Systematic effects

XW Xuejian Wu ZP Zachary Pagel BM Bola S. Malek TN Timothy H. Nguyen FZ Fei Zi DS Daniel S. Scheirer HM Holger Müller

This protocol is extracted from research article:

Gravity surveys using a mobile atom interferometer

**
Sci Adv**,
Sep 6, 2019;
DOI:
10.1126/sciadv.aax0800

Gravity surveys using a mobile atom interferometer

Procedure

The total systematic error of our mobile atomic gravimeter is −8 ± 15 μGal (table S1). Here, we briefly discuss those contributions that produce an error larger than 1 μGal.

Magnetic fields. The cesium ground state hyperfine splitting exhibits a quadratic Zeeman effect of 0.43 kHz/G^{2}. A homogenous magnetic field has no influence on the phase of a Mach-Zehnder atom interferometer, but a magnetic field gradient *B*′ on top of a bias field *B*_{0} can add an effective force that is proportional to *B*_{0}*B*′. To characterize the shift, we varied *B*_{0} and found that the gravity measurement was modified by ~1.8 mGal/G. Wave vector reversal reduces this to −9 ± 28 μGal/G (fig. S6). At a nominal bias field of 0.3 G, this systematic is −3 ± 9 μGal.

Refractive index of background vapor. Operating with small detunings, we had to check whether the refractive index *n* of cesium background atoms generates a substantial systematic error. It is related to the atomic polarizability α and thus to the AC Stark shift ω^{AC}, by (*n* − 1) = 2παρ = 2πcρħω^{AC}/*I*, where ρ is the number density of atoms, *c* is the speed of light in vacuum, and *I* is the laser intensity. The calculation of the AC Stark shift can be found in our previous paper (*28*). This refractive index is averaged over the *F* = 3 and *F* = 4 ground states (background atoms are equally distributed in either state), the two Raman frequencies, and the 270-MHz-wide thermal distribution of the Doppler shifts. With a cesium partial pressure of (3 ± 3) × 10^{−10} torr, the index of refraction is (*n* – 1) ≈ (−7 ± 7) × 10^{−9}, corresponding to an error in the gravity measurement of about −7 ± 7 μGal.

Coriolis effect. The Coriolis effect causes an extra phase shift of 2**Ω**·(**v** × **k**_{eff})*T*^{2} and thus an error Δ*g*/*g* = 2**Ω**·(**v** × **k**_{eff})/(**k**_{eff}·**g**), where **v** is the initial velocity of the atoms and **Ω** is the Earth’s rotation (*42*). The Earth’s rotation at the latitude of Berkeley has a horizontal component of 58 μrad/s. For a transverse velocity component of the atoms of 0 ± 1 mm/s, the systematic is 0 ± 6 μGal.

Vertical alignment after correction. To calibrate the alignment to the gravity axis, we measured gravity with different tilt angles of the retroreflector and fit the data with a bivariate quadratic function to obtain the gravity correction (fig. S7). The SE of the fit residual is ±5 μGal. For each measurement, the gravity values were corrected on the basis of the real-time tilt and the fit function. After correction, the systematic is 0 ± 5 μGal.

Differential AC Stark shift. A differential AC Stark shift between the hyperfine ground states *a* and *b* during Raman transitions adds a phase Δφ^{AC} = (δ_{3}^{AC}/Ω_{3} − δ_{1}^{AC}/Ω_{1}) to the interferometer, where δ_{1,3}^{AC} = ω_{a}^{AC} − ω_{b}^{AC} are the differential AC Stark shifts during the first and the third pulses, and Ω_{1,3} is the two-photon Rabi frequencies (*11*). We worked at a red detuning of 158 MHz relative to four to five transitions and a modulation index β of 1.1, close to the zero crossing of the differential AC Stark shift (*28*). Moreover, the AC Stark shift systematic was suppressed by cancellation between the first and the third pulses and by wave vector reversal. In principle, the AC Stark effect systematic can be eliminated by varying the power *P* of the interferometry beam and extrapolating the measured gravity value to zero power. When we did this, we found no statistically resolved sign for an AC Stark effect, Δ*g* = (0 ± 3) (Δ*P*/*P*) μGal (fig. S6).

