Surface excitation of Rydberg dressed quantum droplet of Bose-Einstein Condensates

Avra Banerjee; Dwipesh Majumder

REVIEW 4 major objections 4 minor 55 references

Surface excitation of Rydberg dressed quantum droplet of Bose-Einstein Condensates

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read Rydberg dressing raises the surface tension of a binary BEC quantum droplet, making it more stable.

desk verdict A clean numerical extension that computes a rising surface tension for Rydberg-dressed droplets, but the 'more stable' conclusion is not supported by any stability diagnostic they actually compute. read the letter →

arxiv 2505.09474 v1 pith:JYTVOPMG submitted 2025-05-14 physics.atom-ph

Avra Banerjee , Dwipesh Majumder This is my paper · ORCID

classification physics.atom-ph PACS 03.75.Kk32.80.Ee67.85.-d

keywords RydbergpolaronquantumdropletsurfaceexcitationtensionBogoliubovtheorybinaryBose-Einsteincondensatebeyond-mean-fieldcorrectionextendedGross-Pitaevskiiequation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper argues that placing a spherical binary Bose-Einstein condensate quantum droplet inside the electron cloud of a Rydberg atom changes the droplet's surface physics in a predictable way. Treating the Rydberg electron as a fixed external potential $V_0|\Psi_R(r)|^2$ added to the extended Gross-Pitaevskii equation with beyond-mean-field quantum fluctuations, the authors compute Bogoliubov surface excitations and find higher mode frequencies than in a free droplet. Fitting the long-wavelength spectrum to $\omega_s = \sqrt{\sigma_s} k^{3/2}$ yields a surface tension $\sigma_s$ that increases with the electron-atom interaction strength $V_0$. If correct, this means Rydberg dressing stiffens the droplet surface and stabilizes the droplet without any external trap, and it gives an experimentally measurable signature in the quadrupole surface mode.

What carries the argument

The central object is the surface-mode dispersion relation $\omega_s = \sqrt{\sigma_s} k^{3/2}$ with $k = [l(l-1)(l+2)]^{1/3}/R$, which turns Bogoliubov excitation energies $\omega_{l0}$ ($l\ge 2$, radial node number $n=0$) into a surface tension $\sigma_s$ through a fit. The calculation is carried by the extended Gross-Pitaevskii equation with a beyond-mean-field stabilizing term and the Rydberg potential $V_0|\Psi_R(r)|^2$, linearized into Bogoliubov equations that are decoupled with $\psi^\pm_j = u_j \pm v_j$ auxiliary functions and solved by matrix diagonalization in the eigenbasis of the single-particle Hamiltonian. The fitting slope $\sqrt{\sigma_s}$ is the quantity plotted against $V_0$.

What would settle it

Compute the same surface spectrum with the Rydberg electron wavefunction updated self-consistently in the potential of the dressed droplet: if the effective potential changes enough that $\sigma_s(V_0)$ saturates or decreases, the one-way-potential result is not the physical one. Experimentally, measuring the $l=2$ quadrupole mode frequency of a Rydberg-dressed self-bound droplet as $V_0$ is varied would settle whether the fitted surface tension really rises.

Watch

Extended reading notes

Core claim

For a self-bound, spherically symmetric binary droplet enclosed by a Rydberg electron, the surface excitation spectrum at low effective wave vectors follows the ripplon law $\omega_s = \sqrt{\sigma_s} k^{3/2}$, and the surface tension extracted from that fit rises with the strength $V_0$ of the Rydberg electron-atom interaction. The Rydberg potential compresses the ground-state density and shifts the surface modes upward, so the droplet's surface energy is enhanced inside the electron cloud. The paper also finds that the maximum number of atoms that can fit inside the Rydberg electron increases with $V_0$, and that the long-wavelength surface modes of dressed and free droplets coincide as $k\to 0$, consistent with the idea that only the surface energy, not the long-wavelength dynamics, is modified by the dressing.

Load-bearing premise

The load-bearing assumption is that the Rydberg electron's wavefunction stays fixed and only acts on the droplet as a one-way potential $V_0|\Psi_R(r)|^2$, so the dense droplet does not feed back and alter the electron.

