Turbulence in Quantum Gases: Vortices, Waves, and Cascades

Ashton S. Bradley; Brian P. Anderson; Tyler W. Neely; Xiaoquan Yu

Turbulence in Quantum Gases: Vortices, Waves, and Cascades

T0 review · 0 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A spectral power law is not proof of a turbulent cascade in quantum gases: cascade claims need a measured flux, not just a slope.

desk verdict A careful, field-organizing review whose real contribution is a stricter cascade-evidence standard; the proposed vortex-flux gold standard has a genuine soft spot in the Helmholtz decomposition, but the central argument holds. read the letter →

arxiv 2607.22244 v1 pith:ROGWHPXL submitted 2026-07-24 cond-mat.quant-gas physics.atom-phquant-ph

Ashton S. Bradley , Tyler W. Neely , Xiaoquan Yu , Brian P. Anderson This is my paper

classification cond-mat.quant-gasphysics.atom-phquant-ph

keywords ultracoldquantumgasesturbulencequantizedvorticeswavekinetic-energyspectraspectralfluxesturbulentcascadesvortexcorrelations

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

Turbulence in ultracold atomic gases has two coupled sectors—quantized vortex motion and compressible sound-like waves—and the paper's organizing claim is that these sectors can be distinguished, and cascade physics identified, only by combining spectra with transport information. The central thesis is that a power law, however clean, is supporting evidence for a cascade rather than a definition of one; a cascade claim must identify what quantity is transported, across which wavenumber range, through which flux, and into which dissipation channel. To make this concrete the review assembles and assesses a diagnostic family: incompressible versus compressible kinetic-energy spectra, wave-occupation spectra, spectral fluxes, vortex-resolved correlations, and velocity structure functions. These diagnostics are shown to separate equilibrium vortex organization, decaying turbulent relaxation, forced cascade dynamics, and weak-wave turbulence. The reader should take away that atomic-gas turbulence now has tools for quantitative transport tests, and that the next decisive step is direct flux measurement rather than exponent fitting.

What carries the argument

The load-bearing tool is the solenoidal/longitudinal decomposition of the density-weighted velocity field w = sqrt(n) v into a divergence-free part carrying vortex kinetic energy and a curl-free part carrying sound-like kinetic energy. The paper stresses that this is a formal projection—diagnostically useful, not a statement that vortices and waves are dynamically independent. From it come the shell-integrated spectra E_i(k) and E_c(k), organized by a spectral-budget equation in which a conversion term tracks energy exchanged between sectors during annihilation, reconnection, and nucleation. The companion tool is the spectral flux Pi(k): a cascade is operationally defined as a wavenumber int

What would settle it

A decisive test would be a single simulation of the mean-field wave equation with independently specified vortex content and phonon content, comparing E_i(k) and E_c(k) with vortex positions and wave amplitude reconstructed by other means: if a vortex-free phonon state yields substantial incompressible spectral weight, or a pure vortex state leaks comparable energy into the compressible sector beyond the known core contribution, the decomposition's diagnostic claim fails. The same simulation could also look for a constant spectral flux through a k^-5/3 range in which no actual vortex transport

Watch

Extended reading notes

Core claim

The central claim is that a finite compressible condensate, modeled by the scalar contact-interaction mean-field wave equation, supports two separable turbulent sectors—solenoidal vortex motion and longitudinal sound-like motion—and that the density-weighted velocity field w = sqrt(n) v, split into divergence-free and curl-free parts, gives the cleanest spectral separation. Shell integration yields incompressible E_i(k) and compressible E_c(k) spectra, the review's core regime diagnostics; together with occupation spectra, fluxes, vortex-resolved correlations, and velocity statistics they separate equilibrium vortex organization, decaying relaxation, forced cascades, and weak-wave turbulence

Load-bearing premise

The load-bearing premise is that splitting the density-weighted velocity field into divergence-free and curl-free parts faithfully attributes kinetic energy to vortices versus sound in finite, inhomogeneous, compressible condensates—a formal projection the paper itself warns should not be read as dynamical independence of vortices and waves.

Editorial extensions

If this is right

Interpreting any quantum-gas spectrum as cascade evidence will require adding a flux measurement or an equivalent transport reconstruction; an exponent alone will no longer suffice.
Box-trap experiments with engineered high-wavenumber dissipation become the decisive setting, because they permit direct comparison of injected power, particle loss, and the particle and energy fluxes through selected shells.
Reported k^-5/3-like ranges in two-dimensional experiments should be read as vortex-organization signatures unless accompanied by vortex-resolved correlations or flux evidence, since same-sign clustering can enhance low-wavenumber energy without any inverse cascade flux.
Regime labels—equilibrium organization, decaying relaxation, forced cascade, weak-wave turbulence—become testable classifications built from dimensionality, forcing amplitude, dissipation scale, compressibility, flux direction, and the measured transported quantity.
Velocity structure functions and turbulent equations of state provide non-spectral evidence that can corroborate or contradict a proposed cascade interpretation.

Reading between the lines

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

The authors leave implicit that adopting this standard would require revisiting earlier 'cascade' claims whose evidence rests on spectral slopes alone; reanalyzing those datasets for flux signatures is a concrete next step.
The diagnostic split suggests a test not emphasized in the paper: in a controlled vortex-annihilation experiment, the integrated conversion term between compressible and incompressible sectors should cancel globally; measuring that cancellation would directly probe whether the solenoidal/longitudinal split is physically faithful.
One can extend the same evidentiary discipline to dipolar, spinor, and fermionic superfluids: if flux-based cascade tests remain clean when the equation of state and internal degrees of freedom change, that would argue that turbulent cascades are universal features of coherent nonlinear quantum fluids rather than platform-specific accidents.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First thing you should know: this is a review, and its value is in the standard it sets, not new results. The authors argue—correctly, I think—that a power-law spectrum is supporting evidence for a cascade, not the definition of one. To claim a cascade you need to say what is transported, over which k range, via what flux, into which dissipation channel. That organizing principle is the main contribution, and it is a good one. Tables 1 and 2 make it concrete by separating baseline scalings (single-vortex k^-3, coarsening) from cascade candidates, and the experimental survey is accurate about what has and has not been measured. The paper is careful: it flags when evidence is only suggestive (the Johnstone intermediate-time k^-5/3, for instance) and distinguishes Onsager clustering from inverse transfer.

Credit where due: the GPE/Madelung/Bogoliubov material is standard but cleanly stated; the spectral-budget Eq. 23 is a useful consolidation; the self-citations appear for specific formulas and are legitimate. No circular reasoning that I can see. The negative thesis—power laws alone are insufficient—is independently supported by counterexamples and does not depend on any suspect diagnostic.

