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REVIEW 3 major objections 5 minor 46 references

Anisotropic quantum polytropes

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read In the pressureless quantum-polytrope limit examined in this paper, positive anisotropic pressure compresses the configuration and a stronger gravitational coupling loosens it, the reverse of classical polytropes.

desk verdict The central sign-flip claim rests on an equation that doesn't follow from the stated model; the paper is a solid idea with an unsupported core. read the letter →

arxiv 2506.08135 v1 pith:2YVOG3AV submitted 2025-06-09 gr-qc astro-ph.GA

Hermano Velten , Felipe S. Escórcio , Nadson J. S. Trindade This is my paper
classification gr-qcastro-ph.GA
keywords polytropeshydrostaticequilibriumanisotropicpressurequantumpotentialGross-Pitaevskii-PoissonsystemLane-Emdenequationdarkmatterhalosbosonstars
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

Polytropes — self-gravitating spheres supported by a power-law pressure-density relation — are a standard tool for stars and dark-matter halos. This paper studies what happens when the supporting pressure is instead the Bohm quantum potential of a condensed boson gas, the “quantum polytrope” regime. It reports that in that regime the usual sign rules reverse: a positive anisotropic pressure (tangential exceeding radial) acts as an inward, clustering force, and a stronger gravitational constant makes the configuration less bound. These claims come from a generalized fourth-order Lane-Emden-like equilibrium equation solved numerically, which the paper interprets as a new class of hydrostatic equilibrium objects.

What carries the argument

The carrying object is the generalized Lane-Emden equation (23), a fourth-order ordinary differential equation for $b(r) = \sqrt{\rho(r)}$ that encodes hydrostatic equilibrium as a competition among five dimensionless parameters: pressure gradient ($\Omega_P$), Bohm quantum potential ($\Omega_q$), boson-boson interaction ($\Omega_s$), Newtonian gravity ($\Omega_G$), and anisotropy ($\Omega_A$). The anisotropy is modeled with the standard ansatz $\Delta = (\rho + P) r^N f$ with $f = 1$ and $N = 2$ set by regularity at the center. The argument proceeds by switching these parameters on and off; the anomalous sign reversal appears in the corner $\Omega_q = 1$, $\Omega_P = \Omega_s = 0$, when the quantum potential alone supports the configuration against gravity.

What would settle it

Symbolically expand the force-balance equation (22) term by term and compare with the solved equation (23); if the gravity term changes from $b^2$ to $b^3$ or the anisotropy terms differ in powers of $b$, then the reported sign reversal and thresholds belong to a different equation than the one stated.

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Extended reading notes

Core claim

Starting from a static Gross-Pitaevskii-Poisson system with a polytropic pressure, the Bohm quantum potential, boson-boson scattering, and an anisotropy parameter $\Delta = P_t - P_r$, the paper packages the force balance into a fourth-order equation for $b = \sqrt{\rho}$ whose five dimensionless coefficients set the magnitudes of pressure, quantum potential, interaction, gravity, and anisotropy (Eq. 23). The numerical solutions show that when pressure and interaction are switched off ($\Omega_P = \Omega_s = 0$, $\Omega_q = 1$), positive values of the anisotropy parameter $\Omega_A$ shrink the equilibrium radius, and bound configurations exist only above $\Omega_A \approx -0.5$. Increasing the gravitational coefficient from $\Omega_G = 1$ to $\Omega_G = +10$ makes the structure less bounded, whereas $\Omega_G = -10$ compresses it. The paper concludes that positive anisotropy and stronger gravity play roles opposite to the classical case in this limit, defining a new class of equilibrium configurations.

Load-bearing premise

The numerical story rests on equation (23) being the correct expanded form of the model's force balance, a step the paper does not demonstrate algebraically.

