A scalable, symmetric atom interferometer for infrasound gravitational wave detection

The direct observation of gravitational waves with laser interferometers^1,2 marks the beginning of a new area in astronomy with new searches targeting signals in a broader frequency band with a variety of detectors. One class of proposed detectors relies on atom interferometers rather than on macroscopic mirrors as inertial references. Indeed, both vertical^3–10 or horizontal^10–12 baselines interrogated by common laser beams propagating along those baselines have been proposed for terrestrial observatories.¹³ In this article, we present a new class of symmetric atom interferometers enabling single and multi-loop^14–16 geometries for broad- and narrow-band detection in the frequency range¹⁷ of 0.3–5 Hz.

Today's detectors are based on laser interferometers and operate in the acoustic frequency band between ten and hundreds of hertz.^1,2,18,19 Future space-borne interferometers, such as LISA,^20–24 are designed to target signals in the range of millihertz to decihertz. Detectors operating in the mid-frequency band^25–27 or improving on the frequency band of current ground-based devices^28,29 have also been proposed and investigated. Moreover, recent results from pulsar timing arrays indicate the existence of a signal at ultralow frequencies.^13,30–33

After first suggestions exploiting mechanical resonators^34–36 and atomic spectroscopy such as hole burning,^37–40 matter-wave interferometers,^{3,4,6,11,12,41–47} and optical clocks⁴⁸ are pursued to search for sources of gravitational waves in the infrasound domain featuring frequencies inaccessible for today's ground-based detectors. Waves in this band are emitted, for example, by inspiraling stellar-mass binaries days or hours before they merge within fractions of a second^49,50 as the first observed event GW150914,^1,2 but also by inspirals around an intermediate-mass black hole or mergers of two intermediate-mass black holes, which never emit significantly in the Advanced LIGO band.^13,17 Hence, atom-interferometric observation in the infrasound band could be combined with standard astronomical observations for improved angular localization and prediction of upcoming events for other detectors.⁵¹ The vast benefits of joint observations manifested itself in the case of a merger of neutron stars.⁵² 

Our proposed detector is based on a type of interferometer where matter waves form a single or several folded loops of symmetric shape.⁵³ Compared to single-loop interferometers, dual- or triple-loop geometries suppress dependencies on input parameters.^11,46,54,55 While a folded double-loop interferometer is also possible,^56,57 we choose a folded triple-loop scheme as the study case of this paper due to its inherent features.

An antenna employing folded multi-loop interferometers shows three distinct advantages: (i) The detector is less susceptible to environmental perturbations, and less restrictive on the expansion rate of the atomic ensemble, which otherwise needs to be at the energy level of femtokelvins. (ii) It combines the scalability of arm lengths of horizontal and the single-laser link of vertical antenna types, (iii) and it is less susceptible to technical noise such as the pointing jitter of the atomic sources.

In this article, we explain the concept of our detector and compare a symmetric single-loop interferometer with a folded multi-loop one, confronting their scaling, spectral responses, and critical parameters. Similar to the matter-wave interferometric gravitational antenna MIGA^12,58,59 and laser interferometers,^1,2 our detector concepts have two perpendicular, horizontal arms, depicted in Fig. 1(a), suppressing laser frequency noise at lower frequencies as in light interferometers.^1,18,21 Two light-pulse atom interferometers, separated by a distance L, and located in each arm, are sensitive to the phase of the light pulses traveling along the x and y axes between the interferometers and are employed for coherent manipulation of the atoms. A gravitational wave, propagating along the z axis, modifies the geodesic of the light connecting the interferometers and modulates its phase which appears in the differential signal of the two atom interferometers.

Our article is organized as follows: in Sec. II, we introduce the symmetric single-loop geometry of our gravitational wave detector based on atom interferometry. Although this arrangement is in principle sensitive to gravitational waves, the need to suppress the spurious Sagnac effect enforces kinetic expansion energies of the atomic ensemble beyond reach. For this reason, we propose in Sec. III the multi-loop geometry and discuss in great detail in Sec. IV the necessary experimental requirements. We conclude by summarizing our results in Sec. V.

In order to focus on the essential points but, at the same time, keep our article self-contained, we have included detailed discussions in four  Appendixes. In  Appendix A, we estimate the parameters for the twin lattice used to re-launch the atoms.  Appendix B is dedicated to a brief analysis of the influence of timing errors on the coherent re-launch. Finally, requirements on the parameters specifying the single-loop and the triple-loop are discussed in  Appendixes C and  D, respectively.

