Pith. sign in

REVIEW 2 major objections 2 minor 69 references

Elastic Modulus in One-Dimensional Quantum Droplets

T0 review · 2 major / 2 minor · reviewed 2026-07-03 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read One-dimensional quantum droplets have an elastic modulus linked quantitatively to breathing-mode frequency, with the ratio to particle number showing intricate dependence on interaction strength due to soliton-droplet crossover.

desk verdict This is a clean extension of the 3D elastic-modulus work to 1D droplets that correctly flags the soliton crossover as the source of non-power-law scaling. read the letter →

arxiv 2607.01863 v1 pith:FMHOGZ6I submitted 2026-07-02 cond-mat.quant-gas nlin.PS

Rui Zhang , Tianmiao Zhang , Huan-Bo Luo , Zibin Zhao This is my paper
classification cond-mat.quant-gasnlin.PS
keywords quantumdropletselasticmodulusone-dimensionalLee-Huang-Yangcorrectionbreathingmodevariationalansatzsoliton-to-dropletcrossoverultracoldatoms
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 derives the elastic modulus B of one-dimensional quantum droplets stabilized by Lee-Huang-Yang corrections using a super-Gaussian variational ansatz. It obtains the dependence of B on interaction strength g and particle number N, validated against imaginary-time evolution and spatial scaling numerics. A direct quantitative relation is established between B and the eigenfrequency of the breathing mode. Corrections beyond the Thomas-Fermi approximation reveal that the ratio η = B/2 depends on g and N in a manner shaped by the soliton-to-droplet crossover. In the large-N limit B saturates to a value set primarily by g, while at small N it depends on both parameters.

What carries the argument

Super-Gaussian variational ansatz for the droplet density profile, from which the elastic modulus B is obtained by systematic variation of the width parameter.

What would settle it

Numerical computation of the breathing-mode frequency for a range of g and N, followed by direct comparison against the predicted relation B = 2 * (frequency)^2 scaled by the appropriate factor, would test the quantitative link; significant deviation outside the variational error would falsify the central relation.

Watch

Extended reading notes

Core claim

Based on a super Gaussian variational ansatz, we systematically derive the elastic modulus B and analyze its dependence on the interaction strength and particle number. The analytical predictions are further validated by numerical simulations based on imaginary time evolution and the spatial scaling method. We also establish a quantitative relation between the elastic modulus and the eigenfrequency of the breathing mode. In addition, by incorporating corrections to the droplet width beyond the Thomas Fermi approximation, we obtain the dependence of the ratio η = B/2 on the control parameters g and N. Unlike the three-dimensional case, where the corresponding ratio follows a simple power-law

Load-bearing premise

The super Gaussian variational ansatz accurately captures the density profile of the one-dimensional quantum droplet, allowing reliable derivation of the elastic modulus B and its relation to breathing-mode frequency.

Editorial extensions

If this is right

  • In the high-particle-number regime the elastic modulus approaches a value set mainly by the interaction strength g.
  • In the low-particle-number regime the elastic modulus depends on both particle number N and interaction strength g.
  • The ratio η = B/2 exhibits a more intricate dependence on g and N than the simple power-law found in three dimensions.
  • The elastic modulus is quantitatively tied to the eigenfrequency of the breathing mode.
  • The soliton-to-droplet crossover modifies the scaling of elastic properties in one dimension.

Reading between the lines

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

  • The breathing-mode relation could enable experimental extraction of the elastic modulus from collective oscillation data without separate compression measurements.
  • The crossover-induced intricacy in η(g,N) may produce observable changes in droplet response when tuning across the mean-field to LHY-dominated boundary in quasi-1D traps.
  • Similar variational methods could be applied to study elastic response in other low-dimensional droplet or soliton systems with competing interactions.
  • Finite-temperature or multi-component extensions would test whether the modulus-breathing link survives when thermal fluctuations or additional degrees of freedom are present.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the elastic modulus B of one-dimensional quantum droplets stabilized by LHY corrections. Using a super-Gaussian variational ansatz, it derives analytic expressions for B and the ratio η = B/2, establishes a quantitative link between B and the breathing-mode eigenfrequency, and reports the dependence of η on interaction strength g and particle number N. The 1D case exhibits intricate behavior due to the soliton-to-droplet crossover, unlike the power-law scaling found in 3D; results are stated to be validated by imaginary-time evolution and spatial scaling.

