  Thanks to common-mode noise rejection, differential configurations are
crucial for realistic applications of phase and frequency estimation with atom
interferometers. Currently, differential protocols with uncorrelated particles
and mode-separable settings reach a sensitivity bounded by the standard quantum
limit (SQL). Here we show that differential interferometry can be understood as
a distributed multiparameter estimation problem and can benefit from both mode
and particle entanglement. Our protocol uses a single spin-squeezed state that
is mode-swapped among common interferometric modes. The mode swapping is
optimized to estimate the differential phase shift with sub-SQL sensitivity.
Numerical calculations are supported by analytical approximations that guide
the optimization of the protocol. The scheme is also tested with simulation of
noise in atomic clocks and interferometers.