Hence, three parameters determine the spectral sensitivity of such a detector: (i) The distance L between the interferometers, (ii) the number of photon momenta transferred differentially between the two interferometer paths corresponding to the wave number k, and (iii) the pulse-separation time T. The ability to change T in this setup allows for adjusting the frequency of the maximum spectral response which scales with the inverse of T.

Unfortunately, spurious effects can mask the phase induced by a gravitational wave. For example, in the single-loop interferometer, the shot-to-shot jitter of the initial velocity v and position r causes noise as it enters the phase terms^75,76 $2(k×v)·ΩT2, k·Γ r T2$⁠, and $k·Γ v T3$ arising from the Sagnac effect and gravity gradients. Here, $Ω$ and Γ denote Earth's angular velocity and the gravity gradient tensor, respectively.⁷⁷ 

Whereas the impact of gravity gradients can be compensated,^78,79 typical parameters targeting a competitive strain sensitivity enforce kinetic expansion energies of the atomic ensembles to suppress the spurious Sagnac effect corresponding to femtokelvin temperatures. Such a regime is beyond reach even with delta-kick collimation.^80–83 

This challenge can be mitigated by a suitably tailored geometry with atomic trajectories following multiple loops inside the interferometer^46,84 as depicted in Fig. 1(c). In this scheme, the interferometer begins after a vertical launch at time t = 0, with the first interaction with the symmetric twin lattice creating a coherent superposition of two momentum states. Subsequently, the atoms are coherently reflected at t = T along the horizontal direction so that they complete their first loop and the paths of the atom interferometer corresponding to the trajectories of the two partial atomic wave functions intersect at the bottom at $t=(9/5)T$⁠. Here, they are launched upwards again,⁷⁰ and their horizontal motion is inverted at the vertex in the middle of the interferometer at $t=3T$⁠. After completion of the second loop, they are again relaunched at $t=(21/5)T$⁠, once more redirected at $t=5T$⁠, and finally recombined at $t=6T$ to close the interferometer.

In order to suppress spurious phase contributions due to the Sagnac effect and gravity gradients, the wave numbers of the five pulses are adjusted⁴⁶ to k, $94k, 52k, 94k$⁠, k, corresponding to the differential momentum between the two paths of the interferometer.

This scheme shares the advantage of a horizontal detector^12,59 enabling us to operate the interferometers separated by up to several ten kilometers similar to light interferometers.^1,2 Moreover, it requires only a single horizontal beam axis in each arm for the coherent manipulation of the atoms. Here, T can be tuned to specific frequencies of the gravitational waves¹⁴ similar to vertical antennas, where the light beam propagates along the direction of the free fall of the atoms. This feature differs from the configuration with two horizontal beams, in which the height difference between them constrains T.

The folded triple-loop interferometer mitigates several important drawbacks of other concepts. For example, the specific combination of three loops and increased momentum transfer by the central pulses compared to the initial and final pulse renders this interferometer insensitive to fluctuations of initial position and velocity,⁴⁶ mitigating the requirement for femtokelvin energies.

Unfortunately, folded multi-loop interferometers are susceptible to pointing noise of the relaunch vector. In a model with a vanishing atom–light interaction time and no timing errors, the requirements remain comparable to those for a single-loop geometry and have to be limited to picoradians. For relaunches not centered on the intersections of the trajectories, see Fig. 1(c), new terms appear if their directions are not properly aligned. See  Appendix B for the impact of timing errors. Table I summarizes the requirements imposed by the two geometries. Mean-field effects constrain the initial density and consequently the size of the atomic ensemble for a given number of atoms.^6,85 In the single-loop scheme, these constrains introduce requirements on the beam-splitting fidelity and gravity-gradient compensation⁷⁹ to avoid other phase errors linked to large size. However, they are relaxed in the multi-loop scheme as briefly discussed in  Appendix D.

