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.