Editorial extensions

If this is right

  • If the sign reversal is correct, positive tangential-pressure anisotropy can make bosonic dark-matter halos more compact, so anisotropy could mimic an attractive self-interaction in halo models.
  • The numerical survey yields thresholds that can be confronted with observation: pressureless quantum polytropes require $\Omega_A \gtrsim -0.5$ for bound configurations, while repulsive Thomas-Fermi configurations lose bound states for $\Omega_A \gtrsim 0.1$.
  • Oscillations in the density profile induced by positive anisotropy resemble oscillations in simulated rotation curves of ultralight dark-matter halos, creating a degeneracy between anisotropic-pressure and axion-like effects.
  • In the quantum-polytrope limit, increasing the gravitational constant weakens binding, so standard structure-formation expectations do not carry over to this regime.

Reading between the lines

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

  • If the sign reversal is confirmed, the quantum stress tensor (12) offers a natural bookkeeping: because the quantum potential is equivalent to an anisotropic pressure, a positive $\Delta$ can be absorbed into a renormalized quantum pressure, potentially linking the anomaly to known stability criteria for Bose-Einstein condensates.
  • A testable extension is to fit the model to dwarf-galaxy rotation curves; unexplained oscillatory features would translate into upper bounds on $\Omega_A$, or could be read as evidence for anisotropic stresses in dark halos.
  • One could also push the same parameter survey into the fully interacting, pressure-supported regime and ask whether the sign reversal persists when both $\Omega_P$ and $\Omega_s$ are nonzero; the reported reversal is demonstrated in the pressureless non-interacting corner.
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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 / 5 minor

Summary. The manuscript extends the Gross-Pitaevskii-Poisson (GPP) description of BEC dark-matter halos and boson stars to include pressure anisotropy, Δ = Pt − Pr. It writes a hydrostatic equilibrium equation combining polytropic pressure, the Bohm quantum potential, boson self-interaction, Newtonian gravity, and anisotropy, then reduces it to Eq. (23) with the auxiliary definitions (24)–(27). The paper solves this equation numerically for classical, quantum, and Thomas–Fermi regimes, and reports that in pressureless 'quantum polytropes' positive anisotropy contracts the configuration while stronger gravity expands it, opposite to the classical polytropic behavior. The authors interpret this as a new class of hydrostatic equilibrium objects and connect the oscillatory density profiles to features in simulated ultralight dark-matter halos.

Significance. If the central derivation and numerics were valid, the paper would offer a useful extension of quantum-polytrope models and could inform modeling of anisotropic BEC dark-matter halos. The paper has clear strengths: it carefully motivates the GPP framework, calibrates the classical polytropic limit in Sec. III.A, and explores a broad parameter space. However, the central equation (23) is not algebraically equivalent to the stated model (22), and the main physical claim—the sign reversal of the anisotropy effect—rests entirely on that equation. No code, convergence tests, or derivational details are provided to compensate. As submitted, the claimed new class of hydrostatic equilibrium objects is not established by the manuscript.

