  We theoretically investigate quantum-mechanical dynamics of
quasi-one-dimensional boson-fermion mixtures of atomic gases trapped in a
toroidal potential, where effective inter-atomic interactions are tunable and
affect the dynamics. We especially focus on effects of quantum statistics and
many-body correlations beyond the Hartree-Fock (HF) mean-field approximation on
the dynamics. In order to predict the dynamics, we utilize the numerical exact
diagonalization method and also reproduce the calculation in the HF
approximation for comparison. The toroidal gases originally have a rotational
symmetry in the toroidal direction. We firstly prepare a deformed ground state
as an initial state by adding a weak potential deformed in the toroidal
direction, and then remove the potential to start the dynamics. In the
dynamics, number densities of the deformed gases exhibit oscillations as
demonstrated in the present paper. As a result, we find out that the bosons and
fermions show quite different behaviors owing to quantum statistics. In
particular, the bosons exhibit a low-frequency oscillation in the strong
boson-boson attraction regime owing to the many-body correlations, and it can
not be reproduced in the HF approximation. The oscillation of the fermions is
strongly influenced by that of the bosons through the boson-fermion interaction
as a forced oscillator. In addition, we also discuss a relationship between the
low-frequency oscillation and restoration of the broken symmetry.
