Given a real symmetric matrix $A$ and a vector $v$ of the same dimension we know that the eigenvalues of $A + vv^T$ are left interlaced by the eigenvalues of $A$.

But do we have any quantitative estimates of the amount of interlacing produced?

Like as a function of $A$ and $v$ if we can say how much will the $k^{th}$ eigenvalue of $A+vv^T$ be ahead of the $k^{th}$ eigenvalue of $A$? (at least under some restrictions about the nature of $A$ and $v$?)