as the natural species populations of the $k$-th species function $\Psi^{\sigma}_k$ of the $\sigma$-species. They serve as a measure of the system’s entanglement or interspecies correlations. Specifically, the system is called entangled or interspecies correlated [@roncaglia2014bipartite] when at least two distinct $\lambda_k(t)$ are nonzero, since in this latter case the total MB state \[Eq. (\[Eq:WF\])\] cannot be expressed as a direct product of two states.

To explicitly incorporate the interparticle correlations each of the species functions $\Psi^{\sigma}_k (\vec x^{\sigma};t)$ is expanded using the determinants of $m^{\sigma}$ distinct time-dependent fermionic single-particle functions (SPFs), $\varphi_1,\dots,\varphi_{m_{\sigma}}$. In particular $$\begin{split}
&\Psi_k^{\sigma}(\vec x^{\sigma};t) = \sum_{\substack{l_1,\dots,l_{m_{\sigma}} \\ \sum l_i=N}} C_{k,(l_1,
    \dots,l_{m_{\sigma}})}(t)\\& \sum_{i=1}^{N_{\sigma}!} {\rm sign}(\mathcal{P}_i) \mathcal{P}_i
 \bigg[ \prod_{\substack{j\in\{1,\dots,m^{\sigma}\} \\ {\rm with}~ l_j=1}} \varphi_j(x_{K(j)};t)\bigg].  
\label{Eq:SPFs}
\end{split}$$ Here, $\mathcal{P}$ refers to the permutation operator which exchanges the particle positions $x_{\mu}$, $\mu=1,\dots,N_{\sigma}$ within the SPFs. Also $K(j)\equiv \sum_{\nu=1}^{j}l_{\nu}$ with $l_{\nu}$ denoting the occupation of the $\nu$th SPF and $j\in\{1,2,\dots,m^{\sigma}\}$. The symbol $\rm{sign}(\mathcal{P}_i)$ denotes the sign of the corresponding permutation and $C_{k,(l_1,\dots,l_{m_{\sigma}})}(t)$ are the time-dependent expansion coefficients of a certain determinant. The eigenfunctions of the one-body reduced density matrix of the $\sigma$-species $\rho_\sigma^{(1)}(x,x^\prime;t)=\langle\Psi_{MB}(t)|\hat{\Psi}^{\sigma,\dagger}(x)\hat{\Psi}^\sigma(x^\prime)|\Psi_{MB}(t)\rangle$ are termed natural orbitals $\phi^{\sigma}_i(x;t)$, where $\hat{\Psi}^{\sigma}(x)$ refers to the fermionic field operator of the $\sigma$-species. The eigenvalues 