Experimental realization of one dimensional helium

Helium isotopes ³He and ⁴He are paradigmatic systems for testing theories of strongly interacting quantum matter and phase transitions. In two and three dimensions, excitations are well described by quasiparticles, but in one spatial dimension (1D) excitations become collective and, at long wavelengths and low energies, follow Tomonaga–Luttinger liquid (TLL) hydrodynamics. Access to both bosonic (⁴He) and fermionic (³He) isotopes makes 1D helium an exciting platform where the distinction between bosons and fermions blurs. Prior experimental and theoretical work in quasi-1D confinement has provided hints of low-dimensional or TLL behavior, but realizing true 1D helium is challenging because neutral helium requires physical confinement to the nanometer scale (comparable to the superfluid coherence length). In this study, the authors introduce a nanoengineered confining environment to achieve effective 1D confinement and use elastic and inelastic neutron scattering to probe the static and dynamic correlations. By analyzing the excitation spectrum within a non-linear Luttinger liquid framework, they demonstrate generalized quantum hydrodynamics in 1D bosonic ⁴He.

The paper situates its contribution within extensive prior work: helium in 2D and 3D has validated quasiparticle-based theories; in contrast, 1D systems are governed by TLL theory. Experimental efforts on quasi-1D helium in nanopores and nanotubes and theoretical studies (including path-integral quantum Monte Carlo and hard-rod models) have suggested low-dimensional behavior but typically in pores too large to achieve strict 1D confinement. Observations in larger pores have shown bulk-like phonon–maxon–roton spectra. Theoretical advances beyond linear TLL predict non-linear effects and edge singularities in S(Q,E) with mobile impurity phenomenology. These developments motivate a platform that enhances dimensional reduction sufficiently to realize a 1D helium quantum liquid and to test predictions such as emergent fermionization, 2k_F features in elastic scattering, and threshold dispersions in S(Q,E).

Confinement platform and adsorption: ⁴He is adsorbed inside MCM-41, a mesoporous silica with cylindrical pores arranged on a hexagonal lattice. The as-synthesized pore diameter is 3.0 ± 0.3 nm (from N₂ isotherms), too large for 1D. Dimensional reduction is enhanced by pre-plating the pores with a single monolayer of argon at 90 K, reducing the effective diameter to ~2 nm. Ar monolayer coverage is 8.994 mmol g⁻¹ (BET), corresponding to an areal coverage 0.59 Å⁻² and monolayer density n_Ar ≈ 0.017 Å⁻³. ⁴He adsorption isotherms at 4.2 K show strong binding to the Ar-plated walls up to ~7.5 mmol g⁻¹, gradual pressure increase between ~7.5 and 13 mmol g⁻¹, and complete pore filling at 13 mmol g⁻¹; capillary condensation between grains occurs only for P/P₀ ≳ 0.9.

Sample characterization: The MCM-41 (Sigma-Aldrich) exhibits a single phase with hexagonal pore lattice constant 4.7 nm (XRD). BET surface area is 915 m²/g. Pore diameter distribution is Gaussian with mean 3.0 nm and FWHM 0.3 nm (Kruk–Jaroniec–Sayari method).

Neutron scattering experiments: Conducted on the Disc Chopper Spectrometer (DCS) at the NIST Center for Neutron Research. The instrument is a direct-geometry TOF chopper spectrometer viewing a cold moderator. The sample comprises 6.13 g of MCM-41 in an aluminum can (OD 1.5 cm, height 6 cm, wall 1 mm), with pellets (1.25 cm diameter, 1 cm high, 0.875 g each) separated by cadmium spacers. The cell is mounted in a top-loading helium cryostat; temperature monitored by a silicon diode. Incident wavevectors used: 4.00 Å⁻¹ (energy resolution 93.6 µeV; flux 1.05×10⁵ n/cm²/s; Q_max 2.9 Å⁻¹), 2.50 Å⁻¹ (772 µeV; 8.75×10⁵ n/cm²/s; Q_max 4.6 Å⁻¹), and 1.71 Å⁻¹ (2370 µeV; 1.89×10⁵ n/cm²/s; Q_max 6.8 Å⁻¹). Data reduction yields S(Q,E). For elastic S(Q,0), measurements at full pore (13 mmol g⁻¹) subtract the Ar and boundary-layer helium background (8.68 mmol g⁻¹). Inelastic measurements focus on full pores; elastic scattering is suppressed in the processed maps.

Quantum Monte Carlo (QMC) modeling: Confined helium is modeled with an N-body Hamiltonian H = Σ p_i²/2m + Σ_{i<j} V_HeHe(r_ij) + Σ U_pore(r_i), with U_pore for Ar-plated MCM-41 from prior work and high-precision V_HeHe potentials. Path-integral QMC samples finite-temperature observables at T = 1.6 K in the grand canonical ensemble for cylindrical pores of radius R = 15.51 Å and length L = 50 Å. Chemical potentials studied: µ/k_B = −47, −27, −19, −7 K, spanning single-layer to fully filled pores; at µ/k_B = −7 K, N ≈ 600. Trotter step τ chosen so errors are O(τ⁴) and below statistical uncertainty (τ = k_B T/1250 K⁻¹). Observables include radial density ρ_r(r), pair correlations g₂(r), and static structure factor S(Q); core-restricted estimators include only atoms with √(x² + y²) < 1.72 Å. For dynamics, a purely 1D ⁴He system with L = 200 Å, density ρ_D = 0.14 Å⁻¹ at T = 1.6 K is simulated, computing the intermediate scattering function F(Q,t) and obtaining S(Q,E) via a parameter-free differential evolution analytic continuation algorithm.

