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Spontaneous Quantum Turbulence in a Newborn Bose-Einstein Condensate via the Kibble-Zurek Mechanism

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Pith's one-line read A finite-time thermal quench alone can generate quantum turbulence in a newborn Bose-Einstein condensate, with vortex energy spectra obeying both Kolmogorov and Kibble-Zurek scaling.

desk verdict A genuinely new protocol for quench-generated quantum turbulence with a beauty KZ collapse across quench times, but the 5/3 claim is weakened by an unresolved internal inconsistency between the spectral fit and the ESS exponent, and by unexamined mean-field exponents in a 2D BKT system. read the letter →

arxiv 2506.21670 v3 pith:BCIKKVQC submitted 2025-06-26 cond-mat.quant-gas cond-mat.stat-mechquant-ph

Seong-Ho Shinn , Matteo Massaro , Mithun Thudiyangal , Adolfo del Campo This is my paper · ORCID
classification cond-mat.quant-gascond-mat.stat-mechquant-ph
keywords quantumturbulenceKibble-ZurekmechanismBose-EinsteincondensationvorticesKolmogorovscalingstochasticprojectedGross-Pitaevskiiequationtwo-dimensionalsuperfluidsuniversal
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that there is a simpler route to quantum turbulence than stirring a superfluid: cool a two-dimensional Bose gas through its condensation transition at a finite rate. The quench forces the order parameter to break symmetry at the Kibble-Zurek freeze-out time, leaving a newborn condensate seeded with a random tangle of vortices and antivortices whose density is set entirely by the quench duration. Using stochastic Gross-Pitaevskii simulations, the authors find that the incompressible kinetic energy of this vortex tangle shows the Kolmogorov $k^{-5/3}$ spectrum, the hallmark of turbulence, and that spectra from different quench times collapse onto one universal curve under the Kibble-Zurek rescaling. If true, the quench rate serves as a single control knob that determines both the vortex density and the turbulent state, and two distinct universality principles, defect scaling at phase transitions and energy-spectrum scaling in turbulence, meet in one system.

What carries the argument

The argument is carried by three objects working together: the stochastic projected Gross-Pitaevskii equation (SPGPE), a classical-field equation of motion for the low-energy coherent modes of a Bose gas in contact with a thermal reservoir, which supplies the ensemble of quench trajectories; the canonical Helmholtz-like decomposition of the density-weighted velocity field $\mathbf{u} = \sqrt{\rho}\,\mathbf{v}$ into divergence-free (incompressible) and curl-free (compressible) parts, which isolates the vortex contribution to the kinetic energy; and the Kibble-Zurek correlation length $\hat{\xi} \propto \tau_Q^{\nu/(1+z\nu)}$, which sets both the vortex density $n_v \propto \hat{\xi}^{-2}$ and the characteristic momentum scale at equilibration. The load-bearing identity is the scaling ansatz $E_i(k;\tau_Q) = A(\tau_Q) F(k\hat{\xi})$ with $A(\tau_Q) \propto \tau_Q^{-(1+\nu)/(1+z\nu)}$, which converts into the concrete collapse rule of Eq. (4), $k \to k \tau_Q^{\nu/(1+z\nu)}$ and $E_i \to E_i \tau_Q^{(1+\nu)/(1+z\nu)}$, evaluated at $\nu=1/2$, $z=2$ so that the exponents become $1/4$ and $3/4$.

What would settle it

Repeat the SPGPE quench simulations (or perform the equivalent experiment in a uniform quasi-2D BEC) over a wider range of $\tau_Q$, extending below 10 and above 1000, and test two things: whether the inertial-range exponent stays at $5/3$ and whether the spectra still collapse under $k \tau_Q^{1/4}$ versus $E_i \tau_Q^{3/4}$. If the collapse breaks down at slow quenches or the fitted exponents drift with $\tau_Q$, the combination of Kolmogorov and Kibble-Zurek scaling for SQT is not universal. A complementary test replaces the power-law correlation length with the BKT exponential scaling and asks whether the collapse variable becomes logarithmic in $\tau_Q$.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that spontaneous quantum turbulence (SQT) is an intrinsic outcome of finite-time Bose-Einstein condensation: a linear quench of the chemical potential through the critical point produces a condensate populated with vortices of winding number $\pm 1$, whose number scales as $n_v \propto \tau_Q^{-1/2}$ and whose density-weighted superfluid velocity splits into compressible and incompressible parts with the incompressible kinetic energy spectrum obeying $E_i(k) \propto k^{-5/3}$ over an inertial range bounded below by the vortex-core scale and above by the mean intervortex distance. The spectral shape is universal across quench rates: under the rescaling $k \to k \tau_Q^{1/4}$ and $E_i \to E_i \tau_Q^{3/4}$ (the Kibble-Zurek exponents obtained from the mean-field values $\nu=1/2$, $z=2$), all spectra collapse onto a single curve, and the total incompressible energy follows $E_i \propto \tau_Q^{-1}$. The same collapse works for the compressible spectrum at low temperature, where phonons emitted by vortex-antivortex annihilation dominate, and the velocity structure functions show extended self-similarity with intermittency corrections described by the refined K62 model.

