Thermal Entanglement and Out-of-Equilibrium Thermodynamics in 1D Bose gases

We investigate entanglement in and out of equilibrium in a one-dimensional Bose gas in its low-energy Bogoliubov regime. In this Gaussian setting, the state is fully characterized by its covariance matrix, which allows us to detect and quantify entanglement using a covariance-based framework and associated entanglement monotones. For thermal states, we determine the optimal entanglement witness arising from the covariance matrix criterion and show that it has a remarkably simple mode-resolved structure: it is diagonal in the normal-mode basis and admits a simple analytic form that can be expressed as a product of only two normal-mode uncertainties. We then study out-of-equilibrium dynamics induced by unitary compression and show that entanglement can be generated even from initially separable thermal states. When the evolution is fully adiabatic, the optimal witness retains the same two-mode structure as in the thermal case. Departing from this regime, i.e., performing increasingly rapid compression, the optimal witness becomes genuinely more intricate. Our methods and results provide a unified and physically intuitive picture of how entanglement emerges and evolves in 1D quantum Bose gases, and identify an optimal witness structure relevant more broadly to the analysis of entanglement in quadratic bosonic models and its role in thermodynamic cycles.