Significance. If the variational ansatz remains quantitatively faithful across the crossover, the work would usefully extend elastic-modulus concepts from 3D to 1D droplets and provide a concrete relation between static elasticity and collective-mode frequency. The emphasis on crossover-induced deviations from simple scaling is a distinguishing feature that could guide experiments in quasi-1D ultracold gases.

major comments (2)
  1. [Variational ansatz and numerical validation] The derivation of B, its relation to breathing frequency, and the non-power-law η(g,N) all originate from energy minimization with the super-Gaussian trial density. The abstract claims validation by imaginary-time evolution, yet no overlap integrals, L2 errors, or density-profile residuals are reported, especially for small N where soliton tails become relevant. This quantitative gap is load-bearing for the central claim of an intricate dependence.
  2. [Definition of B and η] The definition of the droplet width from the same variational minimization that yields B and η raises a circularity risk: it is unclear whether the reported g- and N-dependence of η is an independent physical result or is partly fixed by the ansatz parameters themselves. A direct comparison of the variational B against an independent numerical extraction (e.g., from the second derivative of the energy functional on exact profiles) would resolve this.
minor comments (2)
  1. [Abstract] The abstract mentions validation but omits any mention of error bars, exclusion criteria, or quantitative metrics; these should be added for clarity.
  2. Notation for the super-Gaussian parameters and the precise definition of the elastic modulus B should be introduced with an equation number at first use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the presentation of our results. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Variational ansatz and numerical validation] The derivation of B, its relation to breathing frequency, and the non-power-law η(g,N) all originate from energy minimization with the super-Gaussian trial density. The abstract claims validation by imaginary-time evolution, yet no overlap integrals, L2 errors, or density-profile residuals are reported, especially for small N where soliton tails become relevant. This quantitative gap is load-bearing for the central claim of an intricate dependence.

    Authors: We agree that explicit quantitative error metrics would strengthen the validation section. While the manuscript demonstrates agreement through matching values of B and η obtained from the variational ansatz versus imaginary-time evolution and spatial scaling, we did not report L2 residuals or overlap integrals. In the revised manuscript we will add a new figure (or table) showing density-profile comparisons together with L2-norm differences for representative g and N values, including the small-N soliton regime. This will make the quantitative fidelity of the ansatz explicit. revision: yes

  2. Referee: [Definition of B and η] The definition of the droplet width from the same variational minimization that yields B and η raises a circularity risk: it is unclear whether the reported g- and N-dependence of η is an independent physical result or is partly fixed by the ansatz parameters themselves. A direct comparison of the variational B against an independent numerical extraction (e.g., from the second derivative of the energy functional on exact profiles) would resolve this.

    Authors: The spatial scaling method used for validation extracts B directly from the second derivative of the numerically computed energy functional applied to the imaginary-time-evolved density profiles; this procedure does not rely on the variational width parameter. We will revise the text to emphasize this independence and will include an explicit side-by-side comparison of variational versus numerically extracted B (and η) for several (g,N) points. This comparison will confirm that the intricate dependence survives beyond the variational ansatz. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: variational derivation is independent and numerically validated

full rationale

The paper derives the elastic modulus B from minimization of the energy functional under a super-Gaussian variational ansatz, obtains corrections to droplet width beyond Thomas-Fermi, and extracts η(g,N) dependence. These steps constitute an approximate calculation rather than a reduction by construction. Results are cross-checked against independent numerical methods (imaginary-time evolution and spatial scaling), so the reported relations do not collapse to fitted inputs or self-citations. No quoted equation shows a prediction equivalent to its own ansatz parameters by definition.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Review performed on abstract only; no explicit free parameters, axioms, or invented entities are extractable beyond the standard assumption that LHY corrections stabilize the droplets.

assumptions (1)
  • domain assumption Lee-Huang-Yang quantum-fluctuation correction together with mean-field interaction stabilizes self-bound quantum droplets
    Invoked in the first sentence of the abstract as the physical basis for the system under study.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Elastic Modulus in One-Dimensional Quantum Droplets." pith.science (2026). https://pith.science/paper/FMHOGZ6I

@misc{pith2026260701863,
  author       = {Pith},
  title        = {Pith review of: Elastic Modulus in One-Dimensional Quantum Droplets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FMHOGZ6I}},
  note         = {Machine review of arXiv:2607.01863}
}
read the original abstract