The soft spot is exactly the one the stress-test note names. The 'gold standard' for vortex turbulence, a measured constant incompressible flux, rests operationally on the Helmholtz decomposition of w = sqrt(n) v into incompressible and compressible parts. Near vortex cores, boundaries, and reconnection events, that global projection is nonlocal and can mix quantum-pressure, boundary, and sound contributions into both sectors. The paper itself admits the separation is 'formal' and decomposition-dependent, but then still builds its recommended cascade standard on it. That is a real gap: no atomic-gas experiment has yet measured a vortex-energy flux directly, and the flux evidence in the review's own tables comes from GPE simulations using this decomposition. The fix is straightforward—require robustness tests against decomposition choice, or elevate independent real-space diagnostics (vortex tracking, velocity statistics) to co-equal status. But this is a weakness in the prescription's operational basis, not a fatal flaw: the central argument survives.

Who gets value: anyone reviewing or designing quantum-gas turbulence experiments, and theorists who want a checklist before calling something a cascade. It deserves a serious referee—not because it is groundbreaking, but because it will shape how the field validates cascade claims. Peer review should push on the decomposition issue.

Referee Report

0 major / 5 minor

Summary. This paper reviews turbulence in ultracold quantum gases with the scalar contact-interaction Bose–Einstein condensate as the reference system. After establishing the Gross-Pitaevskii foundation (hydrodynamics, quantum pressure, Bogoliubov phonons, vortex structure), it defines vortex, wave, and mixed turbulent regimes and introduces the standard diagnostics: incompressible/compressible kinetic-energy spectra, wave-occupation spectra, spectral fluxes, vortex-resolved correlations, and velocity statistics. The central methodological claim is stated in Sec. 2.4 and reiterated in Sec. 5: a power-law spectrum is supporting evidence for a cascade only when supplemented by identification of the transported quantity, the k-range, the flux direction, and the dissipation channel. The survey of experiments covers 2D Onsager clustering, 3D vortex-line turbulence, box-trap wave cascades, engineered dissipation, and turbulent equations of state. Two tables summarize power-law predictions with measurement status and classify experiments by diagnostic evidence. The review is careful to separate baseline scalings (single-vortex k^-3, coarsening spectra) from actual cascade evidence.

Significance. The review makes a valuable and timely methodological intervention. If adopted, its standard would raise the evidentiary bar for cascade claims in atomic-gas turbulence and help the community distinguish direct flux measurements from spectral-slope inference. The authors are scrupulous about the current measurement status, explicitly marking open items (e.g., no direct vortex-energy flux measurement in any atomic-gas experiment). The inclusion of Table 1, which separates baseline scalings from cascade evidence, is particularly useful. The paper does not overclaim: it acknowledges that the Helmholtz decomposition is a formal projection and that the separation of transfer and conversion is decomposition-dependent. This intellectual honesty, together with a broad and balanced reference list, makes the review a trustworthy reference for both newcomers and specialists.

minor comments (5)

[Sec. 2.4, Eqs. (21)–(25)] The operational cascade standard relies on the Helmholtz decomposition of w = sqrt(n) v into incompressible and compressible parts. The text correctly warns that this is a formal projection and that the separation of transfer and conversion is decomposition-dependent. I suggest adding a short paragraph (or box) that explicitly mentions the main sources of ambiguity—vortex-core density depletion, nonlocal projection near boundaries, and quantum-pressure contributions—and recommends concrete cross-checks (e.g., varying the projection convention, computing fluxes on sub-domains, or comparing E_i(k) from different numerical schemes). This would make the proposed standard more actionable without changing the central claim.
[References] Several references have malformed or placeholder-looking DOIs: ref. [30] (10.1103/s31t-tjl9), ref. [108] (10.1103/1ppc-pl4k), and possibly refs. [56], [60], [119], [126]. Please verify all DOIs and bibliographic details before publication.
[Sec. 1.1] The rendering of the Reynolds number definition is garbled in the manuscript text (appears as a sequence of Unicode glyphs). Please ensure the final typeset version correctly displays Re = vL/ν and similar inline expressions.
[Table 1] In the 'Classical direct wave cascade' row, the range of reported exponents 'near 2.9–3.5' would benefit from explicit citations of the specific experiments (e.g., Navon et al. and Galka et al.) so readers can trace the values without hunting through the text.
[Sec. 5] The final paragraph repeats some of the abstract's language; consider tightening it to focus on the open problems already listed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the cascade standard is an explicitly stated operational criterion, not a derived prediction, and external experiments carry the load.

full rationale

This is a review with no new parameter-free derivation, so the relevant question is whether its diagnostic claims reduce to their inputs. They do not. The central thesis (Sec. 2.4, Sec. 5) is that a cascade requires more than a power law: one must identify the transported quantity, the k-range, a flux, and the dissipation channel. The paper presents this as an explicit convention rather than as a derived result: 'An inertial-range cascade is indicated by an interval of k between forcing and dissipation scales in which the direct injection and loss terms are negligible and Pi_E(k) is approximately constant. A power law is then supporting evidence for a cascade, not the definition of one.' The spectral budget (Eq. 23) and flux (Eq. 25) introduce transfer, injection, dissipation, and conversion terms by definition; the constancy of Pi is a criterion, not a prediction. The one load-bearing diagnostic, the Helmholtz decomposition (Eq. 21), is explicitly caveated: 'The decomposition is a formal projection... It should therefore not be read as saying that vortices and waves are dynamically independent,' and 'The precise separation of transfer and conversion is decomposition-dependent.' These admissions remove any hidden definitional circularity. The review's survey repeatedly marks experiments as lacking direct flux evidence (e.g., for Neely et al., 'not a direct measurement of a constant inertial-range flux'), showing the standard is applied externally rather than fitted to the data. The self-citations (e.g., [81,83] for point-vortex spectra) support specific spectral formulas but are not load-bearing for the central argument, which also rests on external wave-turbulence monographs [86,87], classical Kolmogorov theory, and independent experiments (Navon et al., Galka et al., Hadzibabic group). No step equates a fitted parameter with a prediction, and no load-bearing claim rests on an unverified self-citation. Hence no circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are introduced by this review; quoted exponents and fluxes are taken from cited experiments. The axioms listed are standard Gross-Pitaevskii and wave-kinetic assumptions necessary for the diagnostic framework to work. No new entities are postulated.

