REVIEW 4 major objections 4 minor 47 references
Crossing over universal scaling laws in two-dimensional driven dissipative condensates
T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash
Pith's one-line read Two-dimensional polariton condensates can be tuned between KPZ and Edwards–Wilkinson universal coherence decays.
desk verdict First real experimental claim of 2D KPZ scaling in a continuous planar polariton condensate, with an EW-KPZ crossover; the evidence is substantial but the partly manual analysis pipeline needs a null test before the identification is air-tight. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the mapping of the condensate phase dynamics onto a two-dimensional Kardar–Parisi–Zhang equation, ∂tθ = ν∇²θ + (λ/2)(∇θ)² + √D η, where ν, λ, D are effective parameters built from the microscopic gain, loss, mass, and interaction coefficients. The dimensionless effective nonlinearity g_KPZ = λ²D/ν³ decides the class: weak nonlinearity gives the Edwards–Wilkinson diffusive regime with logarithmic phase correlations (algebraic coherence decays), and strong nonlinearity gives the KPZ super-diffusive regime with power-law phase correlations (stretched-exponential coherence decays). The link to the experiment is the identity g(1)(Δr,Δt) ∝ exp(−C_θθ(Δr,Δt)/2), relating the measured first-order coherence to the two-point phase correlation C_θθ; the paper verifies numerically that density correlations are negligible in the scaling windows. Interferometric measurement of g(1) followed by dispersion-branch filtering and the κ renormalization converts the measured fringe visibility into a test of the universal scaling functions.
What would settle it
Measure the phase correlation function C_θθ(Δr,Δt) directly in the same sample—for instance by off-axis digital holography that retrieves the optical phase—and check whether it obeys $Δt^{{2β}}$ or logarithmic growth with the fitted exponents; if the directly measured C_θθ does not show the same scaling, the coherence exponents are not evidence for KPZ or EW universality.
Extended reading notes
Core claim
The paper's central claim is that a continuous two-dimensional polariton condensate in a planar microcavity can realize both the KPZ and EW universality classes as its microscopic parameters are varied, and that the crossover between them appears in the spatio-temporal decay of the first-order coherence. In the more excitonic regime (detuning ℏδ=−5.2 meV) the coherence decays as stretched exponentials g(1)∝exp(−A_t $Δt^{{2β}}$) in time and g(1)∝exp(−A_r $Δr^{{2χ}}$) in space, with fitted exponents agreeing with the numerical KPZ values β≈0.24 and χ≈0.39; the full space-time data collapse onto the 2D KPZ scaling function. In the more photonic regime (ℏδ=−12.6 meV) the same measurements show power-law decays, with spatial exponent a_s>0.25 and ratio a_s/a_t≈2, the signature of the EW diffusive phase. A statistical comparison of stretched-exponential versus power-law fits across detuning and pump power yields a phase diagram showing the EW-to-KPZ crossover, and numerical simulations with experimentally calibrated parameters reproduce both regimes and show that vortex-antivortex pairs remain bound, so the quasi-ordered KPZ/EW phases are the nonequilibrium analog of the ordered BKT phase.
Load-bearing premise
The classification stands or falls on whether, after removing the dispersive branches and excluding short-time and short-distance transients, the measured g(1) decay is governed by condensate phase fluctuations through g(1)∝exp(−C_θθ/2), with density fluctuations and uncondensed-polariton emission negligible in the selected windows.
Editorial extensions
If this is right
- Tuning a single accessible parameter—the cavity-exciton detuning—moves a driven-dissipative condensate across a universality-class crossover, giving experimental control over KPZ versus EW behavior.
- The measured collapse onto the KPZ universal scaling function in a continuous 2D system resolves the earlier debate about whether vortices necessarily destroy KPZ order in planar polariton condensates.
- The phase diagram in detuning and pump power provides a map for choosing parameters that exhibit KPZ or EW scaling, which can guide future experiments.
- Because the EW regime is only transient for a finite system (in infinite size, 2D KPZ is the stable fixed point), the observed EW window is a finite-size effect that should shrink when the pump spot is enlarged.
- Vortex-antivortex pairing in numerics implies that topological defects do not spoil the universal scaling in the measured window, so the KPZ/EW phases are the nonequilibrium counterpart of the ordered BKT phase.
