KPZ scaling in one-dimensional arrays of photon condensates

Abstract: Bose-Einstein condensates of photons in dye filled microcavities are known to feature many properties of the ideal bose gas at thermal equilibrium. Nevertheless, the finite photon life time renders them driven dissipative systems where losses are balanced by continuous driving by a pump laser. We show here with simulations based on a classical field model that their driven-dissipative nature causes their first order coherence to feature Kardar-Parisi-Zhang (KPZ) scaling, a phenomenon well known in the related condensates of exciton-polaritons. Remarkably, we find that the KPZ scaling even occurs in a regime with large density fluctuations. We discuss the modification of the scaling by the occupation of excited states and how to recover the universal scaling behavior by spectral filtering.

• The paper demonstrates KPZ scaling in one-dimensional dye-filled photon-condensate arrays through stochastic classical-field simulations, including the grand-canonical regime where the quasi-condensate fraction is nearly zero and temporal fluctuations follow the predicted β = 1/3 law.
• The simulations reproduce the universal Prähofer–Spohn spatiotemporal coherence function without filtering in the deeply grand-canonical regime, with fitted scaling parameters y₀ ≈ 0.83 and C₀ ≈ 3.0 × 10⁻², while analytical weak-fluctuation predictions fail quantitatively.
• In the crossover regime, high-energy propagating modes prevent raw data from collapsing onto the KPZ curve, but spectral filtering restores scaling with y₀ ≈ 0.91 and C₀ ≈ 2.3 × 10⁻², highlighting a practical route for identifying universality experimentally.

The KPZ universality class has been established as the framework governing long-wavelength phase fluctuations of incoherently pumped quantum fluids, with experimental confirmation in exciton-polariton condensates. Whether photon condensates—systems that simultaneously exhibit near-thermal equilibrium properties and strong driven-dissipative character—share this universality has remained an open theoretical question, complicated by their anomalously large density fluctuations. In "KPZ scaling in one-dimensional arrays of photon condensates" (2608.17602), Yzewyn and Wouters answer this question affirmatively through stochastic classical field simulations of a one-dimensional lattice of dye-filled microcavities. Their central finding is twofold: KPZ scaling of the spatiotemporal first-order coherence persists even deep in the grand-canonical regime where the quasi-condensate fraction is essentially zero, and the universal scaling collapse in the crossover regime is recovered only after spectral filtering of high-energy modes.

Model and physical setting

The authors simulate a classical stochastic field $\psi(x,t)$ on a lattice of coupled cavities, governed by a driven-dissipative Gross–Pitaevskii-type equation with complex tunneling (real hopping $J$, imaginary part set by the Kennard–Stepanov-related parameter $\kappa = B_{12}M_1/2k_BT$), gain from excited dye molecules $M_2$, cavity loss $\gamma$, and additive complex Gaussian noise. The photon field is coupled to a rate equation for the excited molecular population with incoherent pump $P = \gamma \bar n$. This model, analogous to truncated Wigner descriptions of polariton condensates, captures both Hamiltonian supercurrent physics and dissipative gain/loss dynamics.

The density fluctuation magnitude is controlled by the ratio $\bar n^2/M_{\rm eff}$, with $M_{\rm eff}$ the effective reservoir size. Two regimes emerge: a grand-canonical regime with $g^{(2)}(0) \approx 2$ and vanishing quasi-condensate density, and a canonical regime with suppressed fluctuations. This distinguishes photon condensates from polariton condensates, where $g^{(2)}(0)$ drops to unity above threshold. The authors note that the steady state is fundamentally nonequilibrium because the real and imaginary parts of the relaxation coefficients do not coincide, even though $J$0-terms drive model-A relaxation toward equilibrium.

Analytical derivation

Adiabatically eliminating density and reservoir fluctuations under the assumption of weak density fluctuations, the authors derive a Kuramoto–Sivashinsky-type equation for the phase that reduces at long wavelengths to the KPZ equation:

$J$1

with KPZ coefficients $J$2, $J$3, and noise $J$4. This derivation yields analytical predictions for the nonuniversal scaling parameters, e.g. $J$5 and $J$6. The derivation also predicts a spatial coherence length matching that of the weakly interacting Bose gas, $J$7—twice the noninteracting value. The authors flag an important inconsistency here: their simulations instead reproduce the noninteracting Bose gas coherence length, indicating that the weak-fluctuation approximations underlying the derivation fail in the regimes of interest, even though the KPZ universality itself survives.