assumptions (4)

domain assumption The Gross-Pitaevskii equation with contact interactions describes the relevant dynamical regimes of dilute atomic BECs.
Invoked throughout Sec. 1.2 and used as the reference model for all diagnostics; the review's claims about spectra and fluxes assume GPE provides a valid description.
domain assumption The Helmholtz decomposition of the density-weighted velocity field w = sqrt(n) v into incompressible and compressible parts (Eq. 21) meaningfully separates vortex and wave kinetic energy.
Central to Sec. 2.4; if this decomposition does not cleanly separate the sectors, E_i(k) and E_c(k) are not reliable regime diagnostics. The paper itself calls it a 'formal projection.'
domain assumption The point-vortex spectral formula with the vortex-gas structure factor (Eq. 26) applies to the Hard-wall GPE vortex gases considered in Fig. 5.
Used to interpret spectra of dipole gas, plasma, and clustered states; assumes well-separated point vortices in a homogeneous disk, while the figure uses a Thomas-Fermi background with boundary images.
domain assumption The four-wave (Kolmogorov-Zakharov) kinetic equation describes the direct energy cascade in the particle-like dispersion range of a driven box gas.
Sec. 2.5 and Table 1 rely on wave-kinetic theory from refs [86,87,88,89] to identify the Navon et al. cascade as a four-wave cascade; this is a standard but nontrivial assumption.

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Cite this review

Pith. "Pith review of Turbulence in Quantum Gases: Vortices, Waves, and Cascades." pith.science (2026). https://pith.science/paper/ROGWHPXL

@misc{pith2026260722244,
  author       = {Pith},
  title        = {Pith review of: Turbulence in Quantum Gases: Vortices, Waves, and Cascades},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ROGWHPXL}},
  note         = {Machine review of arXiv:2607.22244}
}

read the original abstract

We review turbulence in ultracold quantum gases, using the scalar contact-interaction Bose-Einstein condensate as the reference system for quantized circulation, compressibility, vortices, sound, and cascades. We focus on the quantitative diagnostics that connect helium and classical phenomenology to microscopic wave-function dynamics: incompressible and compressible kinetic-energy spectra, wave-occupation spectra, spectral fluxes, vortex-resolved correlations, and velocity statistics. These diagnostics distinguish equilibrium vortex organization, decaying turbulent relaxation, forced cascade dynamics, and weak-wave turbulence, and show why power laws alone are insufficient evidence for a cascade. We survey experiments on two-dimensional Onsager clustering, three-dimensional vortex-line turbulence, box-trap wave cascades, engineered dissipation, and turbulent equations of state. We close by briefly placing the contact-interaction scalar superfluid system in a broader landscape of nonlocal, multicomponent, fermionic, and driven-dissipative quantum fluids, where turbulence concepts can be tested for universality.

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Works this paper leans on

131 extracted references · 88 canonical work pages

[1]

Anderson, J.R

M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science269(5221), 198 (1995). https://doi.org/10.1126/science.269.5221.198

doi:10.1126/science.269.5221.198 1995
[2]

Cornell, C.E

E.A. Cornell, C.E. Wieman, Rev. Mod. Phys. 74(3), 875 (2002). https://doi.org/10.1103/RevModPhys.74.875

doi:10.1103/revmodphys.74.875 2002
[3]

Ketterle, Rev

W. Ketterle, Rev. Mod. Phys. 74(4), 1131 (2002). https://doi.org/10.1103/RevModPhys.74.1131

doi:10.1103/revmodphys.74.1131 2002
[4]

Onsager, Il Nuovo Cimento 1943-1954 6(Suppl 2), 279 (1949)

L. Onsager, Il Nuovo Cimento 1943-1954 6(Suppl 2), 279 (1949). https://doi.org/10.1007/BF02780991

doi:10.1007/bf02780991 1943
[5]

Feynman, in Progress in Low Temperature Physics, ed

R.P. Feynman, in Progress in Low Temperature Physics, ed. by W.P. Halperin (Elsevier, 1955), pp. 17–53. https://doi.org/10.1016/S0079-6417(08)60077-3

doi:10.1016/s0079-6417(08)60077-3 1955
[6]

Vinen, Proc

W.F. Vinen, Proc. R. Soc. Lond. Ser. Math. Phys. Sci. 242(1231), 493 (1957)

1957
[7]

Vinen, J

W.F. Vinen, J. Low Temp. Phys. 145(1-4), 7 (2006). https://doi.org/10.1007/s10909-006-9240-6 28 Ashton S. Bradley, Tyler W. Neely, Xiaoquan Yu, and Brian P . Anderson

work page doi:10.1007/s10909-006-9240-6 2006
[8]

Matthews, B.P

M.R. Matthews, B.P. Anderson, P.C. Haljan, D.S. Hall, C.E . Wieman, E.A. Cornell, Phys. Rev. Lett. 83(13), 2498 (1999). https://doi.org/10.1103/PhysRevLett.83.2498

doi:10.1103/physrevlett.83.2498 1999

Show all 131 references

[9]

Madison, F

K.W. Madison, F. Chevy, W. Wohlleben, J. Dalibard, Phys. R ev. Lett. 84(5), 806 (2000). https://doi.org/10.1103/PhysRevLett.84.806

2000 doi
[10]

Abo-Shaeer, C

J.R. Abo-Shaeer, C. Raman, J.M. Vogels, W. Ketterle, Sci ence 292(5516), 476 (2001). https://doi.org/10.1126/science.1060182

2001 doi
[11]

Neely, E.C

T.W. Neely, E.C. Samson, A.S. Bradley, M.J. Davis, B.P. Anderson, Phys. Rev. Lett.104(16), 160401 (2010). https://doi.org/10.1103/PhysRevLett.104.160401

2010 doi
[12]

Navon, A.L

N. Navon, A.L. Gaunt, R.P. Smith, Z. Hadzibabic, Nature 539(7627), 72 (2016). https://doi.org/10.1038/nature20114

2016 doi
[13]

Gauthier, M.T

G. Gauthier, M.T. Reeves, X. Yu, A.S. Bradley, M.A. Baker , T.A. Bell, H. Rubinsztein-Dunlop, M.J. Davis, T.W. Neely, Science 364(6447), 1264 (2019). https://doi.org/10.1126/science.aat5718

2019 doi
[14]

Johnstone, A.J

S.P. Johnstone, A.J. Groszek, P.T. Starkey, C.J. Billington, T.P. Simula, K. Helmerson, Science 364(6447), 1267 (2019). https://doi.org/10.1126/science.aat5793

2019 doi
[15]

M. Zhao, J. Tao, I.B. Spielman, Phys. Rev. Lett. 134(8), 083402 (2025). https://doi.org/10.1103/PhysRevLett.134.083402

2025 doi
[16]