Reading between the lines
- Beyond the paper, the same coherence-measurement protocol could be applied to other dissipative condensates of light—photon condensates, plasmonic lattices, or vertical-cavity lasers—to test whether the same detuning-controlled crossover appears there.
- The central relation g(1) ≈ exp(−C_θθ/2) is validated here mainly by numerics; a future experiment that directly images the phase field (for example by off-axis holography) could independently extract C_θθ and check the exponent attribution without relying on window selection.
- If the finite-size explanation for the EW regime is right, systematically varying the pump spot diameter should move the onset of the Schawlow–Townes exponential tail and change the extent of the EW window; this is a quantitative prediction the paper does not make explicitly.
- The vortex statistics suggest that increasing the noise or the pump spot beyond the studied range should eventually favor free vortices and a transition to the spiral-vortex or disordered phase that the paper lists as future work.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper reports interferometric measurements of the first-order coherence g(1)(Δr,Δt) of continuous two-dimensional polariton condensates at two cavity-exciton detunings and several pump powers. For a more excitonic detuning (ℏδ=−5.2 meV and −6.8 meV), the authors observe stretched-exponential decays with fitted exponents β≈0.24 and χ≈0.39, which they attribute to the 2D KPZ universality class, and they show collapses of the spatiotemporal data onto the KPZ universal scaling function. For a more photonic detuning (ℏδ=−12.6 meV), they observe algebraic decays with spatial and temporal exponents satisfying a_s/a_t≈2, which they attribute to the EW universality class. A statistical phase diagram as a function of detuning and pump power is constructed by comparing power-law and stretched-exponential fits. Numerical simulations of the generalized Gross-Pitaevskii equation with realistic parameters reproduce the two regimes and show that vortex-antivortex pairs dominate, supporting the interpretation that the observed phases are the nonequilibrium analogue of the ordered BKT phase.
Significance. If the central claim is correct, this is a substantial experimental advance: it would establish continuous two-dimensional driven-dissipative condensates as a platform for observing 2D KPZ universality and a crossover to EW scaling, going beyond prior 1D demonstrations and addressing the debate about vortex-induced destruction of KPZ order. The paper has genuine strengths: multiple independent datasets at several detunings and powers, raw and filtered collapses, comparison with independent theoretical exponents and scaling functions, a parameter-free linearized model for the threshold determination, and numerical simulations with realistic parameters that explicitly check the phase-only approximation and the pairing of vortices. The main weakness is that the identification of the universal regimes depends at several steps on manually chosen analysis choices: the grey-shaded scaling windows, the free-threshold dispersive-branch filter, the renormalization constant κ, and the scale factors C_0 and y_0. These choices are not accompanied by null tests or a systematic sensitivity analysis, so the claimed exponents may partly reflect the flexibility of the analysis pipeline.
major comments (4)
- [Main text Fig. 2d-f; SI S3.5] The dispersive-branch filtering procedure is load-bearing for the KPZ claim, especially at lower powers, but it is not blinded or shown to be harmless. The main text states that filtering can 'reveal otherwise masked KPZ stretched exponential decays' and can 'shift the oblique asymptote and align it with the KPZ universal function'; the SI procedure uses red and green threshold lines chosen by inspection of the Fourier phase maps (Fig. S8c). A null test is needed: the same filtering and window-selection pipeline should be applied to synthetic g(1)(Δr,Δt) maps with known non-universal decays (e.g., simple exponentials, power laws with a_s/a_t≠2, stretched exponentials with β≠0.24) and the authors should show that the pipeline does not produce apparent KPZ exponents and collapses. Without such a test, the filtered spatial scaling in Fig. 2d-e and the asymptotic alignment in Fig. 2f cannot be fully distinguished from artifacts of the filter.
- [Fig. 2a-b and Fig. 3a-b; SI S3.8] The KPZ and EW scaling windows are hand-picked grey-shaded regions, and the temporal KPZ window spans only about 2–35 ps, roughly 1.2 decades. The reported fit uncertainties on β_exp and χ_exp in the insets of Fig. 2 are only statistical fit errors and do not include uncertainty from window choice. This matters because the SI itself shows a systematic effect: at ℏδ=−6.8 meV (Fig. S13), the fitted β is larger than 0.24 and the deviation is attributed to weak oscillations, demonstrating that fit-only error bars understate the systematics. The authors should state the objective criteria for window selection and provide a sensitivity sweep of the fitted exponents as the window bounds are varied by plausible amounts.
