{"id":"60a31be4-04c0-4a72-94dd-718c4109cfad","arxiv_id":"2507.21805","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Axial confinement lowers the critical atom number for binary quantum droplet formation and lowers the rotation frequency at which a central vortex appears.","lead":"This paper maps how a one-dimensional harmonic trap changes the properties of self-bound quantum droplets in a two-component Bose gas. It provides a phase diagram and a formula for the rotation speed at which a central vortex becomes energetically favourable.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The analytical vortex-energy derivation rests on an unverified fitted coefficient and a boundary-condition mismatch between the s=0 and s=1 numerical states; if these biases shift the trends, the central monotonicity claim is not established.","rationale":"The paper makes a clear central claim and supports it numerically, but the analytical part that is supposed to explain the numerics rests on a fitted coefficient and a chain of approximations (Eqs. (34), (35)-(42)). The most load-bearing weakness is that the analytical formula is not independently derived and the final Fig. 8 agreement may be partly fitting the numerics rather than validating a prediction. A second, distinct issue is that the numerical energy difference ΔE = E(1,0) − E(0,0) uses different boundary conditions for s=0 (Neumann) and s=1 (Dirichlet at ρ=0, with an infinite-radius Green's function), so the subtraction can carry a boundary-condition-dependent offset; if this offset varies with ω̃, it could bias the trend. However, the numerical phase diagram, qualitative density profiles, and the benchmarking against Petrov's N_c ≈ 18.65 are strong independent support. A re-check of the vortex-energy derivation against the full two-component system would settle whether the monotonic decrease is a genuine property or an artefact of the locked approximation.","tokens_in":17775,"tokens_out":1554,"duration_ms":18422,"concrete_test":"Recompute Ωc by solving the full two-component eGPE without the density-locking reduction for a representative parameter set, e.g. N=1000 with ω̃=0.25 and a 10% population imbalance or unequal masses, and compare with the density-locked prediction. If Ωc shifts by more than the plotting markers in Fig. 8, the central claim is restricted to the locked case.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central quantitative claim is that Ωc decreases monotonically with stronger axial trapping and larger N. The numerical Fig. 8 supports this, but the analytical derivation backing it has a load-bearing soft spot: Eq. (34) asserts ε_LHY ≈ π E(0) ξ^2 ln(1.078 D) with a numerically fitted coefficient 1.078. Unlike the mean-field case, where 1.464 follows from the well-known Ginzburg-Pitaevskii result, the droplet case substitutes the numerical solution of Eq. (21) into Eq. (31); no closed-form or independent derivation of the 1.078 coefficient is given, and the reported 'verification' over a range of D is not shown. The derivation then substitutes the variational log-density profile Eq. (14) into the vortex energy and, without rigorous justification, replaces R(z) via the Thomas-Fermi-type Eq. (39) and the approximation Eq. (41). Although the final comparison in Fig. 8 is encouraging, the analytical formula is thus a fit dressed as a derivation, and the monotonic decreases in Ωc are not analytically proven. Moreover, a separate concern: the vortex-state energy difference in Eq. (49) is evaluated at s=1 for fixed atom number, whereas the numerical s=1 and s=0 states are obtained on different grids and with different boundary conditions; the Appendix specifies homogeneous Neumann conditions for s=0 and an enforced f=0 at ρ=0 for s=1, so the two energy surfaces may not be directly comparable without careful convergence checks. The paper states data are openly available, which would allow such checks, but the manuscript itself does not document grid-convergence or the energy-difference sensitivity to D and L_z.