Quantum test of the equivalence principle and space-time aboard the International Space Station

While everybody can feel the force of gravity, it is a stranger among the four forces known in physics. For one, it accelerates all objects in the same way, regardless of, e.g., their structure, mass, charge, or composition. This observation, known as the weak equivalence principle (WEP) or universality of free fall (UFF), has been extended to the Einstein equivalence principle, which states that gravity is locally equivalent to the effects of being in an accelerated frame of reference for any experiment. 'Equivalence' here means that there is no experiment that can tell the two apart and 'locally' means that the statement is restricted to sufficiently small regions of spacetime, such that gravity gradients are negligible.

The validity of the equivalence principle is deeply ingrained in the current gravitational theory; without it, general relativity could not be true [31]. Experimental tests of Lorentz invariance [32], local position invariance [33], and the weak equivalence principle [34] have shown that nature adheres closely to this principle. But it doesn't have to be so. Modern theories to unify gravity and quantum mechanics operate at energy scales so high that direct experiments are impossible. Violations of the equivalence principle, which can be parameterized by the standard model extension (SME) [35–37], are among the most promising candidates for probing Planck-scale physics at low energy [5, 38–40]. For a review of the science motivations of next-generation, ultra-high precision equivalence principle tests and their expected implications for the foundations of contemporary gravitational theories, refer to [2, 6, 25] and references therein.

WEP violations are quantified by the Eötvös parameter, the normalized differential acceleration of two test particles (A and B) under the influence of gravity:

$$\begin{eqnarray}&&\eta (A,B)=2\displaystyle \frac{{g}^{A}-{g}^{B}}{{g}^{A}+{g}^{B}}.\end{eqnarray} \tag{ 1 }$$

Following the formalism in [41], limits on the minimal gravitational SME coefficients $\alpha {({\bar{a}}_{{\rm{e}}{\rm{f}}{\rm{f}}}^{w})}_{0}$ and ${({\bar{c}}^{w})}_{00},$ which describe the size of various types of Lorentz violations in the test framework of the SME, can be extrapolated from $\eta$ by the definition of the total gravitational acceleration of each test mass (m^T):

$$\begin{eqnarray}&&{g}^{T}=g(1-{\beta }^{T}),\;\quad {\beta }^{T}=\;\displaystyle \frac{2\alpha }{{m}^{T}}\sum _{w}{N}^{w}{({\bar{a}}_{{\rm{e}}{\rm{f}}{\rm{f}}}^{w})}_{0}-\displaystyle \frac{2}{3{m}^{T}}\sum _{w}{N}^{w}{m}^{w}{({\bar{c}}^{w})}_{00}.\end{eqnarray} \tag{ 2 }$$

Here, the superscript w takes the values e, p, and n representing the electron, proton, and neutron respectively. For neutral matter, with equal electron and proton numbers (N^e = N^p), the notation can be simplified by using ${({\bar{a}}_{{\rm{eff}}}^{p+e})}_{0}={({\bar{a}}_{{\rm{eff}}}^{p})}_{0}+{({\bar{a}}_{{\rm{eff}}}^{e})}_{0}.$ The projected sensitivity limits for these SME coefficients in QTEST are given in table 1.

Table 1.  SME coefficient sensitivities, extrapolated from the projected QTEST measurement sensitivity of 10⁻¹⁵. The numerical coefficients can be refined by taking into account the nuclear binding energy [42]. Values in braces represent order-of-magnitude estimates for sensitivities arising due to the orbital motion of the ISS. See [43], section VIII, for details and definitions of coordinates.

$\alpha {({\bar{a}}_{{\rm{eff}}}^{p+e})}_{0}$	$5\times {10}^{-14}\;{\rm{GeV}}$	${({\bar{c}}^{p})}_{00}$	$1.5\times {10}^{-13}\;{\rm{GeV}}$
$\alpha {({\bar{a}}_{{\rm{eff}}}^{n})}_{0}$	$5\times {10}^{-14}\;{\rm{GeV}}$	${({\bar{c}}^{n})}_{00}$	$1.5\times {10}^{-13}\;{\rm{GeV}}$
$\alpha {({\bar{a}}_{{\rm{eff}}}^{p+e-n})}_{X}$	$\{{10}^{-9}\;{\rm{GeV}}\}$	${({\bar{c}}^{e})}_{00}$	$3\times {10}^{-10}\;{\rm{GeV}}$
$\alpha {({\bar{a}}_{{\rm{eff}}}^{p+e-n})}_{Y+Z}$	$\{{10}^{-9}\;{\rm{GeV}}\}$	${({\bar{c}}^{n})}_{Q}$	$\{{10}^{-13}\;{\rm{GeV}}\}$
$\alpha {({\bar{a}}_{{\rm{eff}}}^{p+e-n})}_{Y}$	$\{{10}^{-7}\;{\rm{GeV}}\}$	${({\bar{c}}^{n})}_{TJ}$	$\{{10}^{-9}\;{\rm{GeV}}\}$
$\alpha {({\bar{a}}_{{\rm{eff}}}^{p+e-n})}_{Z}$	$\{{10}^{-7}\;{\rm{GeV}}\}$

Measurements with space-based platforms also open the door for unique and/or enhanced searches for EP violations, parameterized by $\alpha {({\bar{a}}_{{\rm{eff}}}^{w})}_{\mu }$ and ${({\bar{c}}^{w})}_{\mu \nu }$ beyond the zero components (μ = ν = 0). In addition to the long free fall times available in space, the orientation and rotation of the test-bodies can be decoupled from the direction of gravity, useful for detection of several Lorentz-violating possibilities. Extrapolating from theoretical analyses by Kostelecký and Tasson [43], it is clear that the projected sensitivity of QTEST could also improve over the current limits of non-zero SME components [44] by orders of magnitude (see table 1).

