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Abstract
The laser beam waist has an impact both in the sensitivity and systematic effects present in gravimetry and atom interferometry in general. In this paper we consider how different effects contribute to both aspects in order to make a better selection of the radius of the Raman beam given a particular laser power available. A large beam waist reduces systematic effects coming from wavefront curvature and Gouy phase contributions and improves the fringe contrast due to reduced intensity gradients. On the other hand, a large waist gives a smaller Rabi frequency, which lowers the sensitivity by reducing the fraction of atoms in the selected velocity range. Considering all contributions, we find that systematic effects usually have a dominant role in selecting a beam waist.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years high precision measurements have proven useful as a tool to test fundamental physics and develop new technologies [1–7]. Accelerometers based on atomic interferometry give a clear example of this [8]. They measure gravitational accelerations, something that can be exploited in geophysics and underground exploration [9].
The relative phase acquired for a two-level atom during the gravimetric sequence is given by $\Delta \phi =k_{e}gT^2$, with $k_{e}$ the wave vector corresponding to the momentum transfer, $g$ the local gravitational acceleration, and $T$ the time between pulses. These measurements face complicated experimental challenges to minimize technical noise, contributions such as phase noise for example. Neglecting these technical problems, the sensitivity limit of the measurements is $S\sim 1/k_{e} T^2 C\sqrt {N}$, with $C$ the fringe contrast and $N$ the number of atoms [10,11]. It improves by increasing $T$ as has been done with high free-fall towers [12,13] or in microgravity environments [14–16]. Given a particular laser power available, the contrast and the number of atoms excited in the interferometric sequence depend on the waist chosen for the Raman beams.
The choice of beam size has to consider both the sensitivity and the systematic effects introduced by the Raman beam profile. The effects of the size of the Raman beam include light shifts, intensity gradients, different detunings from the velocity distribution, wavefront curvature or aberrations, and the Gouy phase. The light shift can be eliminated by controlling the intensity ratio of the Raman beams depending on their single-photon detuning [12,17]. One can increase the beam size to improve the contrast by reducing the intensity gradients [18,19], but for a given power, this also reduces the fraction of atoms contributing to the signal due to the reduction in Rabi frequency. The gradients can also be reduced by the implementation of flat-top profiles [20,21], but here we consider only Gaussian profile beams. With respect to systematic effects, several works have studied ways to minimize the contributions of aberrations or wavefront curvature [19,20,22–25].
Instead of looking at a particular contribution separately, we give an integral view of the contributions involved in selecting the best beam waist for the Raman beams. On the one hand, we identify the effects contributing to the sensitivity, and from this we identify the optimum waist size for a particular available power. On the other hand, we provide simple expressions to estimate the dependence of systematic contributions on the waist size. We conclude that the selection of an appropriate waist depends more on systematic contributions than on sensitivity considerations.
2. Level dynamics for gravimetry
We consider a cloud of atoms ($^{87}$Rb in our case) that have been manipulated with the standard laser cooling techniques and are left in free fall. A $\pi$ pulse is applied to the atoms to select a narrower velocity distribution in the vertical ($z$) axis, and then the Mach-Zehnder interferometric sequence is applied to them, consisting of a series of $\pi /2 - \pi - \pi /2$ counter-propagating Raman pulses separated by time $T$. We assume a Raman beam with a radius $w_R$ (we use a 1/$\sqrt {e}$ radius in all the text for the intensity, the atomic trap size, and the velocity width) traveling along the $z$-axis so that its transverse profile introduces a Rabi frequency given by [18,19]
The Hamiltonian for the atoms in free-fall is given by [26]
The coupled equations for the coefficients are given by [26]
In the time between pulses, the coefficients do not change, but the relative phase between the state and the laser does change. This can be taken into account by applying the time translation $t'=t+T$ to Eq. (6). We absorb that extra phase in the Rabi frequency for the application of subsequent pulses.