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies binary quantum droplets in the density-locked single-component extended Gross-Pitaevskii formalism, subject to an axial harmonic trap. The authors solve the eGPE numerically in cylindrical coordinates with imaginary-time propagation, construct a four-parameter super-Gaussian variational ansatz, and map the droplet-gas phase diagram in the effective-atom-number versus axial-trap-frequency plane. They then compute s=1 central-vortex solutions, extract the vortex energy per unit length for a homogeneous droplet from the numerical vortex profile, and combine this with the variational ground-state profile to derive an approximate analytical formula for the critical rotation frequency Ωc. The formula is compared with numerically computed Ωc values for N in {500,1000,1500,2000} and ω in {0,0.125,0.25,0.5,1}. The central claims are that stronger axial trapping monotonically lowers the critical atom number for droplet formation and that both stronger axial trapping and larger atom numbers monotonically lower the critical rotation frequency for vortex nucleation.","tokens_in":18177,"tokens_out":14068,"duration_ms":170264,"significance":"If the analytical derivation is placed on sound footing, this is a useful systematic study of a less-explored regime: binary droplets in one-dimensional confinement, with a phase diagram, vortex profiles, and rotation-frequency estimates that could guide experiments in flattened geometries. The numerical work has clear strengths: the method reproduces Petrov's free-space critical atom number Nc≈18.65 to 18.66, the data are stated to be openly available, and the comparison in Fig. 8 is a genuine comparison rather than a fit of the formula to the numerical Ωc values. The central monotonicity trends are supported by the numerical data even before the analytical formula is invoked. The main weaknesses are that the analytical vortex-energy coefficient 1.078 is not independently derived or shown, the printed Eq. (31) contains an apparent internal inconsistency, and the numerical energy difference in Eq. (49) is computed with different boundary-condition treatments for s=0 and s=1 without convergence checks.","major_comments":[{"comment":"As printed, the two integrals in Eq. (31) cannot be equal. The first integrand contains f^2/ρ^2, which for f→1 gives a logarithmic divergence ∼∫dρ/ρ, while the second integrand vanishes at f=1 (and also at f=0), so the second integral has no logarithmic divergence in D. Since the paper states that substituting the numerical solution of Eq. (21) into Eq. (31) yields the coefficient 1.078 in Eq. (34), the reader cannot tell which expression was actually evaluated. Please correct Eq. (31) or explicitly state the correct energy functional used to obtain Eq. (34).","section":"Section V A, Eq. (31)"},{"comment":"The logarithmic coefficient 1.078 is load-bearing for the analytical Ωc formula through ln(1.078 kR) in Eq. (45), but it is obtained solely by numerically inserting the model's own vortex profile into Eq. (31), with the claimed verification 'across a range of D' not shown. Unlike the mean-field coefficient 1.464, which follows from the known Ginzburg-Pitaevskii result, no independent derivation or consistency check is provided. Please include the coefficient as a function of D, describe the extrapolation procedure, and quantify how much the Ωc curves in Fig. 8 shift when the coefficient is varied within a plausible error bar.","section":"Section V A, Eq. (34)"},{"comment":"The numerical Ωc is an energy difference between s=0 and s=1 states that are obtained with different boundary-condition treatments. For s=0 the decomposition f=f0+f∞ in Eq. (A1) permits a nonzero background at the outer boundary, whereas for s=1 the first-order Hankel transform enforces f(ρ=0)=0 and the outer boundary is handled without such a decomposition. No convergence study with respect to Lρ, Lz, Nρ, or Nz is reported for the energy difference itself. Please provide grid and domain convergence checks for both states and show that the monotonic decrease in Fig. 8 is stable, especially for the smaller droplets near the self-bound regime.","section":"Section V C and Appendix A, Eq. (49)"},{"comment":"The analytical Ωc formula depends on the replacement R(z)≈R exp[-(z/Z)^{nz}/nr] in Eq. (41) and on a small-argument expansion of the exponential integral. The paper also provides an alternative exact-numerical-integral form in Eqs. (47)-(48), but Fig. 8 is made using Eqs. (45)-(46). The error introduced by Eq. (41) is not quantified. Please compare the Ωc values obtained with Eqs. (46) and (47) against the numerical values, or otherwise show that the agreement in Fig. 8 is not an artifact of this approximation.","section":"Section V B, Eqs. (40)-(47)"}],"minor_comments":[{"comment":"The factor exp(nr/γ) in Eqs. (36) and (40) is inconsistent with the preceding result in Eq. (35), where the Euler-Mascheroni constant appears as -γ/nr, and with the final expressions in Eqs. (45)-(46), which use γ/nr with the correct sign. Please correct this typo throughout the derivation.","section":"Section V B, Eqs. (36) and (40)"},{"comment":"There is a duplicated article in the phrase 'a a variational ansatz' near Eq. (14).","section":"Section II B"},{"comment":"The notation '2enzF(nz)' in Eq. (47) is ambiguous; it should be typeset so that the exponential factor (presumably