Editorial extensions

If this is right

The $l=2$ quadrupole mode frequency should be measurably higher in a Rydberg-dressed droplet than in a free droplet of the same atom number, providing an experimental route to read off $\sigma_s(V_0)$.
The atom capacity of the Rydberg container grows linearly with $n_r^6$ for fixed $V_0$, so larger Rydberg states can enclose denser, more strongly dressed droplets.
At larger effective wave vectors the spectrum deviates from the $\sqrt{k^{3/2}}$ law as modes approach the particle-emission threshold; only the low-$k$ portion should be used to define surface tension.
The surface tension extracted from surface-mode fits agrees with the direct thermodynamic surface tension in the undressed limit, confirming that the surface-mode fit measures the same physical quantity.

Reading between the lines

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

The one-way potential approximation likely confines the validity of the prediction to moderate $V_0$; a self-consistent treatment of the Rydberg electron could reveal saturation or reversal of the surface-tension increase, but that goes beyond the paper.
This setup offers an independent control knob, the electron-atom interaction strength, for droplet surface stiffness, which might be used to create trapless droplets with tunable shape fluctuations.
The same Bogoliubov machinery could be applied to Rydberg-dressed droplets with unequal masses or unequal populations, where spherical symmetry and the surface-mode structure would change.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Quick take: the paper is a reasonably clean numerical extension of Hu and Liu's surface-mode formalism to a droplet sitting inside a Rydberg electron orbital, and the newly computed quantity — σ_s(V0) — does increase with V0 in their calculation. But the conclusion they sell, that the droplet is therefore "more stable," does not follow from anything they compute. The paper would be fine with a corrected interpretation.

What's actually new: they add V0|Ψ_R|² to the extended GPE and recompute the Bogoliubov surface modes and the ripplon fit. The Bogoliubov equations, the symmetrized secular equation (14), and the basis expansion follow Hu–Liu/Petrov faithfully, and the numerical solution is standard. Fig. 4 — the overlap between free and dressed excited states going to unity as k→0 — is a nice check that the Rydberg potential does not change the long-wavelength surface physics. Fig. 1's N vs n_r⁶ scaling is a practical setup choice for filling the Rydberg volume.

Where it's soft: the stability claim. Surface tension is a derived property of the energy functional, not a stability criterion for a self-bound droplet. The stability boundary is set by the mean-field attraction vs LHY repulsion, and since V0 compresses the density (Fig. 1[b]), it shifts that balance. A droplet can have larger σ_s while being closer to collapse. They never compute the l=0 breathing mode, the chemical potential, or the energy per particle as functions of V0, so the final sentence is unsupported. This is the load-bearing problem, and it is fixable: report those diagnostics, or drop the stability language and just report σ_s(V0). I'd also want to see the direct formula from ref [1] plotted against the fitted points instead of just asserted, and some check that σ_s is insensitive to the hand-chosen k≤0.4 fit window. The use of an unperturbed Ψ_R and a Fermi pseudopotential at V0 up to 750 is a real assumption; if the dense droplet back-acts on the Rydberg electron, the effective potential is not what they solve. They should at least flag that. Minor: several typos ("Introduntion") and unpolished prose.

Who this is for: people working on Rydberg polarons and droplet surface physics. It's a credible, incremental numerical study. The core computation and the σ_s(V0) trend are worth knowing about; the stability inference is not. I'd send it to a referee and ask for the stability diagnostics or a revised claim, but I would not desk reject it.

Referee Report

4 major / 4 minor

Summary. The manuscript studies surface excitations of a two-component BEC quantum droplet placed inside the electronic wavefunction of a Rydberg atom. The authors model the system with an extended Gross–Pitaevskii equation that includes the LHY correction and an external potential V0|Ψ_R(r)|^2, obtain spherically symmetric ground states numerically, and linearize around them to obtain Bogoliubov excitations. They diagonalize the symmetrized secular equation (14) for angular momenta l=2,...,9 and n=0, extract surface-mode frequencies, and fit them to ω_s = √σ_s k^{3/2} in a low-k window to obtain the surface tension σ_s as a function of V0. They report that σ_s increases with V0 and conclude that the droplet is more stable inside the Rydberg electron.

