Unified theory of attractive and repulsive polarons in one-dimensional Bose gas

Abstract

We present a unified description of attractive and repulsive polarons, formed in a one-dimensional Bose gas hosting an impurity particle, by obtaining all ground and excited state solutions to the Gross-Pitaevskii equation. Modeling the impurity with an attractive square-well potential, we characterize the excited-state energy branches as a function of interaction strength. As the impurity-bath coupling increases, the excited states change from distinct soliton configurations to hybridized soliton-polaron states, eventually crossing over from repulsive to attractive polarons at unitarity. We identify a universal regime near this crossover where the polaron properties are accurately characterized by the zero-energy scattering length.

Figures (3)

• Figure 1: Energy branches for an attractive contact impurity, given as a function of the coupling strength $mG\xi$ and obtained for the ground-state (red) and excited-state (gray and blue) solutions of the Gross–Pitaevskii equation Eq. \ref{['eq:GPl']}.
• Figure 2: Energy branches for an attractive square well potential as a function of coupling constant $g$. The insets display the density profiles $|\phi(x>0)|^2$ for the evolution of the ground-state (red solid), first odd-parity excited state (gray solid), and first even-parity excited state (blue solid) solutions to the GPe \ref{['eq:LGPe0inText']}. The dashed lines indicate the regimes where the energy of the solution can be described by the universal expansions for polaronic Eq. \ref{['eq:dpwc']} or solitonic Eq. \ref{['eq:dpfc']} configurations with the correct choice of $a_s$ and $a_p$ in Eq. \ref{['eq:asap']} for the even or odd states.
• Figure 3: Repulsive-to-attractive polaron crossover near the first unitary point of the odd-parity excited state for an impurity potential of (a) a square well and of (b) an exponential potential. Solid lines show the exact solutions of the GPe in Eq. \ref{['eq:LGPe0inText']}, while dashed lines correspond to solutions of the zero-energy Schrödinger equation.