+\sigma t^{1/3}X+(\gamma_{c}t)^{1/2}Y+...$
(20)

where $X$ is a Baik-Rains-distributed random variable and $Y$ is a Gaussian
variable. According to this formula, the long time decay of the coherence
function is proportional to $\propto\exp\\{-\frac{\gamma_{c}}{2}t+o(t)...\\}$,
where the leading order term is determined by the cumulant of the Gaussian
term of Eq. (20) and is safely obtained by exponentiation; calculation of the
subleading terms denoted as $o(t)$ would instead require a very nontrivial re-
exponentiation of the stochastic variable in Eq. (20). When the logarithm of
the numerically calculated $g^{(1)}(t)$ is fitted to extract the linewidth, no
signatures of a deviation from linear behavior are found: most likely
detection of the subleading terms would require extremely clean data on a very
broad temporal window, which goes beyond our numerical possibilities.

Direct numerical evidence in support of Eq. (20) is displayed in Fig. 3, where
we directly measured the phase of the $g=0$ CGLE and computed the temporal
evolution of its skewness for different system sizes. As usual, the skewness
is defined as the normalized third cumulant

$skew(t)=\frac{\langle(\phi(x,t)-\phi(x,0))^{3}\rangle}{\left[\langle(\phi(x,t)-\phi(x,0))^{2}\rangle\right]^{3/2}}$
(21)

and is a measure of the asymmetry of a distribution. Within the temporal
window corresponding to the KPZ regime, the skewness is finite and has a
magnitude comparable to the one expected from the Baik-Rains distribution. As
expected, it gets smaller at later times, the crossover time depending on the
system size. For system sizes that are too small to support the KPZ regime,
the skewness remains always small.

Similarly to previous studies dealing with the KPZ sub-universality classes in
polariton systems Squizzato  _et al._ (2018), the numerical load of this
calculation makes it difficult to obtain a clean measurement of the skewness.
Nevertheless, the available data confirm the excellent agreement between the
linewidth obtained from $\log g^{(1)}(t)$ and $\Delta{\phi}_{{x},{t}}^{2}$ and
further justify re-exponentiation at very long times.

Figure 4:  (a) Sketch of different boundary conditions for a discrete lattice
geometry. The color code indicates the spatial intensity profile of the steady
state condensate: the enhanced open boundary conditions (eOBC) introduced in
Eq. (22) allow to obtain a uniform density. (b) Coherence time as a function
of the system size in the two cases of PBC (green) and eOBC (orange), plotted
in the same units as in Fig. 1. The system with eOBC supports Bogoliubov modes
with longer lifetime and, thus, displays an earlier departure from the
Schawlow-Townes scaling at smaller system sizes.

##  V The role of boundary conditions

A crucial issue in view of experiments is to understand how the boundary
conditions affect the dynamics of fluctuations and, then, the linewidth. To
highlight the analogy with the most promising device used in Fontaine  _et
al._ (2022) and, at the same time, avoid UV regularizations, in this Section
we focus on the case of a discrete lattice of resonators with hopping $J$,
whose single-particle conservative Hamiltonian will be denoted $H_{latt}$. So
far we were concerned with systems with periodic boundary conditions (PBC), as
in the top sketch of Fig. 4(a). While experimental implementation of a 1D
device with periodic boundary conditions is possible, it may be in practice
not straightforward Contractor  _et al._ (2022).

Standard systems realize in fact open boundary conditions (OBC). From the
point of view of critical systems, this configuration presents serious
drawbacks, since the uniform state is no longer an eigenstate of
$H_{latt}^{OBC}$. This leads to a non-uniform spatial shape of the condensate
mode, involving reflection from the two endpoints and complex interference
phenomena, as shown in the central sketch of Fig. 4(a).

A practical way around, adopted in several current experiments, is to restrict
pumping to the central part of a very long lattice. In spite of the ensuing
outward current Wouters  _et al._ (2008), this configuration allowed to
observe KPZ universality Fontaine  _et al._ (2022) and address the linewidth
problem. While flow can be avoided by imposing an additional harmonic
confinement, as numerically considered in Deligiannis  _et al._ (2020),
quantitative studies of finite size effects would still benefit from using a
spatially uniform condensate and exploting the full length of the avaiable
device.