Quantum droplets (QDs) are self-bound states of ultradilute quantum fluids stabilized by the interplay between the Lee Huang-Yang (LHY) quantum-fluctuation correction and the mean-field interaction, providing a useful platform for exploring macroscopic quantum phenomena. Recent studies on three-dimensional QDs have introduced the concept of bulk modulus and revealed its connection with the breathing-mode frequency, thereby linking the elastic response of QDs to their collective dynamics. Motivated by this progress, we investigate the elastic modulus of one-dimensional QDs. Based on a super Gaussian variational ansatz, we systematically derive the elastic modulus B and analyze its dependence on the interaction strength and particle number. The analytical predictions are further validated by numerical simulations based on imaginary time evolution and the spatial scaling method. We also establish a quantitative relation between the elastic modulus and the eigenfrequency of the breathing mode. In addition, by incorporating corrections to the droplet width beyond the Thomas Fermi approximation, we obtain the dependence of the ratio {\eta} = B/2 on the control parameters g and N. Unlike the three-dimensional case, where the corresponding ratio follows a simple power-law scaling, the one-dimensional system is affected by the soliton-to-droplet crossover, leading to a more intricate dependence of {\eta} on g and N. Our results show that, in the high-particle-number regime, the elastic modulus asymptotically approaches a limiting value determined mainly by the interaction strength, whereas in the low-particle-number regime it depends on both the particle number and the interaction strength.

Figures

Figures reproduced from arXiv: 2607.01863 by the authors.

Figure 1
Figure 1. Radial density distribution of stationary wave fu [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Density plots of (a) the elastic modulus [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Energy-effective-length relation of a quantum droplet with N = 5 and g = 1 under different spatial-scaling factors a = 0.96, 0.98, 1.00, 1.02, and 1.04. The blue filled circles denote the numerical results obtained using the spatial-scaling method. The green solid line represents the quadratic polynomial fit, E = 0.0014W2 − 0.04W − 0.72. the parameter a characterizes the strength of the transformation and satisfies … view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: (a–b) The modulus B versus g and N. The green solid lines represent the results obtained from the VA [Eq. (20)]. The red triangles denote the numerical results. The red dashed line denotes the asymptotic value B∞ = 4/(81g 3 ) ≃ 0.0494. (c–d) The frequency Ω as a functi…
Figure 5
Figure 5. Figure 5: Effective length W as a function of the particle number N for QDs with g = 1. The green solid line denotes the VA prediction WN,g, the blue circles represent the numerical results, and the red dashed line denotes the Thomas-Fermi (TF) approximation result WTF. 5. The r…
Figure 6
Figure 6. Figure 6: Panel (a) presents the ηVA(N, g) density map, where the horizontal axis denotes the particle number N and interaction strength g. The black contour lines indicate the isovalues of ηVA, with ηVA increasing from bottom to top. The two white dashed lines correspond to g =…

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages

  1. [1]

    Theo ry of Elasticity

    Landau LD, Lifshitz EM, Kosevich AM, Pitaevskii LP . Theo ry of Elasticity. 3rd ed. Course of Theoretical Physics, vol

    work page

  2. [2]

    Chapter I, pp

    Oxford: Butterworth-Heinemann; 1986. Chapter I, pp. 1–3 7. https://doi.org/10.1016/C2009-0-25521-8

    work page doi:10.1016/c2009-0-25521-8 1986

  3. [3]

    An Introduction to Continuum Mechanics

    Reddy JN. An Introduction to Continuum Mechanics. 2nd ed . Cambridge: Cambridge University Press; 2013. https://doi.org/10.1017/CBO9781139178952

    work page doi:10.1017/cbo9781139178952 2013

  4. [4]

    Quantum Mechanical Stabilization of a Collap sing Bose-Bose Mixture

    Petrov DS. Quantum Mechanical Stabilization of a Collap sing Bose-Bose Mixture. Phys Rev Lett 2015;115:155302. https://doi.org/10.1103/PhysRevLett.115.155302

    work page doi:10.1103/physrevlett.115.155302 2015

  5. [5]

    Ultradilute Low-Dimensio nal Liquids

    Petrov DS, Astrakharchik GE. Ultradilute Low-Dimensio nal Liquids. Phys Rev Lett 2016;117:100401. https://doi.org/10.1103/PhysRevLett.117.100401

    work page doi:10.1103/physrevlett.117.100401 2016

  6. [6]