- [SI S3.6; Fig. 2 caption and Fig. S13 caption] The collapse onto the KPZ universal scaling function uses the renormalization constant κ, which is obtained by extrapolating stretched-exponential fits to Δt=0, and the scale factors C_0 and y_0, which are adjusted to the experimental data. The SI correctly warns that an incorrect κ 'can lead to misleading conclusions', but the present test has at least two free parameters whose values are not independently measured. The authors should quantify the effective number of independent data points entering each collapse and demonstrate that alternative scaling functions or wrong exponents fail the same collapse test. As it stands, the agreement in Figs. 2c and 2f, while visually good, has an unknown false-positive rate.
- [SI Eq. (14); SI S5.2] The extraction of universal exponents from g(1) relies on the approximation g(1)≈g_n^(1)⟨e^{iΔθ}⟩≈exp(−C_θθ/2), i.e., that density fluctuations and uncondensed-polariton emission are negligible in the selected windows. This is validated only numerically (Fig. S15 and related discussion), not by a direct experimental decomposition of the measured coherence into density and phase contributions. Since the filtering step removes the dispersive branches associated with uncondensed polaritons, the residual filtered signal is assumed to be the condensate phase contribution. This is a reasonable assumption, but it would be strengthened by analyzing the sensitivity of the extracted exponents to the filter strength and by showing that the conclusions are stable when the filtering is varied within a physically motivated range.
minor comments (4)
- [SI Table I] The line for the KPZ dataset lists ℏδ=−15.2 meV, whereas the main text and the rest of the SI use ℏδ=−5.2 meV for the KPZ regime; this appears to be a typographical error and should be corrected.
- [Throughout] The name Berezinskii is misspelled as 'Berezinski' in the abstract and introduction; the standard transliteration should be used consistently.
- [Fig. 4 and SI S3.7] The statistical comparison fixes the stretched-exponential exponent to the KPZ value β=0.24 while leaving the power-law exponent free; this asymmetry should be stated explicitly in the main text so that the phase diagram is not misinterpreted as a model selection over equally parameterized hypotheses.
- [SI S5.3] The statement that 'the KPZ regime is accessed starting from the EW regime by simultaneously increasing the interaction strength g_R via μ and the loss rate γ_2' would benefit from a small quantitative summary of how g_KPZ varies across the two parameter sets in Table II, since this is the physical mechanism behind the observed crossover.
Circularity Check
No significant circularity: the KPZ/EW identification rests on independent free-exponent fits and external universal functions; the fixed-β rescaling and self-citations create selection/robustness concerns but not a by-construction reduction.
full rationale
The paper's central claim—a crossover between KPZ and EW scaling in the coherence of 2D polariton condensates—is not derived from its own inputs. The theoretical mapping from the generalized Gross-Pitaevskii equation to the KPZ equation is cited to the authors' prior Ref. [29], and the EW connection to Ref. [15], but these are separate published derivations, not restatements of the present measurement. The experimental data are new, and the critical exponents are obtained from free stretched-exponential and power-law fits (insets of Figs. 2, 3 and S13), then compared with independent numerical estimates (Ref. [46]) and universal scaling functions from functional renormalization (Refs. [47,48]). The collapse plots tune only non-universal constants C0, y0, and κ, which is standard practice for scaling analyses. The use of Δt^{2β} with β=0.24 to construct the KPZ plots, followed by identification of 'grey-shaded' windows where the data appear straight, introduces a mild selection bias: the window is chosen on axes already rescaled with the hypothesized exponent, and the fits are performed inside that window. This is a methodological weakness rather than a formal circularity, since the free fits can and do deviate from the assumed value (the SI reports β systematically larger than 0.24 at δ=-6.8 meV). Similarly, the phase diagram of Fig. 4 hard-codes β=0.24 in the AIC comparison, so its KPZ-versus-EW coloring is partly constructed from the exponent it claims to identify; however, the direct free-exponent fits provide independent content. The dispersive-branch filter has tunable thresholds and the text states it 'may enable revealing otherwise masked KPZ stretched exponential decays,' but the filter is a deterministic transform, not a fitted parameter renamed as a prediction. No equation in the paper reduces to its own input by construction. The main limitations—window selection, filter thresholds, and absence of a null test—are robustness concerns, not circular derivation.
