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The Davis-Landau Sonic Onset Law: Universal Critical Velocity for Vortex Nucleation in Bose-Einstein Condensates

Bee Rosa Davis

Published Jan 23, 2026• 0 citations
Publisher

Abstract

We report systematic validation of the Davis Law of Critical Velocity governing vortex nucleation in two-dimensional Gross-Pitaevskii dynamics: $$ C = \frac{\tau}{K} \quad \Rightarrow \quad \beta = \frac{v_c}{c_s} = f(K_{\text{eff}}) \quad \text{(species-independent, geometry-indexed)} $$ Through GPU-accelerated Gross-Pitaevskii simulations across three atomic species (⁸⁷Rb, ²³Na, ⁷Li) spanning a 12:1 mass ratio, we demonstrate that the dimensionless critical velocity β = v_c/c_s is species-independent—a universal geometric invariant within each curvature regime, determined solely by obstacle geometry. Key Results Universal Critical Velocity: Hard obstacles (tanh-wall): β_hard = 0.425 ± 0.000 across all three species Semi-hard obstacles (Gaussian, V₀ = 50μ): β_soft = 0.486 ± 0.000 across all species Excellent agreement with Frisch-Pomeau-Rica result (β ≈ 0.4) Perfect Species Independence: The coefficient of variation is 0.0% for both obstacle types across a 12:1 mass ratio, definitively proving that β depends only on obstacle geometry, not atomic properties. Davis Law Support: The relation C = τ/K from the Davis Field Equations correctly predicts that obstacle curvature controls the critical velocity. High-curvature (hard) obstacles nucleate vortices at lower β than low-curvature (soft) obstacles. Quantum vs Classical Wake Dynamics: In the post-critical wake, coherent oscillations occur with phase φ ≈ 0° (symmetric breathing) rather than φ ≈ π (alternating von Kármán), implying suppression of the classical anti-phase shedding mode. The Geometric Trichotomy Regime Obstacle Type K_eff Mechanism β = v_c/c_s Sonic Hard (tanh-wall) High Local Mach = 1 at obstacle rim 0.42–0.49 Sonic Semi-hard (Gaussian) Medium Same mechanism, different K_eff ~0.486 Quantum Truly soft (V₀ ≪ μ) Low Holonomy budget τ = 2π exhausted ~0.95–√2 The functional relationship β = f(K_eff) is universal—independent of atomic mass, scattering length, and density—with distinct numerical values corresponding to distinct geometric classes. Unifying Principle The various measured thresholds are unified by a single principle: Geometry selects the regime (soft vs hard) Compressibility selects the prefactor (local c_s and amplification A) Barrier physics selects the delay (sonic onset vs nucleation threshold) This validates the Davis Law prediction that coherence thresholds are determined by constraint geometry (C = τ/K), not microscopic physics. Evidence Status Claim Status Evidence β is species-independent Proven 0.0% CV across 12:1 mass ratio β_hard = 0.425 ± 0.000 Proven Identical value across Rb-87, Na-23, Li-7 β_soft = 0.486 ± 0.000 Proven Identical value across Rb-87, Na-23, Li-7 K_eff controls regime Proven Curvature scaling law shows monotonic β(K_eff) C = τ/K governs transition Supported Qualitative agreement; mechanism confirmed Wake mode is symmetric Proven Phase φ ≈ 0° measured (not 180° classical VK) Frisch-Pomeau-Rica agreement Proven β_hard = 0.425 matches FPR β ≈ 0.4 Included Files Simulation Code multi_species_trichotomy.py — Main simulation testing β across Rb-87, Na-23, Li-7 curvature_scaling_law.py — Tests C = τ/K scaling, K_eff operational definition Figure Generation plot_davis_landau_law.py — Publication figures (universal β, v_c vs c_s) mach_summary_for_paper.py — M_rim vs v/c_s showing sonic onset mechanism phase_comparison_figure.py — VK vs GP symmetric modes diagnostic stability_islands_extended.py — Stability islands analysis Data trichotomy_data.json — Pre-computed multi-species results superfluid_test_results_corrected.json — Validation data Relation to Prior Work This paper extends the Davis Framework for geometric physics: The Incompressibility of Topological Charge (Yang-Mills Mass Gap) — DOI: 10.5281/zenodo.17846521 Holonomy-First Navier-Stokes Regularity — DOI: 10.5281/zenodo.18216597 Field Equations of Semantic Coherence — DOI: 10.5281/zenodo.14784553 The critical velocity emerges as the threshold where topological defect creation (vortex nucleation) becomes energetically favorable—the "cost of distinguishability" in the information-geometric framework. Keywords Bose-Einstein condensate, critical velocity, superfluidity, Gross-Pitaevskii equation, vortex nucleation, Landau criterion, sonic onset, geometric trichotomy, Davis Law, quantum hydrodynamics, obstacle geometry, Mach number, effective curvature, species independence, universal constant License Creative Commons Attribution 4.0 International (CC BY 4.0) Related Identifiers References: DOI 10.5281/zenodo.17846521 (Yang-Mills Mass Gap) References: DOI 10.5281/zenodo.18216597 (Navier-Stokes Regularity) References: DOI 10.5281/zenodo.14784553 (Field Equations of Semantic Coherence)

