  The out-of-equilibrium quantum dynamics of an interacting Bose gas trapped in
a 1D asymmetric double-well potential is studied by solving the many-body
Schr\"odinger equation numerically accurately. We examine how the loss of
symmetry of the confining trap affects the macroscopic quantum tunneling
dynamics of the system between the two wells. In an asymmetric DW, the two
wells are not equivalent anymore -the left well is deeper than the right one.
Accordingly, we analyze the dynamics by initially preparing the condensate in
both the left and the right well. We examined the frequencies and amplitudes of
the oscillations of the survival probabilities, the time scale for the
development of fragmentation and its degree, and the growth and oscillatory
behavior of the many-body position and momentum variances. There is an overall
suppression of the oscillations of the survival probabilities in an asymmetric
double well. However, depending on whether the condensate is initially prepared
in the left or right well, the repulsive inter-atomic interactions affect the
survival probabilities differently. The degree of fragmentation depends both on
the asymmetry of the trap and the initial well in which the condensate is
prepared in a non-trivial manner. Overall, the many-body position and momentum
variances bear the prominent signatures of the density oscillations of the
system in the asymmetric double well as well as a breathing-mode oscillation.
Finally, a universality of fragmentation for systems made of different numbers
of particles but the same interaction parameter is also found. The phenomenon
is robust despite the asymmetry of the junction and admits a
macroscopically-large fragmented condensate characterized by a diverging
many-body position variance.