3. Selection pulse as a function of the beam radius
We produce a Monte Carlo simulation for the initial position of the atoms in the cloud. We assume a Gaussian distribution for the initial position of the atoms with a (1/$\sqrt {e}$) radius $w_a$, centered with the Raman beam. Only the transverse position ($x$ and $y$) of the atoms with respect to the Raman beam is relevant for the discussion.
We consider a Gaussian distribution for the velocity in the transverse direction ($v_x$ and $v_y$) with a width $\sigma _v = \sqrt {k_B T_0 / m}$ with $T_0$ the temperature (3 $\mu$K is assumed in the calculations). For the vertical ($z$) direction, the selection $\pi$ pulse gives a narrower velocity distribution at the price of losing some of the atoms. The remaining fraction ($f_s$) depends on the initial Maxwell-Boltzmann distribution ($P_v$) and the transfer probability, that is
Fig. 1. Fraction of atoms ($f_s$) after the selection pulse as a function of the normalized Raman beam radius ($s=w_R/w_a$) for different laser powers. The solid line is the full calculation (Eq. (7)), the dashed line assumes that all the atoms feel the peak intensity of the Raman beam independent on their position, and the dotted curve corresponds to the approximated expression when the selected width is much smaller than the initial thermal distribution (Eq. (8)).
The value of the square parenthesis is approximately 0.84. Figure 1 shows the remaining fraction of atoms after the selection pulse as a function of the radius of the Raman beam ($w_R$) normalized by the atomic trap radius ($w_a$), that is, $s=w_R/w_a$, for different laser powers. The remaining fraction is reduced at high $s$ because of the reduction of the Rabi frequency and also at low $s$ since the beam does not illuminate all the atomic cloud. There is a maximum value that contributes to optimizing the signal-to-noise ratio, as shown in Section 6. All the plots go to values up to $s=10$, the expected relevant range as gravimeters are reduced in size, but it is relatively easy to scale them to higher values. Here for instance, Eq. (8) shows that $f_s$ scales approximately as $1/s^2$ (dotted curve in Fig. 1).
4. Decoherence of the Rabi oscillations
In this section, we analyze the different contributions that induce decoherence on the Rabi oscillations of the first pulse (or any other pulse). Here we focus only on contributions to decoherence that have to do with the Raman beam: intensity non-homogeneity, light shifts, different detunings from the velocity distribution, wavefront curvature (or aberrations), and the Gouy phase.
The intensity non-homogeneity (Eq. (1)) introduces decoherence coming from the spatial variations of the Rabi oscillations (inset of Fig. 2) [18,19,27]. The decoherence ($\gamma$) size depends on the normalized Raman beam radius ($s$). We look at this contribution alone by considering co-propagating Raman transitions, and we quantify the decoherence by fitting the Rabi oscillations of the Monte Carlo simulation to [28]
Fig. 2. Number of Rabi oscillations $n_{osc} = |\Omega |/ 3 \pi \gamma$ before the contrast is reduced by $1/e$ as a function of the normalized Raman beam radius $s$. Inset: example of the calculated decoherence on the Rabi oscillations with $s=2$ and a laser power of 60 mW.
Fig. 3. Number of oscillations as a function of the Raman beam misalignment $\rho _0$ normalized to the value with no misalignment $\hat {n}_{osc} = n_{osc} (\rho _0) / n_{osc} (\rho _0 = 0)$. The curves with different $s$ have a similar dependence, and their value reduces to one half at approximately $\rho _0 = w_a$.
Light shifts change the detuning of the oscillations, but can be eliminated by controlling the intensity ratio of the Raman beams depending on their single photon-detuning or by inverting the momentum transfer [12,29]. Another source of detuning for velocity-dependent transitions comes from the velocity distribution obtained after the selection pulse. The velocity width in the $z$-axis depends on the Rabi frequency of the selection pulse (Eqs. (7) and (8)). If we keep the same Rabi frequency for the subsequent pulses, we are going to have atoms contributing to the signal with detunings comparable to the Rabi frequency, giving a reduction in the fringe contrast as we show in Section 5.
