Dynamical fermionization and emergent Bethe rapidity structure in the spatial density of cold quenched Lieb–Liniger gas

We investigate whether Bethe-rapidity information can be extracted directly from real-space density measurements during the non-equilibrium expansion of an interacting one-dimensional Lieb–Liniger Bose gas. Unlike conventional approaches that rely on momentum-space observables, we examine whether the asymptotic spatial density generated by ballistic expansion encodes the underlying rapidity distribution. The system is prepared in the interacting ground state of a hard-wall box and subjected to a geometric quench into a larger box at fixed interaction strength. The ground state and subsequent dynamics are computed using the generalized Feynman–Kac quantum Monte Carlo method, which provides accurate eigenvalues and eigenfunctions of many-body Schrödinger Hamiltonians. We show that, when expressed in the velocity variable x∕t, the density approaches a stationary profile whose interaction-dependent shape closely mirrors the corresponding Bethe-rapidity distribution. The velocity-space density broadens systematically with increasing interaction strength and rapidly approaches the Tonks–Girardeau limit. Our results demonstrate that ballistic expansion provides a practical route to accessing rapidity-related information through spatial-density measurements, establishing a direct connection between real-space observables, dynamical fermionization, and the integrable structure of the Lieb–Liniger model.