Load-bearing premise

The predicted exponents and the spectral collapse assume mean-field critical exponents $\nu = 1/2$ and $z = 2$ for the 2D Bose-gas transition, even though the homogeneous 2D gas has a Berezinskii-Kosterlitz-Thouless transition with exponential rather than power-law scaling; if the effective exponents differ over the simulated quench times, the Kibble-Zurek exponents and the collapse variables change.

Editorial extensions

If this is right

  • No external stirring is required to create a turbulent superfluid: the quench duration $\tau_Q$ alone sets the vortex density and, through it, the energy-injection scale and the width of the Kolmogorov inertial range.
  • The universal collapse of Eq. (4) gives experimenters a concrete diagnostic: measure $E_i(k)$ at equilibration for several quench times and check whether $E_i \tau_Q^{3/4}$ plotted against $k \tau_Q^{1/4}$ falls on one curve.
  • The turbulent state is transient by construction: vortex-antivortex annihilation, atom losses, and coarsening suppress the Kolmogorov scaling after equilibration, so SQT is best probed in a window around the equilibration time.
  • Slower quenches broaden the inertial range (larger intervortex separation at fixed core size), while quenches faster than a threshold ($\tau_Q \lesssim 10$) leave the Kibble-Zurek scaling regime and lose the collapse.
  • The total incompressible kinetic energy at equilibration obeys $E_i \propto \tau_Q^{-1}$ for the mean-field exponents, a scaling that follows from the same argument and is verified in the data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the load-bearing choice of mean-field exponents $\nu=1/2$, $z=2$ deserves scrutiny because the homogeneous 2D Bose gas has a Berezinskii-Kosterlitz-Thouless equilibrium transition with exponential rather than power-law critical scaling; re-running the collapse test with the BKT scaling form over slower and faster quenches would reveal whether the reported exponents and rescali
  • Editorial inference: the same mechanism should generalize to other symmetry-breaking transitions that create vortex-like defects, such as spinor condensates or higher-dimensional superfluids, with the rescaling exponents set by the relevant $\nu$ and $z$; the paper notes a 4D study in passing but does not develop this analogy.
  • Editorial inference: because the inertial range spans barely a decade, the extended self-similarity analysis of velocity structure functions is what actually pins the $5/3$ law; a sharper experimental test would measure $S_p(r)$ directly in a quenched gas rather than relying on the energy spectrum alone.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. This manuscript reports stochastic projected Gross-Pitaevskii simulations of a homogeneous 2D Bose gas driven across condensation by a linear chemical-potential quench. It finds that the vortex number at equilibration scales as tau_Q^-0.534±0.047, the equilibration time as tau_Q^0.541±0.069, and the incompressible kinetic-energy spectrum exhibits a power-law region with fitted exponent alpha≈1.65–1.71 that the authors associate with Kolmogorov k^-5/3 scaling. It further proposes a scaling ansatz leading to collapse of spectra for different quench times under the rescalings E_i tau_Q^{3/4} vs. k tau_Q^{1/4}, and it reports a universal collapse in Fig. 3. The authors interpret these results as spontaneous quantum turbulence generated by the Kibble-Zurek mechanism.

Significance. The paper addresses a question of genuine current interest: whether a finite-time quench can produce quantum turbulence without external stirring. It has clear strengths: the KZ predictions are tested directly against the simulations rather than used as fitting assumptions; the vortex-number and equilibration-time power laws are checked over more than an order of magnitude in tau_Q; the spectral collapse in Fig. 3 is a nontrivial prediction; and the data set is openly available. The single-vortex derivation leading to E_i ∝ tau_Q^{-1} is simple and consistent with the numerical fits in Fig. S4. However, the central claim that the spectrum is Kolmogorov k^-5/3 is weakened by an internal inconsistency with the supplement's own extended-self-similarity analysis, as detailed below; the title and abstract claim of 'spontaneous quantum turbulence' should be evaluated after that inconsistency is resolved.