    Ob servation of quantum droplets in a strongly dipolar Bose gas

    Ferrier-Barbut I, Kadau H, Schmitt M, Wenzel M, Pfau T. Ob servation of quantum droplets in a strongly dipolar Bose gas. Phys Rev Lett. 2016;116:215301. https: //doi.org/10.1103/PhysRevLett.116.215301

    work page doi:10.1103/physrevlett.116.215301 2016

  7. [7]

    Quantum liquid droplets in a mixture of Bose- Einstein condensates

    Cabrera CR, Tanzi L, Sanz J, Naylor B, Thomas P , Cheiney P , et al. Quantum liquid droplets in a mixture of Bose- Einstein condensates. Science 2018;359:301-4. https: //doi.org/10.1126/science.aao5686

    work page doi:10.1126/science.aao5686 2018

  8. [8]

    Self-Bound Quantum Droplets of Atomic Mixtures in Free Space

    Semeghini G, Ferioli G, Masi L, Mazzinghi C, Wolswijk L, M inardi F, et al. Self-Bound Quantum Droplets of Atomic Mixtures in Free Space. Phys Rev Lett 2018;120:235301. http s://doi.org/10.1103/PhysRevLett.120.235301

    work page doi:10.1103/physrevlett.120.235301 2018

Show all 69 references
  1. [9]

    Dynamical Formation of Multiple Quantum Droplets in a Bose-Bose Mixture

    Cavicchioli L, Fort C, Ancilotto F, Modugno M, Minardi F, Burchianti A. Dynamical Formation of Multiple Quantum Droplets in a Bose-Bose Mixture. Phys Rev Lett 2025;134:093 401. https://doi.org/10.1103/PhysRevLett.134.093401

    2025 doi

  2. [10]

    Dynamical formation of quantum droplets in a 39K mixture

    Ferioli G, Semeghini G, Terradas-Briansó S, Masi L, Fatt ori M, Modugno M. Dynamical formation of quantum droplets in a 39K mixture. Phys Rev Res 2020;2:013269. https: //doi.org/10.1103/PhysRevResearch.2.013269

    2020 doi

  3. [11]

    Collisions of Self-Bound Quantum Droplets

    Ferioli G, Semeghini G, Masi L, Giusti G, Modugno G, Ingu scio M, et al. Collisions of Self-Bound Quantum Droplets. Phys Rev Lett 2019;122:090401. https: //doi.org/10.1103/PhysRevLett.122.090401. 10

    2019 doi

  4. [12]

    Rotating Multidimensional Quant um Droplets

    Dong L, Kartashov YV . Rotating Multidimensional Quant um Droplets. Phys Rev Lett 2021;126:244101. https://doi.org/10.1103/PhysRevLett.126.244101

    2021 doi

  5. [13]

    Quantum-fluctuation-driven dynamics of dro plet splashing, recoiling, and deposition in ultracold bin ary Bose gases

    Ma Y , Cui X. Quantum-fluctuation-driven dynamics of dro plet splashing, recoiling, and deposition in ultracold bin ary Bose gases. Phys Rev Res 2023;5:013100. https: //doi.org/10.1103/PhysRevResearch.5.013100

    2023 doi

  6. [14]

    Bulk modulus of three-dimensional quantum droplets

    Zhao Z, Li G, Chen Z, Luo H-B, Liu B, Malomed BA, et al. Bulk modulus of three-dimensional quantum droplets. Phys Rev A 2026;113:043315. https: //doi.org/10.1103/cbg5-9r8v

    2026 doi

  7. [15]

    A new form of liq uid matter: Quantum droplets

    Luo Z-H, Pang W , Liu B, Li Y -Y , Malomed BA. A new form of liq uid matter: Quantum droplets. Front Phys 2020;16:32201. https://doi.org/10.1007/s11467-020-1020-2

    2020 doi

  8. [16]

    Three-d imensional droplets of swirling superfluids

    Kartashov YV , Malomed BA, Tarruell L, Torner L. Three-d imensional droplets of swirling superfluids. Phys Rev A 2018;98:013612. https://doi.org/10.1103/PhysRevA.98.013612

    2018 doi

  9. [17]

    Three-dimensional vortex and multipole quantum droplets in a toroidal potential

    Dong L, Fan M, Malomed BA. Three-dimensional vortex and multipole quantum droplets in a toroidal potential. Chaos, Solitons Fractals 2024;188:115499. https: //doi.org/10.1016/j.chaos.2024.115499

    2024 doi

  10. [18]

