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We present a Gaudin-like determinant expression for overlaps of $q$-raisedN\'eel states with Bethe states of the spin-1/2 XXZ chain in the non-zeromagnetization sector. The former is constructed by applying global $U_q(sl_2)$spin raising operators to the N\'eel state, the ground state of theantiferromagnetic Ising chain. The formulas presented are derived fromrecently-obtained results for the overlap of the N\'eel state with XXZ Bethestates. The determinants as well as their prefactors can be evaluated in thescaling limit of the XXZ spin chain to the Lieb-Liniger Bose gas. Within thislimit a $q$-raised N\'eel state that contains finitely many down spinscorresponds to the ground state of free bosons. This allows for a rigorousproof of the BEC Lieb-Liniger overlap formula for an arbitrary number ofparticles.

Overlaps of $q$-raised N\'eel states with XXZ Bethe states and their relation to the Lieb-Liniger Bose gas