major comments (3)
  1. [Fig. 2(b) and Fig. S3(b)] The direct spectral fit and the extended-self-similarity (ESS) fit are mutually inconsistent for the same tau_Q=350 dataset. The main text reports E_i(k) ∝ k^{-1.654±0.031} in the inertial range, while the supplement reports zeta(2)=0.820±0.022 for S_2 ∝ [S_3]^{zeta(2)} over the corresponding range. Using the paper's own relation (S_3 ∝ r in the inertial range, and E_i(k) ∝ k^{-(1+zeta(2))}), the ESS value implies E_i(k) ∝ k^{-1.820±0.022}; the difference is about 4.4 combined standard deviations. The K62 intermittency correction would make the spectrum steeper, not shallower, so it cannot reconcile the two diagnostics. Please reconcile the fit ranges and definitions, or revise the claim that the spectrum agrees with 5/3.
  2. [Main text, 'Kibble-Zurek dynamics of the BEC transition', and Eq. (4)] The KZ exponents and the collapse variables in Eq. (4) are derived using mean-field critical exponents nu=1/2, z=2, but the manuscript does not justify why these apply to the homogeneous 2D Bose gas, whose equilibrium transition is of Berezinskii-Kosterlitz-Thouless type with exponential, not power-law, scaling. The numerical exponents for n_v and t_eq are consistent with nu=1/2, z=2 within error, but they are not by themselves a proof of the mean-field choice over the simulated range. Please provide a justification for the effective mean-field behavior (for example, the role of the c-field cutoff or finite-size effects), or present the collapse using fitted effective exponents and show the sensitivity of the collapse to their uncertainties.
  3. [Main text, 'Kibble-Zurek universality of SQT'] The derivation of the amplitude A(tau_Q) in Eq. (4) uses the total incompressible energy E_i ∝ tau_Q^{-(1+2nu)/(1+z nu)} together with the single-length-scale ansatz E_i(k; tau_Q)=A(tau_Q) F(k xi_hat). However, the spectrum also contains a k^{-3} vortex-core region whose amplitude is independent of tau_Q, and the authors themselves state that no collapse holds there. The total-energy integration therefore includes a tau_Q-independent contribution. The argument should either restrict the ansatz to the inertial window and show that the core contribution is negligible for the total-energy scaling, or the derivation should be stated as approximate for the inertial range only.
minor comments (3)
  1. [Fig. 2 caption and surrounding text] The fit intervals used to obtain the exponents alpha and beta in Fig. 2 are not stated explicitly; they are only indicated by vertical lines. Please state the k ranges used for each fit.
  2. [Reference [49]] The reference to the supplementary material appears as 'url will be inserted by publisher'; this placeholder should be completed before publication.
  3. [Eq. (S6) in the Supplementary Material] The equilibration-time fit is a two-term expression, t_eq = (3.779±0.693) tau_Q^{0.541±0.069} + (0.078±0.059) tau_Q, but the main text quotes only the power-law part. The linear term is non-negligible at the largest simulated tau_Q (about 78 at tau_Q=1000), so its potential effect on the quoted KZ exponent should be discussed.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the KZ collapse and Kolmogorov scaling are tested against the simulations rather than fitted into existence, and the self-citations are not load-bearing.

full rationale

Walking the derivation chain: the SPGPE simulation (Eq. 1) is the first-principles input, and the KZM exponents nu=1/2, z=2 are stated as mean-field assumptions rather than extracted from the spectra. The vortex-number and equilibration-time scalings in Fig. 1(c) are power-law fits to the simulation data, but the quoted exponents (nv ~ tau^-0.534 and teq ~ tau^0.541) agree with the assumed KZM values within error, so these are genuine tests rather than predictions forced by construction. The total incompressible-energy scaling E_i ~ tau^{-(1+2nu)/(1+z nu)} is derived from the single-vortex integral in the supplement (Eq. S10) combined with the KZM vortex-number scaling, and the spectral collapse in Fig. 3 uses the amplitude A(tau) ~ tau^{-(1+nu)/(1+z nu)} obtained from that derivation; it is not obtained by fitting the spectrum amplitude to each tau_Q. Thus the collapse is a nontrivial consistency check of the ansatz E_i(k;tau)=A(tau)F(k xi_hat). The paper cites prior work by the same authors (refs. 28, 36, 37, 68, and its own supplement 49) for the KZM framework, fast-quench corrections, and simulation parameters, but these citations do not supply the central numerical result or forbid alternatives; the spectra and collapse come from the present simulation dataset. The internal tension between the direct spectral fit alpha=1.654 and the ESS-derived zeta(2)=0.820 (which would imply E_i ~ k^-1.82) is a correctness or self-consistency concern, not a circularity, because neither diagnostic is an input to the other. No step reduces by construction to an input, so no circular step is flagged.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

No new physical entities are introduced. The main external inputs are the SPGPE model, the assumption of mean-field critical exponents in 2D, and the scaling ansatz for the spectrum; the rest is simulation and analysis.

free parameters (3)
  • Damping rate gamma = 0.03
    Controls coupling to the thermal reservoir in the SPGPE (Table S1); dissipation strength can affect vortex dynamics and cascade. Value taken from prior work, not fitted here.
  • Energy cutoff epsilon_cut = 2.5 mu_f
    Defines the projected coherent region in SPGPE (Table S1); a different cutoff changes the C-field description of the transition and the spectrum.
  • K62 intermittency parameter kappa = 0.114 +/- 0.006
    Fitted to velocity structure function exponents in Fig. S3(b); supports the K62 model but is not required for the main KZ-Kolmogorov claim.
assumptions (5)
  • domain assumption Near the critical point, equilibrium correlation length and relaxation time obey xi = xi_0/|epsilon|^nu and tau = tau_0/|epsilon|^{z nu} (standard Kibble-Zurek scaling).
    Invoked in the opening sections to derive n_v and t_eq scaling; standard for continuous transitions but requires a true critical point.
  • domain assumption The 2D BEC transition is described by mean-field exponents nu=1/2 and z=2.
    Used to predict t_eq proportional to tau^{1/2}, n_v proportional to tau^{-1/2}, and the rescaling in Eq. (4); not derived for 2D and no discussion of BKT character.
  • domain assumption The stochastic projected Gross-Pitaevskii equation with parameters in Table S1 is an accurate model of the BEC transition.
    All results rest on SPGPE dynamics; authors cite prior agreement with experiments, but this is a modeling assumption.
  • ad hoc to paper The total incompressible energy can be approximated as a sum of isolated vortex contributions with uniform background density rho = mu/g.
    Used to derive E_i proportional to tau^{-(1+2nu)/(1+z nu)} around Eq. (S10); ignores vortex overlap, sound emission, and density fluctuations.
  • ad hoc to paper The scaling ansatz E_i(k;tau)=A(tau)F(k xi_hat) with xi_hat proportional to tau^{nu/(1+z nu)} holds at equilibration.
    Central to the spectral collapse in Fig. 3; a natural scaling form but not derived from the SPGPE equations.

