  In this study, we conducted a detailed investigation into the time evolution
of the probability density within a 1D double-well potential hosting a
Bose-Fermi mixture. This system comprised spinless bosons and spin one-half
fermions with weak repulsive contact interactions. Notably, even at very low
effective coupling constants, periodic probabilities were observed, indicating
correlated tunneling of both bosons and fermions, leading to complete
miscibility, which disappears when an external electric field is turned on. The
electric field accentuated fermion-fermion interactions due to the Pauli
exclusion principle, altering both boson density and interactions and leading
to spatial redistribution of particles. These findings underscore the complex
interplay between interactions, external fields, and spatial distributions
within confined quantum systems.
  Our exploration of higher interaction strengths revealed conditions under
which probability density functions are decoupled. Furthermore, we observed
that increased fermion interaction, driven by the electric field, led to higher
tunneling frequencies for both species because of the repulsive nature of the
boson-fermion interaction. Conversely, increased boson-boson interaction
resulted in complete tunneling of both species, especially when boson density
was high, leading to effective fermion repulsion. Expanding our analysis to
scenarios involving four bosons demonstrated that higher interaction values
corresponded to increased oscillation frequencies in tunneling probabilities.
Finally, by manipulating interaction parameters and activating the electric
field, we achieved complete tunneling of both species, further increasing
oscillation frequencies and resulting in intervals characterized by overlapping
probability functions.