Significance. If the extraction is robust, the paper offers a concrete prediction for how Rydberg dressing modifies the surface-mode spectrum of a self-bound binary droplet, and it uses a clean numerical implementation of the symmetrized Bogoliubov approach. The ground-state computation and the matrix diagonalization are standard and appear internally consistent. However, the paper's headline stability conclusion is not established by the presented quantities, and the fit-based surface tension lacks uncertainty estimates and independent validation, which limits the current significance.

major comments (4)

[Abstract and Results and discussion] The claim that the droplet becomes 'more stable' because σ_s increases with V0 is not supported by any stability diagnostic in the paper. For a self-bound binary droplet, stability is determined by the balance between mean-field attraction, LHY repulsion, and the Rydberg potential; the l=0 breathing mode frequency, the chemical potential, and the energy per particle as functions of V0 are never reported. Since Figure 1 shows that the Rydberg potential compresses the droplet, a larger surface tension could coexist with a reduced stability. Please compute a stability measure or temper the conclusion to 'surface energy increases'.
[Eq. (17) and Figure 3] The surface tension σ_s is extracted by fitting ω_s = √σ_s k^{3/2} to data with k ≤ 0.4, but the paper does not state how many data points fall in this window, reports no fit uncertainty, and does not test the sensitivity of σ_s to the chosen cutoff. Because the V0 dependence of σ_s is the central quantitative result, please provide the fit curves for each V0, report parameter errors, and discuss the choice of k_max.
[Results and discussion] The sentence 'Surface tension obtained from the surface excitation curve agrees with the calculated surface tension using the direct formula given in ref. [1]' is a key validation, but no direct formula or comparison is shown anywhere in the manuscript. Please display the formula and a comparison table or plot so the reader can verify the extraction independently of the k^{3/2} fit.
[Model and calculation, Eq. (3)] The Rydberg electron is treated as a static external potential V0|Ψ_R(r)|^2 with the hydrogenic wavefunction Ψ_nr00. The paper does not justify neglecting the back-action of up to 10^5 condensed atoms on the Rydberg electron, nor the range of V0 for which the Fermi pseudopotential with a constant scattering length remains accurate. Since the ground-state density changes substantially with V0, this one-way coupling is a load-bearing assumption and should be discussed or its limitations stated.

minor comments (4)

[Introduction] The section heading 'Introduntion' is a typo, and 'Cikojevi' should read 'Cikojević'.
[Model and calculation, Eq. (3)] The sign convention for V0 is not stated: Eq. (3) contains '+V0|Ψ_R|^2' while Figure 5 is plotted against increasing V0. Please clarify whether V0 is meant to be positive or negative and how the attractive or repulsive nature of the electron-atom interaction enters.
[Figure 3] The number of data points and the specific values of N used for each V0 are not listed; please provide the data or a table so that the fitting procedure is reproducible.
[Figure 4 and surrounding text] The statement that the overlap tending to unity 'suggests that the surface energy ... tends to zero as k→0' is not a direct consequence of the wavefunction overlap alone; please explain the logic or rephrase.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: surface modes are computed from the Rydberg-dressed GP/Bogoliubov equations and the surface tension is extracted via the standard ripplon dispersion, not fed back into the calculation.