Theoretical analysis: The inelastic spectra are interpreted using non-linear Luttinger liquid theory with a mobile impurity framework. The dynamic structure factor near low energies exhibits edge singularities above a threshold E_th(Q) with S(Q,E) ∝ (E − E_th(Q))^{|μ(Q) − μ(0)|}. For hard-core bosons, the threshold dispersion is E_th(Q) = K (2k_F − |Q − 2k_F|)², where K is the Luttinger parameter and k_F = πρ₁D the emergent Fermi wavevector. Elastic TLL predictions include algebraically decaying Bragg peaks at 2k_F.

• Elastic scattering S(Q,0) from ⁴He in Ar pre-plated MCM-41 shows two features at T = 1.6 K: (1) a broad peak near Q ≈ 2.1 Å⁻¹ attributed to dense second/third adsorbed layers; (2) a narrow Gaussian peak at Q = 1.60 ± 0.02 Å⁻¹ corresponding to the central core liquid, implying an atomic spacing of 3.92 ± 0.05 Å. QMC reproduces these features and shows weakly decaying oscillations along the pore axis for core atoms, yielding a strong elastic peak at 2k_F = 2πρ₁D, consistent with TLL predictions and indicative of emergent fermionization.
• The core liquid’s linear density is near ρ ≈ 0.25 Å⁻¹ (consistent with 2k_F ≈ 1.6 Å⁻¹).
• Inelastic neutron scattering S(Q,E) reveals a single, well-defined dispersing feature that starts at Q ≈ 1.6 Å⁻¹ at E = 0 and increases smoothly with Q, qualitatively distinct from bulk ⁴He phonon–maxon–roton behavior (which would show an intense roton minimum near Q_r ≈ 1.9 Å⁻¹ and a high-Q plateau near ~1.5 meV).
• Fitting the dispersion maxima of the observed inelastic branch to the non-linear LL threshold E_th(Q) = K (2k_F − |Q − 2k_F|)², with 2k_F fixed at 1.6 Å⁻¹ from elastic data, yields a Luttinger parameter K = 1.18 ± 0.38.
• Temperature dependence (Q_in = 4.00 Å⁻¹): the dispersing inelastic feature remains robust from 1.6 K to 4.2 K, becoming slightly more diffuse with increasing temperature; the relevant TLL scale estimated as T_th ≈ 4 E_p/(K k_B) ≈ 13 K for Q ≥ 2k_F, indicating 1D excitation character persists above the bulk superfluid transition.
• QMC for a purely 1D ⁴He system at comparable K reproduces a similar dynamic structure factor, supporting the interpretation of the experimental spectra as 1D.
• Core-only TLL fits in simulations at higher density (ρ_D ≈ 0.293 Å⁻¹) give K = 0.15(4) and ħv/k_B = 8(3) Å·K, consistent with density-dependent K and the experimental value corresponding to lower density (ρ_D ≈ 0.25 Å⁻¹).
• The data support separation of contributions from adsorbed layers and the 1D core, and demonstrate generalized quantum hydrodynamics beyond linear TLL, consistent with a mobile impurity description.

The experiments realize an effectively one-dimensional quantum liquid of bosonic ⁴He by nanoengineering argon-preplated MCM-41 nanopores that isolate a central core liquid. Elastic scattering reveals a sharp peak at 2k_F, establishing a linear density and the emergent fermionization characteristic of 1D systems. Inelastic neutron scattering shows a single threshold-like dispersing branch inconsistent with bulk phonon–maxon–roton excitations. Interpreting the spectra via non-linear Luttinger liquid theory with a mobile-impurity picture captures the observed dispersion and yields a Luttinger parameter K ≈ 1.2, in line with microscopic predictions. The weak temperature dependence up to 4.2 K and consistency with 1D QMC simulations further validate the 1D hydrodynamic description. These findings address the long-standing challenge of observing 1D helium by demonstrating both static (2k_F peak) and dynamic (threshold dispersion) signatures of a 1D quantum liquid. The platform enables exploration of interaction-tuned regimes (e.g., towards the super Tonks–Girardeau regime) and, with fermionic ³He, potential observation of spin–mass separation where spin and density excitations propagate at different velocities.

The study demonstrates an experimental realization of a one-dimensional quantum liquid of ⁴He via confinement in argon pre-plated MCM-41 nanopores. Combining elastic and inelastic neutron scattering with quantum Monte Carlo and non-linear Luttinger liquid analysis, the authors identify a core 1D liquid exhibiting a 2k_F elastic peak and a threshold-like inelastic dispersion distinct from bulk helium. A Luttinger parameter K = 1.18 ± 0.38 is extracted, consistent with theory and simulations. This nanoengineered platform establishes generalized quantum hydrodynamics in 1D helium and opens avenues to tune density and interactions via pressure, potentially accessing regimes from weakly interacting to super Tonks–Girardeau. Future work includes systematic control of filling to tune K (observable as changes in the inelastic threshold slope) and replacing ⁴He with ³He to search for spin–mass separation.

• Instrumental trade-offs constrained the accessible Q–E space: higher incident wavelength (Q_in = 4.00 Å⁻¹) offered better energy resolution but lower flux and limited Q_max (2.9 Å⁻¹), while shorter wavelengths (1.71 Å⁻¹) provided larger Q range (6.8 Å⁻¹) but significantly worse energy resolution (2370 µeV) and reduced flux.
• Extensive filling-dependent studies were primarily conducted at Q_in = 2.50 Å⁻¹ due to flux and range considerations, potentially limiting resolution of fine spectral features.
• Elastic and inelastic analyses rely on background subtraction of Ar and boundary-layer helium contributions; residual uncertainties after subtraction contribute to statistical errors.
• Simulations assume smooth, idealized pores; differences from real pore roughness and finite-length effects may contribute to quantitative discrepancies (e.g., density-dependent K values between experiment and specific simulation conditions).