Abstract
We study the evolution of a Bose–Einstein condensate 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, as predicted in [, ] and 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 to demonstrate squeezing of Gaussian noise, showing 2.79 ± 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 87Rb. Our scheme can be used for spin-squeezing beyond the standard quantum limit and observing polaron physics close to Feshbach resonances, where interactions diverge, and strong magnetic noises are ever present.
Highlights
We study the evolution of a Bose-Einstein Condensate (BEC) in a two-state superposition due to inter-state interactions
The state of a two-level system can be represented by a vector on the Bloch sphere
In cases where the Bloch sphere represents the average state of an ensemble of two-level systems, interactions can introduce non-linear evolution, represented by torsion and one-axis-twist of the sphere
Summary
Measured phase shifts (circles) on magnetic sensitive transition. A coherent state around the equator is squeezed by one-axis-twist, illustrated by the colored distribution. We use a Bose-Einstein condensate (BEC) of ultracold 87Rb atoms in states |1 = |F = 1, mf and |2 = |F = 2, mf , where F is the total spin and mf is spin projection along the magnetic field axis. The mean-field energy shifts are: 4πh m N1,2 are densities in different states, aij are s-wave scattering lengths, αij are correlation factors accounting for Bose statistics (for thermal clouds αij = 2, and for BEC αij = 1 [2]), m is atomic mass, andh is the reduced. The first term is proportional to the total density n = n1 + n2, while the second is proportional to the density difference δn = n1 − n2
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