Figure 2 compares the spectral strain sensitivities, obtained by the broad-band mode of the single- (green lines) and triple-loop detector (red lines), and the narrow-band mode (cyan lines) of the multi-loop geometry, with a signal (orange dashed-dotted line) generated by a black hole binary.⁴⁹ They are also confronted with the anticipated strain sensitivity of the space-borne detector LISA²¹ (black dotted line) and the operating advanced LIGO^86,87 (brown dashed-dotted line). We emphasize that the sensitivity curves of our detector concepts fill the gap between LISA and advanced LIGO, enabling the terrestrial detection of infrasound gravitational waves. To put our proposed scheme into the context of adjacent frequency bands and third-generation optical gravitational wave detectors, we present in  Appendix E the strain sensitivity over a broader range of frequencies including a rough estimate for the Einstein Telescope.²⁹ 

With respect to the strain sensitivity, our designs share assumptions with other proposals^12,59 based on atom interferometry. For two 10-km-long arms, we foresee an intrinsic total phase noise of 1 µ $rad/Hz$ achieved by 10⁹ atoms starting every 100 ms with an upward velocity of gT/2, where g is the gravitational acceleration. Moreover, we expect 20 dB sub-shot-noise detection,⁸⁸ a maximum of $T∼260 ms$⁠, and the symmetric transfer of 1000 photon recoils at the rubidium D2 line in each direction, see  Appendix A for parameter estimates.^60,89–96

Recent research addresses relevant building blocks for the presented geometry: the anticipated high launch and beam-splitting efficiencies during large momentum transfer are expected to benefit^{60,70,97–100} from delta-kick collimated ensembles of rubidium⁸¹ and Bose–Einstein condensates^80,82,83 which feature residual atomic expansion rates of less than 100 µm/s, corresponding to ∼100 pK for ⁸⁷Rb. Additionally, such expansion rates enable ensemble sizes of about 500 µm after 1.6 s total time of flight in broad-band mode or 1100 µm after 4.7 s total time of flight in resonant mode to ease constraints on the beam diameters for coherent manipulation and the detection process. Relaunches¹⁰¹ can be implemented by two counterpropagating light fields injected from different optical ports^102–104 or two retro-reflected light fields injected from the same optical fiber port.⁷⁰ In the latter case, the relative pointing of the relaunch vector is coupled to the orientation of the retro-reflection mirror, which can be placed on a dedicated isolation system to suppress pointing noise. Moreover, these implementations can be combined with techniques that tailor individual light pulses and enhance their efficiency even further. Such methods of an optimal control theory potentially relax the constraints on the longitudinal atomic momentum distribution, the expansion rate of the atomic cloud, and uncompensated vibrational noise. They have already proven successful in the development of advanced and robust atom-interferometer pulses, primarily in the context of single Bragg diffraction,^{100,105–107} but have recently been extended to double Bragg diffraction schemes.⁶⁹ Combining those methods with Bose–Einstein condensates is compatible^108,109 with the future utilization of entangled atoms.^110,111

In addition, the assumed production rate of atomic ensembles enables an interleaved^112,113 operation of several interferometers. Indeed, for the broad-band mode, we employ for the single- and triple-loop detector three interleaved interferometers with the free-fall times $T1=T, T2=0.9T$⁠, and $T3=0.7T$ to avoid peaks in the transfer function⁴³ resulting in the depicted intrinsic strain sensitivities (green, red dashed line). To prevent a temporal or spatial overlap of the atom–light interaction of the individual interferometers and thus to avoid crosstalk, the duration of each interaction is assumed to be less than 6 ms for the chosen parameter set. This choice is consistent with the achieved symmetrical transfer of 1008 photon recoils in less than 4 ms.¹¹⁴ 

Moreover, we extend the triple-loop scheme to a narrow-band mode¹⁴ featuring 3n loops, with a sensitivity enhanced by a factor $n∈ℕ$ at a specific frequency determined by the free-fall time T. This resonant-mode detection comes at the cost of an decreased bandwidth and an increase in the overall duration 6nT of the interferometer that obviously must not exceed the duration of the signal in the aimed frequency band and must still allow for sufficient integration time. In addition, scaling up the overall duration of the interferometer will also cause problems due to dilute atomic ensembles with large spatial extend. Moreover, T is limited by the vertical height of the chamber and the spatial separation of the atom–light interaction regions, in contrast to vertical setups, where increasing n reduces the available baseline.¹⁵ The cyan line in Fig. 2 shows the sensitivity for the frequency 0.85 Hz, corresponding to $T=260 ms$⁠, and n = 3. According to the Nyquist theorem,¹¹⁵ the interleaved operation with a cycle frequency of 10 Hz implies a high-frequency cut-off at 5 Hz. This cycle frequency assumes a preparation time of 100 ms, for which we anticipate the combination of multiple sources¹¹⁶ or subsequently launching a fraction of an atomic samples. For this set of parameters, the overall duration of one interferometer is 1.6 s. An event in the aimed detection band of 0.85 Hz persists⁴⁹ for 3600 s. Aiming, e.g., for ten data points during that observation time implies a maximum interferometer time of $6nT≤360 s$⁠, implying $n≤240$⁠. Technical constraints will likely imply a lower limit. They may be linked to losses during the beam splitting operations and thereby implicitly to the ensemble size at the end of the interferometer or explicitly to the final-size requirements for an efficient detection of the interferometer output ports. Here, a choice of $6nT·100$ µ $m/s≤1 cm$ as maximum ensemble size¹⁰³ corresponds to n = 30.