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Cite this review

Pith. "Pith review of Spontaneous Quantum Turbulence in a Newborn Bose-Einstein Condensate via the Kibble-Zurek Mechanism." pith.science (2026). https://pith.science/paper/BCIKKVQC

@misc{pith2026250621670,
  author       = {Pith},
  title        = {Pith review of: Spontaneous Quantum Turbulence in a Newborn Bose-Einstein Condensate via the Kibble-Zurek Mechanism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BCIKKVQC}},
  note         = {Machine review of arXiv:2506.21670}
}
read the original abstract

The Kibble-Zurek mechanism (KZM) predicts the spontaneous formation of topological defects in a continuous phase transition driven at a finite rate. We propose the generation of spontaneous quantum turbulence (SQT) via the KZM during Bose-Einstein condensation induced by a thermal quench. Using numerical simulations of the stochastic projected Gross-Pitaevskii equation in two spatial dimensions, we describe the formation of a newborn Bose-Einstein condensate proliferated by quantum vortices. We establish the nonequilibrium universality of SQT through the Kibble-Zurek and Kolmogorov scaling of the incompressible kinetic energy.

Figures

Figures reproduced from arXiv: 2506.21670 by the authors.

Figure 1
Figure 1. Typical condensate density and phase profiles at equilibration time, together with the vortex-number scaling. Panel (a) shows the condensate density |ΨC(r, teq)| 2 at equilibration time teq following a quench of duration τQ = 10, while panel (b) displays the corresponding phase of ΨC. Vortices with topological charge w = +1 are marked by red crosses and those with w = −1 by circles; no vortices with |w| > 1 were obs… view at source ↗
Figure 2
Figure 2. Incompressible and compressible kinetic energy spectra. Panels (a) and (b) show the incompress￾ible kinetic energy spectrum Ei(k) at the equilibration time following quenches with τQ = 100 and τQ = 350, respec￾tively, each averaged over R = 1000 independent noise realiza￾tions. The corresponding fitted power-law exponents are α = 1.710±0.079, β = 3.025±0.030 for (a), and α = 1.654±0.031, β = 3.036 ± 0.047 for (b). P… view at source ↗
Figure 3
Figure 3. Kibble-Zurek universality of the incom￾pressible kinetic energy spectrum. Panel (a) shows Ei (k) at equilibration time for various τQ values, each av￾eraged over R = 1000 stochastic realizations. The dashed vertical line marks k = 2π/dv with the vortex diameter esti￾mated to be dv = 4ξh. Panel (b) displays ⟨Ei (k)⟩ τ 3/4 Q as a function of the scaled momentum kτ 1/4 Q ∝ k ˆξ. Shaded error bands indicate one standard… view at source ↗

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Reference graph

Works this paper leans on

77 extracted references · 73 canonical work pages · cited by 1 Pith paper

  1. [1]

    The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds’ Numbers,

    A. Kolmogorov, “The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds’ Numbers,” Akademiia Nauk SSSR Doklady30, 301 (1941)

    work page 1941

  2. [2]

    Frisch, Turbulence: The Legacy of A

    U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995)

    1995

  3. [3]

    Chapter ii application of quantum me- chanics to liquid helium,

    R. Feynman, “Chapter ii application of quantum me- chanics to liquid helium,” (Elsevier, 1955) pp. 17–53

    work page 1955

  4. [4]

    Scenario of strongly nonequilibrated bose-einstein condensation,

    N. G. Berloff and B. V. Svistunov, “Scenario of strongly nonequilibrated bose-einstein condensation,” Phys. Rev. A66, 013603 (2002)

    work page 2002

  5. [5]

    Kolmogorov spectrum of superfluid turbulence: Numerical analysis of the gross- pitaevskii equation with a small-scale dissipation,

    M. Kobayashi and M. Tsubota, “Kolmogorov spectrum of superfluid turbulence: Numerical analysis of the gross- pitaevskii equation with a small-scale dissipation,” Phys. Rev. Lett.94, 065302 (2005)

    work page 2005

  6. [6]

    Tsubota, Progress in Low Temperature Physics: Quantum Turbulence, ISSN (Elsevier Science, 2008)

    M. Tsubota, Progress in Low Temperature Physics: Quantum Turbulence, ISSN (Elsevier Science, 2008)

    work page 2008

  7. [7]

    Condensation of classical nonlinear waves in a two-component system,

    H. Salman and N. G. Berloff, “Condensation of classical nonlinear waves in a two-component system,” Physica D: Nonlinear Phenomena238, 1482 (2009)

    work page 2009

  8. [8]