Finally, any wavefront curvature or Gouy phase introduces a position-dependent phase for the Rabi frequency (Eq. (1)). The atoms do not move much for the short duration of each pulse, and therefore, this phase contributes after the expansion during the full interferometric sequence [23] as we discuss in the next section.
Different contributions to the decoherence can be analyzed separately by changing the excitation method. Using microwaves, we have none of the effects described here and gives little decoherence (solid blue line in Fig. 4). The decoherence is increased for co-propagating Raman transitions (dotted green line in Fig. 4) that do not have the velocity-dependent contributions yet. Here the Raman beams are obtained from a single Ti:sapphire laser detuned by 3 GHz that goes through a phase modulator and a calcite crystal [30]. We found in our case that the decoherence was dominated by photon scattering of resonant contributions from a pedestal of emission of the laser, presumably coming from problems with our laser. The dashed red line in Fig. 4 shows the decoherence observed using microwave excitation and adding the laser light without modulation. The decoherence was 42 $\pm$ 5 times bigger than expected at that detuning. The decoherence disappeared when we added a heated rubidium cell to filter the resonant frequencies [31]. Unfortunately, the filter must be placed before the modulator to avoid undesired nonlinear effects in the rubidium cell, and the decoherence reappears after the modulator due to the frequency shifting of the emission pedestal. An alternative solution to our laser problem must be found in our case before the decoherence is limited by the effects presented here.
5. Shifts and reduction of contrast in the interferometric fringes
The Rabi oscillations of the previous section are applied in three pulses with the corresponding frequency ramp to obtain the gravimetry fringes given by [26]
with $\eta = | \partial ^2_t \alpha (t) |$ the slope of the frequency ramp and $T$ the time between pulses. The spatial intensity variations of the Raman beams and the different velocities in the atomic cloud lead to imperfect pulses for the interferometric sequence. The selection of the size of the Raman beam introduces both a shift in the fringes (and therefore in the determination of $g$) and a reduction in the signal-to-noise ratio due to, for example, a reduction of the contrast.Fig. 4. Rabi oscillations driven by microwave excitation in the presence of laser light (dashed red line) show greater decoherence than with no light (solid blue line). The laser has an intensity of 17 $\pm$ 1 mW/cm$^2$ and a detuning of -3 GHz. We also show the Rabi oscillations for co-propagating Raman transitions (dotted green line). The lines are a fit to Eq. (9).
The transverse size of the cloud expands in the time between pulses as [18]
and the atoms experience a different magnitude and phase of the Rabi frequency, Eq. (1). We analyze first the case of negligible expansion ($\sigma _v T \ll w_a$) so that $w_a(t) \simeq w_a$. Figure 5 shows the contrast of the interferometric fringes as we increase the radius of the Raman beam for different powers (and therefore different intensities). At low powers (such as 0.5 mW) the shape of the curve is independent of the power. The variations start to appear when there is a reasonably big selected fraction of atoms, Eq. (7), that is, when $\Omega \simeq k_e \sigma _v$. At even higher powers, the selection pulse becomes less relevant since we select basically all the velocity distribution and one reaches high contrast (like the maximum of the 90 mW curve of Fig. 5) because the Doppler shifts become negligible compared to the Rabi frequency. The contrast is strongly reduced for a Raman beam radius smaller than the size of the cloud due to variations of the Rabi oscillations at different positions [18,19], just as in Fig. 2.Fig. 5. Contrast ($C$) of the interference fringes as a function of the normalized size of the Raman beam $s$ (with $w_a$=1.4 mm) for different powers of the Raman beam. The saturation value for the contrast at large beam sizes is limited by the velocity selection.
As the beam radius grows, all curves saturate to a value that is weakly dependent on power. This value is $<1$ due to the velocity distribution in the $z$-axis that produces detuned oscillations of the counter-propagating Raman transitions for a fraction of the atoms [11]. The contrast improves with a selection pulse that has a Rabi frequency ($\Omega _{s.p.}$) smaller than the one used for the interferometric sequence ($\Omega _{i.s.}$) (Fig. 6). An alternative way to improve the contrast is to make use of modulated pulses [32].
