Dispersion relation of a polaron in the Yang-Gaudin Bose gas

Abstract

We study a one-dimensional Bose gas with two internal states described by the Yang-Gaudin model and calculate analytically the dispersion relation of a polaron quasiparticle, which is the lowest excitation branch. We study the dispersion in the thermodynamic limit in the regimes of weak and strong interaction without limitations on the momentum. At weak interaction the polaron dispersion is in the vicinity of the dark soliton one; we calculate the leading deviation in the parametric form. At strong interaction we find an ansatz for the explicit form of the polaron dispersion. It has the form of a power series of the sine function of the momentum with interaction dependent coefficients. By increasing the power of the series, the corresponding coefficients show faster decay and thus one practically needs only a few of them; we give the results for the first three. The coefficients of the series are connected to Maclaurin series of the polaron dispersion and thus can be obtained from the latter quantity. The derived results for the dispersion can be used to obtain explicit expressions for the exponents of the power-law singularities in the response functions at the spectral edge.

Polaron-Depleton Transition in the Yrast Excitations of a One-Dimensional Bose Gas with a Mobile Impurity

“…Although the dynamics of impurities have also been actively studied [28,29,36,41], limited understanding has been achieved on the full dispersion relation of a Bose gas coupled with a mobile impurity. There are analytical results on the dispersion relations restricted to specific models, such as the Yang-Gaudin model [42], and the Luttinger liquid [43].…”