(x)\\\hat{\Psi}^\sigma(x')|\Psi(t)\rangle$ denotes the diagonal two-body reduced density matrix which provides the probability of measuring two particles of species $\sigma$ and $\sigma^\prime$ located at $x$ and $x^\prime$ respectively at time $t$. Referring to the same (different) species, i.e. $\sigma=\sigma'$ ($\sigma \not=\sigma'$), $|g^{(2)}_{\sigma \sigma'}(x,x^\prime;t)|$ accounts for the intraspecies (interspecies) two-body correlations. We remark that if $g^{(2)}_{\sigma\sigma\prime}(x,x^\prime;t)=1$ holds, the state is termed fully second order coherent, while in case that $g^{(2)}_{\sigma\sigma\prime}(x,x^\prime;t)>1$ \[$g^{(2)}_{\sigma\sigma\prime}(x,x^\prime;t)<1$\] it is termed correlated \[anti-correlated\]. $g^{(2)}_{\sigma\sigma\prime}(x,x^\prime;t)$ is experimentally accessible by in-situ density-density fluctuation measurements [@tavares2016chaotic; @tsatsos2017granulation; @endres2013single].

Interaction Quench Dynamics of a Particle Imbalanced Mixture {#sec:particle_imbalanced}
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![One-body density $\rho^{(1)}_\sigma(x)$ of the ground state of the $\sigma$-species of a FF mixture for different interspecies repulsions $g_{AB}$ (see legend) within (a) the MB approach and (b) the HF approximation. The mixture consists of $N_A=3$ and $N_B=1$ fermions and it is trapped in a double-well potential. The rectangles in (a) indicate the intrawell fragmentation (filamentation) of $\rho^{(1)}_\sigma(x)$ occuring for strong interspecies interactions. The inset in (b) shows the corresponding energetically degenerate configuration of $\rho^{(1)}_\sigma(x)$ with the $\rho^{(1)}_\sigma(x)$ of the main panel for $g_{AB}=4.0$ within the HF approximation.[]{data-label="abb:gs3-1"}](Fig1.eps){width="48.00000%"}

Initial State {#in_par_imb}
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We consider an interspecies repulsively interacting ($g_{AB}$) mass balanced ($M_A=M_B$) FF mixture 