# Interferometric measurement of interhyperfine scattering lengths in $^{87}$Rb

## TL;DR

This paper reports precise interferometric measurements of inter-hyperfine scattering lengths in a $^{87}$Rb Bose-Einstein condensate, revealing state-dependent spin-mixing dynamics and demonstrating a method with potential for high-precision quantum measurements.

## Contribution

The authors develop a hyperfine-specific Faraday-rotation interferometric method to accurately determine inter-hyperfine scattering length differences in a single-domain $^{87}$Rb condensate, improving measurement precision.

## Key findings

- Measured inter-hyperfine scattering length differences with uncertainties limited by atom number.
- Demonstrated a technique capable of achieving approximately 0.3% precision with improved atom number control.
- Revealed strong, state-dependent modifications in spin-mixing dynamics due to inter-hyperfine interactions.

## Abstract

We present interferometeric measurements of the $f=1$ to $f=2$ inter-hyperfine scattering lengths in a single-domain spinor Bose-Einstein condensate of $^{87}$Rb. The inter-hyperfine interaction leads to a strong and state-dependent modification of the spin-mixing dynamics with respect to a non-interacting description. We employ hyperfine-specific Faraday-rotation probing to reveal the evolution of the transverse magnetization in each hyperfine manifold for different state preparations, and a comagnetometer strategy to cancel laboratory magnetic noise. The method allows precise determination of inter-hyperfine scattering length differences, calibrated to intra-hyperfine scattering length differences. We report $(a_{3}^{(12)}-a_{2}^{(12)})/(a_{2}^{(1)}-a_{0}^{(1)})=-1.27(15)$ and $(a_{1}^{(12)}-a_{2}^{(12)})/(a_{2}^{(1)}-a_{0}^{(1)})=-1.31(13)$, limited by atom number uncertainty. With achievable control of atom number, we estimate precisions of $ \approx 0.3\%$ should be possible with this technique.