    Quantum turbulence in trapped atomic Bose-Einstein condensates,

    M. C. Tsatsos, P. E. Tavares, A. Cidrim, A. R. Fritsch, M. A. Caracanhas, F. E. A. dos Santos, C. F. Barenghi, and V. S. Bagnato, “Quantum turbulence in trapped atomic Bose-Einstein condensates,” Physics Re- ports622, 1 (2016)

    work page 2016

Show all 77 references
  1. [9]

    Long-range ordering of topological excitations in a two-dimensional superfluid far from equilibrium,

    H. Salman and D. Maestrini, “Long-range ordering of topological excitations in a two-dimensional superfluid far from equilibrium,” Phys. Rev. A94, 043642 (2016)

    2016

  2. [10]

    Re- laxation dynamics of half-quantum vortices in a two- dimensional two-component bose-einstein condensate,

    M. T. Wheeler, H. Salman, and M. O. Borgh, “Re- laxation dynamics of half-quantum vortices in a two- dimensional two-component bose-einstein condensate,” Europhysics Letters135, 30004 (2021)

    2021

  3. [11]

    Quantum dynamics in krylov space: Methods and applications,

    P. Nandy, A. S. Matsoukas-Roubeas, P. Mart ´ ınez- Azcona, A. Dymarsky, and A. del Campo, “Quantum dynamics in krylov space: Methods and applications,” Physics Reports1125, 1 (2025)

    2025

  4. [12]

    Ueda, Fundamentals and New Frontiers of Bose- Einstein Condensation (World Scientific, 2010)

    M. Ueda, Fundamentals and New Frontiers of Bose- Einstein Condensation (World Scientific, 2010)

    2010

  5. [13]

    Quantum fluids of light,

    I. Carusotto and C. Ciuti, “Quantum fluids of light,” Rev. Mod. Phys.85, 299 (2013)

    2013

  6. [14]

    Pitaevskii and S

    L. Pitaevskii and S. Stringari, Bose-Einstein Conden- sation, International Series of Monographs on Physics (Clarendon Press, Oxford, 2003)

    2003

  7. [15]

    Quantum hydrodynamics,

    M. Tsubota, M. Kobayashi, and H. Takeuchi, “Quantum hydrodynamics,” Physics Reports522, 191 (2013)

    2013

  8. [16]

    Direct energy cascade in two-dimensional compressible quantum turbulence,

    R. Numasato, M. Tsubota, and V. S. L’vov, “Direct energy cascade in two-dimensional compressible quantum turbulence,” Phys. Rev. A81, 063630 (2010)

    2010

  9. [17]

    Holographic vortex liquids and superfluid turbulence,

    P. M. Chesler, H. Liu, and A. Adams, “Holographic vortex liquids and superfluid turbulence,” Science341, 368 (2013)

    2013

  10. [18]

    Kolmogorov scaling in turbulent 2d bose-einstein condensates,

    M. Zhao, J. Tao, and I. B. Spielman, “Kolmogorov scaling in turbulent 2d bose-einstein condensates,” Phys. Rev. Lett.134, 083402 (2025)

    2025

  11. [19]

    Observation of vortex-antivortex pairing in decaying 2d turbulence of a superfluid gas,

    S. W. Seo, B. Ko, J. H. Kim, and Y. Shin, “Observation of vortex-antivortex pairing in decaying 2d turbulence of a superfluid gas,” Scientific Reports7, 4587 (2017)

    2017

  12. [20]

    Giant vortex clusters in a two- dimensional quantum fluid,

    G. Gauthier, M. T. Reeves, X. Yu, A. S. Bradley, M. A. Baker, T. A. Bell, H. Rubinsztein-Dunlop, M. J. Davis, and T. W. Neely, “Giant vortex clusters in a two- dimensional quantum fluid,” Science364, 1264 (2019)

    2019

  13. [21]

    Onset of vortex cluster- ing and inverse energy cascade in dissipative quantum fluids,

    R. Panico, P. Comaron, M. Matuszewski, A. S. Lanotte, D. Trypogeorgos, G. Gigli, M. D. Giorgi, V. Ardizzone, D. Sanvitto, and D. Ballarini, “Onset of vortex cluster- ing and inverse energy cascade in dissipative quantum fluids,” Nature Photonics17, 451 (2023)

    2023

  14. [22]

    Relaxation of superfluid turbulence in highly 6 oblate bose-einstein condensates,

    W. J. Kwon, G. Moon, J.-y. Choi, S. W. Seo, and Y.- i. Shin, “Relaxation of superfluid turbulence in highly 6 oblate bose-einstein condensates,” Phys. Rev. A90, 063627 (2014)

    2014

  15. [23]

    Turbulent relaxation to equilibrium in a two-dimensional quantum vortex gas,

    M. T. Reeves, K. Goddard-Lee, G. Gauthier, O. R. Stockdale, H. Salman, T. Edmonds, X. Yu, A. S. Bradley, M. Baker, H. Rubinsztein-Dunlop, M. J. Davis, and T. W. Neely, “Turbulent relaxation to equilibrium in a two-dimensional quantum vortex gas,” Phys. Rev. X12, 011031 (2022)

    2022

  16. [24]

    Topology of cosmic domains and strings,

    T. W. B. Kibble, “Topology of cosmic domains and strings,” Journal of Physics A: Mathematical and Gen- eral9, 1387 (1976)

    1976

  17. [25]

    Some implications of a cosmological phase transition,

    T. Kibble, “Some implications of a cosmological phase transition,” Physics Reports67, 183 (1980)

    1980

  18. [26]

    Cosmological experiments in superfluid helium?