full rationale

The paper's derivation chain is self-contained and does not reduce to its inputs. The ground state is obtained by solving the extended GP equation (3) with the Rydberg potential V0|Psi_R|^2; the Bogoliubov equations (6)-(14) are then diagonalized to obtain surface-mode frequencies omega_ln for l=2..9, n=0. Equation (17) states the ripplon dispersion omega_s = sqrt(sigma_s) k^{3/2}; fitting the computed low-k spectra to this defining dispersion is a standard extraction of the parameter sigma_s, not a circular use of sigma_s as an input. The reported V0-dependence of sigma_s follows from the V0-dependence of the numerically computed spectra (Fig. 3), so no fitted parameter is being relabeled as a prediction. The two author self-citations (refs [8] and [18]) are background references and are not load-bearing for the central result. The final claim that the droplet 'will be more stable' is an inference from the increasing fitted surface tension and is not supported by an independent stability diagnostic, such as the l=0 breathing mode or the energy per particle; however, that is a physical-logic gap rather than a circularity, because no equation in the paper defines stability in terms of the fitted surface tension. Similarly, the statement that the extracted surface tension 'agrees with the calculated surface tension using the direct formula given in ref. [1]' is asserted without displaying the formula or the comparison; this is a missing-support issue, not a circular reduction. No specific equation or step can be exhibited in which the output is equivalent to an input by construction.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central calculation rests on the LHY functional, the Fermi-pseudopotential model of the Rydberg electron as an external potential, and the ripplon dispersion for interpreting the spectrum. The two fitted choices are the range of V0 values and the low-k fitting cutoff. No new physical entities are introduced.

free parameters (2)

V0 (Rydberg electron-atom interaction strength) = varied 0 to 750 (dimensionless scaled units)
The control parameter of the central study; the paper varies it by hand over a wide range to map σ_s(V0), without grounding the values in a specific alkali Rydberg system.
k_fit_max (upper cutoff of the low-k fitting region) = 0.4 (in units of 1/l0)
The surface tension is extracted from a fit of ω_s = √σ_s k^{3/2} to computed data points only up to k = 0.4; the choice of cutoff affects the fitted σ_s and is justified only qualitatively by the particle-emission threshold.

assumptions (5)

domain assumption Lee-Huang-Yang (LHY) energy functional for a binary Bose mixture, with the 5/2|ϕ|³ term in eq. (3)
The ground state, Bogoliubov modes, and the comparison surface-tension formula all rely on the LHY beyond-mean-field correction, which is taken from Petrov (ref [1]).
domain assumption Single-mode approximation for the two-species mixture (equal densities, equal intraspecies couplings)
The two coupled GPEs (1)-(2) are reduced to the single equation (3), which is only valid under the stated symmetry assumptions in the Model section.
domain assumption Rydberg electron-atom interaction modeled as an external potential V0|Ψ_R(r)|² with the bare hydrogenic wavefunction Ψ_{nr00}
The paper inserts V0|Ψ_R(r)|² into the GPE and uses the unperturbed hydrogen wavefunction, neglecting back-action of the droplet on the electron; this is a standard but unverified approximation for large V0.
standard math Ripplon dispersion relation ω_s = √σ_s k^{3/2} (eq. 17) for surface modes of a spherical droplet
This is the classical dispersion from Landau-Lifshitz (ref [53]) and is used as the fitting law to extract σ_s; the paper assumes its validity in the low-k region.
standard math Bogoliubov linearization around the ground state, with the decoupling transformation of Hutchinson, Zaremba, and Griffin (ref [42])
The excitation spectrum is obtained from the linearized time-dependent GPE, a standard method in BEC theory; the eigenvalue problem (14) is derived from this.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Surface excitation of Rydberg dressed quantum droplet of Bose-Einstein Condensates." pith.science (2026). https://pith.science/paper/JYTVOPMG

@misc{pith2026250509474,
  author       = {Pith},
  title        = {Pith review of: Surface excitation of Rydberg dressed quantum droplet of Bose-Einstein Condensates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JYTVOPMG}},
  note         = {Machine review of arXiv:2505.09474}
}

read the original abstract

We have considered a quantum droplet of two components of Bose-Einstein condensate (BEC) inside the electron of a Rydberg atom to study the surface mode of collective excitation using the Bogoliubov theory of excitation. We have calculated the surface excitation spectrum for various Rydberg electron-atom interaction strengths. From the energy spectrum, we calculated the surface tension of the droplet as a function of Rydberg electron-atom interaction strength. Our study shows that the electron-atom interaction enhances the surface energy; hence, the droplet will be more stable inside the electron of a Rydberg atom.

Figures

Figures reproduced from arXiv: 2505.09474 by the authors.

