  We study $B_d$ and $B_s$ mixing in unquenched lattice QCD employing the MILC
collaboration gauge configurations that include u, d, and s sea quarks based on
the improved staggered quark (AsqTad) action and a highly improved gluon
action. We implement the valence light quarks also with the AsqTad action and
use the nonrelativistic NRQCD action for the valence b quark. We calculate
hadronic matrix elements necessary for extracting CKM matrix elements from
experimental measurements of mass differences $\Delta M_d$ and $\Delta M_s$. We
find $\xi = f_{B_s} \sqrt{\hat{B}_{B_s}} / f_{B_d} \sqrt{\hat{B}_{B_d}} =
1.258(33)$, $f_{B_d} \sqrt{\hat{B}_{B_d}} = 216(15)$ MeV and $f_{B_s}
\sqrt{\hat{B}_{B_s}} = 266(18)$ MeV. We also update previous results for decay
constants and obtain $f_{B_d} = 190(13)$ MeV, $f_{B_s} = 231(15)$ MeV and
$f_{B_s}/f_{B_d} = 1.226(26)$. The new lattice results lead to updated values
for the ratio of CKM matrix elements $|V_{td}|/|V_{ts}|$ and for the Standard
Model prediction for $Br(B_s \rightarrow \mu^+ \mu^-)$ with reduced errors. We
determine $|V_{td}|/|V_{ts}| = 0.214(1)(5)$ and $Br(B_s \rightarrow \mu^+
\mu^-) = 3.19(19) \times 10^{-9}$.
