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

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

Pau Gomez, Chiara Mazzinghi, Ferran Martin, Simon Coop, Silvana Palacios, and Morgan W. Mitchell

Phys. Rev. A 100, 032704 – Published 10 September 2019

Abstract

We present interferometeric measurements of the $f = 1$ to $f = 2$ interhyperfine scattering lengths in a single-domain spinor Bose-Einstein condensate of ${}^{87}{Rb}$ . The interhyperfine interaction leads to a strong and state-dependent modification of the spin-mixing dynamics with respect to a noninteracting 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 interhyperfine scattering length differences, calibrated to intrahyperfine 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.

Received 17 April 2019

DOI:https://doi.org/10.1103/PhysRevA.100.032704

Physics Subject Headings (PhySH)

Bose-Einstein condensates Fine & hyperfine structure Mixtures of atomic and/or molecular quantum gases Scattering of atoms, molecules, clusters & ions Ultracold collisions

Atomic, Molecular & Optical

Authors & Affiliations

Pau Gomez ^1,*, Chiara Mazzinghi¹, Ferran Martin^1,2, Simon Coop¹, Silvana Palacios¹, and Morgan W. Mitchell^1,3

¹ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain
²Quside Technologies S.L., C/Esteve Terradas 1, Of. 217, 08860 Castelldefels (Barcelona), Spain
³ICREA–Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain

^*Corresponding author: pau.gomez@icfo.eu

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Vol. 100, Iss. 3 — September 2019

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Figure 1
Faraday probing of a SBEC formed by $f = 1$ (blue, striped) and $f = 2$ (red, solid) atoms precessing around the applied magnetic field $B = B z$ . Hyperfine selective probing is achieved by sending light closely detuned to $1 \leftrightarrow 0^{'}$ or $2 \leftrightarrow 3^{'}$ for $f = 1$ and $f = 2$ , respectively. $D_{2}$ line transitions, probe frequencies, and detunings $δ^{(1)}$ and $δ^{(2)}$ are depicted on the left. The dispersive interaction rotates the probe beam by an angle proportional to the atomic spin projection $ϕ^{(f)} = \frac{1}{2} G_{1}^{(f)} F_{y}^{(f)}$ . The rotation angle is recorded on a differential photodetector, which has been balanced by adjusting the $λ / 2$ wave plate (HWP) before the polarizing beam splitter (PBS). The probe light power $P^{(f)}$ is monitored via a photodetector behind a 50:50 beam splitter (BS) at the beginning of the optical path.
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Figure 2
Time evolution of the transverse magnetization in $f = 1$ , starting from a fully aligned ( $F_{⊥}^{(1)} = 0$ ) state $ξ^{(1)} / \sqrt{N} = {(1 / 2, i / \sqrt{2}, 1 / 2)}^{T}$ . The dynamics are governed by the competition between the QZS and the ferromagnetic interaction. Blue circles show the observed transverse magnetization $F_{⊥}^{(1)}$ after a variable hold time $t_{hold}$ . The solid line and colored area represent the median and 90% confidence interval of the theoretical model. The probing is $δ^{(1)} = - 570 MHz$ red-detuned from the $1 \leftrightarrow 0^{'}$ transition, such that $G_{1}^{(1)} = 1.70 (12) \times 10^{- 7} red / spin$ .
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Figure 3
Dynamical evolution for the initial state in Eq. (18) and under a magnetic field of 119.6 mG. Blue circles and red diamonds, are the mean experimental transverse magnetization in $f = 1$ and $f = 2$ , respectively. The solid lines are the median of the theoretical mean-field evolution for $g_{2}^{(12)} / g_{1}^{(1)} = - 6.4$ , while shaded areas represent the 90% confidence intervals.
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Figure 4
Phase evolution of the difference in the comagnetometer readouts between state preparations $A$ and $B$ . Circles show the experimental mean values and the error bars represent one standard deviation. The solid line and shaded area are the mean and standard deviation of the theoretical phase evolution for $g_{2}^{(12)} / g_{1}^{(1)} = - 6.4$ and $g_{1}^{(12)} / g_{1}^{(1)} = - 1.27$ . Atom numbers are estimated by destructive absorption imaging before the first run of experimental sequence $A$ and $B$ , yielding $N_{A} = 88 (3) \times 10^{3}$ and $N_{B} = 96 (3) \times 10^{3}$ .
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Figure 5
Experimental sequences for the determination of the interhyperfine interaction parameters $g_{2}^{(12)}$ (top) and $g_{1}^{(12)}$ (bottom). The relative spin populations in $f = 1$ and $f = 2$ are represented by the blue and red circles. The oscillatory traces at the end of both sequences illustrate the consecutive Faraday readouts of the transverse magnetization in each hyperfine manifold. Note that the bottom sequence branches into $A$ or $B$ for the many-body (MB) evolution of a SBEC initially prepared in $ξ_{0, A}$ or $ξ_{0, B}$ , respectively. The insets of this later sequence show the angular evolution of the transverse spins $θ_{A}^{(f)}$ and $θ_{B}^{(f)}$ , as well as the corresponding comagnetometer readouts ( $θ_{A}^{(12)}$ and $θ_{B}^{(12)}$ ).
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Physical Review A

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

Pau Gomez, Chiara Mazzinghi, Ferran Martin, Simon Coop, Silvana Palacios, and Morgan W. Mitchell

Phys. Rev. A 100, 032704 – Published 10 September 2019

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Physical Review A

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Interferometric measurement of interhyperfine scattering lengths in Rb87

Pau Gomez, Chiara Mazzinghi, Ferran Martin, Simon Coop, Silvana Palacios, and Morgan W. Mitchell

Phys. Rev. A 100, 032704 – Published 10 September 2019

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