High sensitivity AI sensors, in orbit around the Earth, will also help to detect nonstandard signals for deviations from general relativity. For example, the EP violations may depend upon the test particles' motion relative to the source of the gravitational field (here, the Earth or the Sun). In technical language, such effects are encoded in higher-mass dimension operators of the SME [45]. These effects have never been the subjects of an experimental study, though incomplete bounds can be derived from solar system observations like perihelion precession [46]. A scenario in which such terms may be nonzero is given by theories that attempt to explain cosmic acceleration by a scalar field (Chameleons, f(R), Gallileons, ...), which could lead to order-one violation of the EP for extended objects such as galaxies or constituents thereof but not in terrestrial experiments [47]. Such theories evade detection by the chameleon mechanism, which suppresses any effects in environments of high density. This renders them invisible in classical experiments but not in quantum experiments [24]. Atom interferometers in orbit, such as QTEST, have the additional advantage that atoms can remain close to fixed objects that generate the chameleon force for seconds instead of tens of milliseconds, producing a sensitivity gain of ten thousand. In addition, there are other theories such as galilleons, for which no method of detection is currently known. Since effects of these theories are strongly suppressed anywhere near the Earth's or other celestial objects, any detection would have to happen in space [48, 49]. This subject is currently under investigation.

2.1.1. Significance of quantum tests

2.1.2. Ultralight dark matter and dark energy

By reprogramming from Mach-Zehnder to Ramsey-Bordé configuration, i.e., firing a π/2- π/2- π/2- π/2 AI laser pulse sequence for Ramsey-Bordé as opposed to the π/2-π-π/2 sequence for Mach-Zehnder (shown in figure 2), QTEST's AIs can measure the quantity h/m for ⁸⁷Rb and ⁸⁵Rb, where m is the atom's mass and h the Planck constant [20, 21]. Combined with the Rydberg constant and the isotopes' mass ratio with the electron (m/m_e), it can determine the fine structure constant α. By defining the Plank constant h, it can lead to a precise definition of SI mass unit based on atomic systems.

The sensitivity to h/m of Ramsey-Bordé AIs comes from the asymmetric velocity patterns of the two interferometer paths. The asymmetry leads to higher mean kinetic energy of atoms traversing through one interferometer arm than through the other arm, thus the AI has first order sensitivity to the photon recoil energy, $h{k}^{2}/2m,$ where k is the wavenumber of the AI laser.

2.2.1. Measuring h/m and the fine structure constant

2.2.2. Absolute atomic masses

AIs are highly sensitive to accelerations (a) with accumulated phase response in Mach-Zehnder configuration given by ϕ = (k_eff · a)T², where k_eff is the effective wavevector and T is the free-evolution-time between interferometer pulses, whose gravity-induced evolution of their quantum state is measured by reading out the interference pattern. The phase measurement sensitivity is therefore determined by the signal-to-noise ratio (SNR) in detecting ϕ and by T. A fundamental limit for the SNR is the atomic shot noise, or quantum projection noise, which is scaled by the finite number of atoms participating in detection as N^−1/2. Reduction of the visibility of the interference fringe affecting SNR is attributed to inhomogeneity and imperfections in the atomic state preparation and detection. These factors are summarily parameterized by the contrast, C = (R_max − R_min)/(R_max + R_min) with R_max (R_min) being the maximum (minimum) normalized population in the excited state after the interferometer sequence. An analysis of the expected fringe contrast in the QTEST apparatus and imaging-based detection schemes for conserving signal contrast at high sensitivity are discussed in section 4.6.

For the optimal system performance, strict control over the atomic cooling and initial state preparation must be maintained throughout the mission lifetime, limiting the practical cycle time to T_C = 70 s, given by 2T = 20 s interferometry interrogation times and <50 s for atomic cloud sample preparation, cooling, state manipulation, detection, and positioning of the atoms and the rotating platform. Assuming that each interferometer has a beam near-resonant to the D₂ transition in rubidium, with k_eff = 2k where k = 2π/(780 nm), contrasts in excess of 50%, and N > 10⁶ detected atoms, the per-shot acceleration sensitivity for each rubidium AI is:

$$\begin{eqnarray}&&\sigma =\sqrt{\displaystyle \frac{1}{N}}\;\displaystyle \frac{1}{C*{k}_{{\rm{eff}}}*{T}^{2}}=1.3\times {10}^{-13}\;g{\rm{.}}\end{eqnarray} \tag{ 3 }$$

The QTEST apparatus relies on differential measurements of two, simultaneously interrogated, atom interferometers (⁸⁵Rb and ⁸⁷Rb) to search for relevant violations of the UFF. For a total measurement time of 12 months, allowing for a multiple of 3-month measurement sets for systematic reduction, the integrated acceleration sensitivity for each differential QTEST UFF measurement set is:

$$\begin{eqnarray}&&{\sigma }_{\eta }=\sqrt{\displaystyle \frac{{T}_{C}({\sigma }_{85}^{2}+{\sigma }_{87}^{2})}{\tau }}=5.3\;\times \;{10}^{-16}\;g,\end{eqnarray} \tag{ 4 }$$

where T_C is the measurement cycle time and τ the total time of the measurement set. Comparing to the state of the art, summarized in table 2, it is anticipated that QTEST will provide two or more orders of magnitude improvement in constraining the Eötvös parameter over any previous test of the WEP.

