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Systematic Errors in High-Precision Gravity Measurements by Light-Pulse Atom Interferometry on the Ground and in Space

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Systematic Errors in High-Precision Gravity Measurements by Light-Pulse Atom Interferometry on the Ground and in Space Systematic errors in high-precision gravity measurements by light-pulse atom interferometry on the ground and in space Anna M. Nobili,1, 2 Alberto Anselmi,3 and Raffaello Pegna1 1Dept. of Physics \E. Fermi", University of Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy 2INFN (Istituto Nazionale di Fisica Nucleare), Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy 3Thales Alenia Space Italia, Strada Antica di Collegno 253, 10146 Torino, Italy (Dated: February 18, 2020) We focus on the fact that light-pulse atom interferometers measure the atoms' acceleration with only three data points per drop. As a result, the measured effect of the gravity gradient is system- atically larger than the true one, an error linear with the gradient and quadratic in time almost unnoticed so far. We show how this error affects the absolute measurement of the gravitational acceleration g as well as ground and space experiments with gradiometers based on atom inter- ferometry such as those designed for space geodesy, the measurement of the universal constant of gravity and the detection of gravitational waves. When atom interferometers test the universality of free fall and the weak equivalence principle by dropping different isotopes of the same atom one laser interrogates both isotopes and the error reported here cancels out. With atom clouds of different species and two lasers of different frequencies the phase shifts measured by the interferometer differ by a large amount even in absence of violation. Systematic errors, including common mode acceler- ations coupled to the gravity gradient with the reported error, lead to hard concurrent requirements {on the ground and in space{ on several dimensionless parameters all of which must be smaller than the sought-for violation signal. Light-pulse atom interferometers (AIs) are based on We focus on the fact that AIs measure the atoms' po- quantum mechanics. As the atoms fall, the atomic wave sition along the trajectory only three times per drop (in packet is split, redirected, and finally recombined via correspondence with the three light pulses), unlike laser three atom-light interactions at times 0, T , and 2T . The interferometers in falling corner-cube gravimeters which phase that the atoms acquire during the interferometer make hundreds to a thousand measurements per drop [4]. sequence is proportional to the gravitational acceleration Hence, although predicted exactly, the gravitational ac- that they are subjected to. celeration measured by AIs is the true one only in a uni- form field. Although one might think that the phase shift depends The predicted value of the measured acceleration first on quantum mechanical quantities \. this is merely an appeared in Ref. [5], and it has been confirmed ever illusion since we can write the scale factor [between the since[1, 6{9]. However, none of these works mentions phase shift and the gravitational acceleration] in terms that in the presence of gravity gradient this value is the of the parameters we control experimentally, i.e. Raman average free fall acceleration (at time T of the middle pulse vector k and pulse timing T . It then takes the form pulse) based on three position measurements. This is, of kT 2. We can simply ignore the quantum nature of the course, only an approximation to the true acceleration, atom and model it as a classical point particle that car- expressed mathematically by the second time derivative ries an internal clock and can measure the local phase of of the position, or obtained experimentally with a suffi- the light field." ([1], Sec. 2.1.3). The same reference also ciently large number of measurements per drop. demonstrates that, in the case of a gravitational field with a linear gradient, both the exact path integral approach Initially, the lack of precise measurements made the and the purely classical one lead to the same exact closed difference unimportant, and by the time the precision form for the phase shift and free fall acceleration mea- improved nobody went back to this issue. However, sured by the AI, which is then expanded in power series its physical consequences do deserve to be carefully ad- of the local gravity gradient γ for convenience [1]. The