  Ultracold polyatomic molecules are promising candidates for experiments in
quantum science, quantum sensing, ultracold chemistry, and precision
measurements of physics beyond the Standard Model. A key, yet unrealized,
requirement of these experiments is the ability to achieve full quantum control
over the complex internal structure of the molecules. Here, we establish
coherent control of individual quantum states in a polyatomic molecule, calcium
monohydroxide (CaOH), and use these techniques to demonstrate a method for
searching for the electron electric dipole moment (eEDM). Optically trapped,
ultracold CaOH molecules are prepared in a single quantum state, polarized in
an electric field, and coherently transferred into an eEDM sensitive state
where an electron spin precession measurement is performed. To extend the
coherence time of the measurement, we utilize eEDM sensitive states with
tunable, near-zero magnetic field sensitivity. The spin precession coherence
time is limited by AC Stark shifts and uncontrolled magnetic fields. These
results establish a path for eEDM searches with trapped polyatomic molecules,
towards orders-of-magnitude improved experimental sensitivity to
time-reversal-violating physics.