Grand-canonical regime: unfiltered KPZ scaling

In the deeply grand-canonical regime ($J$8, numerically $J$9), simulations on a 1024-site lattice yield three consistent results. First, the spatial coherence decays exponentially with the NIBG length $\kappa = B_{12}M_1/2k_BT$0, in good agreement with theory. Second, the temporal coherence follows the KPZ growth law with $\kappa = B_{12}M_1/2k_BT$1, i.e. phase fluctuations scaling as $\kappa = B_{12}M_1/2k_BT$2, discriminating against Edwards–Wilkinson scaling ($\kappa = B_{12}M_1/2k_BT$3). Third, and most importantly, the rescaled spatiotemporal coherence collapses onto the exact universal KPZ scaling function of Prähofer and Spohn over the window $\kappa = B_{12}M_1/2k_BT$4, with fitted parameters $\kappa = B_{12}M_1/2k_BT$5 and $\kappa = B_{12}M_1/2k_BT$6.

This result constitutes the paper's strongest claim: KPZ scaling is robust even when the quasi-condensate density is nearly zero and density fluctuations approach grand-canonical values. The implication is that the standard theoretical route to KPZ—expanding around a well-defined condensate with weak fluctuations—is sufficient but not necessary for the emergence of the universality class. Notably, the fitted scaling parameters deviate strongly from the analytical values ($\kappa = B_{12}M_1/2k_BT$7, $\kappa = B_{12}M_1/2k_BT$8), which the authors attribute directly to the breakdown of the weak-fluctuation assumption; the universality class is preserved but the nonuniversal amplitudes are not captured by the linearized theory.

Crossover regime: the need for spectral filtering

At intermediate fluctuations ($\kappa = B_{12}M_1/2k_BT$9, $M_2$0, $M_2$1), the picture changes qualitatively. The spatial coherence decays exponentially with an intermediate length $M_2$2, between the NIBG and WIBG values. More significantly, the raw spatiotemporal data fail to collapse onto the universal KPZ curve: while asymptotic slopes match for small and large values of the scaling variable $M_2$3, no single pair $M_2$4 produces a collapse, with pronounced deviations around $M_2$5.

The energy-momentum spectrum explains why. Unlike the grand-canonical case, where the spectrum is dominated by the $M_2$6 condensate peak, the crossover regime shows substantial luminescence on the gapped particle and ghost branches of the Bogoliubov excitation spectrum, as well as emission following the bare tight-binding dispersion—a feature not captured by the linearized theory and signaling significant nonlinear corrections. Applying an exponential spectral filter $M_2$7 with $M_2$8 and $M_2$9 to suppress high-energy modes restores a very good collapse onto the universal KPZ scaling function, with $\gamma$0 and $\gamma$1.

The authors interpret this contrast through the diffusive character of the spectrum: in the grand-canonical regime the dynamics are fully diffusive, so no filtering is needed, whereas in the crossover regime residual propagating (ballistic) modes contaminate the coherence and must be removed. They candidly leave open whether the unfiltered scaling curve in the crossover regime possesses any universal character of its own.

Limitations and open questions

Several limitations are stated explicitly. The analytical derivation of the KPZ coefficients relies on weak density fluctuations, an assumption violated in the grand-canonical regime where the theory's quantitative predictions for $\gamma$2 fail by large factors. The expected late-time Schawlow–Townes crossover for finite systems is not observed due to insufficient statistical convergence, restricting the accessible scaling window. Deeper in the canonical regime, no parameter set within numerically accessible system sizes produced observable KPZ scaling, so the extent of the KPZ phase toward canonical statistics remains unmapped. The study assumes a perfectly disorder-free lattice, and the effect of disorder on the spatiotemporal scaling is deferred to future work. Finally, the model is specific to dye-filled microcavities; whether the same filtering requirement arises in semiconductor photon condensates is untested.

Conclusion

This work establishes, through a quantitatively constrained classical field model, that one-dimensional photon condensates exhibit KPZ scaling of their spatiotemporal coherence despite—and remarkably, independently of—their large grand-canonical density fluctuations. The successful unfiltered scaling collapse in the grand-canonical regime, together with the filter-dependent collapse in the crossover regime, delineates the conditions under which the universal KPZ scaling function is directly observable. The mismatch between analytical and numerical scaling amplitudes identifies the weak-fluctuation expansion as the limiting approximation, and the open questions concerning unfiltered universality, deeper canonical regimes, and disorder effects define a concrete agenda for subsequent theoretical and experimental work on photonic quantum fluids.