Raman frequency offset. Off-transition Raman frequencies can result in interferometer phase shifts (*43*). In a similar atomic gravimeter, Gillot *et al.* (*43*) reported Δ*g* corresponding to ~1.4 μGal/kHz due to the offset from the Raman frequencies. We scanned the Raman frequencies and found that the resonance has a bandwidth of ~70 kHz. Setting the Raman frequency to the peak has an error of 0 ± 2 kHz. The systematic error to gravity is about 0 ± 3 μGal.

Two-photon light shift. Off-resonant Raman transitions can cause an additional phase shift (*33*). For *T* = 130 ms, the Doppler shift at the first Raman pulse is about 0.46 MHz and gets to about 6.44 MHz for the third pulse. A π pulse of 8 μs corresponds to a Rabi frequency of 62.5 kHz. Using equation 8 of (*33*), the two-photon light shift leads to a phase shift of 32 mrad, corresponding to a down correction to *g* by 3 μGal. We estimated the error with the same magnitude of the correction. The systematic effect of two-photon light shift is 3 ± 3 μGal.

Wavefront aberrations. Wavefront aberrations combined with transverse motion of the atomic cloud lead to systematic phase shifts (*42*). Upon retroreflection, the interferometer laser beam passes a λ/4 plate (specified at λ/4 flatness) and a vacuum viewport (λ/4) twice; each pass causes wavefront aberrations proportional to *n* − 1, where *n* ≈ 1.5 is the index of refraction. The mirror itself has λ/10 flatness. Added in quadrature, the three elements cause 0.4λ total aberration. According to equation 6 of (*42*), if this aberration has a parabolic shape over a 25.4-mm diameter, then the optical phase change can be described as φ = *K*σ_{v}^{2}*T*^{2}, with *K* = *k*_{eff}/*R* ≈ 1.5 × 104 m^{−2} (*R* is the radius of curvature) and a horizontal velocity spread of σ* _{v}* = 1 mm/s (an upper limit obtained by detecting the cloud with a camera). It predicts a systematic effect of Δ

Vertical gravity gradients. VGGs are of the order of 3 μGal/cm, which cause an error if the effective measurement height of the instrument is unknown. The instrument was mounted on adjustable feet, which contributed a change of ±0.5 cm in the effective height when leveling the instrument. Other than that, the effective height is known with better than millimeter precision relative to the pyramid mirror. This systematic error is 0 ± 2 μGal.

Gouy phase. The gradient of Gouy phase causes a relative decrease of Δ*k*_{eff}/*k*_{eff} = λ^{2}/(2π*w*_{0}^{2}) to the wave number and thus a scale factor error that depends on the waist *w*_{0} of the laser beam. For *w*_{0} = 5 mm, it amounts to Δ*k*_{eff}/*k*_{eff} = 1.47 parts per billion, so that the measured value of *g* needs to be corrected up by ~1 μGal. We estimated the error with the same magnitude of the correction. Thus, the systematic effect of Gouy phase is about −1 ± 1 μGal.

Laser frequency stability. Laser frequency drift leads to an error to the wave number Δ*k*_{eff}/*k*_{eff} = Δν/ν, where ν is the laser frequency. As we measured the laser frequency stability by beating with a reference laser over days, the SD is about 300 kHz at an averaging time of 1 s. This systematic error is 0 ± 1 μGal.

Self-gravity. Self-gravity of the instrument is dominated by the metal vacuum components, vibration isolation stage, and pyramid mirror. The metal vacuum components and vibration isolation stage attract the atomic cloud downward, while the pyramid mirror attracts it upward. The total self-gravity is less than 1 μGal, so that we can neglect it.

Aperture effect. The intensity variation of the interferometer beam caused by an aperture effect of the pyramidal hole is about 5% with a correlation length in the range of 100 to 200 μm, measured by a beam profiler. This effect on the accuracy of atomic gravimeters has been analyzed for 15% intensity variations and a similar correlation length (*44*). It shows that these effects decrease rapidly to below 10^{−9} as the thermal atom velocity exceeds a few millimeters per second because atoms average over high- and low-intensity spots. The thermal expansion speed of our atomic cloud is about 1 cm/s, so that the potential position-dependent phase shifts caused by the intensity ripple can be negligible.

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