This article presents an elementary introduction on various aspects of the prototypical integrable model the Lieb-Liniger Bose gas ranging from the cooperative to the collective features of many-body phenomena. In 1963 Lieb and Liniger first solved this quantum field theory many-body problem using the Bethe's hypothesis, i.e. a particular form of wave function introduced by Bethe in solving the one-dimensional Heisenberg model in 1931. Despite the Lieb-Liniger model is arguably the simplest exactly solvable model, it exhibits rich quantum many-body physics in terms of the aspects of mathematical integrability and physical universality. Moreover, the Yang-Yang grand canonical ensemble description for the model provides us with a deep understanding of quantum statistics, thermodynamics and quantum critical phenomena at the many-body physics level. Recently, such fundamental physics of this exactly solved model has been attracting growing interest in experiments. Since 2004, there have been more than 20 experimental papers that report novel observations of different physical aspects of the Lieb-Liniger model in the lab. So far the observed results to date are seen to be in excellent agreement with results obtained using the analysis of this simplest exactly solved model. Those experimental observations reveal the unique beauty of integrability.

We experimentally investigate the quantum criticality and Tomonaga-Luttinger liquid (TLL) behavior within one-dimensional (1D) ultracold atomic gases. Based on the measured density profiles at different temperatures, the universal scaling laws of thermodynamic quantities are observed. The quantum critical regime and the relevant crossover temperatures are determined through the double-peak structure of the specific heat. In the TLL regime, we obtain the Luttinger parameter by probing sound propagation. Furthermore, a characteristic power-law behavior emerges in the measured momentum distributions of the 1D ultracold gas, confirming the existence of the TLL.

A quantum heat engine (QHE) based on the interaction driving of a many-particle working medium is introduced. The cycle alternates isochoric heating and cooling strokes with both interaction-driven processes that are simultaneously isochoric and isentropic. When the working substance is confined in a tight waveguide, the efficiency of the cycle becomes universal at low temperatures and governed by the ratio of velocities of a Luttinger liquid. We demonstrate the performance of the engine with an interacting Bose gas as a working medium and show that the average work per particle is maximum at criticality. We further discuss a work outcoupling mechanism based on the dependence of the interaction strength on the external spin degrees of freedom.npj Quantum Information (2019) 5:88; https://doi.