    Shell-Shaped Quantum Droplet in a Three-Com ponent Ultracold Bose Gas

    Ma Y , Cui X. Shell-Shaped Quantum Droplet in a Three-Com ponent Ultracold Bose Gas. Phys Rev Lett 2025;134:043402. https://doi.org/10.1103/PhysRevLett.134.043402

    2025 doi

  11. [19]

    Quantum droplets in three-dimensional Bo se-Einstein condensates

    Otajonov SR. Quantum droplets in three-dimensional Bo se-Einstein condensates. J Phys B: At Mol Opt Phys 2022;55:085001. https://doi.org/10.1088/1361-6455/ac6365

    2022 doi

  12. [20]

    Lee-Huang- Y ang e ffects in the ultracold mixture of 23Na and 87Rb with attractive interspecies interactions

    Guo Z, Jia F, Li L, Ma Y , Hutson JM, Cui X, et al. Lee-Huang- Y ang e ffects in the ultracold mixture of 23Na and 87Rb with attractive interspecies interactions. Phys Rev Res 2021;3:033247. https://doi.org/10.1103/PhysRevResearch.3.033247

    2021 doi

  13. [21]

    Dynamics of one-dimensi onal quantum droplets

    Astrakharchik GE, Malomed BA. Dynamics of one-dimensi onal quantum droplets. Phys Rev A 2018;98:013631. https://doi.org/10.1103/PhysRevA.98.013631

    2018 doi

  14. [22]

    Quantum droplets in one-dimensio nal Bose mixtures: A quantum Monte Carlo study

    Parisi L, Giorgini S. Quantum droplets in one-dimensio nal Bose mixtures: A quantum Monte Carlo study. Phys Rev A 2020;102:023318. https://doi.org/10.1103/PhysRevA.102.023318

    2020 doi

  15. [23]

    Formation and quench of homonuclear and heteronuclear quantum droplets in one dime nsion

    Mistakidis SI, Mithun T, Kevrekidis PG, Sadeghpour HR, Schmelcher P . Formation and quench of homonuclear and heteronuclear quantum droplets in one dime nsion. Phys Rev Res 2021;3:043128. https://doi.org/10.1103/PhysRevResearch.3.043128

    2021 doi

  16. [24]

    Col lective excitations of a one-dimensional quantum droplet

    Tylutki M, Astrakharchik GE, Malomed BA, Petrov DS. Col lective excitations of a one-dimensional quantum droplet. Phys Rev A 2020;101:051601. https: //doi.org/10.1103/PhysRevA.101.051601

    2020 doi

  17. [25]

    Dipolar droplets at the crossove r from three dimensions to one dimension

    Pylak M, Gajda M, Zin P . Dipolar droplets at the crossove r from three dimensions to one dimension. Phys Rev A 2024;110:063322. https://doi.org/10.1103/PhysRevA.110.063322

    2024 doi

  18. [26]

    Dimensi onal crossover for the beyond-mean-field correction in Bose gases

    Ilg T, Kumlin J, Santos L, Petrov DS, Büchler HP . Dimensi onal crossover for the beyond-mean-field correction in Bose gases. Phys Rev A 2018;98:051604. https: //doi.org/10.1103/PhysRevA.98.051604

    2018 doi

  19. [27]

    Two-dimensional vo rtex quantum droplets get thick

    Lin Z, Xu X, Chen Z, Y an Z, Mai Z, Liu B. Two-dimensional vo rtex quantum droplets get thick. Commun Nonlinear Sci Numer Simul 2021;93:105536. https: //doi.org/10.1016/j.cnsns.2020.105536

    2021 doi

  20. [28]

    Two-dimen sional vortex quantum droplets

    Li Y , Chen Z, Luo Z, Huang C, Tan H, Pang W , et al. Two-dimen sional vortex quantum droplets. Phys Rev A 2018;98:063602. https://doi.org/10.1103/PhysRevA.98.063602

    2018 doi

  21. [29]

    Internal modes of two-dimensiona l quantum droplets

    Dong L, Shi K, Huang C. Internal modes of two-dimensiona l quantum droplets. Phys Rev A 2022;106:053303. https://doi.org/10.1103/PhysRevA.106.053303

    2022 doi

  22. [30]

    Collisional dynamics of sy mmetric two-dimensional quantum droplets

    Hu Y , Fei Y , Chen X-L, Zhang Y . Collisional dynamics of sy mmetric two-dimensional quantum droplets. Front Phys 2022;17:61505. https://doi.org/10.1007/s11467-022-1192-z