Next to atomic shot noise, other technical and environmental noise sources¹¹⁸ deteriorate the strain sensitivity. Their impact is evaluated by the transfer function⁷³ of the interferometers. Figure 2 shows that the intrinsic strain sensitivity of the single-loop⁵⁹ (SL, green dashed line) and folded triple-loop (FTL, red dashed line) interferometer decays with $f−2$ and $f−4$ at lower frequencies. Below 1 Hz, vibration models^59,119,120 of the retroreflecting mirrors and Newtonian noise lead to a limitation of the strain sensitivity (green, red, cyan solid lines). The implementation of several, specifically spaced pairs of atom interferometers in each arm, combined with our proposed geometry, may allow for the suppression of Newtonian noise⁵⁹ (see also  Appendix E), which limits the sensitivity at low frequencies. Mirror vibrations enter in the differential signal of the interferometers due to the finite speed c of light causing a delay of $2L/c≈70$ µs and are modeled with a differential weighting function.⁷³ A suspension system isolating the retroreflection mirror^117,121 against seismic noise⁵⁹ reduces this contribution to a level similar to the Newtonian noise.

We emphasize that the spectral sensitivity, in particular, the low-frequency cut-off, can be tuned by the free-fall time T of the atoms. In the case of the triple-loop interferometer, the maximum value of T is only determined by the free-fall height. For example, in a ∼1 m-high chamber T can be tuned up to 260 ms.

In contrast, in the single-loop interferometer, the tuning capability is restricted by the separation of the two twin lattices as illustrated in Fig. 1(b). With T = 260 ms and adding 12 ms for beam splitting, a height difference of $37 cm$ is needed. In our scenario, the triple-loop detector surpasses the signal-to-noise ratio of the single-loop by a factor of about 2 for the broad-band, and about 9 for the narrow-band mode.

The relaunches introduce additional noise terms to the folded triple-loop interferometer. See  Appendix B for a discussion of the error due to imperfect pointing and timing as well as the approach to adjust the absolute pointing with the interferometer itself. Implementations at the most quiet sites, featuring a residual rotation noise¹²² of $2×10−11 (rad/s)/Hz$ enforce either measures such as mechanical noise dampening by one order of magnitude, or atom interferometric measurements sharing a relaunch pulse for two interleaved cycles^{112,113,123,124} to comply with the requirement on pointing noise. Having the second relaunch of the first cycle as the first of a subsequent one leads to a behavior of $∼1/t$ for an integration time t. For a white pointing noise, assuming a normal distribution with $σ=2×10−11 rad$ at 1 s, the limit set by the intrinsic noise would be reached after an integration time of 400 s. As an example, GW150914 spent approximately 1400 s to chirp from 0.9 to 1.1 Hz,⁴⁹ thus allowing for sufficient integration time to reach the intrinsic noise.

To summarize, present designs of terrestrial detectors of infrasound gravitational waves based on atom interferometry face several major challenges related to scalability and atomic expansion often necessitating femtokelvin temperatures. Therefore, we propose atomic folded-loop interferometers for horizontal antennas, which overcome these stringent requirements at the cost of a very stable relaunch of the atoms.

