that the filament formation of both species is washed out for higher particle numbers \[see also the rectangles in Figs. \[abb:den3-1\] (e) and (f)\]. However, the particle number does not significantly alter the breathing frequency of each species and their miscible character.

To expose the degree of spatial phase separation, namely the degree of miscibility or immiscibility of the mixture, occuring on the one-body level during the quench dynamics, we employ the overlap integral function $\Lambda(t)$ [@mistakidis2017correlation; @bandyopadhyay2017dynamics; @jain2011quantum] between the two species $$\begin{aligned}
\Lambda (t)=\frac{\left[\int d x \, \rho_A^{(1)}(x;t)\rho_B^{(1)}(x;t)\right]^2}{\left[\int d x\, (\rho_A^{(1)}(x;t))^2\right]\left[\int d x\, (\rho_B^{(1)}(x;t))^2\right]}.\end{aligned}$$ This quantity being normalized to unity takes values between $\Lambda=0$ and $\Lambda=1$ corresponding to zero and complete spatial overlap of the two species on the single-particle level. Figures \[abb:den3-1\] (g) and (h) present $\Lambda(t)$ for the setups $N_A=3$, $N_B=1$ and $N_A=5$, $N_B=1$ respectively for different interaction quench amplitudes. Regarding the evolution in the HF approximation, $\Lambda(t)$ drops close to zero at short time scales ($t>6$) for both systems. After this initial drop the overlap remains almost constant exhibiting small amplitude oscillations which reflect the breathing motion of each cloud. Notice that the maxima of these small amplitude oscillations appear at time intervals of the 