Mapping the absolute magnetic field and evaluating the quadratic Zeeman-effect-induced systematic error in an atom interferometer gravimeter

Precisely evaluating the systematic error induced by the quadratic Zeeman effect is important for developing atom interferometer gravimeters aiming at an accuracy in the $\mu \mathrm{Gal}$ regime ($1\mu \mathrm{Gal}={10}^{-8}\mathrm{m}/{\mathrm{s}}^2\approx {10}^{-9}\mathrm{g}$). This paper reports on the experimental investigation of Raman spectroscopy-based magnetic field measurements and the evaluation of the systematic error in the gravimetric atom interferometer (GAIN) due to quadratic Zeeman effect. We discuss Raman duration and frequency step-size-dependent magnetic field measurement uncertainty, present vector light shift and tensor light shift induced magnetic field measurement offset, and map the absolute magnetic field inside the interferometer chamber of GAIN with an uncertainty of 0.72 nT and a spatial resolution of 12.8 mm. We evaluate the quadratic Zeeman-effect-induced gravity measurement error in GAIN as $2.04\mu \mathrm{Gal}$. The methods shown in this paper are important for precisely mapping the absolute magnetic field in vacuum and reducing the quadratic Zeeman-effect-induced systematic error in Raman transition-based precision measurements, such as atomic interferometer gravimeters.

Figure 1

${}^{87}\mathrm{Rb}$ ground-state magnetic sublevels and Raman transition configurations. The green lines with double arrows are possible transitions when a quantization magnetic field parallel to the Raman beams is applied, and the black dashed lines are possible transitions if the quantization axis is absent. $\mathrm{\Delta }E$ is the energy level shift induced by first-order Zeeman effect, and ${\omega }_{m_F,m_F}$ ($m_F=0,\pm 1$) are the resonance frequencies of the $|2,m_F\rangle \to |1,m_F\rangle$ transitions.

Figure 2

Schematic diagram of the experimental setup.

Figure 3

Raman spectra obtained at one fixed height with different Raman duration $\tau$ and frequency step $\mathrm{\Delta }f$. Black points are measured data and red lines are fitting results with a combined function of Eq. (1).

Figure 4

Magnetic field measurement uncertainty as a function of Raman pulse duration τ for different frequency step $\mathrm{\Delta }f$. When the frequency domain sampling theorem is fulfilled [$\mathrm{\Delta }f\le 1/\left(2\tau \right)$], the measurement uncertainty improves with decreasing $\mathrm{\Delta }f$ approximately as $U_B\propto \sqrt{\mathrm{\Delta }f}$ while improving with increasing τ with a smaller steepness than $1/\phantom{1\tau }\tau$. The measured data are represented by the points, and the lines are simply to guide the eye. The data marked with (a)–(f) represent the achieved measurement uncertainties corresponding to the spectrums shown in Figs. 3–3.

Figure 5

The influence of vector and tensor light shift on measured magnetic field intensity. The measured results $B_{m_F}^{\sigma }$ are shown as points, in which the superscript $\sigma$ ($\sigma ={\sigma }^{+},{\sigma }^{-}$) represents the polarization of Raman laser, and the subscript $m_F$ represents the result inferred from the resonance frequency of the $|F=2,m_F\rangle \leftrightarrow |F=1,m_F\rangle$ transition peak, respectively. The amplitude of the fictitious magnetic field induced by VLS is inversely proportional to the Raman duration as $B_{vls}\propto 1/\phantom{1\tau }\tau$. The effect of TLS is one order of magnitude smaller than the effect of VLS.

Figure 6

Measured magnetic field map for the (a) nominal (13 mA) and (b) half (6.5 mA) solenoid current; Inferred field map for the (c) solenoid and (d) background magnetic field. The data are represented by the points, and the lines are simply to guide the eye. Relevant heights for our standard atom interferometer configuration are indicated in (a).