Fig. 6. Contrast ($C$) of the interference fringes as a function of the power of the selection pulse. $\Omega _{s.p.}$ and $\Omega _{i.s.}$ are the Rabi frequencies of the selection and interference sequence pulses. The contrast is improved by reducing the laser power during the selection pulse at the expense of selecting a smaller fraction of atoms.
In the opposite extreme case ($\sigma _v T \gg w_a$, corresponding to $T \gg 80$ ms at a temperature of 3 $\mu K$), the atoms expand radially with $w_a (t) \simeq \sigma _{v} T$ and the normalized beam diameter $s$ decreases with time. Figure 7 shows the contrast of the interferometric fringes in this case with expansion times between 50 ms to 750 ms with no wavefront curvature ($\beta =0$). Reference [19] analyzes the contrast for other values of $\beta$. We plot the contrast as a function of the normalized Raman beam size ($s_2$) for the middle (2nd) pulse in the sequence to connect with the low power, non-expanding case of Fig. 5. The similarity of both figures indicates that we can quantify the contrast limitations in an expanding cloud by looking at the low power, non-expanding case (Fig. 5) with a Raman beam size $s = s_2 = \sigma _v T$. Just as in Fig. 5 the contrast saturates at high values of $s_2$ since the atoms do not feel anymore an intensity gradient. It is possible to further improve the contrast slightly by adjusting the laser intensity depending on the relative Raman beam size ($s$) at each pulse [18].
Fig. 7. Contrast ($C$) of the interference fringes for a cloud with an initial size $w_a=1.4$ mm and temperature of 3 $\mu$K, as a function of the expansion time $T$ for different laser powers. The contrast is also given as a function of the normalized size ($s_2$) of the Raman beam during the middle (2nd) pulse of the interferometric sequence to show the similarity to the 0.5 mW curve of Fig. 5
We move now to the contribution of the wavefront curvature and Gouy phase. The phase of the electric field of a Gaussian beam is given by [33]
where the last term corresponds to the Gouy phase, $z_R=4\pi w_0^2 / \lambda$ is the Rayleigh length for a beam with a waist $w_0$ and is the radius of curvature of the wavefront.The surface quality of the optics, particularly of the retro-reflecting mirror, is critical because the aberrations introduced by the imperfections deform the wavefront and produce shifts on the fringes. A Raman beam with no aberrations still has a phase and a curvature that changes during propagation. The difference in the wavefront curvature ($\beta =(k/2)[1/R(z_1)-1/R(z_2)]$) associated with the Rabi frequency (Eq. (1)) introduces a phase ($\theta _i=\beta \rho _i^2$) for each pulse, with $\rho =\rho _i$ evaluated at time $t_i$, and the fringes (Eq. (10)) get shifted by [24]
As we decrease the size of the Raman beam, the Rayleigh length becomes smaller, and the curvature changes happen over a shorter distance. The value of $\beta$ depends on the particular position along the beam propagation [34]. Here we calculate an upper limit on the change of the phase (Eq. (12)) [25,35] due to the wavefront curvature for a beam that travels a distance $\Delta z$ for the retroreflection
Also, if the atoms fall for a distance $d$, there will be a maximum change due to the Gouy phase of
The corresponding shift in the determination of $g$ is $\Delta g/g = \phi / k_e g T^2$ with $\phi$ given by Eq. (14). An atom starting at $\rho =0$ with no initial vertical velocity would have $\rho = v_T T$ and $d=g T^2 /2$. Integrating over the transverse velocity ($v_T$) distribution corresponds to having twice the phase shift of $v_T= \sigma _{v}$. Including the contributions from the three pulses (Eq. (14)) gives an upper bound for the fractional shift of
for the Gouy phase and6. Choosing a waist for the Raman beam