Assumptions & free parameters
free parameters (4)
- Microscopic parameters m, γ0, γ2/γ0 =
m ≈ 5.9e-5 m_e (KPZ) and 4.7e-5 m_e (EW); ℏγ0 ≈ 35±5 µeV; γ2/γ0 ≈ 0.3±0.05 µm² (KPZ) and 3.0±0.5 µm² (EW)
- γR/R ratio =
8±2×10^4 (KPZ), 12±2×10^4 (EW)
- Normalization constants κ, C0, y0 =
κ ≈ 1.7 to 2.45 depending on dataset; C0 and y0 adjusted by hand
- Effective viscosity ν_e and noise strength D_e in EW collapse =
ν_e ≈ 4 µm²/ps, D_e ≈ 29 µm²/ps
assumptions (4)
- domain assumption The condensate phase obeys a KPZ equation with coefficients derived from the gGPE via weak-density and adiabatic-reservoir assumptions.
- domain assumption The first-order coherence is approximated by exp(−Cθθ/2), with density-phase correlations neglected.
- domain assumption A finite system permits a transient EW regime even though the infinite-size KPZ fixed point is stable.
- standard math The 2D KPZ and EW universal exponents and scaling functions from Refs. [46,47,48] are the correct external benchmarks.
Cite this review
Pith. "Pith review of Crossing over universal scaling laws in two-dimensional driven dissipative condensates." pith.science (2026). https://pith.science/paper/4M4VJZZS
@misc{pith2026260807242,
author = {Pith},
title = {Pith review of: Crossing over universal scaling laws in two-dimensional driven dissipative condensates},
year = {2026},
howpublished = {\url{https://pith.science/paper/4M4VJZZS}},
note = {Machine review of arXiv:2608.07242}
}
read the original abstract
In low dimensional systems, fluctuations are enhanced and prevent the spontaneous breaking of continuous symmetries. As a result, spatial and temporal correlation functions decay at large distances and long times. A well established example is given by two-dimensional bosonic condensates at equilibrium, which do not display long-range order of the coherence but algebraic decay belonging to the Berezinski-Kosterlitz-Thouless universality class. In contrast, the universal behaviors of non-equilibrium bosonic condensates are more diverse and many open questions remain. Here, we explore the spatio-temporal coherence properties of two-dimensional driven-dissipative polariton condensates in semiconductor optical microcavities. By tuning microscopic parameters, we observe a cross-over between two scaling laws that we attribute to the Edwards-Wilkinson (EW) and the Kardar-Parisi-Zhang (KPZ) universality classes. We demonstrate the collapse of the measured first-order correlations onto the EW and KPZ universal scaling functions and obtain critical exponents, well matching the values predicted theoretically. Our results highlight the intrinsic non-equilibrium nature of polariton condensates and establish them as a platform of choice for controlled exploration of the two-dimensional KPZ universality class.
Figures
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Reference graph
Works this paper leans on
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[1]
The pump is spatially homogeneous,
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[2]
The polariton density is low enough so that interactions can be neglectedg|ψ| 2 ≈0,
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[3]
Properties of the condensate coherence The first-order correlation function of the polariton condensate is defined by: g(1)(∆r,∆t) = ψ(∆r+r 0,∆t+t 0)ψ∗(r0, t0) q |ψ(∆r+r 0,∆t+t 0)|2 |ψ(r0,+t 0)|2 ,(13) wheret 0 is a reference time chosen in the non-equilibrium steady state of the condensate dy- namics, andr 0 a reference point chosen at the center of the ...