    W. H. Zurek, “Cosmological experiments in superfluid helium?” Nature317, 505 (1985)

    1985

  19. [27]

    Cosmic strings in laboratory superfluids and the topological remnants of other phase transitions,

    W. H. Zurek, “Cosmic strings in laboratory superfluids and the topological remnants of other phase transitions,” Acta Physica Polonica B24, 1301 (1993)

    1993

  20. [28]

    Universality of phase transition dynamics: Topological defects from symmetry breaking,

    A. del Campo and W. H. Zurek, “Universality of phase transition dynamics: Topological defects from symmetry breaking,” Int. J. Mod. Phys. A29, 1430018 (2014)

    2014

  21. [29]

    Soliton creation dur- ing a bose-einstein condensation,

    B. Damski and W. H. Zurek, “Soliton creation dur- ing a bose-einstein condensation,” Phys. Rev. Lett.104, 160404 (2010)

    2010

  22. [30]

    Spontaneous creation of Kibble-Zurek soli- tons in a Bose-Einstein condensate,

    G. Lamporesi, S. Donadello, S. Serafini, F. Dalfovo, and G. Ferrari, “Spontaneous creation of Kibble-Zurek soli- tons in a Bose-Einstein condensate,” Nature Physics9, 656 (2013)

    2013

  23. [31]

    Spontaneous vortices in the formation of Bose-Einstein condensates,

    C. N. Weiler, T. W. Neely, D. R. Scherer, A. S. Bradley, M. J. Davis, and B. P. Anderson, “Spontaneous vortices in the formation of Bose-Einstein condensates,” Nature 455, 948 (2008)

    2008

  24. [32]

    Emergence of coherence via transverse condensa- tion in a uniform quasi-two-dimensional Bose gas,

    L. Chomaz, L. Corman, T. Bienaim´ e, R. Desbuquois, C. Weitenberg, S. Nascimb` ene, J. Beugnon, and J. Dal- ibard, “Emergence of coherence via transverse condensa- tion in a uniform quasi-two-dimensional Bose gas,” Nat. Comm.6, 6162 (2015)

    2015

  25. [33]

    The inhomogeneous kibble–zurek mechanism: vortex nucle- ation during bose–einstein condensation,

    A. del Campo, A. Retzker, and M. B. Plenio, “The inhomogeneous kibble–zurek mechanism: vortex nucle- ation during bose–einstein condensation,” New Journal of Physics13, 083022 (2011)

    2011

  26. [34]

    Defect formation beyond kibble-zurek mechanism and hologra- phy,

    P. M. Chesler, A. M. Garc ´ ıa-Garc ´ ıa, and H. Liu, “Defect formation beyond kibble-zurek mechanism and hologra- phy,” Phys. Rev. X5, 021015 (2015)

    2015

  27. [35]

    Kibble-zurek dynamics in a trapped ultracold bose gas,

    I.-K. Liu, J. Dziarmaga, S.-C. Gou, F. Dalfovo, and N. P. Proukakis, “Kibble-zurek dynamics in a trapped ultracold bose gas,” Phys. Rev. Res.2, 033183 (2020)

    2020

  28. [36]

    Universal breakdown of kibble-zurek scaling in fast quenches across a phase transition,

    H.-B. Zeng, C.-Y. Xia, and A. del Campo, “Universal breakdown of kibble-zurek scaling in fast quenches across a phase transition,” Phys. Rev. Lett.130, 060402 (2023)

    2023

  29. [37]

    Universal vortex statistics and stochastic geometry of bose-einstein con- densation,

    M. Thudiyangal and A. del Campo, “Universal vortex statistics and stochastic geometry of bose-einstein con- densation,” Phys. Rev. Res.6, 033152 (2024)

    2024

  30. [38]

    Dynamics of a nonequilibrium discontinuous quantum phase tran- sition in a spinor bose–einstein condensate,

    M. T. Wheeler, H. Salman, and M. O. Borgh, “Dynamics of a nonequilibrium discontinuous quantum phase tran- sition in a spinor bose–einstein condensate,” Communi- cations Physics8, 153 (2025)

    2025

  31. [39]

    Kibble-zurek scaling of the superfluid-supersolid tran- sition in an elongated dipolar gas,

    W. Kirkby, H. Salman, T. Gasenzer, and L. Chomaz, “Kibble-zurek scaling of the superfluid-supersolid tran- sition in an elongated dipolar gas,” Phys. Rev. Res.7, 023248 (2025)

    2025

  32. [40]

    Critical dynamics of spontaneous symmetry breaking in a homogeneous Bose gas,

    N. Navon, A. L. Gaunt, R. P. Smith, and Z. Hadzibabic, “Critical dynamics of spontaneous symmetry breaking in a homogeneous Bose gas,” Science347, 167 (2015)

    2015

  33. [41]