**Figure 3.** by the black solid line. FIG. 3: Excitation spectra (ωs) for different values of V0 are obtained by diagonalizing the dispersion relation (14) with n = 0. We have considered the number of particles (N) ranging from 2000 to 100000 to calculate the energy eigenvalue for angular momentum ranging from l = 2 to l = 9. We have plotted the energy as a function of an effective wave vector k = [l(l−1)(l+2)]1/3 /R [49], wh… view at source ↗

**Figure 2.** FIG. 2: Excited state wave functions. The black dash-dot line [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4: For larger values of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

**Figure 5.** Figure 5: FIG. 5: Surface tension ( [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

55 extracted references · 48 canonical work pages

[1]

D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015)

2015
[2]

I. F. Barbut, H. Kadau, M. Schmitt, M. Wenzel, and T. Pfau, Phys. Rev. Lett. 116, 215301 (2016)

work page 2016
[3]

Schmitt, M

M. Schmitt, M. Wenzel, F. Bottcher, I. F. Barbut, and T. Pfau, Nature (London) 539, 259 (2016)

work page 2016
[4]

C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas, P. Cheiney, L. Tarruell, Science359, 301 (2018)

work page 2018
[5]

D’Errico, A

C. D’Errico, A. Burchianti, M. Prevedelli, L. Salasnich, F. Ancilotto, M. Modugno, F. Minardi, and C. Fort, Phys. Rev. Research 1, 033155 (2019)

work page 2019
[6]

Kadau, M

H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. F.Barbut, and T. Pfau, Nat. Phys.530, 194 (2016); I. F. Bar- but, H. Kadau, M. Schmitt, M. Wenzel, and T. Pfau, Phys. Rev. Lett. 116, 215301 (2016)

work page 2016
[7]

Bottcher, J

F. Bottcher, J. N. Schmidt, J. Hertkorn, K. S. H. Ng, S. D. Graham, M. Guo, T. Langen, and T. Pfau, Rep. Prog. Phys. 84, 012403 (2021)

work page 2021
[8]

Banerjee and D

A. Banerjee and D. Majumder, Phys. Scr. 99, 085402 (2024)

work page 2024

Show all 55 references

[9]

Semeghini, G

G. Semeghini, G. Ferioli, L. Masi, C. Mazzinghi, L. Wol- swijk, F. Minardi, M. Modugno, G. Modugno, M. Inguscio, and M. Fattori, Phys. Rev. Lett. 120, 235301 (2018)

2018
[10]

Baillie, R

D. Baillie, R. M. Wilson, R. N. Bisset, and P. B. Blakie, Phys. Rev. A 94, 021602(R) (2016)

2016
[11]

Wachtler and L

F. Wachtler and L. Santos,Phys. Rev. A 94, 043618 (2016)

2016
[12]

Baillie, R

D. Baillie, R. M. Wilson, and P. B. Blakie, Phys. Rev. Lett. 119, 255302 (2017)

2017
[13]

Schmidt, H

R. Schmidt, H. R. Sadeghpour, E. Demler, Phys. Rev. Lett. 116, 105302 (2016)

2016
[14]

Camargo, R

F. Camargo, R. Schmidt, J. D. Whalen, R. Ding, G. Woehl, Jr., S. Yoshida, J. Burgdörfer, F. B. Dunning, H. R. Sadegh- pour, E. Demler and T. C. Killian, Phys. Rev. Lett. 120, 083401 (2018)

2018
[15]

Schmidt, J

R. Schmidt, J. D. Whalen, R. Ding, F. Camargo, G. Woehl, Jr., S. Yoshida, J. Burgdörfer, F. B. Dunning, E. Demler, H. R. Sadeghpour, and T. C. Killian, Phys. Rev. A 97, 022707 (2018)

2018
[16]

C. H. Greene, A. S. Dickinson, and H. R. Sadeghpour, Phys. Rev. Lett. 85, 2458 (2000)

2000
[17]

Brune, P

M. Brune, P. Nussenzveig, F. Schmidt-Kaler, F. Bernardot, A. Maali, J. M. Raimond, and S. Haroche Phys. Rev. Lett. 72, 3339 (1994); M. Saffman, J. Phys. B: At. Mol. Opt. Phys. 49, 202001 (2016); M. Morgado and S. Whitlock, A VS Quantum Science3, 023501 (2021); M. Saffman, T. G...