Two things need to be considered when choosing a waist for the Raman beam: maximize the signal-to-noise ratio (SNR) and minimize the systematic effects. The fundamental limit to the SNR is proportional to $C \sqrt {N}$ [10,11] (Fig. 8), with $C$ the contrast (Fig. 5) and $N$ the number of atoms, which is proportional to the selected fraction of atoms ($f_s$) of Fig. 1. Increasing the beam size improves the contrast at the price of reducing the fraction of atoms selected. In terms of sensitivity, there is an optimum beam size, which grows with increasing power (Fig. 8). As an example, considering the peak value at 30 mW, a conservative number of atoms of $N=10^4$ and an expansion time of $T=0.2$ s gives a fundamental limit to the sensitivity of $S \sim 1/ k_e T^2 C \sqrt {N} = 2 \times 10^{-8}$ m/s$^2$ and fractional sensitivity $S/g \sim 2 \times 10^{-9}$. The decrease in SNR is faster for beam sizes smaller than the optimum, so it is better to choose a radius to the right of that value where it decreases as $1/s$. The sensitivity is often limited by technical phase noise written directly on the beams or due to vibrations in the retro-reflecting mirror. Increasing the beam size reduces the Rabi frequency, requiring longer pulses. The atomic response is characterized by a weight function that acts as a low pass filter with a cutoff frequency proportional to $\Omega$ [36]. This gives a contribution of this technical noise approximately proportional to $1/w_R$.
Fig. 8. Contribution $C \sqrt {f_s}$ to the SNR as a function of the Raman beam size $s$ for different available powers. The plots show the sensitivity scaling and the occurrence of an optimum sensitivity point at a particular beam size $s$ that grows with available power.
The dependence of the SNR on temperature has two contributions. First, it determines the fraction of atoms after the selection pulse ($f_s$). With a selected velocity width much smaller than that of the initial thermal distribution gives an $f_s$ inversely proportional to the temperature (Eq. (8)). Second, to determine the contrast one needs the value of $s$, that corresponds to whatever is larger between the initial size ($w_a$) or the expanded size at the middle (2nd) pulse ($\sigma _v T$), according to what was concluded from comparing Figs. 5 and 7. For this last case, $s_2$ is inversely proportional to the temperature, and there may be a reduction of the contrast if the value of $s_2$ goes below something like 3 (Figs. 5 and 7). Working at higher values of $s_2$ (which is usually the case) gives a contrast that has little variations with temperature. Combining the two contributions, we get a SNR that scales with temperature as $T_0^{-1/2}$.
With respect to systematic effects, we consider contributions coming from the wavefront curvature and Gouy phase. Figure 9 shows an upper limit to the fractional change in the determination of $g$ for different values of the Raman beam radius $w_R$ after adding in quadrature the contributions from Eqs. (17) and (18). Here we consider a distance for Raman beam retroreflection $\Delta z=0.5$ m and a temperature of 3 $\mu$K.
Fig. 9. Fractional shift on the measurement of $g$ as a function of the Raman beam waist $w_R$ including the contributions from the Gouy phase (blue, Eq. (17)), wavefront curvature (red, Eq. (18)) and their sum in quadrature (green).
The systematic shift is dominated by the Gouy phase at large $w_R$, and the crossover between the two contributions depends on the temperature. A beam radius slightly bigger than 3 mm gives a fractional change $\Delta g/g < 10^{-9}$ for the parameters considered. The above expressions give an upper bound on the shifts due to the beam curvature or Gouy phase. These systematic contributions can be further reduced by a proper characterization of the laser beam waist, divergence and aberrations [23,25]. The value of $s$ corresponding to a beam size of $w_R=$3 mm depends on the initial size of the atomic cloud and its expansion. As an example, our atomic cloud size ($w_a=0.3$ mm) corresponds to a value of $s \simeq 10$ without considering the cloud expansion, which is to the right of the optimum sensitivity value of Fig. 8. In this particular case, selecting the most convenient beam size depends more on systematic effects than on statistical limitations. As the temperature or expansion time increases one may have the opposite case where the statistical limitations dominate.