Nemirovskii, Phys

S.K. Nemirovskii, Phys. Rep. 524(3), 85 (2013). https://doi.org/10.1016/j.physrep.2012.10.005

2013 doi
[17]

Pitaevskii, S

L.P. Pitaevskii, S. Stringari, Bose-Einstein Condensation (Clarendon Press, Oxford, 2003)

2003
[18]

Blakie, A

P. Blakie, A. Bradley, M. Davis, R. Ballagh, C. Gardiner, Adv. Phys. 57(5), 363 (2008). https://doi.org/10.1080/00018730802564254

2008 doi
[19]

Proukakis, B

N.P. Proukakis, B. Jackson, J. Phys. B At. Mol. Opt. Phys. 41(20), 203002 (2008). https://doi.org/10.1088/0953-4075/41/20/203002

2008 doi
[20]

Frisch, Y

T. Frisch, Y. Pomeau, S. Rica, Phys. Rev. Lett. 69(11), 1644 (1992). https://doi.org/10.1103/PhysRevLett.69.1644

1992 doi
[21]

Winiecki, J.F

T. Winiecki, J.F. McCann, C.S. Adams, Phys. Rev. Lett. 82(26), 5186 (1999). https://doi.org/10.1103/PhysRevLett.82.5186

1999 doi
[22]

Winiecki, B

T. Winiecki, B. Jackson, J.F. McCann, C.S. Adams, J. Phys . B At. Mol. Opt. Phys. 33(19), 4069 (2000). https://doi.org/10.1088/0953-4075/33/19/317

2000 doi
[23]

dos Santos, Phys

F.E.A. dos Santos, Phys. Rev. A 94(6), 063633 (2016). https://doi.org/10.1103/PhysRevA.94.063633

2016 doi
[24]

Barenghi, H.A.J

C.F. Barenghi, H.A.J. Middleton-Spencer, L. Galantucc i, N.G. Parker, A VS Quantum Sci. 5(2), 025601 (2023). https://doi.org/10.1116/5.0146107

2023 doi
[25]

Reeves, T.P

M.T. Reeves, T.P. Billam, B.P. Anderson, A.S. Bradley, P hys. Rev. Lett. 110(10), 104501 (2013). https://doi.org/10.1103/PhysRevLett.110.104501

2013 doi
[26]

Billam, M.T

T.P. Billam, M.T. Reeves, B.P. Anderson, A.S. Bradley, P hys. Rev. Lett. 112(14), 145301 (2014). https://doi.org/10.1103/PhysRevLett.112.145301

2014 doi
[27]

Reeves, K

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,T.W. Neely, Phys. Rev. X12(1), 011031 (2022). https://doi.org/10.1103/PhysRevX.12.011031

2022 doi
[28]

Onsager, in Int

L. Onsager, in Int. Conf. Theor. Phys. (Science Council of Japan, Kyoto and Tokyo, 1953), pp. 877–880

1953
[29]

Reeves, T.P

M.T. Reeves, T.P. Billam, B.P. Anderson, A.S. Bradley, P hys. Rev. Lett. 114(15), 155302 (2015). https://doi.org/10.1103/PhysRevLett.114.155302

2015 doi
[30]

Christenhusz, A

M.T.M. Christenhusz, A. Safavi-Naini, H. Rubinsztein- Dunlop, T.W. Neely, M.T. Reeves, Phys. Rev. Lett.135(6), 066001 (2025). https://doi.org/10.1103/s31t-tjl9

2025 doi
[31]

Sasaki, N

K. Sasaki, N. Suzuki, H. Saito, Phys. Rev. Lett. 104(15), 150404 (2010). https://doi.org/10.1103/PhysRevLett.104.150404

2010 doi
[32]

Raman, M

C. Raman, M. Kohl, R. Onofrio, D.S. Durfee, C.E. Kuklewic z, Z. Hadzibabic, W. Ketterle, Phys. Rev. Lett.83(13), 2502 (1999)

1999
[33]

Samson, K.E

E.C. Samson, K.E. Wilson, Z.L. Newman, B.P. Anderson, Phys. Rev. A93(2), 023603 (2016). https://doi.org/10.1103/PhysRevA.93.023603

2016 doi
[34]

Henn, J.A

E.A.L. Henn, J.A. Seman, G. Roati, K.M.F. Magalh˜aes, V.S. Bagnato, Phys. Rev. Lett.103(4), 045301 (2009). https://doi.org/10.1103/PhysRevLett.103.045301 Turbulence in Quantum Gases: Vortices, Waves, and Cascades 29

2009 doi
[35]

Scherer, C

D. Scherer, C. Weiler, T. Neely, B.P. Anderson, Phys. Rev . Lett. 98(11), 110402 (2007). https://doi.org/10.1103/PhysRevLett.98.110402

2007 doi
[36]

Hern ´andez-Rajkov, N

D. Hern ´andez-Rajkov, N. Grani, F. Scazza, G. Del Pace, W.J. Kwon, M. Inguscio, K. Xhani, C. Fort, M. Modugno, F. Marino, G. Roati, Nat. Phys. 20(6), 939 (2024). https://doi.org/10.1038/s41567-024-02466-4

2024 doi
[37]

Del Pace, K

G. Del Pace, K. Xhani, A. Muzi Falconi, M. Fedrizzi, N. Gra ni, D. Hernandez Ra- jkov, M. Inguscio, F. Scazza, W.J. Kwon, G. Roati, Phys. Rev. X 12(4), 041037 (2022). https://doi.org/10.1103/PhysRevX.12.041037

2022 doi
[38]

Haljan, Vortices in Bose-Einstein Condensates

P.C. Haljan, Vortices in Bose-Einstein Condensates. Ph .D. thesis, University of Colorado, Colorado, USA (2003)

2003
[39]

Weiler, T.W

C.N. Weiler, T.W. Neely, D.R. Scherer, A.S. Bradley, M.J . Davis, B.P. Anderson, Nature 455(7215), 948 (2008). https://doi.org/10.1038/nature07334

2008 doi
[40]

Hadzibabic, P

Z. Hadzibabic, P. Kruger, M. Cheneau, B. Battelier, J. Da libard, Nature 441(7097), 1118 (2006). https://doi.org/10.1038/nature04851

2006 doi
[41]

Lin, R.L

Y.J. Lin, R.L. Compton, K. Jimenez-Garcia, J.V. Porto, I .B. Spielman, Nature 462(7273), 628 (2009). https://doi.org/10.1038/nature08609

2009 doi
[42]

