  We study the evolution of a Bose-Einstein Condensate (BEC) in a two-state
superposition due to inter-state interactions. Using a population imbalanced
dynamic decoupling scheme, we measure inter-state interactions while canceling
intra-state density shifts and external noise sources. Our measurements show
low statistical uncertainties for both magnetic sensitive and insensitive
superpositions, indicating that we successfully decoupled our system from
strong magnetic noises. We experimentally show that the Bloch sphere
representing general superposition states is "twisted" by inter-state
interactions, and that the twist rate depends on the difference between
inter-state and intra-state scattering lengths $a_{22}+a_{11}-2a_{12}$. We use
the non-linear spin dynamics in our scheme to demonstrate squeezing of gaussian
noise, showing $2.79 \pm 0.43$ dB squeezing when starting with a noisy state
and applying 160 echo pulses. Which can be used to increase sensitivity when
there are errors in state preparation. Our results allow for a better
understanding of inter-atomic potentials in \textsuperscript{87}Rb. Our scheme
can be used for spin squeezing beyond the standard quantum limit and observing
polaron physics close to a Feshbach resonance, where interactions diverge, and
strong magnetic noises are ever present.
