Analytical approach to the Bose-polaron problem in one dimension

We discuss the ground-state properties of a one-dimensional bosonic system doped with an impurity (the so-called Bose polaron problem). We introduce a formalism that allows us to calculate analytically the thermodynamic zero-temperature properties of this system with weak and moderate boson-boson interaction strengths for any boson-impurity interaction. Our approach is validated by comparison to exact quantum Monte Carlo calculations. In addition, we test the method in finite-size systems using numerical results based upon the similarity renormalization group. We argue that the introduced approach provides a simple analytical tool for studies of strongly interacting impurity problems in one dimension.

Figure 1

The solid lines show the energy of the impurity $\epsilon /{\rho }^2$ from Eq. (11) as a function of $c/\rho$ for $\kappa =1$ (an infinitely heavy impurity), $\gamma =0.02,0.2$ and 4 (from the bottom to the top). The points are the corresponding results of Ref. [30].

Figure 2

The solid lines show the energy of the impurity $\epsilon /{\rho }^2$ from Eq. (11) as a function of $c/\rho$ for $\kappa =1/2$ ($m=M$), $\gamma =0.02$, and 0.2 (from the bottom to the top). The points are the corresponding results of Ref. [30]. The solid lines in the inset present the contact parameter $C$ defined in Eq. (12) as a function of $c/\rho$ for $m=M$ and $\gamma =2$ (the upper curve) and $\gamma =0.02$ (the lower curve). The dots are the numerical results of Ref. [30].

Figure 3

The energy of the impurity $\epsilon$ minus the thermodynamic value ${\epsilon }_{\infty }=\sqrt{16\gamma /\left(9\kappa \right)}$ as a function of the particle number $N$, for $\gamma =0.1$ and for $\gamma =0$, in both cases $\kappa =1/2$ ($m=M$). The solid (red) curves depict Eq. (15). The points are calculated numerically using Eq. (17). The dashed lines are to guide the eye. The inset shows $\epsilon$ as a function of $\gamma$ for $N=15$. The solid line is from Eq. (15); points are the numerical results.