e^{nz}) and the product F(nz) are unambiguous.","section":"Section V B, Eq. (47)"},{"comment":"The figure would be easier to interpret if the numerical Ωc values included a marker of numerical uncertainty or at least the convergence data referenced in the major comments, since the claimed monotonicity is the central quantitative result.","section":"Figure 8"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the journal's scope and the numerical component is solid, including the Petrov benchmark and the open-data statement. The main risk is that the analytical section's credibility rests on the unpublished verification of the coefficient 1.078 and on an equation (Eq. (31)) that appears internally inconsistent as printed. These are fixable within the manuscript's scope, so I recommend major revision rather than rejection. I saw no concerns about citation practices or novelty disclosure."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a competent, useful paper rather than a breakthrough. It maps out how axial confinement changes the droplet-gas phase boundary and the critical rotation frequency for a central vortex in a binary Bose mixture, and it provides a practical analytical estimate for that frequency. The numerical eGPE results are carefully benchmarked (they reproduce Petrov's free-space N_c approximately 18.65 as 18.66), and the comparison between the analytical Omega_c and the directly computed numerical energies in Fig. 8 is genuine. The coefficient 1.078 in the vortex line energy comes from a separate homogeneous-vortex calculation, not from fitting the droplet numerics, so the analytical section is on firmer ground than the stress-test note suggests. The phase diagram trend was already predicted via DFT in Ref. [56], but the eGPE confirmation, the variational ansatz, the vortex-droplet solutions, and the Omega_c formula are new and useful. The soft spots are real but mostly cosmetic. First, there is a typo in Eqs. (36) and (40): the factor involving the Euler-Mascheroni constant should be exp(-gamma/n_r), not exp(n_r/gamma); the derivation around Eq. (35) implies the correct sign. Second, the authors say they verified the 1.078 log coefficient over a range of D but do not show that verification; a few sentences with fit values would settle it. Third, and more substantively, the numerical domain is fixed at L_rho = L_z = 40 with no grid-convergence study. At the largest N and highest omega, the quasi-2D droplet radius grows as N^{1/2} and may approach or exceed this domain, which could bias the energy difference in Eq. (49) and hence the Omega_c points in Fig. 8. That is the one place where the central monotonicity claim could be affected, and a convergence check with a larger domain and a different grid size should be requested. The boundary-condition mismatch between the s=0 and s=1 solvers is less alarming because both states are deep in the self-bound regime and decay before the boundary, but the convergence check would also cover that. The density-locked approximation and balanced-population assumption are explicitly flagged by the authors; that is a scope limitation, not an oversight. Who is this for? Groups working on droplet physics and vortex states, especially experimental teams planning flattened-trap geometries; the Omega_c formula is directly usable for estimating rotation frequencies. The paper deserves a serious referee. I would send it to review and, after the typo fix, the log-coefficient verification, and a domain-size test, likely accept it with minor revision.","headline":"Solid, useful subfield paper: axially trapped droplet phase diagram and vortex nucleation frequency, well benchmarked but with a typo and a missing convergence check.","tokens_in":817,"tokens_out":1689,"would_cite":true,"duration_ms":79782,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Axial trapping lowers both the atom threshold for droplet formation and the rotation frequency for nucleating a central vortex in a binary quantum droplet.","keywords":["binary quantum droplets","extended Gross-Pitaevskii equation","axial harmonic trapping","critical atom number","vortex nucleation","critical rotation frequency","self-bound Bose mixtures","quasi-2D droplets"],"falsifier":"If an oblate binary droplet experiment sees the critical rotation frequency for vortex nucleation increase when the axial trap is tightened at fixed atom number, the central claim of monotonic decrease would be falsified; likewise, a two-component imbalanced simulation that shows $\\tilde{N}_c$ rising with $\\tilde{\\omega}$ would break the density-locked prediction.","tokens_in":17580,"feed_emoji":"🌀","tokens_out":6154,"duration_ms":62554,"temperature":0.7,"pith_summary":"Quantum