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Single-atomic impurities immersed in a dilute Bose gas in the sphericallysymmetric harmonic trap potentials are studied at zero temperature. In order tofind the ground state of the polarons, we present a conditional variationalmethod with fixed expectation values of the total angular momentum operators,$\hat{J}^2$ and $\hat{J}_z$, of the system, using a crankinggauge-transformation for bosons to move them in the frame co-rotating with theimpurity. In the formulation, the expectation value $\langle \hat{J^2}\rangle$is shown to be shared in impurity and bosons, but the value $\langle\hat{J}_z\rangle$ is carried by the impurity due to the rotational symmetry. Wealso analyze the ground-state properties numerically obtained in thisvariational method for the system of the attractive impurity-boson interaction,and find that excited boson distributions around the impurity overlap largelywith impurity's wave function in their quantum-number spaces and also in thereal space because of the attractive interaction employed.

Bose polaron in spherically symmetric trap potentials: Ground states with zero and lower angular momenta