1994
[18]

Banerjee and D

A. Banerjee and D. Majumder, J. Low Temp. Phys. 215, 64 (2024)

2024
[19]

Schmid, A

S. Schmid, A. Harter, and J. H. Denschlag, Phys. Rev. Lett. 105, 133202 (2010)

2010
[20]

J. B. Balewski, A. T. Krupp, A. Gaj, D. Peter, H. P. Buchler, R. Low, S. Hofferberth, and T. Pfau, Nature (London) 502, 664(2013)

2013
[21]

A. T. Grier, M. Cetina, F. Orucevic, and V . Vuletic, Phys. Rev. Lett. 102, 223201 (2009)

2009
[22]

Ratschbacher, C

L. Ratschbacher, C. Zipkes, C. Sias, and M. Kohl, Nat. Phys. 8, 649 (2012)

2012
[23]

Krokow, A

A. Krokow, A. Mohammadi, A. Harter, J. Hecker Den- schlag, J. P erez-Rios, and C. H. Greene, Phys. Rev. Lett. 116, 193201 (2016)

2016
[24]

F. H. J. Hall and S. Willitsch, Phys. Rev. Lett. 109, 233202 (2012)

2012
[25]

Zipkes, S

C. Zipkes, S. Palzer, C. Sias, and M. Kohl, Nature (London) 464, 388 (2010)

2010
[26]

Mukherjee, C

R. Mukherjee, C. Ates, W. Li, and S. Wuster, Phys. Rev. Lett. 115, 040401 (2015)

2015
[27]

Karpiuk, M

T. Karpiuk, M. Brewczyk, K. Rzazewski, A. Gaj, J. B Balewski, A. T. Krupp, M. Schlagmuller, R. Low, S. Hof- ferberth and T. Pfau, New J. Phys. 17, 053046 (2015)

2015
[28]

H. T. Ng, S. Bose, Phys. Rev. A 78, 023610 (2008)

2008
[29]

Y . J. Song, L. M. Kuang, Ann. Phys. (Berlin, Ger.) 531, 1800423 (2019)

2019
[30]

M. S. Smith, Phys. Rev. Lett. 117, 100001 (2016)

2016
[31]

F. B. Dunning, J. J. Mestayer, C. O. Reinhold, S. Yoshida and J. Burgdorfer, J. Phys. B: At. Mol. Opt. Phys. 42, 022001 (2009)

2009
[32]

T. D. Lee and C. N. Yang, Phys. Rev. 105, 1119 (1957); T. D. Lee, K. Huang, and C. N. Yang, Phys. Rev. 106, 1135 (1957)

1957
[33]

D. S. Petrov and G. E. Astrakharchik, Phys. Rev. Lett. 117, 100401 (2016)

2016
[34]

Y . Li, Z. Luo, Y . Liu, Z. Chen, C. Huang, S. Fu, H. Tan, and B. A. Malomed, New J. Phys 19, 113043 (2017)

2017
[35]

Hu and X

H. Hu and X. J. Liu, Phys. Rev. Lett. A102, 053303 (2020)

2020
[36]

P. V . Pikhitsa and U. R. Fischer, Phys. Rev. B 99, 184504 (2019)

2019
[37]

pink curve

to solve the GP equation (3). One can find the ground state wave function and, hence, the ground state energy and chemical potential. The excitations of the system can be obtained using the Bogoliubov method by considering the fluctuation over the ground state (ϕ0) asϕ(r,τ ) =...