Leanhardt, A

A. Leanhardt, A. Gorlitz, A.P. Chikkatur, D. Kielpinski , Y. Shin, D. Pritchard, W. Ketterle, Phys. Rev. Lett. 89(19), 190403 (2002). https://doi.org/10.1103/PhysRevLett.89.190403

2002 doi
[43]

Moon, W.J

G. Moon, W.J. Kwon, H. Lee, Y.i. Shin, Phys. Rev. A 92(5), 051601 (2015). https://doi.org/10.1103/PhysRevA.92.051601

2015 doi
[44]

Abo-Shaeer, C

J.R. Abo-Shaeer, C. Raman, W. Ketterle, Phys. Rev. Lett. 88(7), 070409 (2002). https://doi.org/10.1103/PhysRevLett.88.070409

2002 doi
[45]

Haljan, I

P. Haljan, I. Coddington, P. Engels, E. Cornell, Phys. Re v. Lett. 87(21), 210403 (2001). https://doi.org/10.1103/PhysRevLett.87.210403

2001 doi
[46]

Engels, I

P. Engels, I. Coddington, P.C. Haljan, V. Schweikhard, E.A. Cornell, Phys. Rev. Lett.90(17), 170405 (2003). https://doi.org/10.1103/PhysRevLett.90.170405

2003 doi
[47]

Coddington, P.C

I. Coddington, P.C. Haljan, P. Engels, V. Schweikhard, S . Tung, E.A. Cornell, Phys. Rev. A 70(6), 063607 (2004). https://doi.org/10.1103/PhysRevA.70.063607

2004 doi
[48]

Castin, R

Y. Castin, R. Dum, Eur. Phys. J. D 7(3), 399 (1999). https://doi.org/10.1007/s100530050584

1999 doi
[49]

Inouye, Phys

S. Inouye, Phys. Rev. Lett. 87(8), 080402 (2001). https://doi.org/10.1103/PhysRevLett.87.080402

2001 doi
[50]

Neely, Formation, Dynamics, and Decay of Quantized Vortices in Bose-Einstein Con- densates: Elements of Quantum Turbulence

T.W. Neely, Formation, Dynamics, and Decay of Quantized Vortices in Bose-Einstein Con- densates: Elements of Quantum Turbulence. Ph.D. thesis, Un iversity of Arizona, Tucson, Arizona (2010)

2010
[51]

Kwon, J.H

W.J. Kwon, J.H. Kim, S.W. Seo, Y. Shin, Phys. Rev. Lett. 117(24), 245301 (2016). https://doi.org/10.1103/PhysRevLett.117.245301

2016 doi
[52]

Wilson, E.C

K.E. Wilson, E.C. Samson, Z.L. Newman, B.P. Anderson, Ph ys. Rev. A 106(3), 033319 (2022). https://doi.org/10.1103/PhysRevA.106.033319

2022 doi
[53]

W.J. Kwon, G. Del Pace, K. Xhani, L. Galantucci, A. Muzi Fa l- coni, M. Inguscio, F. Scazza, G. Roati, Nature 600(7887), 64 (2021). https://doi.org/10.1038/s41586-021-04047-4

2021 doi
[54]

Wilson, Z.L

K.E. Wilson, Z.L. Newman, J.D. Lowney, B.P. Anderson, Phys. Rev. A91(2), 023621 (2015). https://doi.org/10.1103/PhysRevA.91.023621

2015 doi
[55]

Gertjerenken, P.G

B. Gertjerenken, P.G. Kevrekidis, R. Carretero-Gonz´alez, B.P. Anderson, Phys. Rev. A93(2), 023604 (2016). https://doi.org/10.1103/PhysRevA.93.023604

2016 doi
[56]

Neely, G

T.W. Neely, G. Gauthier, C. Glasspool, Matthew J. Davis, M.T. Reeves. Melting of a vortex matter Wigner crystal (2024). https://doi.org/10.48550/arXiv.2402.09920

2024 doi arXiv
[57]

S.W. Seo, B. Ko, J.H. Kim, Y. Shin, Sci. Rep. 7(1), 4587 (2017). https://doi.org/10.1038/s41598-017-04122-9

2017 doi
[58]

Middelkamp, P.J

S. Middelkamp, P.J. Torres, P.G. Kevrekidis, D.J. Frant zeskakis, R. Carretero-Gonz ´alez, P. Schmelcher, D.V. Freilich, D.S. Hall, Phys. Rev. A 84(1), 011605 (2011). https://doi.org/10.1103/PhysRevA.84.011605 30 Ashton S. Bradley, Tyler W. Neely, Xiaoquan Yu, and Brian P . Anderson

2011 doi
[59]

Navarro, R

R. Navarro, R. Carretero-Gonz ´alez, P.J. Torres, P.G. Kevrekidis, D.J. Frantzeskakis, M.W. Ray, E. Altuntas ¸, D.S. Hall, Phys. Rev. Lett. 110(22), 225301 (2013). https://doi.org/10.1103/PhysRevLett.110.225301

2013 doi
[60]

Grani, D

N. Grani, D. Hern ´andez-Rajkov, C. Daix, P. Pieri, M. Pini, P. Magierski, G. Wl az lowski, M. Fr´ometa Fern´andez, F. Scazza, G. Del Pace, G. Roati, Nat. Commun. 16, 10245 (2025). https://doi.org/{10.1038/s41467-025-64992-w}

2025 doi
[61]

Raman, J.R

C. Raman, J.R. Abo-Shaeer, J. Vogels, K. Xu, W. Ketterle, Phys. Rev. Lett. 87(21), 210402 (2001). https://doi.org/10.1103/PhysRevLett.87.210402

2001 doi
[62]

Rosenbusch, V

P. Rosenbusch, V. Bretin, J. Dalibard, Phys. Rev. Lett. 89(20), 200403 (2002). https://doi.org/10.1103/PhysRevLett.89.200403

2002 doi
[63]

Freilich, D.M

D.V. Freilich, D.M. Bianchi, A.M. Kaufman, T.K. Langin, D.S. Hall, Science 329(5996), 1182 (2010). https://doi.org/10.1126/science.1191224

2010 doi
[64]

Seraﬁni, L

S. Seraﬁni, L. Galantucci, E. Iseni, T. Bienaim ´e, R.N. Bisset, C.F. Barenghi, F. Dalfovo, G. Lamporesi, G. Ferrari, Phys. Rev. X 7(2), 021031 (2017). https://doi.org/10.1103/PhysRevX.7.021031

2017 doi
[65]

