A Zoology of Quantum Turing Patterns

Abstract: We explore quantum Turing pattern zoology, where the same Lindblad equation supports a morphology atlas of stripes, spots, holes, labyrinths, and defects. The stable stripe species provides a quantitatively controlled case in which morphology and Gaussian witness loss separate parametrically. In particular, visible Turing stripes can remain after two Gaussian witness margins associated with the same $k_*$ mode cross zero in a completely positive Lindblad lattice. The witness thresholds on the exact shell fall as $\mathcal{N}<sup>{-1}$. The stripe nematic threshold tends to a nonzero value at fixed lattice size, time window, and morphology criterion. The ratio of the morphology threshold to either witness threshold therefore grows with $\mathcal{N}$. Imaging and momentum-resolved covariance measurements probe these sectors separately.

• The paper establishes, using numerical and analytical Lindblad-model analysis, that stripe morphology persists at a finite noise threshold while Gaussian witness thresholds scale as N⁻¹, producing an approximately 54-fold separation at N=10⁴.
• The paper maps stripes, spots, holes, labyrinths, and defect-rich textures with momentum-resolved diagnostics, finding that translation breaking and localized responses near 2k* distinguish patterned states from uniform comparisons.
• The paper shows that multimode logarithmic negativity survives after pairwise shell witnesses vanish, implying that covariance-based pair tests can miss collective nonclassicality even when visible Turing order remains.

This paper presents a systematic numerical and analytical study of quantum Turing patterns in a single completely positive Lindblad master equation, establishing a parametric separation between two distinct noise scales on the same ordering mode $k_*$: the noise level at which visible stripe morphology is lost, and the (much smaller) noise levels at which Gaussian witnesses of the same mode cease to certify nonclassicality. The central result is that resolved Turing stripes persist well beyond the point at which both pairwise entanglement witnesses on the exact $k_*$ shell have crossed zero, with the separation ratio growing linearly in the local inverse-noise parameter.

Lindblad model and the two noise sectors

The system is a periodic square lattice of bosonic modes governed by

$$\dot\rho = -i[H,\rho] + \sum_\alpha \mathcal{D}[L_\alpha]\rho,$$

with one-photon gain/loss ($\kappa=2$), two-photon loss ($\gamma=8$), dissipative and parametric bonds providing quadrature-asymmetric transport, and a symmetrized cubic Hamiltonian that breaks parity. Dissipative anisotropy $\delta\eta$ selects a stripe orientation; choosing $\rho_D = 3+2\sqrt{3}$ places the operating point inside the finite-wave-number wedge and fixes $k_*=\pi/6$ (period 12). The principal branch parameters are $$(\lambda,\delta\eta,\chi_2,\nu)=(0.38,-0.28,-0.32,4)$$.

The essential structural feature is that first moments and second moments respond to different noise variables. Writing $$\langle a_{\bm r}\rangle=\sqrt{N}(q_{\bm r}+ip_{\bm r})/\sqrt2$$, the drift sees the scaled inverse noise

$k_*$0

whereas the canonical covariance obeys a Lyapunov equation $k_*$1 containing the bath occupation $k_*$2 itself. Consequently, Gaussian witness crossings occur at bath occupations of order unity — $k_*$3 and $k_*$4 on the exact shell, stable to numerical precision across commensurate lattices $k_*$5 through $k_*$6 — which map to physical thresholds $k_*$7. The morphology threshold, by contrast, tends to a finite limit $k_*$8 at fixed lattice size, time window, and criterion, so that $k_*$9. This is the paper's quantitative core claim, derived from a simple-zero perturbative argument with smooth covariance feedback entering only at $$\dot\rho = -i[H,\rho] + \sum_\alpha \mathcal{D}[L_\alpha]\rho,$$0 in $$\dot\rho = -i[H,\rho] + \sum_\alpha \mathcal{D}[L_\alpha]\rho,$$1 and $$\dot\rho = -i[H,\rho] + \sum_\alpha \mathcal{D}[L_\alpha]\rho,$$2 in $$\dot\rho = -i[H,\rho] + \sum_\alpha \mathcal{D}[L_\alpha]\rho,$$3.