The presented approaches are benefiting from symmetric manipulation of the atomic ensemble in which identical amounts of photon recoils are transferred along both interferometer paths. The schemes studied in this article take the form of either symmetric single or multi-loop geometries. In particular, the folded triple loop interferometer has the advantage that it is insensitive to fluctuations in the atomic initial position and velocity due to a cancelation of Sagnac phases by a suitable choice of transferred momenta. In addition, they also combine the advantages of horizontal detectors and vertical setups. As horizontal antennas, they display a scalability of the arm length, and do not rely on deep shafts. Furthermore, the tunability of the spectral response of vertical detectors is maintained, especially the low-frequency cutoff, so that broadband and resonant detection modes are possible. Combining our proposed scheme with techniques that suppress the strict limitations given by Newtonian noise through correlated interferometer arrays⁵⁹ opens a new pathway to reach strain sensitivities of the order of $7×10−21/Hz$ at 1 Hz in terrestrial detectors.

The authors thank Víctor J. Martínez-Lahuerta for valuable and fruitful discussions. This work is supported by “Niedersächsisches Vorab” through the “Quantum- and Nano-Metrology (QUANOMET)” initiative within Project No. QT3, the CRC 1227 DQmat within Project Nos. B07 and B09, the CRC 1128 geo-Q within the project A02, the QUEST-LFS, the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy—EXC-2123-B2, and the German Space Agency at the German Aerospace Center (Deutsche Raumfahrtagentur im Deutschen Zentrum für Luft- und Raumfahrt, DLR) with funds provided by the Federal Ministry for Economic Affairs and Climate Action (Bundesministerium für Wirtschaft und Klimaschutz, BMWK) due to an enactment of the German Bundestag under Grant Nos. DLR 50WM1556, 50WM1641, 50WM1952, 50WM1956, 50WP1700, 50WM2250A-2250E (QUANTUS+), 50WM2450A-E (QUANTUS-VI), 50NA2106 (QGyro+), and 50WM2177-2178 (INTENTAS). D.S. gratefully acknowledges funding by the Federal Ministry of Education and Research (BMBF) through the funding program Photonics Research Germany under contract number 13N14875. E.G. thanks the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) for a Mercator Fellowship within CRC 1227 (DQ-mat). A.R. is supported by the Q-GRAV Project within the Space Research and Technology Program of the German Aerospace Center (DLR) W.P.S. is grateful to Texas A&M University for a Faculty Fellowship at the Hagler Institute of Advanced Study at Texas A&M University, and to Texas A&M AgriLife Research for the support of his work.

The authors have no conflicts to disclose.

C. Schubert: Conceptualization (lead); Formal analysis (lead); Writing – original draft (equal); Writing – review & editing (equal). D. Schlippert: Formal analysis (supporting); Writing – review & editing (equal). M. Gersemann: Formal analysis (equal); Writing – review & editing (equal). S. Abend: Formal analysis (supporting); Writing – review & editing (equal). E. Giese: Formal analysis (equal); Writing – review & editing (equal). A. Roura: Formal analysis (supporting); Writing – review & editing (equal). W. P. Schleich: Writing – review & editing (equal). W. Ertmer: Writing – review & editing (equal). E. M. Rasel: Conceptualization (equal); Formal analysis (supporting); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article.

Initially, sequential double Bragg diffraction^61,63,65,66 pulses couple the zero-momentum state to the $±8 ℏks$ momentum states where λ denotes the wavelength of the atomic transition and $ks≡2π/λ$ is the single-photon wave number. This momentum separation ensures an efficient transfer to higher momentum states with subsequent Bloch oscillations induced by counter-propagating lattices.⁶⁰ We expect a transfer of $1000 ℏks$ differential momentum in 3 ms, leading to a total duration of 6 ms for a composite π/2 pulse. The two trajectories separate with a velocity of $ℏk/mRb=2000 ℏks/mRb$ and are later deflected toward each other in a similar way in about 12 ms with $(5/4) ℏk/mRb=2500 ℏks/mRb$ where $mRb$ denotes the atomic mass of rubidium. Symmetric de-acceleration by Bloch lattices, sequential double Bragg π pulses, and symmetric acceleration by Bloch lattices invert the momentum.

In case of ⁸⁷Rb we have $λ=780 nm$ and the maximum wave packet separation would reach $∼4.0$ m.

The additional relaunch, required in the folded triple-loop interferometer, can be based on the same combination of Bloch lattices and double Bragg diffraction with a duration of 15 ms for the upward deflection.⁷⁰ 

The differential measurement scheme of our detectors suppresses timing errors of the beam splitters to first order, because they are common to both atom interferometers contributing to the signal. On the contrary, the relaunches affect both atom interferometers individually. Relaunches that are not centered around the intersections of the trajectories induce a phase shift if in addition the projection of the relaunch vector onto the effective wave vector of the beam splitters is nonzero.

