$\hbar  \omega_{\perp}$. Thus, the corresponding length, time, and interaction strength scales are expressed in terms of $\sqrt{\frac{\hbar}{M \omega_{\perp}}}$, $\omega_{\perp}^{-1}$ and $\sqrt{\frac{\hbar^3 \omega_{\perp}}{M}}$ respectively. Moreover, the amplitude of the Gaussian barrier $V_0$, its width $w$ and the frequency of the harmonic oscillator $\omega$ are given in units of $\sqrt{ \frac{ \hbar^3 \omega_{\perp} }{ M }}$, $\sqrt{\frac{\hbar}{M \omega_{\perp}}}$, and $\omega_{\perp}$. To limit the spatial extension of our system we impose hard-wall boundary conditions at $x_\pm=\pm40$.

Throughout this work, our system is initially prepared in the MB ground state of the Hamiltonian (\[Hamilt\]) within the weak interspecies interaction regime, namely $g_{AB}=0.1$. The corresponding double-well potential is characterized by the harmonic oscillator frequency $\omega=0.1$, barrier height $V_0=2$ and width $w=1$. Thus, in a non-interacting single-particle picture four doublets are included below the maximum of the barrier. To induce the nonequilibrium dynamics of the FF mixture in the double-well we quench at $t=0$ the interspecies interaction strength towards the strongly correlated regime, e.g. $g_{AB}=4.0$, and let the system evolve in time. Quenching the interspecies repulsion towards the strongly interacting regime favors the occurrence of a breathing mode [@mistakidis2017correlation] and the appearance of strong intra- and interspecies correlations \[see Secs. \[sec:particle\_imbalanced\] and \[sec:particle\_balanced\]\] due to the 