In what follows, we propose enhanced open boundary conditions (eOBC) which
allow for a uniform condensate in a spatially finite system. In this scheme,
the energies of the two extremal resonators are lowered by an amount $J$, so
the Hamiltonian for the $N$-site lattice system reads

$H_{latt}^{eOBC}=-J\sum_{i=2}^{N}\left[\psi^{\dagger}_{i}\psi_{i-1}+\psi^{\dagger}_{i-1}\psi_{i}\right]-J\left[\psi^{\dagger}_{1}\psi_{1}+\psi^{\dagger}_{N}\psi_{N}\right].$
(22)

In such a configuration, it is straightforward to verify that the uniform
state is an eigenvector $H_{latt}^{eOBC}$ and a steady-state for the lasing
system, as sketched in the last row of Fig. 4(a). We expect that the required
design of the endpoint resonators can be straightforwardly realized in
polariton micropillar systems.

Let us now consider the dynamics of fluctuations in these eOBC systems. As it
can be checked from numerical diagonalization, the second slowest Bogoliubov
mode after the Goldstone consists of a cosine-like wavefunction of wavelength
$2L$. While such a wavelength would not fit in a PBC system, it is allowed in
our eOBC lattice thanks to the relaxed wavelength quantization constraint. As
a consequence, in eOBC one requires half the length $L$ to display an equally
long-lived mode as in PBC. In particular, this reduction affects the critical
size at which the linewidth departs from the Bogoliubov-Schawlow-Townes
prediction and starts showing KPZ features. In Fig. 4(b), the normalized
coherence time is plotted as a function of the system length in natural units.
Here, the prediction for the CGLE in PBC already shown in Fig. 1 (green
points) is compared to the one of the CGLE with eOBC (orange points): in
agreement with our expectations, the Bogoliubov-Schawlow-Townes theory breaks
down at a smaller size $L/\ell^{*}\sim 2.5$ for eOBC compared to the size
$L/\ell^{*}\sim 5$ for PBC. Beyond the crossover, KPZ physics sets in in both
cases and the universal properties are the same.

##  VI Conclusions

In this work, we have studied the long-time decay of the emission coherence of
a one-dimensional non-equilibrium condensate. An exponential decay always
overtake at long enough times in finite-length systems, so that the emission
linewidth can be seen as a finite size effect. Depending on the system length,
two regimes can be identified: for short systems, one can apply a linearized
Bogoliubov theory and find the usual linear scaling of the coherence time with
the system size originally predicted by Schawlow and Townes. On the other
hand, for long wires the scaling of the linewidth is dominated by effects
beyond Bogoliubov and displays a Kardar-Parisi-Zhang critical behaviour,
leading to a square-root dependence on the length.

Markedly different roles of the optical nonlinearities on the linewidth are
highlighed. On the one hand, optical nonlinearities increase the damping rate
of the Bogoliubov modes belonging to the diffusive Goldstone branch and
correspondingly reinforce the Laplacian term of the KPZ equation: this
enhances the spatial stiffness of the phase dynamics and tends to prolonge the
coherence time. On the other hand, the same optical nonlinearities are
responsible for a Henry broadening effect, which reinforces noise and thus
tends to reduce the coherence time. The interplay of these effects leads to a
very nontrivial scaling of the linewidth with the optical nonlinearity
strength and the system size, the optimal coherence being achieved for
intermediate values of the nonlinearity.

This behaviour was demonstrated numerically by solving the full field equation
in the form of a stochastic Complex Ginzburg Landau Equation, and then
explained in terms of a finite-size scaling hypothesis. Successful comparison
of our results with a simulation of the Kuratomoto-Sivashinsky equation
confirms that our conclusion are due to the phase dynamics. We then show how
the linewidth extracted the logarithm of $g^{(1)}(t)$ matches with the
diffusion rate in the phase-phase correlator. This suggests that the cumulant
approximation is legitimate, at least at large times and is explained by
monitoring the temporal evolution of the skewness of the phase, which is shown
to decay in time. On this basis, we conjecture that, due to the finite size of
the system, the phase distribution shifts from a Baik-Rains to a Gaussian form
at very long time, which justifies the accuracy of the re-exponentiation
procedure.

We finally discuss the different experimental platforms where our predictions
can be investigated. In particular, lattice geometries with enhanced open
boundary conditions are proposed, which support a spatially uniform steady-
state lasing mode and thus facilitate experimental investigations of the
scaling with the system size.

Open theoretical questions include a full understanding of the non-monotonic
feature visible in Fig. 1 at the crossover between the Schawlow-Townes and KPZ
scaling, a general study of the scaling of the linewidth with system size in
higher dimensions and the impact of multi-mode laser operation on the
linewidth.

## Acknowledgements

We are grateful to Leonie Canet, Erwin Frey, Anna Minguzzi and Davide
Squizzato for useful discussions. We acknowledge financial support from the
H2020-FETFLAG-2018-2020 project "PhoQuS" (n.820392), from the Provincia
Autonoma di Trento, and from the PNRR MUR project PE0000023-NQSTI. All
numerical calculations were performed using the Julia Programming Language
Bezanson  _et al._ (2017).

