Effective simultaneous rational approximation to pairs of real quadratic numbers

Let $\xi ,\zeta$ be quadratic real numbers in distinct quadratic fields. We establish the existence of effectively computable, positive real numbers $\tau$ and $c$, such that, for every integer $q$ with $q>c$ we have $$\max \{\mathrm{\parallel }q\xi \mathrm{\parallel },\mathrm{\parallel }q\zeta \mathrm{\parallel }\}>q^{-1+\tau },$$ where $\mathrm{\parallel }\cdot \mathrm{\parallel }$ denotes the distance to the nearest integer.