major comments (3)
  1. [II.C, Eqs. (22)–(25)] Equation (23) is not algebraically equivalent to Eq. (22) as claimed. Direct expansion of Eq. (22) with ρ = b² and P = K b^{2γ} gives the Bohm term (Ωq/r) d²/dr²[(1/b) d²(rb)/dr²] = Ωq [b''''/b + 4b'''/(rb) − 2b'''b'/b² − 8b'b''/(rb²) − (b'')²/b² + 4(b')³/(rb³) + 2b''(b')²/b³], whereas Eq. (23) displays b times this expression. The gravity term in Eq. (22) is −ΩG b², but Eq. (23) has −ΩG b³. The pressure term δP in Eq. (24), with powers b^{2γ−3} and coefficient (2γ−3), does not match the left side of Eq. (22), which expands to terms with b^{2γ−1} and coefficient (2γ−1). The anisotropy term δA in Eq. (25) differs by powers of b from the expansion of (ΩA/r²) d/dr(2r∆/b²), while the self-interaction term is unchanged. No single multiplication or redefinition reconciles all terms. Since the numerical solutions and the sign-reversal claim in Fig. 3 and Sec. IV are obtained from Eq. (23), they do not describe the anisotropic quantum polytrope defined by Eqs. (15)–(22).
  2. [III.B, Fig. 3] The central physical claim—that for Ωq = 1 positive anisotropy acts as an inward force and stronger gravity expands the configuration—rests entirely on numerical solutions of Eq. (23). The paper provides no derivation of Eq. (23) from Eq. (22), no code, no convergence tests, and no demonstration that the reported radii are independent of r0, b''(r0), and b'''(r0). For a fourth-order equation, the statement that b'''(r0) 'can assume any arbitrary value without changing the results' is nontrivial and unverified. Without these, the quoted thresholds, such as the bound configurations existing only for ΩA ≳ −0.5, and the comparisons with ΩG = ±10 cannot be assessed.
  3. [II.C, Eq. (20)] The anisotropy model ∆ = (ρ + Pr) r^N f with f = 1 and N = 2 is introduced as an ad hoc choice, and the text states that this 'does not compromise the generality of the model.' That statement is too strong: fixing f = 1 and N = 2 selects a specific radial dependence of the anisotropy, and different choices would change the equilibrium structure. The paper should either justify this choice more concretely or present it explicitly as a modeling restriction rather than a general ansatz.
minor comments (5)
  1. [II.C] There is a typo in the sentence introducing the general hydrostatic equilibrium condition: 'hydrostatic equilIbrIum' should read 'hydrostatic equilibrium.'
  2. [III.A, after Fig. 1] The sentence beginning 'We have checked that assuming γ = 4 /3 There are bound structures...' is grammatically incomplete and unclear; it should be rewritten for readability.
  3. [References] References [22] and [29] appear to be the same paper (G. Abellán, E. Fuenmayor, and L. Herrera, Physics of the Dark Universe 28, 100549 (2020)); duplicate citations should be consolidated.
  4. [III.B, Fig. 3] The caption and text refer to ΩG = −10 as a 'repulsive interaction,' but in Newtonian gravity a negative coupling is not a standard repulsive interaction; this terminology should be clarified or avoided.
  5. [II.C, Eqs. (20) and (26)] The paper calls ΩA dimensionless and writes ∆ with pressure units, but the term (ΩA/r²)d/dr(2r∆/b²) in Eq. (22) requires ΩA to carry dimensions unless the equations are fixed in a particular unit system; this should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the paper solves a stated anisotropic Gross-Pitaevskii-Poisson equilibrium equation with no fitted parameters; its thresholds and sign effects are numerical outputs, not inputs.

full rationale

The central result—that positive anisotropy compresses pressureless quantum polytropes—is obtained by numerically integrating Eq. (23), presented as the expanded hydrostatic equilibrium condition. No parameter is fitted to data, and no prediction is used to fix constants; the anisotropy model Delta=(rho+P_r)r^2 is explicitly introduced as an ad hoc choice (Eq. (20) with N=2, f=1), not as a quantity inferred from the output. Prior works cited (Refs. [6,21,24]) are used for background or as the isotropic baseline, and the only author self-citation (Ref. [10]) is not load-bearing for the derivation. The algebraic mismatch between Eq. (22) and Eqs. (23)-(25) noted in the skeptic material is a consistency/correctness concern, not a circularity: it does not make the numerical result equivalent to the model input by construction. Therefore the paper is not circular.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The generalized Lane-Emden equation has five dimensionless couplings (ΩP, Ωq, Ωs, ΩG, ΩA) and boundary parameters set by hand; none is fitted to data. The load-bearing ad hoc input is the anisotropy profile ∆=(ρ+P_r)r², and the least-supported step is the algebraic reduction to Eq. (23). No new particles or forces are introduced.

