Response functions of atom gravimeters

Atom gravimeters are equivalent to non-multi-level corner-cube gravimeters in translating the gravity signal into the measurement result. This enables description of atom gravimeters as LTI systems. The system’s impulse responses by acceleration, velocity, and displacement are found to have the shape of triangle, meander, and the Dirac comb resp. The effects of inhomogeneous gravity field are studied for constant and linear vertical gradients and self-attraction of the instrument. For the constant gradient the effective measurement height is below the top of the trajectory at 1/6 and 7/24 of its length for the fountain and the release types of the instruments resp. The analysis is expanded to the gravimeters implementing the Bloch oscillations at the apex of the trajectory. In filtering the vibrations these instruments are equivalent to the first-order low-pass filters, while other atom gravimeters are equivalent to the second-order low-pass filters.

💡 Research Summary

The paper presents a comprehensive analysis of atom gravimeters by modeling them as linear time‑invariant (LTI) systems. Starting from the basic three‑pulse atom interferometer, the authors show that the measured phase shift Δφ = k g T² + φ₁−2φ₂+φ₃ can be interpreted as a linear functional of the time‑dependent acceleration g(t) experienced by the falling atoms. By expressing the measurement result as a convolution of g(t) with an impulse response h(t), the gravimeter is placed within the well‑established framework of LTI systems, allowing the use of classical signal‑processing tools.

The core of the analysis is the derivation of the impulse response (or weighting function) for three different physical quantities: acceleration, velocity, and displacement. Because the measurement involves two successive integrations of the acceleration, the acceleration weighting function w_g(t) is a triangular function that rises linearly from zero to a maximum at t = T and then symmetrically falls to zero at t = 2T. The velocity weighting function w_V(t) = –dw_g/dt is a “meander” (alternating positive and negative rectangular pulses), while the displacement weighting function w_z(t) = d²w_g/dt² reduces to a Dirac‑comb: δ(t) – 2δ(t–T) + δ(t–2T). These three functions are successive derivatives of one another, each changing sign at the integration boundaries, and they embody how the instrument averages the input signal over the measurement interval.

Using these weighting functions, the authors evaluate systematic effects that arise from non‑uniform gravity fields. For a constant vertical gradient γ, the measured gravity can be written as g = g₀ ± γ h_eff, where the effective measurement height h_eff lies below the apex of the atomic trajectory. In a fountain‑type gravimeter (atoms launched upward and detected on the way down) h_eff is 1/6 of the total trajectory length; in a release‑type gravimeter (atoms released from rest) h_eff is approximately 7/24 of the trajectory length. This result clarifies why atom gravimeters, unlike traditional corner‑cube gravimeters, do not measure gravity exactly at the apex.

When the gradient varies linearly with height (γ₁ + γ₂ z), the acceleration signal can be expanded into a polynomial in time. The contribution of each term to the measured gravity is obtained by replacing the time powers with the corresponding moments C_n of the acceleration weighting function, where C_n = ∫₀^{2T} tⁿ w_g(t) dt = 2^{n+2} – 2 (n+1)(n+2) Tⁿ. This compact moment‑based formulation enables rapid estimation of higher‑order gradient effects without resorting to full numerical integration.

The paper also addresses the self‑attraction of the instrument, a complex disturbance that cannot be captured by low‑order polynomials. By fitting the self‑attraction profile with a 6‑th‑degree polynomial and then using moments up to n = 12, the authors compute a correction of –1.27 µGal for a typical instrument, in agreement with independent finite‑element analyses. They further show that the residual error from the polynomial approximation never exceeds the maximum absolute residual, providing a rigorous bound on the uncertainty introduced by the approximation.

A special class of atom gravimeters that employ Bloch oscillations at the trajectory apex is examined next. In these devices the atoms are held stationary for a time T₀ before falling back, effectively adding an extra integration interval. The resulting impulse response is nearly rectangular, meaning the instrument performs an almost uniform average of the acceleration over the total time T + T₀. Consequently, the Bloch‑oscillation gravimeter behaves like a first‑order low‑pass filter in the frequency domain. By contrast, the conventional three‑pulse atom gravimeter corresponds to a second‑order low‑pass filter, and a multi‑level corner‑cube gravimeter behaves as a third‑order low‑pass filter. This filter‑order perspective clarifies the relative vibration‑rejection capabilities of the different designs.

In the concluding section the authors summarize their findings: (1) atom gravimeters are equivalent to non‑multi‑level corner‑cube gravimeters in the way they map the gravity signal to a measurement; (2) the weighting functions are directly obtained from the impulse response of a cascade of double integrators; (3) systematic disturbances can be evaluated by substituting polynomial terms with the appropriate moments of the weighting functions; (4) the effective measurement height is 1/6 of the trajectory for fountain types and about 7/24 for release types; (5) the error from approximating a disturbance never exceeds the maximum approximation error; and (6) with respect to vibration, Bloch‑oscillation gravimeters act as first‑order low‑pass filters, whereas standard three‑pulse atom gravimeters act as second‑order filters. This unified LTI framework offers a powerful, analytically tractable tool for designing, calibrating, and comparing atom gravimeters and for integrating them into high‑precision metrological networks.