droplets are self-bound liquid states of ultracold bosonic mixtures, stabilized by quantum fluctuations against collapse. This paper asks how those droplets change when a harmonic trap squeezes them along one axis, the geometry relevant to flattened, quasi-two-dimensional experiments. The central result is that stronger axial confinement monotonically lowers the critical effective atom number needed to form a droplet, and also monotonically lowers the critical rotation frequency at which a central vortex becomes energetically favorable. A variational ansatz reproduces the ground states at stronger trapping, and an analytic formula for the vortex threshold agrees well with numerical solutions of the extended Gross-Pitaevskii equation. If correct, the work provides a direct route to controlling droplet formation and vortex nucleation by tuning the axial trap.","feed_headline":"Axial trapping lowers atom and vortex thresholds in quantum droplets","feed_subtitle":"Squeezing a binary droplet along one axis lowers the atom count needed to bind it and the rotation rate that nucleates a central vortex.","key_machinery":"The central object is the density-locked extended Gross-Pitaevskii equation (eGPE), Eq. (4), an effective single-component equation for a binary mixture whose components keep a fixed local density ratio and whose masses and trap frequencies satisfy $m_1\\omega_1=m_2\\omega_2$. The argument is carried by a variational super-Gaussian ansatz for the order parameter, Eq. (14), with separate radial and axial widths and exponents, which supplies the parameters used in an analytic formula for the vortex critical rotation frequency, Eqs. (45)–(46). The vortex energy is built from a logarithmic excitation energy per unit length, $\\varepsilon_v^{\\mathrm{LHY}}\\approx \\pi E^{(0)}\\xi^2 \\ln(1.078 D)$, whose coefficient 1.078 replaces the mean-field value 1.464 and encodes the beyond-mean-field vortex profile.","core_discovery":"The paper establishes that, in the density-locked single-component description of a binary Bose mixture, a harmonic trap along the z-axis acts as a control knob on both the liquid-gas transition and the vortex transition. Numerically solving the extended Gross-Pitaevskii equation and fitting a phase diagram shows that the critical effective atom number $\\tilde{N}_c$ drops monotonically as the effective axial trapping frequency $\\tilde{\\omega}$ increases, while the droplet's radial width grows and its axial width is set by the trap. Embedding a unit-circulation vortex along the trap axis, the paper finds the vortex-core profile differs qualitatively from the mean-field gas vortex, and derives an analytic estimate for the critical rotation frequency $\\Omega_c$ in which variational ground-state parameters enter through super-Gaussian widths. The formula predicts $\\Omega_c$ decreases with both increasing $\\tilde{N}$ and increasing $\\tilde{\\omega}$, matching the numerical values well, especially for large droplets.","pith_inferences":["An immediate experimental probe would be to stir an oblate binary droplet at controlled rotation rates and compare the vortex onset with the predicted $\\Omega_c(\\tilde{N},\\tilde{\\omega})$; the monotonic decrease is a clear signature to look for.","The density-locked assumption excludes population-imbalanced mixtures, so the predicted threshold shifts may be modified in imbalanced droplets; extending the vortex calculation to imbalanced mixtures is a natural test.","Because the eGPE vortex has a distinct core profile, vortex–antivortex annihilation and vortex reconnection dynamics in droplets may differ from mean-field condensates, a question the stationary analysis here does not settle.","The analytic formula for $\\Omega_c$ could be adapted to multiply-charged vortices, connecting the three-dimensional results to the better-studied two-dimensional vortex droplet thresholds."],"forward_implications":["Stronger axial trapping lowers $\\tilde{N}_c$, so a droplet can be formed from fewer atoms than in free space, a feature experiments can exploit by tightening the axial confinement.","The critical rotation frequency $\\Omega_c$ decreases with both atom number and axial trap frequency, meaning central vortices should appear at slower stirring rates in oblate droplets.","As $\\tilde{\\omega}$ grows, the large-$\\tilde{N}$ radial width scaling crosses from $\\tilde{N}^{1/3}$ toward $\\tilde{N}^{1/2}$, signaling the crossover from three-dimensional to quasi-two-dimensional droplet behavior.","The vortex core in the droplet is governed by the beyond-mean-field nonlinearity and has a different shape from the usual mean-field vortex, so the standard Padé approximant for the vortex profile does not transfer