    2022 doi

  23. [31]

    Breathing mode in two-dimensional binary self-bound Bose-gas droplets

    Stürmer P , Tengstrand MN, Sachdeva R, Reimann SM. Breathing mode in two-dimensional binary self-bound Bose-gas droplets. Phys Rev A 2021;103:053302. https: //doi.org/10.1103/PhysRevA.103.053302

    2021 doi

  24. [32]

    Ground state and rotation al properties of two-dimensional self-bound quantum dropl ets

    Examilioti P , Kavoulakis GM. Ground state and rotation al properties of two-dimensional self-bound quantum dropl ets. J Phys B At Mol Opt Phys 2020;53. https: //doi.org/10.1088/1361-6455/ab9766

    2020 doi

  25. [33]

    Bose-Einstein Condensatio n and Superfluidity

    Pitaevskii L, Stringari S. Bose-Einstein Condensatio n and Superfluidity. Oxford University Press; 2016. https://doi.org/10.1093/acprof:oso/9780198758884.001.0001

    2016 doi

  26. [34]

    Multidimensional Solitons

    Malomed BA. Multidimensional Solitons. AIP Publishin g LLC; n.d. https: //doi.org/10.1063/9780735425118

    doi

  27. [35]

    S tatistical mechanics of one-dimensional quantum droplets

    Mithun T, Mistakidis SI, Schmelcher P , Kevrekidis PG. S tatistical mechanics of one-dimensional quantum droplets . Phys Rev A 2021;104:033316. https: //doi.org/10.1103/PhysRevA.104.033316

    2021 doi

  28. [36]

    Stationary and dyn amical properties of one-dimensional quantum droplets

    Otajonov SR, Tsoy EN, Abdullaev FKh. Stationary and dyn amical properties of one-dimensional quantum droplets. Phys Lett A 2019;383:125980. https: //doi.org/10.1016/j.physleta.2019.125980

    2019 doi

  29. [37]

    Quantum droplets of quasi- one-dimensional dipolar Bose-Einstein condensates

    Edmonds M, Bland T, Parker N. Quantum droplets of quasi- one-dimensional dipolar Bose-Einstein condensates. J Phys Commun 2020;4:125008. https: //doi.org/10.1088/2399-6528/abcc3b

    2020 doi

  30. [38]

    Discrete quan tum droplets in one-dimensional optical lattices

    Zhao F, Y an Z, Cai X, Li C, Chen G, He H, et al. Discrete quan tum droplets in one-dimensional optical lattices. Chaos, Solitons Fractals 2021;152:111313. https: //doi.org/10.1016/j.chaos.2021.111313

    2021 doi

  31. [39]

    Universality of qua ntum liquids and droplets in one dimension

    Morera I, Juliá-Díaz B, V aliente M. Universality of qua ntum liquids and droplets in one dimension. Phys Rev Res 2022;4:L042024. https://doi.org/10.1103/PhysRevResearch.4.L042024

    2022 doi

  32. [40]

    Two-component dropl et phases and their spectral stability in one dimension

    Charalampidis EG, Mistakidis SI. Two-component dropl et phases and their spectral stability in one dimension. Phy s Rev A 2025;111. https: //doi.org/10.1103/PhysRevA.111.013318

    2025 doi

  33. [41]

    In- teractions and Dynamics of One-Dimensional Droplets, Bubb les and Kinks

    Katsimiga GC, Mistakidis SI, Malomed BA, Frantzeskaki s DJ, Carretero-Gonzalez R, Kevrekidis PG. In- teractions and Dynamics of One-Dimensional Droplets, Bubb les and Kinks. Condensed Matter 2023;8:67. https://doi.org/10.3390/condmat8030067. 11

    2023 doi

  34. [42]

    Breather excitations o n the one-dimensional quantum droplet

    Lv L-Z, Gao P , Y ang Z-Y , Y ang W-L. Breather excitations o n the one-dimensional quantum droplet. Phys Lett A 2022;438:128124. https://doi.org/10.1016/j.physleta.2022.128124

    2022 doi

  35. [43]

    Formation and fragmentati on of quantum droplets in a quasi-one-dimensional dipolar Bose gas

    De Palo S, Orignac E, Citro R. Formation and fragmentati on of quantum droplets in a quasi-one-dimensional dipolar Bose gas. Phys Rev B 2022;106:014503. https: //doi.org/10.1103/PhysRevB.106.014503

