Observation of nonlinear spin dynamics and squeezing in a BEC using dynamic decoupling

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.