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[4]
Indeed, under this condition, the microscopic model Eqs
Linearized model In this section, we derive analytical results describing the emission of uncondensed po- laritons, valid at low pumping power,i.e., far below threshold. Indeed, under this condition, the microscopic model Eqs. (6)-(7) can be linearized assuming that:
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[5]
The reservoir dynamics is adiabatically eliminated usingR|ψ| 2 ≪γ R, yieldingn R ≃ P γR+R|ψ|2 ≃ P γR . The polariton field in Fourier space then reads ψ(k, ω) = ξ(k, ω)/ℏ [ω− ω(k)] + iℏ 2 [γ0(1−p) +γ 2k2],(19) where⟨ξ(k, ω)ξ∗(k′, ω′)⟩= 2ℏ 2σ×(2π) 2+1δ(2)(k−k ′)δ(ω−ω ′) and ω(k) = 2g RnR +ℏ 2k2/2m is the dispersion in the vicinity of the ground state. We a...
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[6]
Sample structure and initial characterization The sample is a semiconductor heterostructure grown by molecular beam epitaxy. It con- sists of a 5λ/2 Ga 0.05Al0.95As microcavity surrounded by two Al 0.20Ga0.80As/Al0.05Ga0.95As distributed Bragg reflectors with 26 (38) pairs in the top (bottom) mirror. Five stacks of pairs of 20 nm thick GaAs quantum wells ...
work page 2000
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[7]
Experimental setup and data processing In this section, we describe the experimental techniques employed in our experiments to measure the coherence of polariton condensates
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[8]
A sketch of the optical setup is shown in Fig
Experimental setup The sample described previously is kept at 4 K in a closed-cycle cryostat. A sketch of the optical setup is shown in Fig. S2. Polaritons are generated by off-resonant, quasi-cw excitation of the microcavity at 735 nm. A top-hat beam shaper converts the excitation into a nearly collimated flat-top beam, which is subsequently de-magnified...
Show all 47 references
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[9]
A PID controller stabilizes the intensity at the desired level for both long and short pulses
Experimental sequence An acousto-optic modulator (AOM) driven by a waveform generator (WG) shapes the temporal intensity profile of the excitation beam into the sequence shown in the insetb1: alternating 170 ms and 10 ms pulses separated by 10 ms dark intervals, with a 5 Hz re...
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[10]
A retroreflector mounted on a high-precision motorized translation stage controls the path-length difference between arms
Michelson interferometry for coherence measurements The spatiotemporal coherence of the microcavity emission is measured by inserting a Michelson interferometer betweenL 3 andL 4, where the light is collimated. A retroreflector mounted on a high-precision motorized translation...
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[11]
Experimental determination of the microscopic parameters In this section, we experimentally determine the microscopic parameters characterizing the polariton condensate, and entering Eq. (6). Namely we provide estimated values for the polariton massm, the bare polariton linewi...
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[12]
It slightly varies with the exciton-photon detuning:m= 5.9× 10−5 me forℏδ=−5.2 meV (Fig
Determination of the polariton massm The polariton effective mass is extracted by fitting the bottom of the lower polariton branch with a parabola. It slightly varies with the exciton-photon detuning:m= 5.9× 10−5 me forℏδ=−5.2 meV (Fig. 2 of the main text) andm= 4.7×10 −5 me f...
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[13]
S3 2 3, the first-order coherence function, g(1)(∆x,∆y,∆t) , is ex- tracted at each time delay ∆tfrom the fringe visibility of the corresponding interferogram
Determination of the zero-momentum linewidthγ 0 As detailed in Sec. S3 2 3, the first-order coherence function, g(1)(∆x,∆y,∆t) , is ex- tracted at each time delay ∆tfrom the fringe visibility of the corresponding interferogram. By closing the spectrometer slit aroundx= 0 and i...
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[14]
S4.Measurement of the gain-curvature coefficientγ 2
Determination of the gain-curvature coefficientγ 2 The gain-curvature coefficientγ 2 is measured experimentally through the fit of the conden- sate momentum distribution,n(k), below transparency threshold to the analytical prediction 45 a1 a2 b1 b2 c 0 2 Bottleneck 4 10-2 100 ...
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[15]
Determination of the reservoir coefficientγ R/R In this paragraph, we experimentally determine the microscopic parameter characterizing the exciton reservoir in Eq. (7). Namely we provide estimated values for the ratioγ R/R through the I-P curve. A minimal kinetic model for a ...
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[16]
(24) of Sec
Determination of the condensation thresholdP th Table I lists the experimentally extracted microscopic parameters entering the linearized model described in Eq. (24) of Sec. S2 4. In the present section, we compare the model pre- dictions with the measured spatiotemporal coher...