We estimate the impact on the detector by determining the mean trajectory of the atoms and subsequent calculation of the phase shift.⁷⁶ Here, we neglect rotations and gravity gradients which are discussed later on without timing errors.

We insert $t1=(9/5)T−(τ1/2)+δτ1, t2=(9/5)T+(τ1/2)+δτ1, t3=(21/5)T−(τ2/2)+δτ2, t4=(21/5)T+(τ2/2)+δτ2$ reflecting our interferometer. Here, $τ1=t2−t1$ and $τ2=t4−t3$ denote the durations of the respective relaunch pulses, displaced by $δτ1$ and $δτ2$ in time from the intersection of the trajectories.

Gravity gradients, rotations, and other quantities can induce phase shifts in the differential signal of the two atom interferometers if the starting conditions are not matched. The estimation of phase terms to leading order^75,76,125 in the limit of infinitely short pulses gives rise to the requirements for a single-loop interferometer summarized in Table II. Since some of these contributions intrinsically of higher order, the phase expressions listed in the table have to be seen as an order-of-magnitude estimate.

Ideally, instabilities in the starting conditions are shot-noise limited. Assuming that other noise contributions can be neglected, the instability of the mean position $rm(n)=σr/n·N$ and velocity $vm(n)=σv/n·N$ of the wave packet for n cycles depend on the number N of atoms per cycle, the initial radius σ_r, and expansion rate σ_v of the wave packet.

Gravity gradients Γ impose a limit on the instabilities of the center and the velocity of the wave packet to $4.3×10−10 m$ at 1 s and $1.7×10−9 m/s$ at 1 s, respectively, pointing instabilities of the initial launch to $∼10−9 rad$ at 1 s, constraining initial radius and expansion rate of the wave packet to $4.3×10−5 m$ and $1.7×10−4 m/s$ for 10¹⁰ atoms per second, respectively. If the gravity gradients are equal and known at the two light-pulse atom interferometers interrogated by the same laser beam, an adjustment of the wave number at the second pulse can be introduced to relax these requirements.⁷⁹ 

Significantly more stringent conditions are set by the Sagnac effect caused by the rotation of the earth Ω which restricts instabilities in the pointing of the initial launch to the level of $2.2×10−12 rad$ at 1 s, and in the center of wave packet to $5.6×10−12 m/s$ at 1 s, implying a residual expansion rate of at most $5.6×10−7 m/s$ corresponding to few femtokelvins. The latter is challenging even for delta-kick collimated, dilute Bose–Einstein condensates.

Beam splitters which are not exactly perpendicular to gravity by an angle β induce a phase shift in a single interferometer that is suppressed in the differential signal. If the gravitational acceleration g differs at the locations of the two atom interferometers by $δg$⁠, a spurious phase shift remains. Assuming $δg=10−7g$⁠, instabilities in β have to be limited to $10−10 rad$⁠. A misalignment at the level of $10−10 rad$ implies less stringent requirements on initial radius and expansion of the wave packet than gravity gradients and rotations, even when assuming a compensation of gravity gradients⁷⁹ by a factor of 10³.

In addition, instabilities in density shifts $∝δ N/(σr)3$ for an ensemble with N atoms and radius $σr$ restrict the maximum initial density and fluctuations in the beam splitting fidelity δ which affects the relative density of atoms in the two arms.^6,85 Assuming $δ∼3×10−5$⁠, a compensation of the phase shift induced by gravity gradients⁷⁹ has to be implemented to at least a factor of 100 to simultaneously comply with the position requirement scaling as $∝σr/N$⁠.

Analogously to  Appendix C, we rely on established methods^46,75,76,125 to estimate phase terms to leading order for a triple-loop interferometer, deduce the requirements, and summarize them in Table III.

Compared to a single-loop arrangement, a coupling of the starting conditions to rotations and accelerations only appears in higher order terms, and leads to requirements which are relaxed by several orders of magnitude. Beam splitter tilts imply tighter bounds on the initial radius and expansion of the wave packet of $1.8×10−2 m$ and $1.6×10−2 m/s$⁠, still less stringent by orders of magnitude than for the single-loop geometry. Due to the modest requirement on the initial radius, the initial density can be adjusted to reduce the impact of interactions.^6,85