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 $Br(B_s rightarrow mu^+ mu^-) = 3.19(19) times 10^{-9}$.
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