Dynamics of quantum double dark-solitons and an exact finite-size scaling of Bose-Einstein condensation

We show several novel aspects in the exact non-equilibrium dynamics of quantum double dark-soliton states in the Lieb-Liniger model for the one-dimensional Bose gas with repulsive interactions. We also show an exact finite-size scaling of the fraction of the quasi-Bose-Einstein condensation (BEC) in the ground state, which should characterize the quasi-BEC in quantum double dark-soliton states that we assume to occur in the weak coupling regime. First, we show the exact time evolution of the density profile in the quantum state associated with a quantum double dark-soliton by the Bethe ansatz. Secondly, we derive a kind of macroscopic quantum wave-function effectively by exactly evaluating the square amplitude and phase profiles of the matrix element of the field operator between the quantum double dark-soliton states. The profiles are close to those of dark-solitons particularly in the weak-coupling regime. Then, the scattering of two notches in the quantum double dark-soliton state is exactly demonstrated. It is suggested from the above observations that the quasi-BEC should play a significant role in the dynamics of quantum double dark-soliton states. If the condensate fraction is close to 1, the quantum state should be well approximated by the quasi-BEC state where the mean-field picture is valid.

Automated summary

We present new findings on the non-equilibrium dynamics of quantum double dark-soliton states in the Lieb-Liniger model for one-dimensional Bose gas. Additionally, we demonstrate the exact finite-size scaling of the quasi-Bose-Einstein condensation (BEC) fraction, which characterizes the quasi-BEC in the weak coupling regime. Our results include the exact time evolution of the density profile, the derivation of a macroscopic quantum wave function, and the demonstration of scattering in the quantum double dark-soliton state. These observations suggest that quasi-BEC plays a significant role in the dynamics of quantum double dark-soliton states.