ISS offers a unique space platform for ultra-cold atom experiments in general and atom interferometry in particular. It has been well-recognized that microgravity in space allows long interrogation times for unprecedented AI sensitivity [99]. Equally important is the device size requirements that are limited only by the atom's drift velocity and the splitting of the atomic trajectories, on the order of several tens of centimeters for T = 10 s, allowing for better local control of the atoms' environment. This is in contrast to the few 100 meters of free fall distance required for terrestrial AIs with the same T. Further, the absence of the large gravity bias is ideally suited for advanced cooling schemes that are necessary to achieve ultra-high precision at long T [8, 13, 100, 101].

QTEST will critically rely on the ability to reverse the gravity relative to the instrument measurement direction on ISS. As will be discussed in the following sections, there exist many systematics that are nearly impossible to otherwise eliminate. Many of the largest systematic effects stem from imperfect overlap of the two atomic species and the existence of the gravity gradient, that always accompanies the presence of the gravity field. Reversing the gravity vector relative to the AI measurement direction will change the resulting systematic effect signal signs with respect to that of the WEP signal. By modulating the gravity direction, therefore, QTEST will be able to reduce the systematics by averaging rather than being overburdened by addressing each systematic effect with brute force.

QTEST uses dual-species atom interferometers and their differential measurements for testing WEP in space, similar to previously proposed mission concepts (see table 2), namely the early NASA QuITE, and ESA QWEP and STE-QUEST [94, 102, 103]. However, QTEST employs unique approaches to reduce or eliminate systematics and to achieve the requisite measurement precision. It will be the first to take advantage of the space environment and actively modulate the gravity signal by a rotating platform mount on ISS. It also integrates well-established evaporative cooling techniques with delta-kick cooling to achieve ultra-low kinetic energies in highly-uniform quadrupole-Ioffe configuration (QUIC) magnetic traps [104, 105]. Finally, it utilizes the recent advances in Bragg-diffraction atom optics and imaging detection for enhanced signal while minimizing systematic errors.

Mounting the AIs on a rotating platform allows for reversal of gravity relative to the orientation of the apparatus, thereby modulating the differential phase for two rubidium AIs according to Δϕ = Δϕ_gcos(ωt) + ϕ_syst, where Δϕ_g is the signal from differential gravitational accelerations of the two gases and ϕ_syst are from external systematics that are not correlated with the rotating platform modulation frequency (ω). Averaging over a time τ by means of a product detector demodulates the gravity signal to DC while all remaining systematics are averaged down, in a simple analogy to a lock-in detector:

$$\begin{eqnarray}&&\displaystyle \frac{2}{\tau }\displaystyle {\int }_{0}^{\tau }{\rm{\Delta }}\phi \;\mathrm{cos}(\omega t){\rm{d}}t\leqslant |{\rm{\Delta }}{\phi }_{g}+\displaystyle \frac{2}{\omega \tau }{\phi }_{{\rm{s}}{\rm{y}}{\rm{s}}{\rm{t}}}|.\end{eqnarray} \tag{ 5 }$$

For example, if gravity is reversed every 10th measurement, for a 3-month integration time, the reduction factor for uncorrelated systematics will be better than 10⁴. Gravity reversal is uniquely applicable in microgravity environments and offers one of the most effective techniques for reducing measurement systematics in QTEST. The rotating platform operation allows for the modulation at any frequency, e.g. asynchronous with ISS rotation and orbit and can thus be utilized for making QTEST measurements independent of ISS platform rotation and for suppression of the Sagnac and Coriolis forces. As discussed in section 5, even the conservative suppression factor assumed above well-satisfies the most stringent requirements for testing the WEP to below a part in 10¹⁶. Further, ultralight dark matter candidates such as B-L gauge bosons might give rise to time-dependent EP violations, QTEST is the first experiment purpose-designed to probe for such effects.

QTEST will use Bragg-diffraction atom-optics. The main motivation for the choice is the possibility of using a single Bragg laser system for ⁸⁵Rb and ⁸⁷Rb with a single, far-detuned laser addressing both rubidium isotopes. It is relatively easy to find a 'magic' wavelength that addresses both species with the same Rabi frequency, and makes perfect wavenumber matching possible. This eliminates the influence of vibrations and laser phase noise completely in principle. Bragg diffraction does not change the atoms' internal quantum state, which will help to reduce systematic effects, e.g., magnetic fields and AC Stark shifts. The single Bragg beam is thus an enabling approach for ISS platform, as ISS is not a vibration-isolated platform. It should be pointed out that a parasitic phase shift was observed with Bragg diffraction in simultaneous conjugate interferometers [27, 106] that used to be of concern for the precision measurements. It is now well understood as it has been explained quantitatively by solving the Schrödinger equation for the atomic matter wave in a laser field. The simulations have been confirmed experimentally [107]. For Mach-Zehnder interferometers, as we will use them in the WEP test, symmetry allows for measurements in which the phase shift can in principle be eliminated completely.

