Quantum correlation of two non-interaction spinless particles: exact solution of the time-dependent Schrodinger equation

I would like to get some comment about the following obtained result.

I consider the one-dimensional time-dependent Schrodinger equation for two non-interaction particles with masses $m_1$ and $m_2$:
$$
i hbar frac{partial Psi}{partial t} = – frac{hbar^2}{2m_1} frac{partial^2 Psi}{partial x_1^2} – frac{hbar^2}{2m_2} frac{partial^2 Psi}{partial x_2^2}.
$$
Let us assume that the wave-function $Psi(x_1, x_2, t)$ has the gaussian form at the initial instant $t=0$:
$$
Psi(x_1,x_2,0) = frac{1}{sqrt{2pi sigma_1 sigma_2},(1-r^2)^{1/4}}, exp left[ – frac{1}{1-r^2} left( frac{x_1^2}{4sigma_1^2} + frac{(x_2-x_0)^2}{4sigma_2^2} – frac{r x_1 (x_2 – x_0)}{2sigma_1sigma_2}right) + frac{i}{hbar}, p_0 x_2 right].
$$
In my understanding, it describes the situation when the first particle is placed at the origin, while the second one is moving along $x$-axis with the velocity $v_0 = p_0/m_2$. Here $sigma_1$ and $sigma_2$ are the initial standard deviations of the particle coordinates, and $r$ is the initial correlation coefficient, $|r|<1$. Thus, if $r neq 0$, then we have a two-particle entangled state at $t=0$. Without loss of generality, we will assume that $r>0$.

I have exactly solved the time-dependent Schrodinger equation with the given initial condition and obtained an explicit form of the wave-function $Psi(x_1,x_2,t)$ (the corresponding expression is rather cumbersome and I do not write it here). Then I have calculated the time-dependent correlation coefficient:
$$
r_t := frac{langle x_1 x_2 rangle – langle x_1 rangle langle x_2 rangle}{delta x_1 delta x_2} = frac{r sigma_1 sigma_2 – frac{r hbar^2 t^2}{4 m_1 m_2 (1-r^2) sigma_1 sigma_2}}{sqrt{sigma_1^2 + frac{hbar^2 t^2}{4 m_1^2 (1-r^2)sigma_1^2}} , sqrt{sigma_2^2 + frac{hbar^2 t^2}{4 m_2^2 (1-r^2) sigma_2^2}}},
$$
where $langle hat{A} rangle := int_{-infty}^{+infty} int_{-infty}^{+infty} overline{Psi(x_1, x_2, t)} , hat{A} Psi(x_1, x_2, t) dx_1 dx_2$, $delta x_i = sqrt{langle x_i^2 rangle – langle x_i rangle^2}$, $i=1,2$. An analysis of this expression shows that: 1) the function $r_t$ has a maximum at $t=0$, which equals $r$; 2) for $t>0$ the function $r_t$ monotonically decreases and asymptotically approaches the value $r_{infty} = -r$; 3) the function $r_t$ vanishes at the instant $t_0 = 2 sigma_1 sigma_2 sqrt{m_1 m_2(1-r^2)}/hbar$. I would like to get comments on these results in the physical context. Especially, what means the change of sign for the correlation coefficient? I would be grateful for any clarifications and references.