Capuzzi, F

P. Capuzzi, F. Federici, M. Tosi, Phys. Rev. A 78(2), 023604 (2008). https://doi.org/10.1103/PhysRevA.78.023604

2008 doi
[66]

Prabhakar, R.P

S. Prabhakar, R.P. Singh, S. Gautam, D. Angom, J. Phys. B: At. Mol. Opt. Phys. 46(12), 125302 (2013). https://doi.org/10.1088/0953-4075/46/12/125302

2013 doi
[67]

Ogawa, M

S.i. Ogawa, M. Tsubota, Y. Hattori, J. Phys. Soc. Jpn. 71(3), 813 (2002). https://doi.org/10.1143/JPSJ.71.813

2002 doi
[68]

Fonda, D.P

E. Fonda, D.P. Meichle, N.T. Ouellette, S. Hormoz, D.P. L athrop, Proc. Nat. Acad. Sci. 111(Supplement 1), 4707 (2014). https://doi.org/10.1073/pnas.1312536110

2014 doi
[69]

Kozik, N

E. Kozik, N. Prokof’ev, B. Svistunov, Phys. Rev. B 73(9), 092501 (2006). https://doi.org/10.1103/PhysRevB.73.092501

2006 doi
[70]

Barenghi, R

C.F. Barenghi, R. H ¨anninen, M. Tsubota, Phys. Rev. E 74(4), 046303 (2006). https://doi.org/{10.1103/PhysRevE.74.046303}

2006 doi
[71]

Fedichev, G.V

P.O. Fedichev, G.V. Shlyapnikov, Phys. Rev. A 60(3), R1779 (1999). https://doi.org/10.1103/PhysRevA.60.R1779

1999 doi
[72]

Zhuravlev, A.E

O.N. Zhuravlev, A.E. Muryshev, P.O. Fedichev, Phys. Rev . A 64(5), 053601 (2001). https://doi.org/10.1103/PhysRevA.64.053601

2001 doi
[73]

Kim, W.J

J.H. Kim, W.J. Kwon, Y. Shin, Phys. Rev. A 94(3), 033612 (2016). https://doi.org/10.1103/PhysRevA.94.033612

2016 doi
[74]

Stockdale, M.T

O.R. Stockdale, M.T. Reeves, X. Yu, G. Gauthier, K. Godda rd-Lee, W.P. Bowen, T.W. Neely, M.J. Davis, Phys. Rev. Res. 2(3), 033138 (2020). https://doi.org/10.1103/PhysRevResearch.2.033138

2020 doi
[75]

Mehdi, J.J

Z. Mehdi, J.J. Hope, S.S. Szigeti, A.S. Bradley, Phys. Re v. Res. 5(1), 013184 (2023). https://doi.org/10.1103/PhysRevResearch.5.013184

2023 doi
[76]

Simula, M.J

T.P. Simula, M.J. Davis, K. Helmerson, Phys. Rev. Lett. 113(16), 165302 (2014). https://doi.org/10.1103/PhysRevLett.113.165302

2014 doi
[77]

X. Yu, T.P. Billam, J. Nian, M.T. Reeves, A.S. Bradley, Phys. Rev. A 94(2), 023602 (2016). https://doi.org/10.1103/PhysRevA.94.023602

2016 doi
[78]

Smith, Phys

R.A. Smith, Phys. Rev. Lett. 63(14), 1479 (1989). https://doi.org/10.1103/PhysRevLett.63.1479

1989 doi
[79]

Sharma, D

R. Sharma, D. Rey, L. Longchambon, A. Perrin, H. Perrin, R . Dubessy, Phys. Rev. Lett. 133(14), 143401 (2024). https://doi.org/10.1103/PhysRevLett.133.143401

2024 doi
[80]

C. Nore, M. Abid, M.E. Brachet, Phys. Rev. Lett. 78(20), 3896 (1997). https://doi.org/10.1103/PhysRevLett.78.3896

1997 doi
[81]

Bradley, B.P

A.S. Bradley, B.P. Anderson, Phys. Rev. X 2(4), 041001 (2012). https://doi.org/10.1103/PhysRevX.2.041001

2012 doi
[82]

Kusumura, H

T. Kusumura, H. Takeuchi, M. Tsubota, J. Low Temp. Phys. 171(5), 563 (2013). https://doi.org/10.1007/s10909-012-0827-9

2013 doi
[83]

Bradley, R.K

A.S. Bradley, R.K. Kumar, S. Pal, X. Yu, Phys. Rev. A 106(4), 043322 (2022). https://doi.org/10.1103/PhysRevA.106.043322 Turbulence in Quantum Gases: Vortices, Waves, and Cascades 31

2022 doi
[84]

Nowak, J

B. Nowak, J. Schole, D. Sexty, T. Gasenzer, Phys. Rev. A 85(4), 043627 (2012). https://doi.org/10.1103/PhysRevA.85.043627

2012 doi
[85]

M. Gazo, A. Karailiev, T. Satoor, C. Eigen, M. Ga lka, Z. H adzibabic, Science 389(6762), 802 (2025). https://doi.org/10.1126/science.ado3487

2025 doi
[86]

Zakharov, V

V. Zakharov, V. Lvov, G.E. Falkovich,Kolmogorov Spectra of Turbulence 1: Wave Turbulence (Springer-Verlag, New York, 1992)

1992
[87]

Nazarenko, Wave Turbulence, Lecture Notes in Physics, vol

S. Nazarenko, Wave Turbulence, Lecture Notes in Physics, vol. 825 (Springer, Berlin, Heidel- berg, 2011). https://doi.org/10.1007/978-3-642-15942-8

2011 doi
[88]

Navon, C

N. Navon, C. Eigen, J. Zhang, R. Lopes, A.L. Gaunt, K. Fuji moto, M. Tsubota, R.P. Smith, Z. Hadzibabic, Science 366(6463), 382 (2019). https://doi.org/10.1126/science.aau6103

2019 doi
[89]

Y. Zhu, B. Semisalov, G. Krstulovic, S. Nazarenko, Phys.Rev. Lett.130(13), 133001 (2023). https://doi.org/10.1103/PhysRevLett.130.133001

2023 doi
[90]

Novikov, Zhurnal Eksp

E.A. Novikov, Zhurnal Eksp. Noi Teor. Fiz. 68, 1868 (1975)

1975
[91]

Skaugen, L

A. Skaugen, L. Angheluta, Phys. Rev. E 95(5), 052144 (2017). https://doi.org/10.1103/PhysRevE.95.052144

2017 doi
[92]

Kolmogorov, Akad

A. Kolmogorov, Akad. Nauk SSSR Dokl. 30, 301 (1941)

1941
[93]