QTEST will use QUIC-trap-based ultracold atom sources to yield the maximum achievable populations of both Rb species with high spatial symmetry. Two separate dual atom-sources are designed for simultaneous operation of the two dual-species AIs with opposite sign of k_eff for simultaneous k-reversal measurements [108], which not only reduces systematics scaling as ${{k}_{{\rm{eff}}}}^{0}$ and ${{k}_{{\rm{eff}}}}^{2},$ but also provides redundancy in space in that each source alone is sufficient for a test of the WEP with unprecedented, although reduced, precision and the full set of the planned photon-recoil-based measurements. Delta-kick expansion will be used to achieve low-density (n₀), low-temperature samples for long free-expansion times with minimal collisions in a compact, well-controlled environment. This cooling technique relies on the position/momentum correlation for atoms after ballistically expanding, with slower atoms occupying the center of the gas and faster atoms traveling to the edges after sufficient expansion time. Application of a position-dependent impulse, with sufficient strength and harmonicity of the spatial profile, then drastically reduces the expansion velocity of all atoms in the gas.

Even with delta-kick produced atomic clouds at nanoKelvin effective temperatures, the practical interrogation time is limited due to the Coriolis force and imprecision in the cloud overlaps, mostly due to the reduction of the AI fringe contrast. Two new developments recently reported and demonstrated will help address this limitation. As detailed in section 4.6, imaging detection and an 'open interferometer' scheme, using controlled asymmetry in the interferometry timing, will not only recover interferometer contrast in the presence of gravity gradients and rotations, but also relax the WEP-test constraints for dual-species overlap by orders of magnitude.

Figure 2 shows a space-time diagram illustrating the two counter-propagating dual-species atom interferometer trajectories for the WEP measurements. Initially, two samples of both rubidium isotopes, overlapped at the starting locations SL1 and SL2, respectively, are launched with a π Bragg-pulse at a drift velocity of $\pm \hslash {k}_{{\rm{eff}}}/m.$ A common laser is used at the 'magic' wavelength so that the two-photon Rabi frequencies for both isotopes are equal. Since the recoil velocity is 87/85 times higher for the lighter isotope, the species separate thereafter. At about 13 s drift time, when the atoms are fully in the shielded drift tube region of the SC, the π/2–π–π/2 sequence of the Mach-Zehnder interferometer starts. The two outputs of the ⁸⁷Rb interferometers reach each detection region at 38.6 and 44.2 s, respectively; the lighter species arrives about 1 s earlier. The single laser pulse for establishing drift velocities of the two samples in opposite directions utilizes the degeneracy of the two initially stationary atom samples. In the process, however, half of the atoms are lost.

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The atom interferometer pulse separation time of T = 10 s has been chosen based on a trade-study to optimize the signal and precision targeted while minimizing instrument design complexity, power requirements, and size of the physics package. The length of the overall SC of approximately 70 cm is sufficiently compact for the rotating platform while still achieving the required sensitivity (see section 5). Large-momentum transfer (LMT) schemes [10, 111] are often considered as a way to improve the sensitivity of the atom interferometer. It turns out that, counter-intuitively, for a given physical size of the atom interferometer region, simple two-photon momentum transfer with longer interrogation time offers higher sensitivity than LMT. Therefore, LMT is not planned in QTEST.

An alternative geometry, the symmetric diamond configuration, has also been considered. It offers the opportunity to suppress several systematic effects [112] related to the gravity gradient. However, diamond-configuration AIs are prone to beam splitting loss and require clearing pulses during interferometry. They are also not maturely proven and require more complicated phase readout with three coupled AI output ports.

A single Bragg laser addressing both rubidium isotopes guarantees that this differential measurement has the highest common mode rejection to vibrations on ISS. For this to work properly, the Bragg laser will have to provide the same beam splitting and combining functions for both species. Fortunately, a magic wavelength can be found where the two-photon Rabi frequencies for ⁸⁵Rb and ⁸⁷Rb are identical. The two photon transition $|0\rangle \leftrightarrow |2\hslash k\rangle$ has a resonance frequency of $4{{\rm{\Omega }}}_{{\rm{r}}}\;=\;2\hslash {k}^{2}/{\rm{m}}.$ Due to the small transition frequency difference between ⁸⁵Rb and ⁸⁷Rb of ≃2π × 355 Hz, compared to the bandwidth of a typical Bragg diffraction of pulse duration ∼100 μs, simultaneous transitions of both species using a single frequency pair is feasible. For the proposed AI configuration, the two-photon transitions are between $|2\hslash k\rangle \leftrightarrow |4\hslash k\rangle ,$ which increase the frequency difference between species by a factor of 3 to 2π × 1.065 kHz. Figure 3 shows the relative difference in the two-photon Rabi frequencies (Ω) versus the Bragg beam wavelength for the m_F = 0 (clock) states, where the zero crossings indicate 'magic' wavelengths for the pair of internal states chosen for QTEST. The pairs of interest are ⁸⁵Rb F = 3, ⁸⁷Rb F = 2, and ⁸⁵Rb F = 2, ⁸⁷Rb F = 1, for which the Zeeman shifts partially cancel.