7. Conclusions
The selection of a laser beam waist for Raman transitions in atomic gravimetry must take into account the sensitivity as well as systematic effects. Large beams introduce smaller intensity gradients and give a higher contrast at the price of working with a smaller fraction of atoms. There is a beam size that gives the best sensitivity, and this optimum size grows with the available power. We provide simple expressions to estimate the size of systematic contributions coming from the wavefront curvature and Gouy phase that establish a minimum beam size for the desired accuracy. From these results we conclude that for typical configurations the beam radius selection is guided more by systematic effects than by sensitivity considerations.
Funding
Consejo Nacional de Ciencia y Tecnología (CB 254460, CB A1-S-18696); Universidad Autónoma de San Luis Potosí.
Acknowledgments
We thank Luis Orozco for useful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. N. Bezginov, T. Valdez, M. Horbatsch, A. Marsman, A. C. Vutha, and E. A. Hessels, “A measurement of the atomic hydrogen Lamb shift and the proton charge radius,” Science 365(6457), 1007–1012 (2019). [CrossRef]
2. C. Smorra, S. Sellner, M. J. Borchert, J. A. Harrington, T. Higuchi, H. Nagahama, T. Tanaka, A. Mooser, G. Schneider, M. Bohman, K. Blaum, Y. Matsuda, C. Ospelkaus, W. Quint, J. Walz, Y. Yamazaki, and S. Ulmer, “A parts-per-billion measurement of the antiproton magnetic moment,” Nature 550(7676), 371–374 (2017). [CrossRef]
3. I. A. Robinson and S. Schlamminger, “The watt or Kibble balance: a technique for implementing the new SI definition of the unit of mass,” Metrologia 53(5), A46–A74 (2016). [CrossRef]
4. H. Muller, A. Peters, and S. Chu, “A precision measurement of the gravitational redshift by the interference of matter waves,” Nature 463(7283), 926–929 (2010). [CrossRef]
5. R. H. Parker, C. Yu, W. Zhong, B. Estey, and H. Muller, “Measurement of the fine-structure constant as a test of the Standard Model,” Science 360(6385), 191–195 (2018). [CrossRef]
6. G. W. Biedermann, X. Wu, L. Deslauriers, S. Roy, C. Mahadeswaraswamy, and M. A. Kasevich, “Testing gravity with cold-atom interferometers,” Phys. Rev. A 91(3), 033629 (2015). [CrossRef]
7. D. O. Sabulsky, I. Dutta, E. A. Hinds, B. Elder, C. Burrage, and E. J. Copeland, “Experiment to Detect Dark Energy Forces Using Atom Interferometry,” Phys. Rev. Lett. 123(6), 061102 (2019). [CrossRef]
8. R. Charriere, M. Cadoret, N. Zahzam, Y. Bidel, and A. Bresson, “Local gravity measurement with the combination of atom interferometry and Bloch oscillations,” Phys. Rev. A 85(1), 013639 (2012). [CrossRef]
9. Y. Bidel, N. Zahzam, C. Blanchard, A. Bonnin, M. Cadoret, A. Bresson, D. Rouxel, and M. F. Lequentrec-Lalancette, “Absolute marine gravimetry with matter-wave interferometry,” Nat. Commun. 9(1), 627 (2018). [CrossRef]
10. Q. Hu, J. Yang, Y. Luo, A. Jia, C. Wei, and Z. Li, “A theoretical analysis and determination of the technical requirements for a Bragg diffraction-based cold atom interferometry gravimeter,” Optik 131, 632–639 (2017). [CrossRef]
