Quantum Monte Carlo study of the Bose-polaron problem in a one-dimensional gas with contact interactions

We present a theoretical study based upon quantum Monte Carlo methods of the Bose polaron in one-dimensional systems with contact interactions. In this instance of the problem of a single impurity immersed in a quantum bath, the medium is a Lieb-Liniger gas of bosons ranging from the weakly interacting to the Tonks-Girardeau regime, whereas the impurity is coupled to the bath via a different contact potential, producing both repulsive and attractive interactions. Both the case of a mobile impurity, having the same mass as the particles in the medium, and the case of a static impurity with infinite mass are considered. We make use of numerical techniques that allow us to calculate the ground-state energy of the impurity, its effective mass, and the contact parameter between the impurity and the bath. These quantities are investigated as a function of the strength of interactions between the impurity and the bath and within the bath. In particular, we find that the effective mass rapidly increases to very large values when the impurity gets strongly coupled to an otherwise weakly repulsive bath. This heavy impurity hardly moves within the medium, thereby realizing the “self-localization” regime of the Landau-Pekar polaron. Furthermore, we compare our results with predictions of perturbation theory valid for weak interactions and with exact solutions available when the bosons in the medium behave as impenetrable particles.

Figure 1

Binding energy (in units of the scale ${\epsilon }_F$) of the mobile impurity with mass ratio $w=1$ as a function of the impurity-boson interaction parameter $\eta$ and for different values of the coupling strength $\gamma$ within the bath. (a) Negative and (b) positive values of $\tilde{g}$; in (a) we subtracted from $\mu$ the contribution ${\mu }_d=-\frac{{\eta }^2}{2{\pi }^2}$ from the dimer bound state. The $\gamma =\infty$ curve corresponds to the exact result of McGuire [Eq. (10)] in the TG limit. Dotted lines refer instead to the result of perturbation theory given in Eq. (7).

Figure 2

Binding energy (in units of the scale ${\epsilon }_F$) of the static impurity with mass ratio $w=0$ as a function of the impurity-boson interaction parameter $\eta$ and for different values of the coupling strength $\gamma$ within the bath. (a) Negative and (b) positive values of $\tilde{g}$; in (a) we subtracted from $\mu$ the contribution ${\mu }_d=-\frac{{\eta }^2}{4{\pi }^2}$ from the dimer bound state. The $\gamma =\infty$ curve corresponds to the exact result of Eq. (13) in the TG limit. Dotted lines refer to the result of perturbation theory given in Eq. (9), and dashed lines to the dark-soliton excitation energy, (14).

Figure 3

Inverse binding energy (in units of the scale ${\epsilon }_F$) of the mobile impurity with $w=1$ as a function of the parameter $\gamma$ and for the fixed value $\eta =-1$ of the impurity-boson coupling constant. The line is a linear fit to the data and the shaded region shows the statistical uncertainty of the fit. The dashed line is the prediction from perturbation theory given in Eq. (7).

Figure 4

Inverse effective mass of the mobile impurity with mass ratio $w=1$ as a function of the impurity-boson interaction parameter $\eta$ and for different values of the coupling strength $\gamma$ within the bath. Values corresponding to both $\eta$ positive and $\eta$ negative are shown. The $\gamma =\infty$ curve corresponds to the exact results of McGuire for Eqs. (11) and (12) in the TG limit. Dotted lines refer to the result of perturbation theory given in Eq. (8).

Figure 5

Normalized density profile $\frac{n\left(x\right)}{n}$ of the bath as a function of the distance from the repulsive mobile impurity with mass ratio $w=1$. Distances are in units of the inverse 1D density $1/n$. Different curves correspond to different values of the parameter $\eta$ characterizing the strength of impurity-bath repulsive interactions. Values of the coupling constant within the bath: (a) $\gamma =\infty$, (b) $\gamma =2$, and (c) $\gamma =0.02$.

Figure 6

Normalized density profile $\frac{n\left(x\right)}{n}$ of the bath as a function of the distance from the attractive mobile impurity with mass ratio $w=1$. Distances are in units of the inverse 1D density $1/n$. Different curves correspond to different values of the parameter $\left|\eta \right|$ characterizing the strength of impurity-bath attractive interactions. Values of the coupling constant within the bath: (a) $\gamma =\infty$, (b) $\gamma =2$, and (c) $\gamma =0.02$.

Figure 7

Contact parameter $C$ [see Eq. (15)] of a mobile impurity with mass ratio $w=1$ as a function of the impurity-boson interaction parameter $\eta$ and for different values of the coupling strength $\gamma$ within the bath. Values corresponding to both $\eta$ positive and $\eta$ negative are shown. The solid line corresponds to the exact result, (16), holding in the TG limit. Dotted lines refer to the perturbative result holding for small values of $\eta$.