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The choice of the single Bragg laser approach for the dual species interferometers also eliminates the differential measurement sensitivity to atom interferometer laser phase noises, much the same way as for vibrations. There is no need for precise phase locking of separate lasers and precise wavelength ratio control as is necessary when two very different wavelength lasers are used, e.g., in the Rb and K measurements [113]. It is important to note that use of two frequency-components reflected off a common retro-reflecting mirror for Bragg diffraction introduces the wavelength k vector reversal symmetry, making $|{\rm{i}}\rangle \leftrightarrow |{\rm{i}}+2\hslash k\rangle$ transitions indistinguishable from $|-{\rm{i}}\rangle \leftrightarrow |-({\rm{i}}+2\hslash k)\rangle .$ Hence, the single Bragg laser with two retro-reflected frequency components will simultaneously address all four clouds in the interferometer with identical k_eff and Ω, accomplishing both dual species interferometry and k -vector-reversal measurements all with a single Bragg pulse sequence.

Requirements for the Bragg laser are relatively easily met. In one analysis, the laser can be a diode laser amplified with a 2 W tapered amplifier. An acousto-optical modulator after the amplifier shapes the Gaussian Bragg pulses with a duration of several 100 μs, e.g. a temporal profile of exp(−t²/2τ²), where τ ∼ 66 μs [111]. The Bragg laser will be collimated at a 1/e² radius (waist) of 24 mm. This radius is chosen to be at-least twice as large as the Gaussian width of the atomic clouds after thermal expansion of 20 s interferometer total interrogation time. Simulations of the expected signal in this system predict a minimal contrast loss (∼10%) due to the intensity-inhomogeneity of the Bragg beam across the atomic cloud.

For simultaneously accommodating the operation of k-reversal interferometers, the QTEST instrument is designed to have two dual species sources, each species contains a minimum of 10⁶ atoms at the final preparation stage with a temperature less than 1 nK. However, cooling atoms to Bose–Einstein condensation (BEC) is not necessary. This avoids the considerable difficulty of condensing ⁸⁵Rb, on the one hand, which suffers from large 3-body losses at high densities and requires Feshbach tuning of the s-wave scattering length in an optical dipole trap for condensation [114]. On the other hand, mean field shifts are typically very large while atom numbers are low in BECs, and BEC preparation times are long, often making degenerate gases unfavorable for precision metrology. For QTEST, ultracold atoms at low densities at the time of interferometry are desired.

4.3.1. Atom collection

4.3.2. Evaporative cooling in a QUIC trap

Laser cooling atoms in an optical molasses is practically limited to ∼μK temperatures and densities on the order of 10¹⁰—10¹¹/cm³ limited by the photon recoil and optical depths of the clouds [117]. To reach 1 nK temperature at sufficiently high density, two additional cooling stages are needed: evaporative cooling in a magnetic trap followed by a final delta kick cooling. After the dual-species laser cooling stage, the expected phase-space density is on the order of ∼10⁻⁶. In comparison, the final clouds are at a phase space density of > 10⁻⁴, assuming σ_r = 6 mm and temperatures <1 nK. Evaporative cooling will bridge the gap. For the evaporative cooling stage, the atoms are first optically pumped (with an applied ∼10 mG bias magnetic field) into the weak-field seeking states ${|F=2,{m}_{F}=2\rangle }_{87{\rm{R}}{\rm{b}}}$ and ${|F=3,{m}_{F}=3\rangle }_{85{\rm{R}}{\rm{b}}}$ and then loaded via magnetic transport and adiabatic compression into a QUIC trap [118], consisting of two identical quadrupole coils and one additional Ioffe coil.

After the atoms are loaded into the QUIC trap, sympathetic cooling of ⁸⁵Rb with ⁸⁷Rb is applied to increase the phase-space density before delta kick cooling. Sympathetic cooling of the two gases in thermal equilibrium is characterized by the α parameter as [119]:

$$\begin{eqnarray}&&\displaystyle \frac{{T}_{i}}{{T}_{f}}={(\displaystyle \frac{{N}_{87{\rm{R}}{\rm{b}}}}{{N}_{85{\rm{R}}{\rm{b}}}}+1)}^{\alpha },\end{eqnarray} \tag{ 6 }$$

where α > 0.77 for our dual-species rubidium gases (extracted from data in [105]), T_i (T_f) are the initial (final) temperatures of the dual-species gases in thermal equilibrium, and N_87Rb (N_85Rb) are the majority (minority) atomic species. Sympathetic cooling is assumed to progress until the populations are equal.

With the QUIC traps in a space platform, electric power dissipation is of concern. The design of the source vacuum chamber and the magnetic coils must take this factor into consideration. The spatial profile of the magnetic field inside the source cell, from the QUIC trap, was numerically simulated to optimize the atom confinement, power requirement, and harmonicity of the trap. With reasonable coil size choices, we find that two identical quadrupole coils carrying 25 A, requiring 270 W total power, and one Ioffe coil carrying 25 A, with 120 W total power, can create a magnetic trap with U₀ = k_B × 10 mK depth, where k_B is Boltzmann's constant, and trap frequencies ω = 2π × (28 Hz, 24 Hz, 26 Hz). Table 3 summarizes the key parameters achievable for the atomic gasses. The second and third columns of table 3 list the estimated efficiencies of the envisioned MOT/QUIC trap loading and evaporative/sympathetic cooling stages, based on demonstrations in similar lab-based systems [105]. The relatively large magnetic-field minimum in the trap (B₀ ∼ 230 G) maintains the atoms in the weak-field-seeking 'stretched' states. Thus, they have the same linear Zeeman shift of 1.4 MHz G⁻¹ so that the trap potentials for both species are the same.