free parameters (5)
  • ΩA (anisotropy magnitude) = varied by hand: 0, ±0.01, ±0.5, ±1.0, ±1.4, ±2.0 depending on figure
    Introduced in Eqs. (22)-(23) to quantify ∆; the claimed sign inversion is a scan over this parameter.
  • ΩG (gravitational coupling) = 1, 10, -10
    Set by hand in Fig. 3 to demonstrate the anomalous gravity response.
  • ΩP (pressure magnitude) = 0, 1, 10
    Set by hand; the quantum polytrope limit uses ΩP=0.
  • b''(r0) = -1
    Boundary condition at r0=0.001 chosen to enforce a local maximum of density; no derivation from the physical SP eigenvalue.
  • N in ∆=(ρ+P_r)r^N f = 2
    Ad hoc choice for regularity at the origin and slowly rotating systems; the central result may depend on this choice.
assumptions (6)
  • domain assumption The matter density equals |Ψ|² (quantum equilibrium postulate)
    Stated in footnote 1 of Sec. II.B; needed to connect the wave function to astrophysical density.
  • domain assumption Static, spherically symmetric hydrostatic equilibrium describes boson stars and dark matter halos
    Assumed throughout Sec. II; rotation, time dependence, and relativistic corrections are discarded.
  • domain assumption The Gross-Pitaevskii-Poisson system with Madelung decomposition is a valid model for self-gravitating BEC dark matter
    Sec. II.A; the whole derivation starts from this identification.
  • ad hoc to paper Anisotropy has the form ∆=(ρ+P_r)r^N f with f=1 and N=2
    Eq. (20) and the sentence 'taken as f=1... to simplify the algebraic treatment'; no microphysical derivation is provided.
  • ad hoc to paper Eq. (23) is the correct reduced equilibrium equation
    Eqs. (22)-(25); the reduction is not shown and appears algebraically inconsistent.
  • domain assumption Polytropic equation of state P=Kρ^γ with γ=1+1/n for the macroscopic pressure
    Eq. (9); a modeling assumption inherited from stellar structure literature.

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Cite this review

Pith. "Pith review of Anisotropic quantum polytropes." pith.science (2026). https://pith.science/paper/2YVOG3AV

@misc{pith2026250608135,
  author       = {Pith},
  title        = {Pith review of: Anisotropic quantum polytropes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2YVOG3AV}},
  note         = {Machine review of arXiv:2506.08135}
}
read the original abstract

The structure of astrophysical objects is usually modeled under the assumption of hydrostatic equilibrium. However, actual configurations may deviate from perfect spherical or isotropic properties. Consequently, cosmic objects are expected to exhibit some degree of anisotropy. This consideration also extends to hypothetical dark structures, such as dark stars and dark matter halos. Although the nature of dark matter remains unknown, axion-like particles (ALPs) are strong candidates, suggesting that dark matter halos may have originated from bosonic configurations undergoing gravitational collapse, sustained by boson-boson interactions in the condensate state. This system is described by the Gross-Pitaevskii-Poisson equation. Furthermore, within the framework of the Bohm-de Broglie approach, quantum effects,encapsulated in the so-called quantum potential, may play a significant role in equilibrium astrophysical configurations. In this study, we examine a class of static anisotropic boson stars which are non-minimally coupled to gravity. By including all these factors, we derive a generalized Lane-Emden-like equation and conduct a detailed analysis of the maximum degree of anisotropy that such systems can sustain, thereby identifying physically viable equilibrium configurations. Apart from focusing on the impact of anisotropic contributions, we find that for the so-called Quantum Polytropes (when the quantum potential is the main responsible for the equilibrium condition), the anisotropic factor and the gravitational field have opposite roles compared to the classical case. This leads to a new class of hydrostatic equilibrium objects.

Figures

Figures reproduced from arXiv: 2506.08135 by the authors.

Figure 1
Figure 1. FIG. 1: Top panels: Radial density profile in the non-interacting case with [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Radial density profile in the non-interacting case with [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Radial density profiles for different parameter combinations in the pressureless case ( [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Radial density profiles for different parameter combinations in the pressureless case ( [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Radial density profile in the interacting case with [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

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Works this paper leans on

46 extracted references · 24 canonical work pages

  1. [1]

    Peters, K

    A. Peters, K. Y. Chung, and S. Chu, Metrologia 38, 25 (2001), URL https://dx.doi.org/10.1088/ 0026-1394/38/1/4

    work page 2001

  2. [2]

    Asenbaum, C

    P. Asenbaum, C. Overstreet, M. Kim, J. Curti, and M. A. Kasevich, Phys. Rev. Lett.125, 191101 (2020), URL https://link.aps.org/doi/10.1103/PhysRevLett.125.191101

    doi:10.1103/physrevlett.125.191101 2020

  3. [3]