    2022 doi

  36. [44]

    Ground-state properties a nd Bogoliubov modes of a harmonically trapped one- dimensional quantum droplet

    Du X, Fei Y , Chen X-L, Zhang Y . Ground-state properties a nd Bogoliubov modes of a harmonically trapped one- dimensional quantum droplet. Phys Rev A 2023;108:033312. h ttps://doi.org/10.1103/PhysRevA.108.033312

    2023 doi

  37. [45]

    Bose-Einstein Condensation in Dil ute Gases

    Pethick CJ, Smith H. Bose-Einstein Condensation in Dil ute Gases. 2nd ed. Cambridge: Cambridge University Press

  38. [46]

    https://doi.org/10.1017/CBO9780511802850

    doi

  39. [47]

    V ariational analysis of flat-top solitons in Bose- Einstein condensates

    Baizakov BB, Bouketir A, Messikh A, Benseghir A, Pumaro v BA. V ariational analysis of flat-top solitons in Bose- Einstein condensates. Int J Mod Phys B 2011;25:2427-2440. h ttps://doi.org/10.1142/S0217979211101521

    2011 doi

  40. [48]

    Nonlinear Waves in Integrable and Nonintegrabl e Systems

    Y ang J. Nonlinear Waves in Integrable and Nonintegrabl e Systems. Society for Industrial and Applied Mathematics

  41. [49]

    https://doi.org/10.1137/1.9780898719680

    doi

  42. [50]

    Mechanical Vibrations in SI Units

    Rao S. Mechanical Vibrations in SI Units. Pearson Deuts chland; 2017

    2017

  43. [51]

    Cold Dipolar Gases in Quasi-One-Dime nsional Geometries

    Sinha S, Santos L. Cold Dipolar Gases in Quasi-One-Dime nsional Geometries. Phys Rev Lett 2007;99:140406. https://doi.org/10.1103/PhysRevLett.99.140406

    2007 doi

  44. [52]

    Stro ngly Anisotropic V ortices in Dipolar Quantum Droplets

    Li G, Zhao Z, Jiang X, Chen Z, Liu B, Malomed BA, et al. Stro ngly Anisotropic V ortices in Dipolar Quantum Droplets. Phys Rev Lett 2024;133:053804. https: //doi.org/10.1103/PhysRevLett.133.053804

    2024 doi

  45. [53]

    Two-dimensi onal anisotropic vortex quantum droplets in dipolar Bose-Einstein condensates

    Li G, Jiang X, Liu B, Chen Z, Malomed BA, Li Y . Two-dimensi onal anisotropic vortex quantum droplets in dipolar Bose-Einstein condensates. Front Phys 2024;19:22202. htt ps://doi.org/10.1007/s11467-023-1338-7

    2024 doi

  46. [54]

    V ortex patterns and the critical rotational frequency in rotating dipolar Bose-Einstein condensates

    Cai Y , Y uan Y , Rosenkranz M, Pu H, Bao W . V ortex patterns and the critical rotational frequency in rotating dipolar Bose-Einstein condensates. Phys Rev A 2018;98:023610. htt ps://doi.org/10.1103/PhysRevA.98.023610

    2018 doi

  47. [55]

    Ground-state properties and elem entary excitations of quantum droplets in dipolar Bose-Ein stein condensates

    Wächtler F, Santos L. Ground-state properties and elem entary excitations of quantum droplets in dipolar Bose-Ein stein condensates. Phys Rev A 2016;94:043618. https: //doi.org/10.1103/PhysRevA.94.043618

    2016 doi

  48. [56]

    Ground-stat e phase diagram of a dipolar condensate with quantum fluctuations

    Bisset RN, Wilson RM, Baillie D, Blakie PB. Ground-stat e phase diagram of a dipolar condensate with quantum fluctuations. Phys Rev A 2016;94:033619. https: //doi.org/10.1103/PhysRevA.94.033619

    2016 doi

  49. [57]

    Probing the Roton Excitation Spectrum of a Stable Dipolar Bose Gas

    Petter D, Natale G, van Bijnen RMW , Patscheider A, Mark MJ, Chomaz L, et al. Probing the Roton Excitation Spectrum of a Stable Dipolar Bose Gas. Phys Rev Lett 2019;122. https: //doi.org/10.1103/PhysRevLett.122.183401

    2019 doi

  50. [58]

    Ground state and collective excitations of a dipolar Bose-Einstein condensate in a bubble trap

