Fig. 3: Inelastic scattering of a 1D quantum liquid.

a The dynamic structure factor, S(Q, E), from helium confined in the Ar plated pores of MCM-41. The scattering was measured at a filling of 13 mmol g⁻¹, corresponding to full pores, and the background from the Ar/MCM-41 matrix has been removed with the elastic scattering suppressed. The most prominent feature is the strong excitation that begins at ~1.6 Å⁻¹ at E = 0 and extends to high energy. This is quite distinct from the scattering that would be expected from either bulk ⁴He or that of ⁴He confined in larger pores where the scattering would be described by the phonon-maxon-roton excitation curve (gray dashed line). For the bulk liquid, the scattering would be most intense around the roton minimum at Q_r ≃ 1.9 Å⁻¹ and would plateau at twice the roton energy (~1.5 meV). The purple line shows a theoretical prediction for the threshold energy of a purely 1D quantum liquid of hard spheres with emergent Fermi wavevector 2k_F = 1.6 Å⁻¹ and dimensionless Tomonaga-Luttinger liquid parameter K ≃ 1.2. The value for k_F has been obtained from S(Q, 0), while K comes from fitting the inelastic branch of S(Q, E) (see Fig. 4). b The dynamic structure factor of a purely 1D system of ⁴He at T = 1.6 K obtained via numerical analytic continuation of the imaginary time scattering function measured via quantum Monte Carlo for a system with L = 200 Å. The density was chosen to produce a value of K similar to that seen in the experiment.