11. S. S. Szigeti, J. E. Debs, J. J. Hope, N. P. Robins, and J. D. Close, “Why momentum width matters for atom interferometry with Bragg pulses,” New J. Phys. 14(2), 023009 (2012). [CrossRef]
12. S. M. Dickerson, J. M. Hogan, A. Sugarbaker, D. M. S. Johnson, and M. A. Kasevich, “Multiaxis Inertial Sensing with Long-Time Point Source Atom Interferometry,” Phys. Rev. Lett. 111(8), 083001 (2013). [CrossRef]
13. T. van Zoest, N. Gaaloul, Y. Singh, H. Ahlers, W. Herr, S. T. Seidel, W. Ertmer, E. Rasel, M. Eckart, E. Kajari, S. Arnold, G. Nandi, W. P. Schleich, R. Walser, A. Vogel, K. Sengstock, K. Bongs, W. Lewoczko-Adamczyk, M. Schiemangk, T. Schuldt, A. Peters, T. Konemann, H. Muntinga, C. Lammerzahl, H. Dittus, T. Steinmetz, T. W. Hansch, and J. Reichel, “Bose-Einstein Condensation in Microgravity,” Science 328(5985), 1540–1543 (2010). [CrossRef]
14. L. Liu, D.-S. Lu, W.-B. Chen, T. Li, Q.-Z. Qu, B. Wang, L. Li, W. Ren, Z.-R. Dong, J.-B. Zhao, W.-B. Xia, X. Zhao, J.-W. Ji, M.-F. Ye, Y.-G. Sun, Y.-Y. Yao, D. Song, Z.-G. Liang, S.-J. Hu, D.-H. Yu, X. Hou, W. Shi, H.-G. Zang, J.-F. Xiang, X.-K. Peng, and Y.-Z. Wang, “In-orbit operation of an atomic clock based on laser-cooled 87Rb atoms,” Nat. Commun. 9(1), 2760 (2018). [CrossRef]
15. H. Muntinga, H. Ahlers, M. Krutzik, A. Wenzlawski, S. Arnold, D. Becker, K. Bongs, H. Dittus, H. Duncker, N. Gaaloul, C. Gherasim, E. Giese, C. Grzeschik, T. W. Hansch, O. Hellmig, W. Herr, S. Herrmann, E. Kajari, S. Kleinert, C. Lammerzahl, W. Lewoczko-Adamczyk, J. Malcolm, N. Meyer, R. Nolte, A. Peters, M. Popp, J. Reichel, A. Roura, J. Rudolph, M. Schiemangk, M. Schneider, S. T. Seidel, K. Sengstock, V. Tamma, T. Valenzuela, A. Vogel, R. Walser, T. Wendrich, P. Windpassinger, W. Zeller, T. van Zoest, W. Ertmer, W. P. Schleich, and E. M. Rasel, “Interferometry with Bose-Einstein Condensates in Microgravity,” Phys. Rev. Lett. 110(9), 093602 (2013). [CrossRef]
16. S. Kulas, C. Vogt, A. Resch, J. Hartwig, S. Ganske, J. Matthias, D. Schlippert, T. Wendrich, W. Ertmer, E. M. Rasel, M. Damjanic, P. Webels, A. Kohfeldt, E. Luvsandamdin, M. Schiemangk, C. Grzeschik, M. Krutzik, A. Wicht, A. Peters, S. Herrmann, and C. Lammerzahl, “Miniaturized Lab System for Future Cold Atom Experiments in Microgravity,” Microgravity Sci. Technol. 29(1-2), 37–48 (2017). [CrossRef]
17. J. Ye, H. J. Kimble, and H. Katori, “Quantum State Engineering and Precision Metrology Using State-Insensitive Light Traps,” Science 320(5884), 1734–1738 (2008). [CrossRef]
18. Q.-Q. Hu, Y.-K. Luo, A.-A. Jia, C.-H. Wei, S.-H. Yan, and J. Yang, “Scheme for suppressing atom expansion induced contrast loss in atom interferometers,” Opt. Commun. 390, 111–116 (2017). [CrossRef]
19. B. Cheng, Z. Wang, B. Wu, A. Xu, Q. Wang, and Q. Lin, “The effects of the non-uniformity of Raman laser beams on an atom interferometer,” Eur. Phys. J. D 68(11), 343 (2014). [CrossRef]
20. N. Mielec, M. Altorio, R. Sapam, D. Horville, D. Holleville, L. A. Sidorenkov, A. Landragin, and R. Geiger, “Atom interferometry with top-hat laser beams,” Appl. Phys. Lett. 113(16), 161108 (2018). [CrossRef]