Morphology atlas and diagnostics

A single generator supports a "zoology" of morphologies: stripes, spots, holes, labyrinths, and defect-rich textures, mapped as a continuation atlas from a common initial state over $$\dot\rho = -i[H,\rho] + \sum_\alpha \mathcal{D}[L_\alpha]\rho,$$4. Stripe order is quantified by the nematic moment $$\dot\rho = -i[H,\rho] + \sum_\alpha \mathcal{D}[L_\alpha]\rho,$$5, an angular Fourier component of the spectral weight $$\dot\rho = -i[H,\rho] + \sum_\alpha \mathcal{D}[L_\alpha]\rho,$$6 (with $$\dot\rho = -i[H,\rho] + \sum_\alpha \mathcal{D}[L_\alpha]\rho,$$7) on an annulus about $$\dot\rho = -i[H,\rho] + \sum_\alpha \mathcal{D}[L_\alpha]\rho,$$8, supplemented by independent real-space contrast and Bragg-concentration criteria. Preparation-dependence checks over eight initial conditions, cutoffs from $$\dot\rho = -i[H,\rho] + \sum_\alpha \mathcal{D}[L_\alpha]\rho,$$9 to $\kappa=2$0, longer trajectories, and the orthogonal stripe branch all leave the qualitative ordering intact, though the static atlas is explicitly a continuation map from one preparation rather than an ensemble of basins.

Momentum-resolved Gaussian structure and translation breaking

For opposite-momentum pairs the paper computes two witnesses: $\kappa=2$1 (logarithmic-negativity-based Gaussian entanglement) and $\kappa=2$2 (second-moment Glauber–Sudarshan nonclassicality). A key methodological control is a family of 38 Hurwitz-stable uniform comparison states, amplitude-matched to the stripe and rendered stationary by constant drives; these reproduce most of the broad NPT profile over momentum pairs, leaving only a localized contrast near $\kappa=2$3 attributable to the stripe itself. The last pairwise boundary across the Brillouin zone approaches the inversion-invariant $\kappa=2$4 point.

Two further findings deserve emphasis. First, a full Bloch-sector calculation shows multimode logarithmic negativity remaining nonzero after both pairwise shell crossings have gone negative — the pairwise witnesses do not exhaust the sector's nonclassicality. Second, broken translation symmetry produces a genuinely asymmetric covariance: after removing the nearly neutral $\kappa=2$5 translation sector, the registry asymmetry $\kappa=2$6 takes values $\kappa=2$7 (principal stripe) and $\kappa=2$8 (horizontal branch) at zero bath occupation, while uniform comparison states are exactly translation invariant.

Scale separation and robustness of the crossover

The stripe nematic crossover is defined by $\kappa=2$9. The long-time ensemble gives $\gamma=8$0 with a 95% bootstrap interval of $\gamma=8$1, while direct integration of the limiting first-moment equation gives $\gamma=8$2 with interval $\gamma=8$3 — consistent estimates from trajectory statistics and deterministic dynamics respectively. At $\gamma=8$4, the morphology threshold exceeds the NPT threshold by a factor of approximately 54, with propagated uncertainty narrowing this to roughly 52–56. Including self-consistent covariance feedback shifts the exact-shell crossings by less than 0.3%, weak pinning at the 1% deformation boundary preserves the sign of the localized $\gamma=8$5 response against detuning- and transport-tuned controls, and raw, isotonic, and logistic crossover estimators agree to within 0.1%. The practical implication is that spatial imaging or Bragg measurements and covariance reconstruction near $\gamma=8$6 probe physically distinct sectors, so loss of certifiable Gaussian entanglement does not imply loss of macroscopic Turing order.

Limitations and open questions

Several caveats are stated plainly by the author. The morphology threshold is defined at fixed lattice size, time window, and criterion; whether it remains finite under joint thermodynamic scaling is not addressed. The witness analysis is restricted to Gaussian (covariance-level) criteria for opposite-momentum pairs and Bloch sectors — non-Gaussian nonclassicality is untested. The atlas's preparation dependence is characterized but not eliminated, and the four displayed snapshots come from stored arrays whose stream identifiers were partly not retained, with one snapshot selected post hoc by proximity to the cutoff. Whether the observed separation generalizes beyond this specific generator to photonic arrays or polariton fluids is asserted as plausible but not demonstrated.

Conclusion

The paper provides a controlled instance, within a fully positive Lindblad lattice model, where visible quantum Turing order and its Gaussian witnesses fail at parametrically separated noise scales — witness thresholds falling as $\gamma=8$7 while the stripe morphology threshold approaches a nonzero constant. The accompanying multimode negativity and translation-asymmetry results indicate that pairwise shell witnesses understate the surviving quantum correlations. The open questions are concrete: the scaling of $\gamma=8$8 with system size, the fate of non-Gaussian witnesses, and experimental realization in driven bosonic lattices where first and second moments can be probed independently.