##  Appendix A Bogoliubov modes

Figure 5: Real (left) and imaginary (right) part of the Bogoliubov dispersion
of collective excitations Eq. (23) as a function of momentum $k$. Blue (red)
points refer to vanishing (finite) values of the optical nonlinearity $g$.

In this Appendix, we briefly review the Bogloliubov theory of the collective
excitations on top of a non-equilibrium condensate Wouters and Carusotto
(2007); Chiocchetta and Carusotto (2013). The linearized perturbations on top
of the spatially uniform mean-field steady-state $\psi=\sqrt{n_{0}}$ have a
complex frequency dispersion

$\omega_{\pm}(k)=-i\frac{\Gamma}{2}\pm\sqrt{\frac{k^{2}}{2m}\left(\frac{k^{2}}{2m}+2\mu\right)-\left(\frac{\Gamma}{2}\right)^{2}+i0^{+}}$
(23)

as a function of momentum $k$. The real and imaginary parts of the Bogoliubov
spectrum are displayed in the left and right panels of Fig. 5, respectively.
For stronger optical nonlinearities $\mu=g\,n_{0}$, the size of the diffusive
region shrinks in $k$.

The soft mode with $\omega_{+}(k=0)=0$ is the Goldstone mode associated to the
spontaneous symmetry breaking mechanism that fixes the condensate phase. The
low-$k$ long-wavelength region of the Goldstone branch consists of phase-like
modes and recovers the linear part of the KSE (2), with the usual Laplacian
and quartic derivative terms. The amplitude mode corresponds instead to
density fluctuations and shows a finite damping rate $\Gamma$ in the $k\to 0$
long-wavelength limit. The amplitude and Goldstone branches merge at larger
momenta to yield single-particle-like modes with a parabolic dispersion and a
constant decay rate $\Gamma/2$.

As long as the Bogoliubov theory holds, only the free diffusion of the
Goldstone mode contributes to the linewidth. All other modes have in fact a
finite lifetime and do not play any significant role at very long times Amelio
and Carusotto (2020). On this basis, we refer to the Schawlow-Townes linewidth
(that only involves the single condensed mode) as the Bogoliubov linewidth
Amelio and Carusotto (2022). A geometric explanation of the Henry broadening
is that the nonlinearity $\alpha$ makes the the Goldstone and amplitude modes
non-orthogonal Amelio and Carusotto (2022).

