Analytical study of a finite-range impurity in a one-dimensional Bose gas

One-dimensional Bose gases present an interesting setting to study the physics of Bose polarons, as density fluctuations play an enhanced role due to reduced dimensionality. Theoretical descriptions of this system have predominantly relied on contact pseudopotentials to model the impurity-bath interaction, leading to unphysical results in the strongly coupled limit. In this work, we analytically solve the Gross-Pitaevskii equation, using a square-well potential instead of a zero-range potential, for the ground-state wave function of a static impurity. We compute perturbative corrections arising from infinitesimally slow impurity motion. The polaron energy and effective mass remain finite in the strongly coupled regime, in contrast to the divergent behavior obtained using a contact potential. In this limit, we characterize the polaron properties in terms of the dimensionless ratio $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{w}\ensuremath{\equiv}w/\ensuremath{\xi}$ between the interaction range $w$ of the impurity-bath potential and the coherence length $\ensuremath{\xi}$ of the Bose gas. The effective mass exhibits a $1/\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{w}$ scaling. The energy of the attractive polaron scales as $\ensuremath{-}1/{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{w}}^{3}$, whereas the repulsive polaron features subleading corrections to the dark soliton energy at the order ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{w}}^{3}$.