Repulsive-to-attractive interaction quenches of a one-dimensional Bose gas in a harmonic trap

We consider quantum quenches of harmonically trapped one-dimensional bosons from repulsive-to-attractive interactions. We study the resulting breathing dynamics of the so-called ``super-Tonks-Girardeau'' (STG) state at large attraction, or of its analog at moderate or small attraction. This state is highly excited compared to the ground state of the attractive gas, and is the lowest eigenstate where the particles are not bound or clustered. We analyze the dynamics from a spectral point of view, identifying the relevant eigenstates of the interacting trapped many-body system, and analyzing the nature of these quantum eigenstates. To obtain explicit eigenspectra, we use Hamiltonians with finite-dimensional Hilbert spaces to approximate the Lieb-Liniger system. We employ two very different approximate approaches: an expansion in a truncated single-particle harmonic-trap basis and a lattice (Bose-Hubbard) model. We show how the breathing frequency, identified with the energy difference between the sTG state and another particular eigenstate, varies with interaction.

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