## References

  * Huang (1963) K. Huang,  _Statistical Mechanics_ (J. Wiley and Sons, 1963).
  * Brezin (2010) E. Brezin,  _Introduction to Statistical Field Theory_ (Cambridge University Press, 2010).
  * Fisher and Barber (1972) M. E. Fisher and M. N. Barber, Phys. Rev. Lett. 28, 1516 (1972).
  * Cardy (2012) J. Cardy,  _Finite-Size Scaling_, ISSN (Elsevier Science, 2012).
  * Bak  _et al._ (1988) P. Bak, C. Tang, and K. Wiesenfeld, Physical review A 38, 364 (1988).
  * Zhang  _et al._ (1995) J. P. Zhang, D. Y. Chu, S. L. Wu, S. T. Ho, W. G. Bi, C. W. Tu, and R. C. Tiberio, Phys. Rev. Lett. 75, 2678 (1995).
  * Wertz  _et al._ (2010) E. Wertz, L. Ferrier, D. Solnyshkov, R. Johne, D. Sanvitto, A. Lemaître, I. Sagnes, R. Grousson, A. V. Kavokin, P. Senellart, _et al._ , Nature physics 6, 860 (2010).
  * Amo and Bloch (2016) A. Amo and J. Bloch, Comptes Rendus Physique 17, 934 (2016), polariton physics / Physique des polaritons.
  * Fontaine  _et al._ (2022) Q. Fontaine, D. Squizzato, F. Baboux, I. Amelio, A. Lemaître, M. Morassi, I. Sagnes, L. L. Gratiet, A. Harouri, M. Wouters, I. Carusotto, A. Amo, M. Richard, A. Minguzzi, L. Canet, S. Ravets, and J. Bloch, Nature 608, 687 (2022).
  * Grabherr  _et al._ (1999) M. Grabherr, M. Miller, R. Jager, R. Michalzik, U. Martin, H. Unold, and K. Ebeling, IEEE Journal of Selected Topics in Quantum Electronics 5, 495 (1999).
  * Cui  _et al._ (2009) J. Cui, Y. Ning, Y. Zhang, P. Kong, G. Liu, X. Zhang, Z. Wang, T. Li, Y. Sun, and L. Wang, Appl. Opt. 48, 3317 (2009).
  * Harari  _et al._ (2018) G. Harari, M. A. Bandres, Y. Lumer, M. C. Rechtsman, Y. D. Chong, M. Khajavikhan, D. N. Christodoulides, and M. Segev, Science 359, eaar4003 (2018).
  * Bahari  _et al._ (2017) B. Bahari, A. Ndao, F. Vallini, A. El Amili, Y. Fainman, and B. Kanté, Science 358, 636 (2017), https://science.sciencemag.org/content/358/6363/636.full.pdf .
  * Loirette-Pelous  _et al._ (2021) A. Loirette-Pelous, I. Amelio, M. Seclì, and I. Carusotto, Phys. Rev. A 104, 053516 (2021).
  * Amelio and Carusotto (2020) I. Amelio and I. Carusotto, Phys. Rev. X 10, 041060 (2020).
  * Schawlow and Townes (1958) A. L. Schawlow and C. H. Townes, Phys. Rev. 112, 1940 (1958).
  * Gladilin  _et al._ (2014) V. N. Gladilin, K. Ji, and M. Wouters, Phys. Rev. A 90, 023615 (2014).
  * Altman  _et al._ (2015) E. Altman, L. M. Sieberer, L. Chen, S. Diehl, and J. Toner, Phys. Rev. X 5, 011017 (2015).
  * He  _et al._ (2015) L. He, L. M. Sieberer, E. Altman, and S. Diehl, Phys. Rev. B 92, 155307 (2015).
  * Squizzato  _et al._ (2018) D. Squizzato, L. Canet, and A. Minguzzi, Phys. Rev. B 97, 195453 (2018).
  * Kardar  _et al._ (1986) M. Kardar, G. Parisi, and Y.-C. Zhang, Physical Review Letters 56, 889 (1986).
  * Takeuchi (2018) K. A. Takeuchi, Physica A: Statistical Mechanics and its Applications 504, 77 (2018), lecture Notes of the 14th International Summer School on Fundamental Problems in Statistical Physics.
  * Note (1) Note how our Eq. (2) differs from the equation considered in Gladilin  _et al._ (2014) in that we are not making the near-threshold approximation $\alpha\ll 1$.
  * Henry (1982) C. Henry, IEEE Journal of Quantum Electronics 18, 259 (1982).
  * He  _et al._ (2017) L. He, L. M. Sieberer, and S. Diehl, Phys. Rev. Lett. 118, 085301 (2017).
  * Ueno  _et al._ (2005) K. Ueno, H. Sakaguchi, and M. Okamura, Phys. Rev. E 71, 046138 (2005).
  * Prähofer and Spohn (2004) M. Prähofer and H. Spohn, Journal of Statistical Physics 115, 255 (2004).
  * Baik and Rains (2001) J. Baik and E. M. Rains, Duke Mathematical Journal 109, 205 (2001).
  * Wouters and Carusotto (2007) M. Wouters and I. Carusotto, Phys. Rev. Lett. 99, 140402 (2007).
  * Amelio and Carusotto (2022) I. Amelio and I. Carusotto, Phys. Rev. A 105, 023527 (2022).
  * Sieberer  _et al._ (2013) L. M. Sieberer, S. D. Huber, E. Altman, and S. Diehl, Phys. Rev. Lett. 110, 195301 (2013).
  * Kloss  _et al._ (2012) T. Kloss, L. Canet, and N. Wschebor, Phys. Rev. E 86, 051124 (2012).
  * Zamora  _et al._ (2017) A. Zamora, L. M. Sieberer, K. Dunnett, S. Diehl, and M. H. Szymańska, Phys. Rev. X 7, 041006 (2017).
  * Ferrier  _et al._ (2020) A. Ferrier, A. Zamora, G. Dagvadorj, and M. H. Szymańska, (2020), arXiv:2009.05177 [cond-mat.quant-gas] .
  * Contractor  _et al._ (2022) R. Contractor, W. Noh, W. Redjem, W. Qarony, E. Martin, S. Dhuey, A. Schwartzberg, and B. Kanté, Nature 608, 692 (2022).
  * Wouters  _et al._ (2008) M. Wouters, I. Carusotto, and C. Ciuti, Phys. Rev. B 77, 115340 (2008).
  * Deligiannis  _et al._ (2020) K. Deligiannis, D. Squizzato, A. Minguzzi, and L. Canet, Europhysics Letters 132, 67004 (2020).
  * Bezanson  _et al._ (2017) J. Bezanson, A. Edelman, S. Karpinski, and V. B. Shah, SIAM review 59, 65 (2017).
  * Chiocchetta and Carusotto (2013) A. Chiocchetta and I. Carusotto, EPL (Europhysics Letters) 102, 67007 (2013).