    Kibble–zurek univer- sality in a strongly interacting fermi superfluid,

    B. Ko, J. W. Park, and Y. Shin, “Kibble–zurek univer- sality in a strongly interacting fermi superfluid,” Nature Physics15, 1227 (2019)

    2019

  34. [42]

    Suppression of spontaneous defect formation in inhomo- geneous bose gases,

    M. Kim, T. Rabga, Y. Lee, J. Goo, D. Bae, and Y. Shin, “Suppression of spontaneous defect formation in inhomo- geneous bose gases,” Phys. Rev. A106, L061301 (2022)

    2022

  35. [43]

    The stochastic gross– pitaevskii equation: Ii,

    C. W. Gardiner and M. J. Davis, “The stochastic gross– pitaevskii equation: Ii,” Journal of Physics B: Atomic, Molecular and Optical Physics36, 4731 (2003)

    2003

  36. [44]

    Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques,

    P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh and C. W. Gardiner, “Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques,” Advances in Physics57, 363 (2008)

    2008

  37. [45]

    Stochastic projected gross-pitaevskii equation,

    S. J. Rooney, P. B. Blakie, and A. S. Bradley, “Stochastic projected gross-pitaevskii equation,” Phys. Rev. A86, 053634 (2012)

    2012

  38. [46]

    Numer- ical method for the stochastic projected gross-pitaevskii equation,

    S. J. Rooney, P. B. Blakie, and A. S. Bradley, “Numer- ical method for the stochastic projected gross-pitaevskii equation,” Phys. Rev. E89, 013302 (2014)

    2014

  39. [47]

    Low- dimensional stochastic projected gross-pitaevskii equa- tion,

    A. S. Bradley, S. J. Rooney, and R. G. McDonald, “Low- dimensional stochastic projected gross-pitaevskii equa- tion,” Phys. Rev. A92, 033631 (2015)

    2015

  40. [48]

    Dynamics of hot Bose-Einstein condensates: stochastic Ehrenfest relations for number and energy damping,

    R. G. McDonald, P. S. Barnett, F. Atayee, and A. S. Bradley, “Dynamics of hot Bose-Einstein condensates: stochastic Ehrenfest relations for number and energy damping,” SciPost Phys.8, 029 (2020)

    2020

  41. [49]

    url will be inserted by publisher,

  42. [50]

    Quantum turbulence,

    M. Tsubota, “Quantum turbulence,” Journal of the Physical Society of Japan77, 111006 (2008)

    2008

  43. [51]

    Kolmogorov turbulence in low-temperature superflows,

    C. Nore, M. Abid, and M. E. Brachet, “Kolmogorov turbulence in low-temperature superflows,” Phys. Rev. Lett.78, 3896 (1997)

    1997

  44. [52]

    Decaying kol- mogorov turbulence in a model of superflow,

    C. Nore, M. Abid, and M. E. Brachet, “Decaying kol- mogorov turbulence in a model of superflow,” Physics of Fluids9, 2644 (1997)

    1997

  45. [53]

    Study of re- connection and acoustic emission of quantized vortices in superfluid by the numerical analysis of the gross- pitaevskii equation,

    S.-i. Ogawa, M. Tsubota, and Y. Hattori, “Study of re- connection and acoustic emission of quantized vortices in superfluid by the numerical analysis of the gross- pitaevskii equation,” Journal of the Physical Society of Japan71, 813 (2002)

    2002

  46. [54]

    Kolmogorov spectrum of quantum turbulence,

    M. Kobayashi and M. Tsubota, “Kolmogorov spectrum of quantum turbulence,” Journal of the Physical Society of Japan74, 3248 (2005)

    2005

  47. [55]

    Energy spectra of vortex distributions in two-dimensional quantum turbu- lence,

    A. S. Bradley and B. P. Anderson, “Energy spectra of vortex distributions in two-dimensional quantum turbu- lence,” Phys. Rev. X2, 041001 (2012)

    2012

  48. [56]

    Char- acteristics of two-dimensional quantum turbulence in a compressible superfluid,

    T. W. Neely, A. S. Bradley, E. C. Samson, S. J. Rooney, E. M. Wright, K. J. H. Law, R. Carretero-Gonz´ alez, P. G. Kevrekidis, M. J. Davis, and B. P. Anderson, “Char- acteristics of two-dimensional quantum turbulence in a compressible superfluid,” Phys. Rev. Lett.111, 235301 (2013)

    2013

  49. [57]

    C. F. Barenghi, L. Skrbek, and K. R. Sreenivasan, Quan- tum Turbulence (Cambridge University Press, Cam- bridge, 2023)

    2023

  50. [58]

    Quantum turbulence,

    W. F. Vinen and J. J. Niemela, “Quantum turbulence,” Journal of Low Temperature Physics128, 167 (2002)

    2002

  51. [59]

    Dissipation of Energy in Locally Isotropic Turbulence,

    A. N. Kolmogorov, “Dissipation of Energy in Locally Isotropic Turbulence,” Akademiia Nauk SSSR Doklady 32, 16 (1941)

    1941

  52. [60]

    Theoretical analysis of quantum turbu- lence using the onsager ideal turbulence theory,