2000
[38]

S. K. Adhikari, Phys. Rev. A 95, 023606 (2017)

2017
[39]

J. Wang, M. Gacesa, and R. Cote, Phys. Rev. Lett. 114, 243003 (2015)

2015
[40]

S. K. Tiwari and S. Wuster, Phys. Rev. A99, 043616 (2019)

2019
[41]

Fermi, Nuovo Cimento 11, 157 (1934)

E. Fermi, Nuovo Cimento 11, 157 (1934)

1934
[42]

Skov, Magnus G

Thomas G. Skov, Magnus G. Skou, Nils B. Jorgensen, and Jan J. Arlt, Phys. Rev. Lett. 126, 230404 (2021)

2021
[43]

D. A. W. Hutchinson, E. Zaremba, and A. Griffin, Phys. Rev. Lett. 78, 1842 (1997)

1997
[44]

Cikojevic, L

V . Cikojevic, L. V . Markic, M. Pi, M. Barranco, and J. Boronat, Phys. Rev. A 102, 033335 (2020)

2020
[45]

Ferioli, G

G. Ferioli, G. Semeghini, S. T. Brianso, L. Masi, M. Fattori, and M. Modugno, Phys. Rev. Research 2, 013269 (2020)

2020
[46]

Matrix diagonalization subroutine is available here (Linear Algebra PACKage) https://www.netlib.org/
[47]

Pu and N

H. Pu and N. P. Bigelow, Phys. Rev. Lett. 80, 1130 (1998)

1998
[48]

T. R. Prisk, N. C. Das, S. O. Diallo, G. Ehlers, A. A. Podlesnyak, N. Wada, S. Inagaki, and P. E. Sokol, Phys. Rev. B 88, 014521 (2013)

2013
[49]

Onofrio, D

R. Onofrio, D. S. Durfee, C. Raman, M. Köhl, C. E. Kuk- lewicz, and W. Ketterle, Phys. Rev. Lett.84, 810 (2000)

2000
[50]

S. A. Chin and E. Krotscheck, Phys. Rev. Lett. 74, 1143 (1995)

1995
[51]

Gautam, S

S. Gautam, S. K. Adhikari, J. Phys. B 52, 055302 (2019);Annals of Phys. 409, 167917 (2019)

2019
[52]

Y . V . Kartashov, B. A. Malomed, L. Tarruell, and L. Torner, Phys. Rev. A 98, 013612 (2018)

2018
[53]

Muruganandam, S

P. Muruganandam, S. K. Adhikari, Comput. Phys. Com- mun. 180, 1888 (2009)

2009
[54]

L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Perg- amon Press, Oxford, 1987), §61 and §62

1987
[55]

J. K. Jain and R. K. Kamilla, Int. J. Modern Phys. B 11, 2621(1997); D. Majumder, S. S. Mandal, J. K. Jain, Nat. Phys. 5, 403 (2009)

1997

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

D. S. Petrov, Phys. Rev. Lett. 115, 155302 (2015)

2015

[2] [2]

I. F. Barbut, H. Kadau, M. Schmitt, M. Wenzel, and T. Pfau, Phys. Rev. Lett. 116, 215301 (2016)

work page 2016

[3] [3]

Schmitt, M

M. Schmitt, M. Wenzel, F. Bottcher, I. F. Barbut, and T. Pfau, Nature (London) 539, 259 (2016)

work page 2016

[4] [4]

C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas, P. Cheiney, L. Tarruell, Science359, 301 (2018)

work page 2018

[5] [5]

D’Errico, A

C. D’Errico, A. Burchianti, M. Prevedelli, L. Salasnich, F. Ancilotto, M. Modugno, F. Minardi, and C. Fort, Phys. Rev. Research 1, 033155 (2019)

work page 2019

[6] [6]

Kadau, M

H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. F.Barbut, and T. Pfau, Nat. Phys.530, 194 (2016); I. F. Bar- but, H. Kadau, M. Schmitt, M. Wenzel, and T. Pfau, Phys. Rev. Lett. 116, 215301 (2016)

work page 2016

[7] [7]

Bottcher, J

F. Bottcher, J. N. Schmidt, J. Hertkorn, K. S. H. Ng, S. D. Graham, M. Guo, T. Langen, and T. Pfau, Rep. Prog. Phys. 84, 012403 (2021)

work page 2021

[8] [8]

Banerjee and D

A. Banerjee and D. Majumder, Phys. Scr. 99, 085402 (2024)

work page 2024