Let 0<θ<1 be an irrational number with continued fraction expansion θ=[a1,a2,a3,…], and consider the quadratic polynomial Pθ:z↦e2πiθz+z2. By performing a trans-quasiconformal surgery on an associated Blaschke product model, we prove that if
then the Julia set of Pθ is locally connected and has Lebesgue measure zero. In particular, it follows that for almost every 0<θ<1, the quadratic Pθ has a Siegel disk whose boundary is a Jordan curve passing through the critical point of Pθ. By standard renormalization theory, these results generalize to the quadratics which have Siegel disks of higher periods.