    Dimopoulos, P

    S. Dimopoulos, P. W. Graham, J. M. Hogan, and M. A. Kasevich, Phys. Rev. Lett.98, 111102 (2007), URL https://link.aps.org/doi/10.1103/PhysRevLett.98.111102

    doi:10.1103/physrevlett.98.111102 2007

  4. [4]

    Müntinga, H

    H. Müntinga, H. Ahlers, M. Krutzik, A. Wenzlawski, S. Arnold, D. Becker, K. Bongs, H. Dittus, H. Duncker, N. Gaaloul, et al., Phys. Rev. Lett. 110, 093602 (2013), URL https://link.aps.org/doi/10.1103/ PhysRevLett.110.093602

    work page 2013

  5. [5]

    Kovachy, P

    T. Kovachy, P. Asenbaum, C. Overstreet, C. A. Donnelly, S. M. Dickerson, A. Sugarbaker, J. M. Hogan, and M. A. Kasevich, Nature528, 530 (2015)

    2015

  6. [6]

    Pierre-Henri, Phys

    C. Pierre-Henri, Phys. Rev. D84, 043531 (2011), URLhttps://link.aps.org/doi/10.1103/PhysRevD.84. 043531

    doi:10.1103/physrevd.84 2011

  7. [7]

    P. H. Chavanis and L. Delfini, Phys. Rev. D84, 043532 (2011), 1103.2054

    arXiv 2011

  8. [8]

    Haegeman, D

    J. Haegeman, D. Draxler, V. Stojevic, J. I. Cirac, T. Osborne, and F. Verstraete, SciPost Physics3, 006 (2017)

    work page 2017

Show all 46 references
  1. [9]

    F. S. Guzman and L. A. Urena-Lopez, The Astrophysical Journal645, 814 (2006)

    2006

  2. [10]

    Velten and E

    H. Velten and E. Wamba, Phys. Lett. B709, 1 (2012), 1111.2032

    2012 arXiv

  3. [11]

    Schwabe, J

    B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D94, 043513 (2016), 1606.05151

    2016 arXiv

  4. [12]

    Calabrese and D

    E. Calabrese and D. N. Spergel, Mon. Not. Roy. Astron. Soc.460, 4397 (2016), 1603.07321

    2016 arXiv

  5. [13]

    L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Physical Review D95, 043541 (2017)

    2017

  6. [14]

    N. Bar, D. Blas, K. Blum, and S. Sibiryakov, Phys. Rev. D98, 083027 (2018), 1805.00122

    2018 arXiv

  7. [15]

    Korshynska, Y

    K. Korshynska, Y. Bidasyuk, E. Gorbar, J. Jia, and A. Yakimenko, The European Physical Journal C83, 451 (2023)

    2023

  8. [16]

    Chavanis, The European Physical Journal Plus132, 1 (2017)

    P.-H. Chavanis, The European Physical Journal Plus132, 1 (2017)

    2017

  9. [17]

    J. C. Niemeyer (2019), 1912.07064

    2019 arXiv

  10. [18]

    Chavanis, Phys

    P.-H. Chavanis, Phys. Rev. D100, 083022 (2019), URLhttps://link.aps.org/doi/10.1103/PhysRevD.100. 083022

    2019 doi

  11. [19]

    Hui, Ann

    L. Hui, Ann. Rev. Astron. Astrophys.59, 247 (2021), 2101.11735

    2021 arXiv

  12. [20]

    E. G. M. Ferreira, Astron. Astrophys. Rev.29, 7 (2021), 2005.03254

    2021 arXiv

  13. [21]

    C. G. Boehmer and T. Harko, JCAP06, 025 (2007), 0705.4158

    2007 arXiv

  14. [23]

    Herrera and N

    L. Herrera and N. O. Santos, Phys. Rept.286, 53 (1997)

    1997

  15. [24]

    J. Heyl, M. W. Choptuik, and D. Shinkaruk, Phys. Rev. D96, 103010 (2017), 1709.09291

    2017 arXiv

  16. [25]

