$N_B=1$. []{data-label="abb:den3-1"}](Fig2.eps){width="49.00000%"}

Single-Particle Density Evolution {#one_body_par_imb}
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To visualize the nonequilibrium dynamics of the particle imbalanced FF mixture on the one-body level we employ the single-particle density evolution $\rho_\sigma^{(1)}(x;t)$ for each of the species after the quench. Focusing on the HF approximation, see Figs. \[abb:den3-1\] (a) and (b), we observe that an overall breathing mode [@abraham2012quantum; @abraham2014quantum; @abraham2014quantum] of both fermionic clouds takes place manifested as a contraction and expansion dynamics of $\rho_\sigma^{(1)}(x;t)$. The frequency of this breathing mode is $\omega_{br}=0.2=2\omega$ which is in accordance with the corresponding theoretical prediction [@abraham2014quantum]. Most importantly, a phase separation process between the two species occurs and each $\rho_\sigma^{(1)}(x;t)$ exhibits an intrawell fragmentation. This phase separation is a consequence of the Stoner instability that exists in this strongly interacting regime even in the ground state of the system \[see also our discussion in Sec. \[in\_par\_imb\].\] Regarding the intrawell fragmentation we observe that $\rho_A^{(1)}(x;t)$ forms two filaments in each well, while $\rho_B^{(1)}(x;t)$ exhibits one filament in each well and one (of lower amplitude) located at the position of the barrier of the double-well \[see also the dashed rectangles in Figs. \[abb:den3-1\] (a) and (b)\]. In sharp contrast to the above, utilizing the correlated approach the 