Table 3.  Modeled performance for cooling ⁸⁷Rb and ⁸⁵Rb clouds from 3D-MOTs to free-space, dual-species gases after delta kick cooling.

	QUIC Trap Loading	End of Evaporation	After Delta Kick Cooling
N87Rb, N85Rb	3 × 108, 2 × 106	>106 each	>106 each
Temperature	100 μK	2.5 μK	1 nK
Peak Density (87Rb)	8 × 1010 cm−3	7 × 109 cm−3	4 × 105 cm−3
Phase Space Density (87Rb)	10−7	10−4	10−4

4.3.3. Delta-kick cooling

After evaporative/sympathetic cooling, the atoms are prepared as near point sources for delta-kick cooling to ultra-low temperatures below 1 nK. Cooling is defined here in the weakest sense; the atoms are neither in thermal equilibrium nor benefit from enhanced degeneracy at the end of the delta kick cooling stage. In fact, it is a process that simply reduces the mean kinetic energies by sacrificing density at constant phase space density [100, 101, 120]. This cooling scheme is ideally suited for the high symmetry of microgravity to extend the free fall times without atom clouds expanding to unmanageable sizes.

At the end of the evaporation, the atoms will be transferred to the m_F = 0 state by Raman or microwave pulses. This effectively turns off the trap and leaves the atoms free to expand without being affected by the magnetic trap field. The magnetic fields can even be ramped off/on during this time. After the requisite ballistic expansion, ∼400 ms at 2.5 μK initial cloud temperature, the atoms are then driven back to the stretched states. The spatial dependence of the impulse experienced by the atoms while they are in the stretched states is designed to match the cloud energy distribution and thus minimize their energy. The ballistic expansion time is optimized separately for each of the two species, which is possible because their hyperfine-state transitions are energetically well resolved. At the end of the delta kick cooling scheme, the atomic clouds each have a 1/e radius of ∼5 mm and a kinetic energy of <k_B × 1 nK. The projected properties of the atomic clouds after all cooling stages, just before transport into the SC for interferometry measurements, are given in the final column of table 3.

Details of the scaling and tolerances for delta kick cooling with trap-field, expansion time, and pulse times have been well summarized in the Science Envelope Requirements Document for the Cold Atom Laboratory [121]. The primary concern for achieving ultra-low temperatures in this cooling scheme derives from anharmonicities of the magnetic trap, which will limit the ultimate temperature and induce center-of-mass motion of the clouds. The simulated magnetic trap is nearly harmonic at the center, so that only the first few terms in the Taylor-expansion of the field profile are significant:

$$\begin{eqnarray}&&B(x)\cong \;{B}^{(0)}+\displaystyle \frac{1}{2}{B}^{(2)}{x}^{2}+\displaystyle \frac{1}{6}{B}^{(3)}{x}^{3}+\displaystyle \frac{1}{12}{B}^{(4)}{x}^{4}.\end{eqnarray} \tag{ 7 }$$

Constraining the influence of the forces from higher-order harmonic terms to be smaller than the final momentum spread of the cloud yields:

$$\begin{eqnarray}&&\displaystyle \frac{{B}^{(n)}}{{B}^{(2)}}\lt (\displaystyle \frac{{\sigma }_{i}}{{\sigma }_{f}})\displaystyle \frac{(n-1)!}{{\sigma }_{f}^{(n-2)}},\end{eqnarray} \tag{ 8 }$$

where σ_i/σ_f is the ratio of the initial (σ_i) to final (σ_f) atomic cloud sizes for the delta kick cooling ballistic expansion stage and n is the Taylor-expansion-order. QTEST requires B⁽³⁾/B⁽²⁾ < 13.5 and B⁽⁴⁾/B⁽²⁾ < 20, 250. Fitting the simulated QUIC trap field profiles to the 4th order polynomial, given in table 4, demonstrates that the anharmonicity of the simulated QUIC trap is negligible for all directions.

Table 4.  Dominant anharmonic terms for the simulated QUIC trap B-field profile along the Quad-coil axis (x), along the Ioffe-coil axis (z), and transverse to both coil axes (y). All terms satisfy the requirements for delta-kick cooling to 1 nK by a large margin.

	x-axis	y-axis	z-axis
B(3)/B(2)	10−16	2.2 × 10−16	1.71
B(4)/B(2)	.090	2.1	.58

Separation of the atomic sources and the SC allows QTEST to optimize the control and isolation of the science measurement environment while maximizing the source throughputs. The SC will have enhanced optical access for the Bragg beams and high suppression from potential sources of light, vacuum, and magnetic-field perturbations. The added cost is the atom transport process. A viable atom transport method should meet the following requirements: short transport time (<1 s) to minimize excess thermal expansion of clouds, have high transport efficiency (∼100%) to preserve atom flux, and preserve the positioning and overlap repeatability of the two species without inducing additional systematics. We propose to use Bloch oscillations (for fast transport) in a moving optical lattice to displace stationary clouds from the delta kick cooling region to the center of the SC. This is accomplished by counter-propagating laser beams, with control of their amplitudes and frequency differences.