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[17]
These dispersive branches are most pro- nounced close to the condensation threshold, where the condensed fraction only marginally exceeds the non-condensed one
Filtering of the dispersive branches As mentioned in the main text, the residual population of the polariton dispersion near the ground state gives rise to dispersive branches in the space-time correlations, appearing as alternating regions of enhanced/reduced coherence. These...
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[18]
The coherence heat map g(1)(∆r,∆t) is padded with zero such as dim g(1) p = 2 dim g(1) + 1
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[19]
We compute the fast Fourier transform (FFT) of g(1) p : FFT(k, E) = fftshift fft2 g(1) p (∆r,∆t)
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[20]
S8a 1) and amplitudeA 0 =|FFT| (see Fig
We retrieve its phaseθ 0 = angle (FFT) (see Fig. S8a 1) and amplitudeA 0 =|FFT| (see Fig. S9a). 53 →STEP 2 - Phase filtering
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[21]
We start by computingθ= fftshift (θ 0) (Fig. S8b 1)
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[22]
The result is shown in Fig
We remove the phase rampθ(1,:) from the phase heat mapθ: ∆θ=θ−θ(1,:). The result is shown in Fig. S8c 1
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[23]
We average all rows in ∆θlying in between the red lines in Fig. S8c 1). We call ∆θ the resulting mean vector
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[24]
bellow) the upper (resp
All rows in ∆θlying above (resp. bellow) the upper (resp. lower) green lines are replaced by the mean vector ∆θ, thereby removing most of the dispersive features visible in the upper-left and lower-right corners of Fig. S8c 1. The resulting heat map is shown in Fig. S8c 2. We ...
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[25]
We finally retrieve the corrected phaseθ c by performing another fftshift (see Fig. S8a2). →STEP 3 - Amplitude filtering We now aim at filtering the fft amplitudeA 0 (see Fig. S9a)
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[26]
We replicate and flip this quadrant to replace{k≤0, E≥0} and{k≥0, E≤0}(see Fig
The quadrant{k≥0, E≥0}(as well as{k≤0, E≤0}) inA 0 does not show any dispersive feature. We replicate and flip this quadrant to replace{k≤0, E≥0} and{k≥0, E≤0}(see Fig. S9b). The corrected amplitude is denotedA c. Fig. S9c shows the difference betweenA 0 andA c, highlighting t...
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[27]
The initial FFT phase and amplitude are replaced by the corrected ones: FFT(k, E) =A0(k, E) exp (iθ0(k, E))− →FFTc(k, E) =Ac(k, E)×exp (iθc(k, E))
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[28]
The result is shown in Fig
We compute the 2D inverse Fourier transform (ifft2) and crop the resulting g(1) c heat map to its original dimensions: g(1) c (∆r,∆t) =|ifft2 (ifftshift (FFT c(k, E)))|. The result is shown in Fig. S9b. 54 a b 10-2 10-1 100 0 20 40 0 10 20 30 40 50 0 20 40 0 10 20 30 40 50 Fig...
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[29]
We plot the spatial (temporal) experimental decay in logarithmic scales as a function of the expected KPZ scaling ∆r 2χ (∆t2β) (see Fig
Normalization of the coherence To identify KPZ scaling in the condensate spatio temporal coherence decay, we first examine the temporal decay of g(1)(∆r,∆t) for a fixed value of ∆r≃0 and the spatial decay of g(1)(∆r,∆t) for ∆t= 0. We plot the spatial (temporal) experimental de...
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[30]
4 The phase diagram in Fig
Computation of the phase diagram in Fig. 4 The phase diagram in Fig. 4 is obtained by comparing power-law and stretched-exponential fits to the temporal (r≃0) coherence decay over the same fitting window using the Akaike information criterion (AIC), which balances goodness of ...
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[31]
We first demonstrate additional data collapses onto the universal KPZ scaling function for different pump powers, using the dataset presented in Fig
Additional data demonstrated 2D KPZ scaling in polariton condensates In this section, we provide additional results in the KPZ regime. We first demonstrate additional data collapses onto the universal KPZ scaling function for different pump powers, using the dataset presented ...
