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Eigendecomposition problem?

Hi! I'm having some trouble with this. In the code I define two
matrices, then calculate its eigenvalues/eigenvectors, and finally I
plot the absolute values of the components of the eigenvectors. The
results are consistent with (I'm quite sure) the analytical ones but
only below "t approx 30". Above this value, (only) the first plot
corresponding to one of the vectors shows some strange behaviour.

Any idea? Thanks!

    This is the code:

Clear[a, b, w, g, t]
\[Rho]t = {{a, b*Exp[-I*w*t - g*t]}, {b\[Conjugate]*Exp[I*w*t - g*t],
    1 - a}};
{e1, e2} = Eigenvalues[\[Rho]t];
{v1, v2} = Eigenvectors[\[Rho]t];

a = 0.6`1000; b = 1`1000 + 2`1000*I; w = 1`1000; g = 1`1000;
v1n = Normalize[N[v1, 1000]]; v2n = Normalize[N[v2, 1000]];
Abs1 = {Abs[v1n[[1]]], Abs[v1n[[2]]]};
Abs2 = {Abs[v2n[[1]]], Abs[v2n[[2]]]};

Plot[Abs1, {t, 0, 50}, Filling -> Bottom]
Plot[Abs2, {t, 0, 50}, Filling -> Bottom]

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