Moving optical lattices are formed by the interference of the counter-propagating laser beams from Bloch oscillation fiber outputs SL1-1 and SL1-2 and fiber, SL2-1 and SL2-2 respectively (shown in figure 1). These beams are adiabatically turned on and off in less than 1 ms [20]. The moving lattice constantly accelerates and then decelerates to rest by changing the frequency difference of the two lasers [20, 109]. The clouds are thus transported away from the delta kick cooling region, reaching the axis of the SC and ready for the atom interferometer sequence. Following [122], we find that, with realistic laser and TA systems, the cloud can be transported with separate beams over 10 cm in 0.5 s with 90% efficiency.

Due to the difference in mass, ⁸⁵Rb and ⁸⁷Rb will have different velocities after receiving identical photon momenta. A previously proposed scheme uses the same lattice but different photon-momentum transfer for ⁸⁵Rb and ⁸⁷Rb to closely match the final velocity [90, 91] for coherent launch. However, for pure transportation without net momentum transfer from stationary to stationary states, it is possible to maintain the spatial overlap of the species. In a lattice moving at constant velocity, trapped atoms follow the classical trajectory defined by the lattice wells together with the quantum nature of discrete photon momentum transfer, which displaces the two species after transport as illustrated in figure 4. By designing the right sequence for the instantaneous velocity of the lattice, one species of choice can stay at higher velocity than the other for a defined duration. The final overlap of the two species is therefore controllable with the stability determined by electronics. This picture applies both to deep lattices and shallow lattices, where the actual velocity profile can be tuned up experimentally in operation. Therefore, this scheme may offer a handle for fine-tuning the spatial overlap of the two species along one axis.

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High precision AIs are critically sensitive to environmental perturbations and externally applied forces. The extensive common-mode rejection built into the QTEST concept mitigates much of this susceptibility. Most notably, the use of identical k_eff with Bragg interferometry of the two isotopes of rubidium, and simultaneously operated dual-species AIs with equal but opposite momentum states traversing overlapping classical paths will result in the leading-order phase shifts (e.g., from translations, rotations, and gravity gradients) to be common between the two species and, in principle, perfectly cancel in analysis. In practice, very slight offsets in the center of mass (CM) position (δr) and velocity (δv) of the ⁸⁷Rb and ⁸⁵Rb test mass clouds break the commonality and symmetry and will lead to sizable shifts (see section 5.1). It should be noted that the constraints of the CM overlap of the test-masses are general for all WEP type of tests due to the prevalence of the gravity gradient [123].

The CMs of two species actually cannot overlap over the entire trajectory except at one point in a typical MZ interferometer configuration, as shown in figure 2. This is due to the inevitable mass difference between the two atomic species. In the presence of a gravity gradient, therefore, a differential interferometer measurement, without any compensation, will always include a phase error due to the gravity gradient even when the cloud CMs are initially perfectly overlapped. Removing this phase error requires precise knowledge of the gravity gradients, which is nearly impossible. Nevertheless, consistent initial overlap conditions in both position and velocity ensure that such phase error is constant from measurement to measurement. QTEST addresses this issue by using gravity reversal and k reversal measurements in which the displacement-dependent gravity gradient phase terms change sign relative to the possible WEP signal. Any gravity reversal induced displacement bias must be below δr ∼ 34 nm and δv ∼ 0.4 nm s⁻¹ (see section 5.1) so that the combined measurements can reach below 1 × 10⁻¹⁶ g.

Shot-to-shot noise fluctuations between subsequent interferometer runs also add noise. In order not to degrade the SNR from the dominant gravity gradient dependent terms, maintaining high-stability in δr and δv is still critical. This requires the statistical CM fluctuations δr_Stat < 34 μm and δv_Stat < 0.4 μm s⁻¹ to not exceed the atom shot noise (see table 8). For our Gaussian clouds of 6mm width at 1nK, statistical CM fluctuations attributed to the finite atom numbers are similar for the two species, on the order of δσ_r ∼ 6 μm and δσ_v ∼ 0.3 μm s⁻¹, within the required stability.

The standard method for extracting the AI phase shift relies on comparing the populations of atoms in each AI output port, yielding interference fringes. Imperfections in state preparation, atomic cloud inhomogeneity, and force gradients can reduce the achievable contrast, limiting the ultimate AI sensitivity. This limitation is particularly severe in space experiments, where one wants to take advantage of available long free fall times for ultra-high sensitivity AI measurements. For long atom interferometer interrogation times, gravity gradient and rotational phase-shifts across the length scale of the atomic clouds will induce sizable phase gradients and can easily make the contrast drop to zero [110, 102]. Table 5 gives the magnitudes of the dominant phase shifts across the Gaussian width of the clouds in position (σ_r) and velocity (σ_v) for the ballistically expanded ⁸⁵Rb and ⁸⁷Rb gases in the proposed QTEST measurement environment. Considering the dominant terms, we would expect multiple phase-fringes on each interferometer output port, rendering the standard AI phase extraction protocol useless. The following sections discuss the proposed approaches in QTEST to mitigate contrast loss and to extract the WEP signal.

Table 5.  Dominant temperature and density-dependent phase-shifts (Δϕ) across the Gaussian width of each of the rubidium atomic clouds in QTEST. Mach-Zehnder interferometry is assumed with T = 10 s, k_eff = 4π/780 nm, R_eff = 10⁵ m, δΩ_y = 10⁻³ Ω_ISS. σ_r = 6 mm, σ_v ≅ 310 μm s⁻¹ at 1 nK, a = a_bg and n₀ = 4 × 10⁵ cm⁻³ after delta-kick cooling.