re- dressed. Using the classical approach [1] we point them arXiv:1806.04397v5 [physics.atom-ph] 17 Feb 2020 coil velocity is the only part of the atom-light interaction out when AIs are used to measure the absolute value of which is not found in the classical model; however, it does the gravitational acceleration g, for gravity gradiometry not appear in the phase shift actually measured by AIs, and for testing the universality of free fall (UFF), both because they are operated symmetrically so as to cancel on the ground and in space. it out [2, 3]. Thus, the classical approach gives excellent Since UFF tests are included, we allow from the start predictions of the phase shift measured by the interfer- the possibility that the equivalence of inertial and grav- ometer, while including the quantum mechanical details itational mass may be violated for atoms of different related to the internal degrees of freedom is needed to species A; B in the field of Earth (violation of the weak account for smaller effects, such as the finite length of equivalence principle, WEP) hence violating UFF [10]. g i the light pulses. We therefore write the masses as mA;B = mA;B(1+ηA;B), 2 g i M⊕ = M⊕(1 + η⊕), where superscripts i, g refer to in- If ηA;B = 0 (WEP and UFF hold) this is the same as ertial or gravitational mass and the E¨otv¨osparameters in [1]; it is the expansion to order γ of an exact result ηA, ηB , η⊕ may not be exactly zero (although they must which can be obtained in closed form by an exact path be smaller than 1 by many orders of magnitude [11, 12]). integral treatment or within a purely classical descrip- The equation of motion for atoms A or B reads tion. In a gravitational field with a linear gradient the free GM i ⊕ fall acceleration of the atoms at time T is obtained from z¨A;B = − 2 (1 + η⊕ + ηA;B) (1) (R⊕ + zA;B) (3) while the AI measurement gives, at the same time T , the value (7), which is systematically larger (in modulus) where R⊕ is the Earth's radius and the z axis points upward. UFF is tested by measuring the differential ac- than the true one by the amount: celerationz ¨B − z¨A, then η⊕ cancels out, and there is a 1 2 violation if, with identical initial conditions and no noise, ∆a = γg◦T (8) 12 the ratio with a relative error ∆a = 1 γT 2. The discrepancy was z¨B − z¨A g◦ 12 = ηB − ηA ≡ η (2) (¨zA +z ¨B)=2 pointed out in Ref. [17] where it was explained with the simple algebra involved in computing (6) from (5) and differs from zero; thus, what matters is the different com- (4). In physical terms, it is due to the limitation, intrin- position of the atoms under test, which should be maxi- sic to the AI instrument, of making only three position g i mized [13{15]. Hence we assume M⊕ = M⊕ ≡ M⊕. measurements per drop. Whether it can be neglected or Using a perturbative approach for a gravity field with not will depend on the specific experiment. If not, appro- a linear gradient γ (see, e.g., Ref. [16]) the equation of priate systematic checks are required in order to partially motion reads model this term, while any remaining unknown fraction 1 of it must be too small to matter. z¨ ' −g (1 + η ) − γ g t2 − v◦ t − z◦ ; (3) A;B ◦ A;B 2 ◦ A;B A;B With the same perturbative approach the calculation 2 2 −2 can be extended to order γ by using the solution zA;B(t) where g◦ = GM⊕=R⊕ ' 9:8 ms , γ = 2g◦=R⊕ ' 3:1 × 10−6 s−2, and z◦ and v◦ are the initial position and to first order in γ as given by (4), rather than to order A;B A;B zero (i.e. z j = − 1 g t2 + v◦ t + z◦ ), as used in velocity errors of the atoms at release (the exact values A;B γ=0 2 ◦ A;B A;B are assumed to be zero). The solution is (3). The new equation of motion reads 1 h 1 2 ◦ ◦ ◦ ◦ 2 z¨A;B ' − g◦(1 + ηA;B) + γ g◦t − v t − z zA;B(t) ' zA;B + vA;Bt − g◦(1 + ηA;B)t A;B A;B 2 2 (9) (4) 1 1 1 i 2 1 2 1 ◦ 1 ◦ 2 4 ◦ 3 ◦ 2 −γt g t − v t − z : +γ g◦t − vA;Bt − zA;Bt : 24 ◦ 6 A;B 2 A;B 24 6 2 We compute the phase shift δφA;B measured by the AI Its solution leads to the phase shift, hence to the accel- following the step-by-step algorithm outlined in Ref. [1]. eration measured by the AI to order γ2: Assuming the same k for all three pulses (~k is the mo- g (T ) ' g (1 + η ) mentum transfer, with } the reduced Planck constant) A;B meas ◦ A;B and the same time interval T between subsequent pulses, 7 +γ g T 2 − v◦ T − z◦ it is 12 ◦ A;B A;B (10) 2 31 4 1 ◦ 3 7 ◦ 2 δφA;B = φA;B(2T ) − 2φA;B(T ) + φA;B(0) +γ g T − v T − z T : (5) 360 ◦ 4 A;B 12 A;B = k[zA;B(2T ) − 2zA;B(T ) + zA;B(0)] The result is the same as Eq.
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