The one-dimensional Bose gas with repulsive contact interactions (the Lieb-Liniger model) is integrable: its eigenstates are labelled by rapidities, the conserved asymptotic momenta of infinitely long-lived quasiparticles, and the local rapidity distribution ρ(x,θ)\rho(x,\theta) is the central object of generalized hydrodynamics (GHD). A proof-of-concept experiment (Dubois et al., Phys. Rev. Lett. 133, 113402 (2024)) demonstrated a local probe of ρ(x0,θ)\rho(x_0,\theta): a laser 'slice' of the cloud is selected with a shaped pushing beam, let expand one-dimensionally, and the density profile is mapped onto rapidities. The experiment itself, however, flags three unresolved issues that jointly determine whether this probe is quantitatively trustworthy: (i) realistic expansion times do not reach the asymptotic position-to-rapidity mapping (about 12% deviation at τ=50\tau=50 ms, with 2% convergence requiring an impractical ∼400\sim400 ms); (ii) extracting ρ(x0,θ)\rho(x_0,\theta) from finite-time data relies on few-parameter ansätze (a homogeneous thermal state), whose ability to capture non-thermal or spatially varying rapidity structure is unknown; and (iii) fit temperatures from expansion profiles (Tfit=230T_{\rm fit}=230 nK at the trap center) exceed the in situ Yang–Yang temperature (TYY=90T_{\rm YY}=90 nK) by a factor ≃2.5\simeq2.5, with a spatial dependence incompatible with the time-invariant cloud shape, for reasons that remain unexplained. The problem is to establish the slice-expansion (or an equivalent local-selection) protocol as a quantitative measurement of ρ(x0,θ)\rho(x_0,\theta): controlled finite-time systematic errors, faithful inference beyond thermal ansätze, and an elucidated origin of the fitted-temperature anomaly. A same-group follow-up (SciPost Phys. 20, 008 (2026)) partially extends the ansatz family but re-encounters unexplained fit discrepancies, so the full agenda remains open.

Background

A one-dimensional (1D) gas of bosons with repulsive contact interactions is described by the Lieb–Liniger Hamiltonian, an exactly solved ('integrable') model: its many-body eigenstates are Bethe wavefunctions labelled by a set of momenta called rapidities, and the infinite family of corresponding conserved quantities prevents conventional thermalization. In the thermodynamic description, a macrostate of the homogeneous gas is fixed by its rapidity distribution ρ(θ)\rho(\theta) (the density of quasiparticle momenta per unit length); at thermal equilibrium this distribution is given by the Yang–Yang thermodynamic Bethe ansatz, whose two parameters (temperature and chemical potential) can be obtained by fitting a measured in situ density profile — so-called Yang–Yang thermometry. For a gas inhomogeneously trapped in a slowly varying potential V(x)V(x), the local density approximation (LDA) treats each position xx as a fluid cell in a homogeneous macrostate, giving a local rapidity distribution ρ(x,θ)\rho(x,\theta); its slow out-of-equilibrium evolution is governed by generalized hydrodynamics (GHD), an Euler-scale hydrodynamic theory built on the rapidity distribution.

Operationally, rapidities are the asymptotic velocities of the atoms after a long 1D expansion: once the gas has expanded enough that interactions are negligible, an atom sits at xj≃θjτx_j\simeq\theta_j\tau. This underlies the measurement of global rapidity distributions of whole clouds (in arrays of 1D tubes, e.g. by observing dynamical fermionization and testing GHD with strongly interacting gases), but no method existed to probe the rapidity distribution of a chosen local fluid cell inside a single inhomogeneous cloud. Dubois et al. (Phys. Rev. Lett. 133, 113402 (2024)) introduced such a local probe on an atom-chip experiment (a chip-based current-wire magnetic waveguide holding the gas): a pushing laser beam, spatially shaped by a digital micro-mirror device (DMD), removes all atoms outside an interval of length ℓ\ell centered at x0x_0 ('slicing'), the remaining slice expands along the waveguide, and the measured density gives ρ(x0,θ)≃τn(x0+θτ,τ)/ℓ\rho(x_0,\theta)\simeq\tau n(x_0+\theta\tau,\tau)/\ell in the asymptotic limit. For a cloud at equilibrium in a harmonic trap in the weakly interacting, low-temperature 'quasi-condensate' regime (dimensionless Lieb–Liniger coupling γ=mg1D/(ℏ2n(0))\gamma=mg_{\rm 1D}/(\hbar^{2}n(0)) between 0.4×10−20.4\times10^{-2} and 0.7×10−20.7\times10^{-2} depending on the data set, the ratio of the 1D contact repulsion strength g1Dg_{\rm 1D} to the kinetic-energy scale set by the central density n(0)n(0), with mm the atomic mass), the measured rescaled profiles agree fairly with the GHD prediction at the Yang–Yang parameters (TYY=90T_{\rm YY}=90 nK, μYY/kB=49\mu_{\rm YY}/k_{\rm B}=49 nK) obtained from the in situ profile, and the probe resolves a doubly-peaked (non-thermal) local rapidity distribution in an out-of-equilibrium situation.