DOI
10.5281/zenodo.18343935

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We investigate analytically and numerically the dynamics of a two-dimensional superflow governed by the Gross-Pitaevskii equation passing over finite-size rectangular obstacles: an impenetrable wall and an impenetrable rectangular well. Extending classical studies of vortex nucleation around smooth obstacles, we focus on the role of sharp corners in determining the onset of vortex nucleation. Using a combination of analytical techniques based on the Schwarz-Christoffel methods for potential flow and on numerical simulations, we show that local velocity amplification near sharp corners crucially controls the critical flow velocity for vortex nucleation. For both wall and well configurations, we identify analytically and theoretically the critical velocities as a function of the obstacle width and its height or depth, finding an excellent agreement between the theory and our numerical simulations. Our results provide a simple framework for understanding superflow stability past finite-size obstacles with sharp features and are directly relevant to experimentally realizable configurations in atomic Bose-Einstein condensates and related superfluid systems.
We present measurements of the critical velocity for vortex shedding in a highly oblate Bose-Einstein condensate with a moving repulsive Gaussian laser beam. As a function of the barrier height V0{V}_{0}V0​, the critical velocity vc{v}_{c}vc​ shows a dip structure having a minimum at ${V}_{0}\ensuremath{\approx}\ensuremath{\mu}$, where $\ensuremath{\mu}$ is the chemical potential of the condensate. At fixed ${V}_{0}\ensuremath{\approx}7\ensuremath{\mu}$, we observe that the ratio of vc{v}_{c}vc​ to the speed of sound cs{c}_{s}cs​ monotonically increases for decreasing $\ensuremath{\sigma}/\ensuremath{\xi}$, where $\ensuremath{\sigma}$ is the beam width and $\ensuremath{\xi}$ is the condensate healing length. We explain our results with the density reduction effect of the soft boundary of the Gaussian obstacle, based on the local Landau criterion for superfluidity. The measured value of vc/cs{v}_{c}/{c}_{s}vc​/cs​ with our stiffest obstacle is about 0.4, which is in good agreement with theoretical predictions for a two-dimensional superflow past a circular cylinder.
When a superfluid flows past an obstacle, quantized vortices can be created in the wake above a certain critical velocity. In the experiment by Kwon et al. [Phys. Rev. A 91, 053615 (2015)], the critical velocity vc{v}_{c}vc​ was measured for atomic Bose-Einstein condensates (BECs) using a moving repulsive Gaussian potential, and vc{v}_{c}vc​ was minimized when the potential height V0{V}_{0}V0​ of the obstacle was close to the condensate chemical potential $\ensuremath{\mu}$. Here we numerically investigate the evolution of the critical vortex shedding in a two-dimensional BEC with increasing V0{V}_{0}V0​ and show that the minimum vc{v}_{c}vc​ at the critical strength ${V}_{0c}\ensuremath{\approx}\ensuremath{\mu}$ results from the local density reduction and vortex-pinning effect of the repulsive obstacle. The spatial distribution of the superflow around the moving obstacle just below vc{v}_{c}vc​ is examined. The particle density at the tip of the obstacle decreases as V0{V}_{0}V0​ increases to Vc0{V}_{c0}Vc0​, and at the critical strength, a vortex dipole is suddenly formed and dragged by the moving obstacle, indicating the onset of vortex pinning. The minimum vc{v}_{c}vc​ exhibits power-law scaling with the obstacle size $\ensuremath{\sigma}$ as ${v}_{c}\ensuremath{\sim}{\ensuremath{\sigma}}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}\ensuremath{\approx}1/2$.
We apply the Halperin-Mazenco formalism within the Gross-Pitaevskii theory to characterise the kinematics and nucleation of quantum vortices in a two-dimensional stirred Bose Einstein condensate. We introduce a smooth defect density field measuring the superfluid vorticity and is a topologically conserved quantity. We use this defect density field and its associated current density to study the precursory pattern formations that occur inside the repulsive potential of an obstacle and determine the onset of vortex nucleation and shedding. We demonstrate that phase slips form inside hard potentials even in the absence of vortex nucleation, whereas for soft potentials they occur only above a critical stirring velocity leading to vortex nucleation. The Halperin-Mazenco formalism provides an elegant and accurate method of deriving the point vortex dynamic directly from the Gross-Pitaevskii equation.
The cooling of a Bose gas in finite time results in the formation of a Bose-Einstein condensate that is spontaneously proliferated with vortices. We propose that the vortex spatial statistics is described by a homogeneous Poisson point process (PPP) with a density dictated by the Kibble-Zurek mechanism (KZM). We validate this model using numerical simulations of the two-dimensional stochastic Gross-Pitaevskii equation (SGPE) for both a homogeneous and a hard-wall trapped condensate. The KZM scaling of the average vortex number with the cooling rate is established along with the universal character of the vortex number distribution. The spatial statistics between vortices is characterized by analyzing the two-point defect-defect correlation function, the corresponding spacing distributions, and the random tessellation of the vortex pattern using the Voronoi cell area statistics. Combining the PPP description with the KZM, we derive universal theoretical predictions for each of these quantities and find them in agreement with the SGPE simulations. Our results establish the universal character of the spatial statistics of point-like topological defects generated during a continuous phase transition and the associated stochastic geometry. Published by the American Physical Society 2024
In classical fluid dynamics, the von Kármán vortex street behind a bluff body is the canonical example used to study vortex formation and dynamics. This is governed by a local Hopf bifurcation, which drives the continuous and asynchronous release of vortical structures in the wake. In superfluid systems, however, the quantization of vorticity fundamentally alters this picture. Several aspects of quantum vortex shedding remain unresolved, including the absence of a unified description spanning the pre- and post-threshold regimes. We show that the onset of vortex emission is governed by a global saddle node on invariant circle bifurcation. This mechanism produces a discrete transition characterized by a macroscopic bottleneck in phase space and a universal inverse-square-root divergence of the emission period near the critical velocity. These features are robust across obstacles of different shapes, revealing a universal mechanism underlying vortex shedding in quantum fluids.