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 Schrodinger 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.
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