Bose-Einstein-condensate polaron in harmonic trap potentials in the weak-coupling regime: Lee-Low-Pines–type approach

We have calculated the zero-temperature binding energy of a single impurity atom immersed in a Bose-Einstein condensate (BEC) of ultracold atoms. The impurity and the condensed atoms are trapped in the respective axially symmetric harmonic potentials, where the impurity interacts with bosonic atoms in the condensate via low-energy $s$-wave scattering. In this case, bosons are excited around the impurity to form a quasiparticle, namely, a BEC polaron. We have developed a variational method, a la Lee-Low-Pines (LLP), for description of the polaron that has a conserved angular momentum around the symmetric axis. We find from numerical results that the binding between the impurity and the excited bosons breaks the degeneracy of the impurity energy with respect to the total angular momentum of the polaron. The angular momentum is partially shared by the excited bosons in a manner that is similar to the drag effect on the polaron momentum by a phonon cloud in the LLP theory for the electron-phonon system.

Figure 1

Mean-field energy (33) and interaction energy (34), plotted as a function of ${\overline{n}}_t$. The number affixed to each bar denotes the value of $|\overline{m}|$.

Figure 2

Drag parameter (31) and the number of excited bosons (40), plotted as a function of ${\overline{n}}_t$. The number affixed to each bar denotes the value of $|\overline{m}|$.

Figure 3

The polaron's binding energy (41) and the number of excited bosons (40) for $\overline{n}=0,1,2$, and ${\overline{n}}_t=0,1,2,3,4,5$, plotted as a function of the inverse scattering length ${\left(a_{bf}{k}_{\mathrm{ref}}\right)}^{-1}=-4.813,-4,-3,-2$. Indication of the quantum numbers $({\overline{n}}_t,\overline{m})$ for each bar is omitted.

Figure 4

The polaron's binding energy (41) and the number of excited bosons (40) for the boson mass of $m_b$ and $m_b/4,{\left(a_{bf}{k}_{\mathrm{ref}}\right)}^{-1}=-4.813,\overline{n}=0$, plotted as a function of ${\overline{n}}_t$. The number affixed to each bar denotes the value of $|\overline{m}|$.