directly."],"supporting_citations":[{"why":"Supplies the density-locked extended Gross-Pitaevskii theory and the free-space critical atom number $\\tilde{N}_c\\approx 18.65$ that anchors the phase diagram.","marker":"[1]"},{"why":"Previous theoretical study of axially trapped binary droplets via density functional theory that predicted the monotonic decrease of $\\tilde{N}_c$ with trap frequency, which this work verifies with the eGPE.","marker":"[56]"},{"why":"Two-dimensional vortex droplet study whose critical rotation frequency results are compared and connected to the present three-dimensional axially confined case.","marker":"[65]"},{"why":"Provides the method for estimating the critical rotation frequency from the vortex excitation energy in a harmonically trapped condensate, adapted here to the density-locked droplet.","marker":"[68]"},{"why":"Standard reference for the Gross-Pitaevskii vortex energy per unit length and the Padé approximant used to contrast the mean-field vortex profile with the droplet vortex profile.","marker":"[59]"},{"why":"Rotating trapped-BEC analysis whose critical rotation frequency behavior is used as a comparison point for the droplet vortex thresholds.","marker":"[67]"}],"fun_headline_variants":["Axial trap eases quantum droplet binding and vortex nucleation","Axial compression lowers thresholds for droplet binding and vortices","Tighter axial trap reduces atom count and rotation speed for droplets","Axial confinement eases droplet formation and vortex onset","Axial squeeze lowers barrier to droplet binding and vortex rotation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire analysis assumes the two components remain density-locked in the ratio $n_2/n_1=\\sqrt{g_{11}/g_{22}}$ and satisfy $m_1\\omega_1=m_2\\omega_2$, which rules out population imbalance, unequal-component masses, and droplet-superfluid compound phases.","fun_headline_variants_meta":{"raw":{"variants":["Axial trap eases quantum droplet binding and vortex nucleation","Axial compression lowers thresholds for droplet binding and vortices","Tighter axial trap reduces atom count and rotation speed for droplets","Axial confinement eases droplet formation and vortex onset","Axial squeeze lowers barrier to droplet binding and vortex rotation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00066,"raw_usage":{"total_tokens":2965,"prompt_tokens":841,"completion_tokens":2124,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":457,"completion_tokens_details":{"reasoning_tokens":2043}},"tokens_in":457,"tokens_out":2124,"duration_ms":15175,"temperature":1.0,"reasoning_tokens":2043,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T12:20:45.848641+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"If an oblate binary droplet experiment sees the critical rotation frequency for vortex nucleation increase when the axial trap is tightened at fixed atom number, the central claim of monotonic decrease would be falsified; likewise, a two-component imbalanced simulation that shows $\\tilde{N}_c$ rising with $\\tilde{\\omega}$ would break the density-locked prediction.","supporting_citations":[{"cited_title":"In (b), the numerically determined contours such that n(r) = 0.1n0 are plotted for ρ ≥ 0, z ≥ 0, as well as the corresponding variationally determined contour","cited_arxiv_id":null,"evidence_quote":"Supplies the density-locked extended Gross-Pitaevskii theory and the free-space critical atom number $\\tilde{N}_c\\approx 18.65$ that anchors the phase diagram."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Previous theoretical study of axially trapped binary droplets via density functional theory that predicted the monotonic decrease of $\\tilde{N}_c$ with trap frequency, which this work verifies with the eGPE."},{"cited_title":"Stürmer, M","cited_arxiv_id":null,"evidence_quote":"Two-dimensional vortex droplet study whose critical rotation frequency results are compared and connected to the present three-dimensional axially confined case."},{"cited_title":"Dalfovo and S","cited_arxiv_id":null,"evidence_quote":"Provides the method for estimating the critical rotation frequency from the vortex excitation energy in a harmonically trapped condensate, adapted here to the density-locked droplet."},{"cited_title":"Ho and V","cited_arxiv_id":null,"evidence_quote":"Standard reference for the Gross-Pitaevskii vortex energy per unit length and the Padé approximant used to contrast the mean-field vortex profile with the droplet vortex profile."},{"cited_title":"Dong and M","cited_arxiv_id":null,"evidence_quote":"Rotating trapped-BEC analysis whose critical rotation frequency behavior is used as a comparison point for the droplet vortex thresholds."}],"review_version":1}