Kobayashi, M

M. Kobayashi, M. Tsubota, Phys. Rev. Lett. 94(6), 065302 (2005). https://doi.org/10.1103/PhysRevLett.94.065302

2005 doi
[94]

Ga lka, P

M. Ga lka, P. Christodoulou, M. Gazo, A. Karailiev, N. Do gra, J. Schmitt, Z. Hadzibabic, Phys. Rev. Lett. 129(19), 190402 (2022). https://doi.org/10.1103/PhysRevLett.129.190402

2022 doi
[95]

Karailiev, M

A. Karailiev, M. Gazo, M. Ga lka, C. Eigen, T. Satoor, Z. Hadzibabic, Phys. Rev. Lett.133(24), 243402 (2024). https://doi.org/10.1103/PhysRevLett.133.243402

2024 doi
[96]

Morris, M

S.J. Morris, M. Gazo, S.M. Fischer, H. Zhang, C.J. Ho, N.R . Cooper, C. Eigen, Z. Hadzibabic. Observation of Vinen turbulence during far- from-equilibrium Bose- Einstein condensation. Preprint at https://arxiv.org/abs/2604.28191 (2026). https://doi.org/10.48550/arXiv.2604.28191

2026 doi arXiv
[97]

Navon, R.P

N. Navon, R.P. Smith, Z. Hadzibabic, Nat. Phys. 17(12), 1334 (2021). https://doi.org/10.1038/s41567-021-01403-z

2021 doi
[98]

Fischer, A.S

T.Z. Fischer, A.S. Bradley, Phys. Rev. A 111(2), 023308 (2025). https://doi.org/10.1103/PhysRevA.111.023308

2025 doi
[99]

Kwon, S.W

W.J. Kwon, S.W. Seo, Y.i. Shin, Phys. Rev. A 92(3), 033613 (2015). https://doi.org/10.1103/PhysRevA.92.033613

2015 doi
[100]

Neely, A.S

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, B.P. Anderson, Phys. Rev . Lett. 111(23), 235301 (2013). https://doi.org/10.1103/PhysRevLett.111.235301

2013 doi
[101]

Baggaley, N.G

A.W. Baggaley, N.G. Parker, Phys. Rev. A 97(5), 053608 (2018). https://doi.org/10.1103/PhysRevA.97.053608

2018 doi
[102]

Simjanovski, G

S. Simjanovski, G. Gauthier, H. Rubinsztein-Dunlop, M .T. Reeves, T.W. Neely, Phys. Rev. A 111(2), 023314 (2025). https://doi.org/10.1103/PhysRevA.111.023314

2025 doi
[103]

Gaunt, T.F

A.L. Gaunt, T.F. Schmidutz, I. Gotlibovych, R.P. Smith , Z. Hadzibabic, Phys. Rev. Lett. 110(20), 200406 (2013). https://doi.org/10.1103/PhysRevLett.110.200406

2013 doi
[104]

Chomaz, L

L. Chomaz, L. Corman, T. Bienaim ´e, R. Desbuquois, C. Weitenberg, S. Nascimb `ene, J. Beugnon, J. Dalibard, Nat. Commun. 6, 6162 (2015). https://doi.org/10.1038/ncomms7162

2015 doi
[105]

Gauthier, I

G. Gauthier, I. Lenton, N.M. Parry, M. Baker, M.J. Davis , H. Rubinsztein-Dunlop, T.W. Neely, Optica 3(10), 1136 (2016). https://doi.org/10.1364/OPTICA.3.001136

2016 doi
[106]

Ville, R

J.L. Ville, R. Saint-Jalm, ´E. Le Cerf, M. Aidelsburger, S. Nascimb `ene, J. Dalibard, J. Beugnon, Phys. Rev. Lett. 121(14), 145301 (2018). https://doi.org/10.1103/PhysRevLett.121.145301

2018 doi
[107]

Dogra, G

L.H. Dogra, G. Martirosyan, T.A. Hilker, J.A.P. Glidde n, J. Etrych, A. Cao, C. Eigen, R.P. Smith, Z. Hadzibabic, Nature 620(7974), 521 (2023). https://doi.org/10.1038/s41586-023-06240-z 32 Ashton S. Bradley, Tyler W. Neely, Xiaoquan Yu, and Brian P . Anderson

2023 doi
[108]

Martirosyan, K

G. Martirosyan, K. Fujimoto, N. Navon, Phys. Rev. Lett. 136(15), 153401 (2026). https://doi.org/10.1103/1ppc-pl4k

2026 doi
[109]

Martirosyan, C.J

G. Martirosyan, C.J. Ho, J. Etrych, Y. Zhang, A. Cao, Z. H adzibabic, C. Eigen, Phys. Rev. Lett. 132(11), 113401 (2024). https://doi.org/10.1103/PhysRevLett.132.113401

2024 doi
[110]

Amette Estrada, M.E

J. Amette Estrada, M.E. Brachet, P.D. Mininni, Phys. Re v. A 105(6), 063321 (2022). https://doi.org/10.1103/PhysRevA.105.063321

2022 doi
[111]

Barenghi, Phys

C.F. Barenghi, Phys. Nonlinear Phenom. 237(14), 2195 (2008). https://doi.org/10.1016/j.physd.2008.01.010

2008 doi
[112]

Chomaz, I

L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe -Tolra, B.L. Lev, T. Pfau, Rep. Prog. Phys. 86(2), 026401 (2022). https://doi.org/10.1088/1361-6633/aca814

2022 doi
[113]

Ticknor, R.M

C. Ticknor, R.M. Wilson, J.L. Bohn, Phys. Rev. Lett. 106(6), 065301 (2011). https://doi.org/10.1103/PhysRevLett.106.065301

2011 doi
[114]

Bland, G

T. Bland, G. Lamporesi, M.J. Mark, F. Ferlaino, Comptes Rendus Phys. 24(S3), 133 (2023). https://doi.org/10.5802/crphys.160

2023 doi
[115]

Casotti, E

E. Casotti, E. Poli, L. Klaus, A. Litvinov, C. Ulm, C. Politi, M.J. Mark, T. Bland, F. Ferlaino, Nature 635(8038), 327 (2024). https://doi.org/10.1038/s41586-024-08149-7

2024 doi
[116]

Stamper-Kurn, M

D.M. Stamper-Kurn, M. Ueda, Rev. Mod. Phys. 85(3), 1191 (2013). https://doi.org/10.1103/RevModPhys.85.1191

2013 doi
[117]

