Precision measurement of spin-dependent inter-hyperfine scattering lengths in 87 Rb Pau Gomez, 1 Chiara Mazzinghi, 1 Ferran Martin, 1, 2 Simon Coop, 1 Silvana Palacios, 1 and Morgan W. Mitchell 1, 3 1 ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 2 Quside Technologies S.L., C/Esteve Terradas 1, Of. 217, 08860 Castelldefels (Barcelona), Spain 3 ICREA – Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain (Dated: November 11, 2022) We present precision measurements of the f =1 to f =2 inter-hyperfine scattering lengths in a single-domain 87 Rb spinor Bose-Einstein condensate. 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 evo- lution of the transverse magnetization in each hyperfine manifold for different state preparations, and a co-magnetometer strategy to cancel laboratory magnetic noise. We find the scattering length ratios a (12) 32 /a (1) 20 = -1.27(15) and a (12) 12 /a (1) 20 = -1.31(13), limited by atom number fluctuations. Using better control of atom number, we estimate precisions of ∼ 0.3% should be possible with this technique. I. INTRODUCTION Since the advent of Bose-Einstein condensation (BEC) in ultracold quantum gases, experimental access to the spin degrees of freedom and resulting spin-dependent in- teractions have expanded greatly. The pioneering 87 Rb, 23 Na and 7 Li experiments [1–3] used magnetic trapping that restricted their studies to scalar BECs in low field seeking Zeeman sublevels. By introducing optical trap- ping techniques [4, 5], the spin degree of freedom became accessible, enabling the study of spin-mixing dynamics [6–9], spontaneous magnetic symmetry breaking [10–12], domain formation [4, 10, 13] and exotic topological spin excitations [14, 15]. These rich dynamics arise from the interplay between superfluidity and magnetism, which for a single, spin-f species and s-wave binary contact interactions are de- scribed by f +1 parameters, the intra-hyperfine scatter- ing lengths. In the case of 87 Rb, these have been sepa- rately determined for the f =1 and f =2 ground-state manifolds [8, 16, 17]. Inter-hyperfine interactions are less well studied, but nonetheless play an important role in determining the miscibility of multiple BEC species [18– 20], and have been used to produce spin-squeezing with its attendant entanglement, and Bell-type correlations [21–25]. For 87 Rb, the full set of inter-hyperfine spin interaction parameters has recently been measured [26] with intriguing results. The current best values indicate that in an equal f =1, f =2 ground-state mixture, the f =1 component manifests a polar ground state at zero magnetic field [27] even though the f =1 component alone is ferromagnetic [6]. In this work we report precision measurements on the 87 Rb inter-hyperfine f =1 ↔ f =2 scattering lengths, using a novel co-magnetometer strategy. We use a single- domain spinor BEC [28], with non-destructive Faraday probing [29] for simultaneous readout of amplitude and phase of the transverse magnetization in f =1 and f =2. The observed dynamics are compared to mean field sim- ulations under the single-mode approximation (SMA) [30, 31], yielding the two spin-dependent inter-hypefine interaction parameters [27]. The presentation is orga- nized as follows: Section II describes the inter-hyperfine interaction for 87 Rb. It discusses the simplifications un- der the rotating wave approximation (section II A) and the implementation of the numerical simulations (sec- tion II B). Data interpretation and error sources are de- tailed in section II C. Section III and section IV intro- duce the experimental setup and required classical cal- ibrations. Section V describes the measurement of the spin-dependent interaction parameters. In section VI we present the resulting inter-hyperfine scattering lengths and compare against literature values. II. MEAN-FIELD DESCRIPTION A spinor BEC can be described by a vectorial order parameter, which in the SMA can be written [31] Ψ (f ) m (r)=Ψ SMA (r)ξ (f ) m (t) , (1) where f =1, 2 and m = -f, ..., f . The spin-independent spatial wavefunction Ψ SMA (r) and the relative spin am- plitudes ξ (f ) m are normalized as follows:  d 3 r|Ψ SMA (r)| 2 = N (2a)  f,m |ξ (f ) m | 2 =1 , (2b) where N is the number of atoms. For BECs significantly larger than the density healing length, the kinetic contri- bution to the total energy is negligible and the density distribution is described by a Thomas-Fermi profile [32– 34]: |Ψ SMA (r)| 2 =  μ−V (r) g (1) 0 when V (r) <µ 0 otherwise (3) arXiv:1904.07617v2 [physics.atom-ph] 17 Apr 2019