Impurity in a Bose-Einstein condensate: Study of the attractive and repulsive branch using quantum Monte Carlo methods

We investigate the properties of an impurity immersed in a dilute Bose gas at zero temperature using quantum Monte Carlo methods. The interactions between bosons are modeled by a hard-sphere potential with scattering length $a$, whereas the interactions between the impurity and the bosons are modeled by a short-range, square-well potential where both the sign and the strength of the scattering length $b$ can be varied by adjusting the well depth. We characterize the attractive and the repulsive polaron branch by calculating the binding energy and the effective mass of the impurity. Furthermore, we investigate the structural properties of the bath, such as the impurity-boson contact parameter and the change of the density profile around the impurity. At the unitary limit of the impurity-boson interaction, we find that the effective mass of the impurity remains smaller than twice its bare mass, while the binding energy scales with ${\hslash }^2{n}^{2/3}/m$, where $n$ is the density of the bath and $m$ is the common mass of the impurity and the bosons in the bath. The implications for the phase diagram of binary Bose-Bose mixtures at low concentrations are also discussed.

Figure 1

Polaron energy $\mu$ as a function of the ratio $a/b$ of scattering lengths for both the repulsive and the attractive branch. The gas parameter of the bosonic bath is $n{a}^3={10}^{-5}$. Symbols are the DMC results obtained with the following impurity-boson interaction potentials: hard sphere [(blue) squares]; square well with $a/R_0=5$ [(green) circles]; and square well with $a/R_0=20$ [(red) diamonds]. The dashed line is the result, (13), of perturbation theory for the two branches. Inset: Polaron energy along the attractive branch on the positive side of the resonance value for the impurity-boson scattering length. The solid line corresponds to the binding energy ${\epsilon }_b$ in the square-well potential with $a/R_0=5$.

Figure 2

Polaron energy $\mu$ with the mean-field contribution ${\mu }_{\mathrm{MF}}$ subtracted as a function of the ratio $a/b$. Symbols are as in Fig. 1. The dashed line is the second-order contribution to the perturbation expansion, (13) (second term in parentheses). Note that the results for $a/b>0$ refer to the repulsive branch only.

Figure 3

Effective mass of the polaron as a function of the ratio $a/b$ for the repulsive [(blue) circles] and attractive [(green) circles] branches. The gas parameter is given by $n{a}^3={10}^{-5}$. The dashed line corresponds to the perturbation expansion, (14).

Figure 4

Density profile of the bath surrounding the impurity for $b/a=\pm 10$. Inset: Integrated number of particles of the bath at a distance $r$ from the impurity.

Figure 5

Density profile of the bath surrounding the impurity for $b/a=\pm 20$. Inset: Integrated number of particles of the bath at a distance $r$ from the impurity.

Figure 6

Density profile of the bath surrounding the impurity for $b/a=\pm 30$. Inset: Integrated number of particles of the bath at a distance $r$ from the impurity.

Figure 7

Contact parameter, Eq. (28), as a function of the ratio $a/b$ for the repulsive [(blue) circles] and attractive [(green) circles] branches. The gas parameter is given by $n{a}^3={10}^{-5}$. Lines correspond to the determination of $C$ from the attractive and repulsive branches of the equation of state [see Eq. (29)].

Figure 8

Polaron binding energy at unitarity $\left(a/b=0\right)$ in units of $\frac{{\hslash }^2{n}^{2/3}}{m}$ as a function of the gas parameter of the bath.

Figure 9

Effective mass of the polaron at unitarity $\left(a/b=0\right)$ as a function of the gas parameter of the bath.

Figure 10

Density profile of the bath surrounding the impurity at unitarity $\left(a/b=0\right)$ for three values of the gas parameter. Inset: Contact parameter $C$ at unitarity as a function of ${\left(n{a}^3\right)}^{1/3}$.

Figure 11

Energy of clusters with $N+1$ particles at the unitary point $\left(a/b=0\right)$ as a function of the number $N$ of bosons. Energies are in units of $\frac{{\hslash }^2}{2m{a}^2}$. The value at $N=1$ refers to the two-body binding energy. Inset: Same as the figure, but at $a/b=0.1$.

Figure 12

Energy shift between a system with a given concentration $x$ of impurities and a bath without impurities as a function of the ratio of scattering lengths along the repulsive branch. The density of the bath is $n{a}^3={10}^{-5}$ and results are shown for three concentrations. Solid lines refer to the perturbation result $E_0-E_B$ from Eq. (33).

Figure 13

Energy shift $\mathrm{\Delta }E(M,N)$ as a function of the concentration of impurities for the two values $\frac{b}{a}=\pm 5$ of the ratio of scattering lengths. The density of the bath is as in Fig. 12. Dashed lines refer to the perturbation result $E_0-E_B$ from Eq. (33).