    Del Popolo and P

    A. Del Popolo and P. Kroupa, Astronomy & Astrophysics502, 733 (2009)

    2009

  17. [26]

    R. L. Bowers and E. P. T. Liang, Astrophys. J.188, 657 (1974)

    1974

  18. [27]

    A. A. Isayev, Phys. Rev. D96, 083007 (2017), 1801.03745

    2017 arXiv

  19. [28]

    Dev and M

    K. Dev and M. Gleiser, International Journal of Modern Physics D13, 1389 (2004), URL https://arxiv. org/abs/astro-ph/0401546

    2004 arXiv

  20. [29]

    Abellán, E

    G. Abellán, E. Fuenmayor, and L. Herrera, Physics of the Dark Universe28, 100549 (2020)

    2020

  21. [30]

    P. León, E. Fuenmayor, and E. Contreras, Phys. Rev. D107, 064010 (2023), 2303.04211

    2023 arXiv

  22. [31]

    Kumar and P

    J. Kumar and P. Bharti, New Astron. Rev.95, 101662 (2022)

    2022

  23. [32]

    Andrade and D

    J. Andrade and D. Santana, Eur. Phys. J. C83, 523 (2023)

    2023

  24. [33]

    L. M. Becerra, E. A. Becerra-Vergara, and F. D. Lora-Clavijo, Phys. Rev. D109, 043025 (2024), 2401.10311

    2024 arXiv

  25. [34]

    M. K. Mak and T. Harko, Proceedings of the Royal Society of London A459, 393 (2003), URL https: //arxiv.org/abs/gr-qc/0110103

    2003 arXiv

  26. [35]

    Raposo, P

    G. Raposo, P. Pani, M. Bezares, C. Palenzuela, and V. Cardoso, Physical Review D99, 104072 (2019), URL https://arxiv.org/abs/1811.07917

    2019 arXiv

  27. [36]

    Cadogan and E

    T. Cadogan and E. Poisson, Gen. Rel. Grav.56, 118 (2024), 2406.03185

    2024 arXiv

  28. [37]

    Cadogan and E

    T. Cadogan and E. Poisson, Gen. Rel. Grav.56, 119 (2024), 2406.03191

    2024 arXiv

  29. [38]

    Cadogan and E

    T. Cadogan and E. Poisson, Gen. Rel. Grav.56, 120 (2024), 2406.03196. 12

    2024 arXiv

  30. [39]

    S. A. Ngubelanga and S. D. Maharaj, Eur. Phys. J. Plus130, 211 (2015), 1512.09003

    2015 arXiv

  31. [40]

    Suleiman, M

    L. Suleiman, M. Fortin, J. L. Zdunik, and C. m. c. Providência, Phys. Rev. C106, 035805 (2022), URL https://link.aps.org/doi/10.1103/PhysRevC.106.035805

    2022 doi

  32. [41]

    Godani and G

    N. Godani and G. C. Samanta, New Astron.107, 102148 (2024)

    2024

  33. [42]

    R. F. Tooper, Astrophysical Journal, vol. 140, p. 434140, 434 (1964)

    1964

  34. [43]

    Toomre, Astrophysical Journal, vol

    A. Toomre, Astrophysical Journal, vol. 139, p. 1217-1238 (1964).139, 1217 (1964)

    1964

  35. [44]

    Inouye, M

    S. Inouye, M. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn, and W. Ketterle, Nature392, 151 (1998)

    1998

  36. [45]

    C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Reviews of Modern Physics82, 1225 (2010)

    2010

  37. [46]

    Schive, T

    H.-Y. Schive, T. Chiueh, and T. Broadhurst, Nature Phys.10, 496 (2014), 1406.6586

    2014 arXiv

  38. [47]

    Schive, M.-H

    H.-Y. Schive, M.-H. Liao, T.-P. Woo, S.-K. Wong, T. Chiueh, T. Broadhurst, and W. Y. P. Hwang, Phys. Rev. Lett.113, 261302 (2014), 1407.7762

    2014 arXiv

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