We develop an interferometric technique for making time-resolved measurements

of field-quadrature operators for nonequilibrium ultracold bosons in optical

lattices. The technique exploits the internal state structure of magnetic atoms

to create two subsystems of atoms in different spin states and lattice sites. A

Feshbach resonance turns off atom-atom interactions in one spin subsystem,

making it a well-characterized reference state, while atoms in the other

subsystem undergo nonequilibrium dynamics for a variable hold time. Interfering

the subsystems via a second beam-splitting operation, time-resolved quadrature

measurements on the interacting atoms are obtained by detecting relative spin

populations. The technique can provide quadrature measurements for a variety of

Hamiltonians and lattice geometries (e.g., cubic, honeycomb, superlattices),

including systems with tunneling, spin-orbit couplings using artificial gauge

fields, and higher-band effects. Analyzing the special case of a deep lattice

with negligible tunneling, we obtain the time evolution of both quadrature

observables and their fluctuations. As a second application, we show that the

interferometer can be used to measure atom-atom interaction strengths with

super-Heisenberg scaling n^(-3/2) in the mean number of atoms per lattice site

n, and standard quantum limit scaling M^(-1/2) in the number of lattice sites

M. In our analysis, we require M >> 1 and for realistic systems n is small, and

therefore the scaling in total atom number N = nM is below the Heisenberg

limit; nevertheless, measurements testing the scaling behaviors for

interaction-based quantum metrologies should be possible in this system.