The Letter's own analysis, however, leaves three linked issues open. First, at realistic expansion times (τ=40\tau=40–5050 ms) the density profile has not converged to the rapidity distribution: GHD simulations show deviations of about 12% in the central part, and convergence to better than 2% would require about 400 ms, unrealistic in the experiment. GHD calculations can model the finite-time dynamics, but turning this into a measurement with a quantified systematic-error budget (finite-time corrections, imperfect slicing, heating by the pushing beam, transverse excitations) is not done. Second, extracting ρ(x0,θ)\rho(x_0,\theta) from finite-time data currently requires fitting the measured profiles with GHD simulations based on a low-parameter ansatz for the initial slice state — the paper uses a homogeneous thermal state with the temperature as fit parameter — while the number of parameters is limited by finite signal-to-noise and computation cost; whether such restricted families can capture non-thermal or spatially varying rapidity features, and how to enlarge them without overfitting, is unresolved. Third, the fits yield Tfit=230T_{\rm fit}=230 nK for the central slice, about 2.5 times TYYT_{\rm YY}, and the ratio Tfit/TYYT_{\rm fit}/T_{\rm YY} decreases away from the trap center — a spatial dependence incompatible with the fact that the trapped cloud shape is time-invariant. The paper states that the origin of this inconsistency 'still needs to be elucidated': it might signal that the local rapidity distribution is genuinely non-thermal, but experimental or modelling artefacts cannot be excluded. Adjacent literature makes the non-thermal hypothesis plausible but unproven: atom-chip quasicondensates subject to losses exhibit long-lived non-thermal states (Johnson et al., Phys. Rev. A 96, 013623 (2017)), and lossy 1D gases display rapidity distributions that deviate from thermal form (Bouchoule and Dubail, Phys. Rev. Lett. 126, 160603 (2021)).

Problem Statement

Establish the slice-expansion protocol (or an equivalent local-selection-plus-expansion protocol) as a quantitative measurement of the local rapidity distribution ρ(x0,θ)\rho(x_0,\theta) of a single inhomogeneous 1D Bose gas, on the platform of the source experiment (weakly interacting quasi-condensate regime, γ≃0.4\gamma\simeq0.4–0.7×10−20.7\times10^{-2}, TYY=90T_{\rm YY}=90 nK, μYY/kB=49\mu_{\rm YY}/k_{\rm B}=49 nK, slice width ℓ≃37\ell\simeq37 μ\mum, expansion times τ=40\tau=40–5050 ms). The three requested, jointly necessary results are:

(1) Finite-time extraction with controlled errors. Provide a procedure to extract ρ(x0,θ)\rho(x_0,\theta) from finite-time expansion data with a quantified systematic-error budget — covering the residual ∼12%\sim12\% deviation from the asymptotic mapping at τ=50\tau=50 ms (percent-level convergence otherwise requiring ∼400\sim400 ms), imperfect slicing, pushing-beam-induced heating and perturbations, transverse (3D) excitations, and detection noise — such that the reported ρ(x0,θ)\rho(x_0,\theta) carries certified error bars rather than only the current fair agreement with GHD predictions.

(2) Faithful inference beyond thermal ansätze. Demonstrate an inference scheme (regularized inversion, enlarged ansatz families, or equivalent) that recovers non-thermal and spatially varying rapidity structure from finite-signal-to-noise, finite-time data without overfitting, replacing the current few-parameter homogeneous-thermal-state fits whose faithfulness for non-equilibrium states is unknown.

(3) Origin of the fitted-temperature anomaly. Elucidate the origin of the discrepancy between expansion-fit temperatures and the in situ Yang–Yang temperature — Tfit=230T_{\rm fit}=230 nK versus TYY=90T_{\rm YY}=90 nK at the trap center (ratio ≃2.5\simeq2.5), with a spatial dependence of Tfit/TYYT_{\rm fit}/T_{\rm YY} incompatible with the time-invariant trapped cloud shape — determining whether it reflects genuinely non-thermal local rapidity distributions (e.g. loss- or evaporation-induced), a breakdown of the homogeneous-slice assumption, or experimental/model artefacts, with evidence that discriminates among these hypotheses and accounts for both the magnitude and the spatial dependence.

These three results are scientifically linked as the uncertainty sources of one and the same measurement chain, and together they decide whether the extracted ρ(x0,θ)\rho(x_0,\theta) is the true local macrostate of the gas. An answer addressing only one component (for example, an improved ansatz without a systematics budget or an elucidated anomaly) does not complete the objective.

References

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