21. A. López-Vázquez, Y. M. Torres, M. S. Billión, W. M. Pimenta, J. A. Franco-Villafañe, and E. Gomez, “Experimental generation of a flat-top beam profile in a stable ring cavity,” Opt. Lett. 44(17), 4428 (2019). [CrossRef]
22. A. Trimeche, M. Langlois, S. Merlet, and F. Pereira Dos Santos, “Active Control of Laser Wavefronts in Atom Interferometers,” Phys. Rev. Appl. 7(3), 034016 (2017). [CrossRef]
23. R. Karcher, A. Imanaliev, S. Merlet, and F. Pereira Dos Santos, “Improving the accuracy of atom interferometers with ultracold sources,” New J. Phys. 20(11), 113041 (2018). [CrossRef]
24. M.-K. Zhou, Q. Luo, L. L. Chen, X.-C. Duan, and Z.-K. Hu, “Observing the effect of wave-front aberrations in an atom interferometer by modulating the diameter of Raman beams,” Phys. Rev. A 93(4), 043610 (2016). [CrossRef]
25. A. Louchet-Chauvet, T. Farah, Q. Bodart, A. Clairon, A. Landragin, S. Merlet, and F. Pereira Dos Santos, “The influence of transverse motion within an atomic gravimeter,” New J. Phys. 13(6), 065025 (2011). [CrossRef]
26. G. A. Olivares-Rentería, D. A. Lancheros-Naranjo, E. Gomez, and J. A. Franco-Villafañe, “Quantum gravimetry in the same internal state using composite light Raman pulses,” Phys. Rev. A 101(4), 043613 (2020). [CrossRef]
27. D. L. Butts, J. M. Kinast, B. P. Timmons, and R. E. Stoner, “Light pulse atom interferometry at short interrogation times,” J. Opt. Soc. Am. B 28(3), 416 (2011). [CrossRef]
28. M. Fox, Quantum Optics: An Introduction (Oxford Master Series in Physics, 2006).
29. M.-K. Zhou, L.-L. Chen, Q. Luo, K. Zhang, X. C. Duan, and Z.-K. Hu, “Effect of the Gaussian distribution of both atomic cloud and laser intensity in an atom gravimeter,” Phys. Rev. A 93(5), 053615 (2016). [CrossRef]
30. N. Arias, V. Abediyeh, S. Hamzeloui, and E. Gomez, “Low phase noise beams for Raman transitions with a phase modulator and a highly birefringent crystal,” Opt. Express 25(5), 5290 (2017). [CrossRef]
31. V. M. Valenzuela, L. Hernandez, and E. Gomez, “High power rapidly tunable system for laser cooling,” Rev. Sci. Instrum. 83(1), 015111 (2012). [CrossRef]
32. Y. Luo, S. Yan, Q. Hu, A. Jia, C. Wei, and J. Yang, “Contrast enhancement via shaped Raman pulses for thermal cold atom cloud interferometry,” Eur. Phys. J. D 70(12), 262 (2016). [CrossRef]
33. G. A. Reider, Photonics An Introduction by Georg A Reider., Springer eBook Collection (Springer International Publishing, 2016).
34. Z.-Y. Wang, T. Chen, X.-L. Wang, Z. Zhang, Y.-F. Xu, and Q. Lin, “A precision analysis and determination of the technical requirements of an atom interferometer for gravity measurement,” Front. Phys. China 4(2), 174–178 (2009). [CrossRef]
35. D. S. Weiss, B. C. Young, and S. Chu, “Precision measurement of $\hbar /m_Cs$ based on photon recoil using laser-cooled atoms and atomic interferometry,” Appl. Phys. B 59(3), 217–256 (1994). [CrossRef]
36. P. Cheinet, B. Canuel, F. Pereira Dos Santos, A. Gauguet, F. Yver-Leduc, and A. Landragin, “Measurement of the Sensitivity Function in a Time-Domain Atomic Interferometer,” IEEE Trans. Instrum. Meas. 57(6), 1141–1148 (2008). [CrossRef]
