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Characterising transport in a quantum gas by measuring Drude weights

Philipp Schüttelkopf, Mohammadamin Tajik, Bazhan, Nataliia, Cataldini, Federica, Ji, Si-Cong, Jörg Schmiedmayer, Frederik Møller

Published Jan 1, 2025• 0 citations
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Abstract

We measure the Drude weights of an ultracold gas of interacting bosonic atoms confined to one dimension, characterising atomic and energy currents induced by applying a constant force and by joining two subsystems prepared in different equilibrium states. We demonstrate dissipationless transport signifying ballistic propagation of conserved quantities corresponding to lowest order hydrodynamics. Our protocols facilitate systematic testing of non-ballistic, higher-order hydrodynamic terms and non-linear responses, finding both negligible within the scales probed.

DOI
10.34726/11411

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Characterising transport in a quantum gas by measuring Drude weights Authors: Philipp Schüttelkopf, Mohammadamin Tajik, Nataliia Bazhan, Federica Cataldini, Si-Cong Ji, Jörg Schmiedmayer, and Frederik Møller arXiv.2406.17569, DOI: 10.48550/arXiv.2406.17569 Recommended with a commentary by Carlo Beenakker , Instituut-Lorentz, Leiden University |View Commentary (pdf)| This commentary may be cited as: DOI: 10.36471/JCCM_November_2024_01 https://doi.org/10.36471/JCCM_November_2024_01

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We measure the Drude weights of an ultracold gas of interacting bosonic atoms confined to one dimension, characterising atomic and energy currents induced by applying a constant force and by joining two subsystems prepared in different equilibrium states. We demonstrate dissipationless transport signifying ballistic propagation of conserved quantities corresponding to lowest order hydrodynamics. Our protocols facilitate systematic testing of non-ballistic, higher-order hydrodynamic terms and non-linear responses, finding both negligible within the scales probed.
Transport properties define materials as insulators, metals, or superconductors. A fundamental parameter is the Drude weight, which quantifies the ballistic transport of charge carriers. In this study, we measured the Drude weights of an ultracold gas of interacting bosonic atoms confined to one dimension, characterizing atomic and energy currents induced by applying a constant force and by joining two subsystems prepared in different equilibrium states. We demonstrate dissipationless transport, even in the presence of interactions and finite temperature, signifying ballistic propagation of conserved quantities corresponding to lowest-order hydrodynamics. Our approach provides a robust and transparent framework for characterizing transport in strongly correlated quantum matter, applicable in regimes where theory remains incomplete.
The Drude weight characterizes ballistic transport in quantum many-body systems. Although analytical calculations of Drude weights have been extensively studied in integrable models, their direct connections to finite-temperature macroscopic state functions remain unestablished, especially for continuous multicomponent quantum gases. In the present work, we use generalized hydrodynamics and the thermodynamic Bethe ansätz to calculate exactly the Drude weights for one-dimensional continuous integrable systems, including the Lieb-Liniger and Bose-Fermi mixture models. We derive universal exact relations between Drude weight matrix components and key thermodynamic densities (particle density, enthalpy, entropy). Analytic expressions for Drude weight are obtained across different physical regimes, i.e. strong- and weak-coupling regimes in addition to universal scaling laws near the quantum phase transitions. To bridge theory and experiment, we simulate two experimental protocols, linear potential quench and bipartitioning quench, to enable reliable measurements of the Drude weights. Using these protocols, we calculate the charge and energy Drude weight for Lieb-Liniger gas and compare with recent measurements reported in [Science 391, 290 (2026)], showing excellent agreement with particle density and enthalpy, respectively, thus offering deeper physical insights into experimental observations. Our findings directly link ballistic transport properties to thermodynamics, providing rigorous theoretical benchmarks for future ultracold atomic gas experiments.
Owing to the fact that the particle current operator in nonrelativistic gases is proportional to the total momentum operator, the particle transport in such systems is always ballistic and fully characterized by a Drude weight Δ. The Drude weight can be calculated within linear response theory. It is given by the formula Δ=2πD, where D is the density of the gas. This holds in any dimension and for every equilibrium ensemble, in particular for generalized Gibbs ensembles that describe possible equilibrium states of isolated integrable quantum systems. In the canonical ensemble case, the Drude weight can be equivalently obtained from a generalized susceptibility related to the fluctuations of the conserved particle current. Such susceptibility can be rigorously calculated for the integrable Lieb-Liniger Bose gas in any generalized Gibbs ensemble using a generalized Yang-Yang thermodynamic formalism. The resulting expression agrees with a prediction made within the context of generalized hydrodynamics. It also allows us to see explicitly that, within truly generalized Gibbs ensembles, the conductivity related with the particle current is not determined by the corresponding current-current autocorrelation function.
Building on the Bethe ansatz and generalized hydrodynamics, we rigorously establish universal relations between Drude weights governing charge, spin, and energy transport and thermodynamic properties for the one-dimensional repulsive two-component continuum Fermi gas - the paradigmatic integrable Yang-Gaudin model. We analytically derive thermodynamic expressions for charge Drude weights: DnnD_{nn}Dnn​, DnmD_{nm}Dnm​, DneD_{ne}Dne​ (responses of particle, magnetization and energy currents to a chemical potential gradient) equal the particle density, magnetization, and temperature times entropy density, respectively. These relations constitute a hallmark of ballistic transport, and can be generalized to other 1D continuum integrable systems. Furthermore, we investigate spin Drude weight (DmmD_{mm}Dmm​, DmeD_{me}Dme​) at zero and low temperatures. They characterize the magnetization and energy current responses to a magnetic field gradient, revealing an essential spin-charge coupling feature in quantum transport. We find that the Drude weights DnmD_{nm}Dnm​, DmmD_{mm}Dmm​, and DmeD_{me}Dme​ across the μ-Hμ\text{-}Hμ-H plane remarkably map out the zero-temperature phase boundary. This work fills a critical gap in the understanding of transport within integrable quantum gases and furnishes a direct theoretical foundation for future ultracold-atom experiments.
We outline a general formalism of hydrodynamics for quantum systems with multiple particle species which undergo completely elastic scattering. In the thermodynamic limit, the complete kinematic data of the problem consist of the particle content, the dispersion relations, and a universal dressing transformation which accounts for interparticle interactions. We consider quantum integrable models and we focus on the one-dimensional fermionic Hubbard model. By linearizing hydrodynamic equations, we provide exact closed-form expressions for Drude weights, generalized static charge susceptibilities, and charge-current correlators valid on the hydrodynamic scale, represented as integral kernels operating diagonally in the space of mode numbers of thermodynamic excitations. We find that, on hydrodynamic scales, Drude weights manifestly display Onsager reciprocal relations even for generic (i.e., noncanonical) equilibrium states, and establish a generalized detailed balance condition for a general quantum integrable model. We present exact analytic expressions for the general Drude weights in the Hubbard model, and explain how to reconcile different approaches for computing Drude weights from the previous literature.