    Diniz PC, Oliveira EAB, Lima ARP , Henn EAL. Ground state and collective excitations of a dipolar Bose-Einstein condensate in a bubble trap. Sci Rep 2020;10. https: //doi.org/10.1038/s41598-020-61657-0

    2020 doi

  51. [59]

    Collective Excitation s of Self-Bound Droplets of a Dipolar Quantum Fluid

    Baillie D, Wilson RM, Blakie PB. Collective Excitation s of Self-Bound Droplets of a Dipolar Quantum Fluid. Phys Rev Lett 2017;119. https: //doi.org/10.1103/PhysRevLett.119.255302

    2017 doi

  52. [60]

    Phonon Stability of Quantum Droplets in D ipolar Bose Gases

    Zhang F, Yin L. Phonon Stability of Quantum Droplets in D ipolar Bose Gases. Chin Phys Lett 2022;39:060301. https://doi.org/10.1088/0256-307X/39/6/060301

    2022 doi

  53. [61]

    Scissors Mode of Dipolar Quantum Droplets of Dysprosium Atoms

    Ferrier-Barbut I, Wenzel M, Böttcher F, Langen T, Isoar d M, Stringari S, et al. Scissors Mode of Dipolar Quantum Droplets of Dysprosium Atoms. Phys Rev Lett 2018;120:16040 2. https://doi.org/10.1103/PhysRevLett.120.160402

    2018 doi

  54. [62]

    Axial Collective Mode of a Dipolar Quantum Dr oplet

    Blakie PB. Axial Collective Mode of a Dipolar Quantum Dr oplet. Photonics 2023;10:393. https://doi.org/10.3390/photonics10040393

    2023 doi

  55. [63]

    Cold Bosonic Atoms in Optical Lattices

    Jaksch D, Bruder C, Cirac JI, Gardiner CW , Zoller P . Cold Bosonic Atoms in Optical Lattices. Phys Rev Lett 1998;81:3108-11. https://doi.org/10.1103/PhysRevLett.81.3108

    1998 doi

  56. [64]

    A hybrid Lagrangian vari- ational method for Bose-Einstein condensates in optical la ttices

    Edwards M, DeBeer LM, Demenikov M, Galbreath J, Mahaney TJ, Nelsen B, et al. A hybrid Lagrangian vari- ational method for Bose-Einstein condensates in optical la ttices. J Phys B: At Mol Opt Phys 2005;38:363. https://doi.org/10.1088/0953-4075/38/4/004

    2005 doi

  57. [65]

    Spectra and dynamics of quantum droplets in an optical lattice

    Nie Y , Zheng J-H, Y ang T. Spectra and dynamics of quantum droplets in an optical lattice. Phys Rev A 2023;108. https://doi.org/10.1103/PhysRevA.108.053310

    2023 doi

  58. [66]

    Quantum droplets with particle imbalance in one- dimensional optical lattices

    V allés-Muns J, Morera I, Astrakharchik GE, Juliá-Díaz B. Quantum droplets with particle imbalance in one- dimensional optical lattices. SciPost Physics 2024;16:07 4. https://doi.org/10.21468/SciPostPhys.16.3.074

    2024 doi

  59. [67]

    Dynamics of quantum dropletsin a one-dimensional optical lattice

    Zhou Z, Y u X, Zou Y , Zhong H. Dynamics of quantum dropletsin a one-dimensional optical lattice. Commun Nonlinear Sci Numer Simul 2019;78:104881. https: //doi.org/10.1016/j.cnsns.2019.104881

    2019 doi

  60. [68]

    Controlla ble dissipative quantum droplets in one-dimensional optical lattices

    Zhou Z, Shi Y , Tang S, Deng H, Wang H, He X, et al. Controlla ble dissipative quantum droplets in one-dimensional optical lattices. Chaos, Solitons Fractals 2021;150:1111 93. https://doi.org/10.1016/j.chaos.2021.111193

    2021 doi

  61. [69]

    Controlling quantum vortex dynamics and v ortex-antivortex annihilation in Bose-Einstein condensa tes with optical lattices

    Ancilotto F. Controlling quantum vortex dynamics and v ortex-antivortex annihilation in Bose-Einstein condensa tes with optical lattices. Phys Rev A 2024;110. https: //doi.org/10.1103/PhysRevA.110.013302. 12

    2024 doi

Pith tools

Reviewed July 3, 2026 · model on record in the stance chip above.