  We study spontaneous symmetry breaking (SSB), Josephson oscillation, and
self-trapping in a stable, mobile, three-dimensional matter-wave spherical
quantum ball self-bound by attractive two-body and repulsive three-body
interactions. The SSB is realized by a parity-symmetric (a) one-dimensional
(1D) double-well potential and (b) a 1D Gaussian potential, both along the $z$
axis and no potential along the $x$ and $y$ axes. In the presence of each of
these potentials, the symmetric ground state dynamically evolves into a
doubly-degenerate SSB ground state. If the SSB ground state in the double well,
predominantly located in the first well ($z>0$), is given a small displacement,
the quantum ball oscillates with a self-trapping in the first well. For a
medium displacement one encounters an asymmetric Josephson oscillation. The
asymmetric oscillation is a consequence of SSB. The study is performed by a
variational and numerical solution of a non-linear mean-field model with 1D
parity-symmetric perturbations.