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[32]
The numerical integration is performed relying on two numerical schemes, used to propagate in time its deterministic and stochastic contributions respectively
Numerical scheme and parameters The generalized Gross-Pitaevskii equation (gGPE) (6) is solved with periodic boundary conditions on a symmetric square lattice of parameterdx=dyand surfaceL x ×L y. The numerical integration is performed relying on two numerical schemes, used to...
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[33]
V ortex tracking Vortices and antivortices are topological defects in the condensate phaseθ(r, t) associated with a quantized±2πcharge. Their proliferation in two-dimensional polariton condensates have already been studied in numerical simulations, which have characterized the...
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[34]
Within Bogoliubov theory, it is signaled by a positive imaginary part of the excitation spectrum [33]
Stability of the condensate Modulation instability fragments a homogeneous condensate into smaller mutually inco- herent domains. Within Bogoliubov theory, it is signaled by a positive imaginary part of the excitation spectrum [33]. At short momenta, this is realized for a neg...
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[35]
2 and 3 of the main text are closely reproduced
Accessing KPZ and EW regimes In this section, we discuss the emergence of KPZ and EW regimes in numerical simulations and show that experimental results of Figs. 2 and 3 of the main text are closely reproduced. The first-order correlation functiong (1)(∆r,∆t) is computed accor...
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[36]
We found that both KPZ and EW regimes are robust against variations of the pumping rateptrp, up to the onset of modula- tion instability
Robustness of the KPZ and EW regimes Let us now comment on the robustness of our numerical observations against variations of the microscopic parameters, with respect to those of Table II. We found that both KPZ and EW regimes are robust against variations of the pumping ratep...
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[37]
Nature of the vortices In this section, we show that the EW and KPZ scaling behaviors are found in a quasi- ordered regime, in which topological defects of opposite charges are bound in pairs. These defects therefore differ from the so-called spiral vortices of the two-dimensi...
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[38]
Vortex clustering In order to understand the microscopic arrangement of phase defects, we implement the vortex clustering algorithm described in Ref. [80]. This method classifies defects into three categories according to their spatial proximity and charge: free defect, dipole...
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[39]
The corresponding distance is denotedR NOS
For each defect, we locate the nearest opposite sign (NOS) defect. The corresponding distance is denotedR NOS
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[40]
We then search for other same sign defects within a circle of radiusR NOS
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[41]
If no additional same sign defect is found within this circle, the vortex–antivortex pair is labeled as a dipole candidate
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[42]
If other same sign defects are found within the circle, they are labeled as cluster candidates
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[43]
For instance, if two opposite charge defects are mutually the nearest neighbors of each other, they are classified as forming a dipole
Candidates are checked sequentially, to determine the subset of mutually agreeing dipoles and clusters candidates. For instance, if two opposite charge defects are mutually the nearest neighbors of each other, they are classified as forming a dipole. If not, they are left unclassified
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[44]
A typical instantaneous phase heat map after clustering is displayed in Fig
Any defects that remain unclassified after this procedure are labeled as free defects. A typical instantaneous phase heat map after clustering is displayed in Fig. S19.afor σf = 0.25µm. It immediately indicates that most defects are forming dipoles. Average over time and indep...
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[45]
[49], which investigates the BKT tran- sition in semiclassical Monte Carlo simulations of 2D quasi-BECs at thermal equilibrium
Spatial coarse-graining In this section, we build upon the work of Ref. [49], which investigates the BKT tran- sition in semiclassical Monte Carlo simulations of 2D quasi-BECs at thermal equilibrium. Specifically, we numerically coarse-grain the condensate wavefunction using a...
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[46]
The probability distribution of vortex (respectively antivortex) lifetimes is displayed in Fig
Defect lifetime We define the lifetimeτof tracked defects from the length of each individual trajecto- ries. The probability distribution of vortex (respectively antivortex) lifetimes is displayed in Fig. S21.a(respectivelyb) for differentσ f . In each case, the distributions ...
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[47]
Results in the KPZ regime are presented in Figs
Defect pair correlation and discussion of the KPZ mapping In this section, we investigate the behavior of the defect pair correlation function in both space and time. Results in the KPZ regime are presented in Figs. S22.b-d, while those in the EW regime are shown in Figs. S22....
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