  Let $A=(a_1,\ldots,a_n)$ be a vector of integers with $d=\sum_{i=1}^n a_i$.
By partial resolution of the classical Abel-Jacobi map, we construct a
universal twisted double ramification cycle $\mathsf{DR}^{\mathsf{op}}_{g,A}$
as an operational Chow class on the Picard stack $\mathfrak{Pic}_{g,n,d}$ of
$n$-pointed genus $g$ curves carrying a degree $d$ line bundle. The method of
construction follows the log (and b-Chow) approach to the standard double
ramification cycle with canonical twists on the moduli space of curves
[arXiv:1707.02261, arXiv:1711.10341, arXiv:1708.04471].
  Our main result is a calculation of $\mathsf{DR}^{\mathsf{op}}_{g,A}$ on the
Picard stack $\mathfrak{Pic}_{g,n,d}$ via an appropriate interpretation of
Pixton's formula in the tautological ring. The basic new tool used in the proof
is the theory of double ramification cycles for target varieties
[arXiv:1812.10136]. The formula on the Picard stack is obtained from
[arXiv:1812.10136] for target varieties $\mathbb{CP}^n$ in the limit $n
\rightarrow \infty$. The result may be viewed as a universal calculation in
Abel-Jacobi theory.
  As a consequence of the calculation of $\mathsf{DR}^{\mathsf{op}}_{g,A}$ on
the Picard stack $\mathfrak{Pic}_{g,n,d}$, we prove that the fundamental
classes of the moduli spaces of twisted meromorphic differentials in
$\overline{\mathcal{M}}_{g,n}$ are exactly given by Pixton's formula (as
conjectured in the appendix to [arXiv:1508.07940] and in [arXiv:1607.08429]).
The comparison result of fundamental classes proven in [arXiv:1909.11981] plays
a crucial role in our argument. We also prove the set of relations in the
tautological ring of the Picard stack $\mathfrak{Pic}_{g,n,d}$ associated to
Pixton's formula.