Kawaguchi, M

Y. Kawaguchi, M. Ueda, Phys. Rep. 520(5), 253 (2012). https://doi.org/10.1016/j.physrep.2012.07.005

2012 doi
[118]

Fujimoto, M

K. Fujimoto, M. Tsubota, Phys. Rev. A 88(6), 063628 (2013). https://doi.org/10.1103/PhysRevA.88.063628

2013 doi
[119]

J. Lee, J. Kim, D. Lee, Y.i. Shin. Energy spectra and cascade in the spin turbulence of a driven spinor Bose-Einstein condensate. Preprint at https://arxiv.org/abs/2606.00766 (2026). https://doi.org/10.48550/arXiv.2606.00766

2026 doi arXiv
[120]

Giorgini, L.P

S. Giorgini, L.P. Pitaevskii, S. Stringari, Rev. Mod. P hys. 80(4), 1215 (2008). https://doi.org/10.1103/RevModPhys.80.1215

2008 doi
[121]

Liu, X.C

X.P. Liu, X.C. Yao, Y. Deng, X.Q. Wang, Y.X. Wang, C.J. Hu ang, X. Li, Y.A. Chen, J.W. Pan, Phys. Rev. Lett. 126(18), 185302 (2021). https://doi.org/10.1103/PhysRevLett.126.185302

2021 doi
[122]

Boulier, M.J

T. Boulier, M.J. Jacquet, A. Ma ˆıtre, G. Lerario, F. Claude, S. Pigeon, Q. Glorieux, A. Amo, J. Bloch, A. Bramati, E. Giacobino, Adv. Quantum Technol. 3(11), 2000052 (2020). https://doi.org/10.1002/qute.202000052

2020 doi
[123]

Jacquet, T

M.J. Jacquet, T. Boulier, F. Claude, A. Maˆıtre, E. Cancellieri, C. Adrados, A. Amo, S. Pigeon, Q. Glorieux, A. Bramati, E. Giacobino, Philos. Trans. R. Soc. Math. Phys. Eng. Sci.378(2177), 20190225 (2020). https://doi.org/10.1098/rsta.2019.0225

2020
[124]

Lerario, A

G. Lerario, A. Ma ˆıtre, R. Boddeda, Q. Glorieux, E. Giacobino, S. Pigeon, A. Bramati, Phys. Rev. Res. 2(2), 023049 (2020). https://doi.org/10.1103/PhysRevResearch.2.023049

2020 doi
[125]

Vocke, K

D. Vocke, K. Wilson, F. Marino, I. Carusotto, E.M. Wrigh t, T. Roger, B.P. Anderson, P. ¨Ohberg, D. Faccio, Phys. Rev. A 94(1), 013849 (2016). https://doi.org/10.1103/PhysRevA.94.013849

2016 doi
[126]

Baker-Rasooli, T

M. Baker-Rasooli, T. Aladjidi, N.A. Krause, A.S. Bradl ey, Q. Glorieux, Phys. Rev. Lett. 134(23), 233401 (2025). https://doi.org/10.1103/PhysRevLett.134.233401

2025 doi
[127]

Baker-Rasooli, W

M. Baker-Rasooli, W. Liu, T. Aladjidi, A. Bramati, Q. Glorieux, Phys. Rev. A108(6), 063512 (2023). https://doi.org/10.1103/PhysRevA.108.063512

2023 doi
[128]

Ferreira, J

T.D. Ferreira, J. Garwo la, N.A. Silva, Phys. Rev. A 109(4), 043704 (2024). https://doi.org/10.1103/PhysRevA.109.043704

2024 doi
[129]

Glorieux, C

Q. Glorieux, C. Piekarski, Q. Schibler, T. Aladjidi, M.Baker-Rasooli, in Advances In Atomic, Molecular, and Optical Physics, vol. 74, ed. by L.F. Dimauro, H. Perrin, S. Yelin (Academic Press, 2025), pp. 157–241. https://doi.org/10.1016/bs.aamop.2025.04.002

2025 doi
[130]

Onofrio, C

R. Onofrio, C. Raman, J.M. Vogels, J.R. Abo-Shaeer, A.P. Chikkatur, W. Ketterle, Phys. Rev. Lett. 85(11), 2228 (2000)

2000
[131]

W.J. Kwon, G. Moon, S.W. Seo, Y. Shin, Phys. Rev. A 91(5), 053615 (2015). https://doi.org/10.1103/PhysRevA.91.053615

2015 doi

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Anderson, J.R

M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, E.A. Cornell, Science269(5221), 198 (1995). https://doi.org/10.1126/science.269.5221.198

doi:10.1126/science.269.5221.198 1995

[2] [2]

Cornell, C.E

E.A. Cornell, C.E. Wieman, Rev. Mod. Phys. 74(3), 875 (2002). https://doi.org/10.1103/RevModPhys.74.875

doi:10.1103/revmodphys.74.875 2002

[3] [3]

Ketterle, Rev

W. Ketterle, Rev. Mod. Phys. 74(4), 1131 (2002). https://doi.org/10.1103/RevModPhys.74.1131

doi:10.1103/revmodphys.74.1131 2002

[4] [4]

Onsager, Il Nuovo Cimento 1943-1954 6(Suppl 2), 279 (1949)

L. Onsager, Il Nuovo Cimento 1943-1954 6(Suppl 2), 279 (1949). https://doi.org/10.1007/BF02780991

doi:10.1007/bf02780991 1943

[5] [5]

Feynman, in Progress in Low Temperature Physics, ed

R.P. Feynman, in Progress in Low Temperature Physics, ed. by W.P. Halperin (Elsevier, 1955), pp. 17–53. https://doi.org/10.1016/S0079-6417(08)60077-3

doi:10.1016/s0079-6417(08)60077-3 1955

[6] [6]

Vinen, Proc

W.F. Vinen, Proc. R. Soc. Lond. Ser. Math. Phys. Sci. 242(1231), 493 (1957)

1957

[7] [7]

Vinen, J

W.F. Vinen, J. Low Temp. Phys. 145(1-4), 7 (2006). https://doi.org/10.1007/s10909-006-9240-6 28 Ashton S. Bradley, Tyler W. Neely, Xiaoquan Yu, and Brian P . Anderson

work page doi:10.1007/s10909-006-9240-6 2006

[8] [8]

Matthews, B.P

M.R. Matthews, B.P. Anderson, P.C. Haljan, D.S. Hall, C.E . Wieman, E.A. Cornell, Phys. Rev. Lett. 83(13), 2498 (1999). https://doi.org/10.1103/PhysRevLett.83.2498

doi:10.1103/physrevlett.83.2498 1999