    T. Tanogami, “Theoretical analysis of quantum turbu- lence using the onsager ideal turbulence theory,” Phys. Rev. E103, 023106 (2021)

    2021

  53. [61]

    On the theory of 7 superfluidity,

    V. L. Ginzburg and L. P. Pitaevskii, “On the theory of 7 superfluidity,” Sov. Phys. JETP7, 858 (1958)

    1958

  54. [62]

    Extended self-similarity in turbu- lent flows,

    R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Mas- saioli, and S. Succi, “Extended self-similarity in turbu- lent flows,” Phys. Rev. E48, R29 (1993)

    1993

  55. [63]

    Intermittency in fully developed turbu- lence: Log-poisson statistics and generalized scale covari- ance,

    B. Dubrulle, “Intermittency in fully developed turbu- lence: Log-poisson statistics and generalized scale covari- ance,” Phys. Rev. Lett.73, 959 (1994)

    1994

  56. [64]

    Extended self similarity,

    S. Ciliberto, “Extended self similarity,” in Turbulence: A Tentative Dictionary, edited by P. Tabeling and O. Car- doso (Springer US, Boston, MA, 1994) pp. 27–30

    1994

  57. [65]

    Grid superfluid turbulence and intermit- tency at very low temperature,

    G. Krstulovic, “Grid superfluid turbulence and intermit- tency at very low temperature,” Phys. Rev. E93, 063104 (2016)

    2016

  58. [66]

    A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high reynolds number,

    A. N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high reynolds number,” Journal of Fluid Mechanics13, 82 (1962)

    1962

  59. [67]

    Comment on “super- fluid turbulence from quantum kelvin wave to classical kolmogorov cascades

    G. Krstulovic and M. Brachet, “Comment on “super- fluid turbulence from quantum kelvin wave to classical kolmogorov cascades”,” Phys. Rev. Lett.105, 129401 (2010)

    2010

  60. [68]

    Kibble-zurek mechanism and beyond: Lessons from a holographic superfluid disk,

    C.-Y. Xia, H.-B. Zeng, A. Grabarits, and A. del Campo, “Kibble-zurek mechanism and beyond: Lessons from a holographic superfluid disk,” (2024), arXiv:2406.09433 [cond-mat.stat-mech]

    2024 arXiv

  61. [69]

    Non-equilibrium dynamics and univer- sality of 4d quantum vortices and turbulence,

    W. can Yang, “Non-equilibrium dynamics and univer- sality of 4d quantum vortices and turbulence,” (2025), arXiv:2504.04409 [cond-mat.quant-gas]

    2025 arXiv

  62. [70]

    Spontaneous quantum turbulence in a newborn bose- einstein condensate via the kibble-zurek mechanism data,

    S.-H. Shinn, M. Massaro, M. THUDIYANGAL, and A. del Campo, “Spontaneous quantum turbulence in a newborn bose- einstein condensate via the kibble-zurek mechanism data,” (2025)

    2025

  63. [71]

    DifferentialEquations.jl - A Performant and Feature-Rich Ecosystem for Solving Dif- ferential Equations in Julia,

    C. Rackauckas and Q. Nie, “DifferentialEquations.jl - A Performant and Feature-Rich Ecosystem for Solving Dif- ferential Equations in Julia,” Journal of Open Research Software (2017), 10.5334/jors.151

    2017 doi

  64. [72]

    Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory,

    C. Rackauckas and Q. Nie, “Adaptive methods for stochastic differential equations via natural embeddings and rejection sampling with memory,” Discrete and Con- tinuous Dynamical Systems - B22, 2731 (2017)

    2017

  65. [73]

    Stability-optimized high or- der methods and stiffness detection for pathwise stiff stochastic differential equations,

    C. Rackauckas and Q. Nie, “Stability-optimized high or- der methods and stiffness detection for pathwise stiff stochastic differential equations,” in 2020 IEEE High Performance Extreme Computing Conference (HPEC) (2020) pp. 1–8

    2020

  66. [74]

    Bose-einstein condensation and liquid helium,

    O. Penrose and L. Onsager, “Bose-einstein condensation and liquid helium,” Phys. Rev.104, 576 (1956)

    1956

  67. [75]

    A vortex filament tracking method for the gross- pitaevskii model of a superfluid,

    A. Villois, G. Krstulovic, D. Proment, and H. Salman, “A vortex filament tracking method for the gross- pitaevskii model of a superfluid,” Journal of Physics A: Mathematical and Theoretical49, 415502 (2016)

    2016

  68. [76]

    Watanabe, Integral Transform Techniques for Green’s Function, Lecture Notes in Applied and Computational Mechanics (Springer International Publishing Switzer- land, Cham, 2015)

    K. Watanabe, Integral Transform Techniques for Green’s Function, Lecture Notes in Applied and Computational Mechanics (Springer International Publishing Switzer- land, Cham, 2015)

    2015

  69. [77]

    The design and imple- mentation of FFTW3,

    M. Frigo and S. G. Johnson, “The design and imple- mentation of FFTW3,” Proceedings of the IEEE93, 216 (2005), special issue on “Program Generation, Optimiza- tion, and Platform Adaptation”. —Supplementary Material— Spontaneous Quantum Turbulence in a Newborn Bose-Einstein Con...

    2005

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