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Dynamics of quantum double dark-solitons and an exact finite-size scaling of Bose-Einstein condensation. (arXiv:2207.00293v2 [cond-mat.quant-gas] UPDATED)

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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 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.

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