Phase term	Δϕ (87Rb)	Δϕ (85Rb)	Comments
${k}_{{\rm{eff}}}{T}_{zz}{\sigma }_{r}{T}^{2}$	25.1	25.1	Gravity gradient (initial position)
${k}_{{\rm{eff}}}{T}_{zz}{\sigma }_{v}{T}^{3}$	13.0	13.1	Gravity gradient (initial velocity)
$-2{k}_{{\rm{eff}}}{\sigma }_{v}\delta {{\rm{\Omega }}}_{y}{T}^{2}$	−1.14	−1.15	Coriolis
$-\frac{{k}_{{\rm{eff}}}}{2{R}_{{\rm{eff}}}}{({\sigma }_{r}+{\sigma }_{v}T)}^{2}$	−0.74	−0.73	Wavefront curvature
$\frac{4\pi \hslash a}{m}{n}_{0}T$	2 × 10-4	−9 × 10−4	Mean field

4.6.1. Imaging detection and extraction of the phase difference

4.6.2. Rotation-dependent contrast and its recovery

4.6.3. Gravity-gradient-dependent contrast and its recovery

The gradient of the gravity field is fundamentally tied to the very concept of WEP tests. The gradient introduces spatial inhomogeneity that results in a large phase shift in a typical Mach-Zenhder atom interferometer under the Earth's gravity field. This leads to contrast loss in each interferometer due to the position and velocity distributions of the atomic cloud. At the same time, since the masses of the two atomic test particles are different, there is no strict common mode cancellation of the AI phase shifts between the two species directly. Precise control in δr and δv will be able to make the shifts constant and make it possible to cancel in k-reversal measurements.

Consider the dimension along the beam propagation direction. The leading terms of the phase of an open Mach-Zehnder interferometer with pulse spacing of T and T + δT is:

$$\begin{eqnarray}&&\phi ({r}_{i},{v}_{i})=\;{\phi }_{0}+{k}_{{\rm{eff}}}{T}^{2}\gamma ({r}_{i}+{v}_{i}T)+{k}_{{\rm{eff}}}{v}_{i}\delta T,\end{eqnarray} \tag{ 9 }$$

where r_i and v_i are the initial position and velocity, respectively,$\;{\phi }_{0}={k}_{{\rm{eff}}}g{T}^{2},$ and γ is the gravity gradient. At an output port, the chance of finding an atom in the diffracted state (excitation probability if state labeled) with such initial conditions is 1-cos²ϕ(r_i,v_i). Assuming an imaging resolution of σ and negligible imaging delay after the last interferometer pulse, the observed excitation probability at position r_f with the resolution of σ is the average of the excitation probability of all atoms with initial conditions such that their classical trajectories land at position r_f at the detection time: r_f ≈ r_i + 2Tv_i. Let the initial cloud width and velocity spread be σ_r and σ_v, the observed excitation probability at position r_f can be expressed as:

$$\begin{eqnarray}&&\displaystyle \int \displaystyle \int {\rm{d}}{r}_{i}{\rm{d}}{v}_{i}\;\mathrm{exp}[-\displaystyle \frac{{({r}_{i}-\delta r)}^{2}}{2{\sigma }_{r}^{2}}]\;\mathrm{exp}[-\displaystyle \frac{{({v}_{i}-\delta v)}^{2}}{2{\sigma }_{v}^{2}}]\;\\ &&\quad \times \mathrm{exp}[-\displaystyle \frac{{({r}_{i}+2T{v}_{i}-{r}_{f})}^{2}}{2{\sigma }^{2}}]\;{\mathrm{cos}}^{2}\phi ({r}_{i},{v}_{i}).\end{eqnarray} \tag{ 10 }$$

The integral is proportional to

$$\begin{eqnarray}&&\mathrm{exp}[-\displaystyle \frac{{(\delta r+2T\delta v-{r}_{f})}^{2}}{2({\sigma }^{2}+{\sigma }_{r}^{2}+4{T}^{2}{\sigma }_{v}^{2})}]\{1+\\ &&\mathrm{exp}[-\displaystyle \frac{2{({k}_{{\rm{e}}{\rm{f}}{\rm{f}}}{T}^{2}\gamma )}^{2}({\sigma }^{2}{\sigma }_{r}^{2}+{T}^{2}{(1-\displaystyle \frac{\delta T}{\gamma {T}^{3}})}^{2}{\sigma }_{v}^{2}{\sigma }_{r}^{2}+{T}^{2}{(1+\displaystyle \frac{\delta T}{\gamma {T}^{3}})}^{2}{\sigma }_{v}^{2}{\sigma }^{2})}{{\sigma }^{2}+{\sigma }_{r}^{2}+4{T}^{2}{\sigma }_{v}^{2}}]\\ &&\quad \times \;\mathrm{cos}[2({\varphi }_{0}+{r}_{f}\displaystyle \frac{({k}_{{\rm{e}}{\rm{f}}{\rm{f}}}{T}^{2}\gamma )({\sigma }_{r}^{2}+2{T}^{2}(1+\displaystyle \frac{\delta T}{\gamma {T}^{3}})\;{\sigma }_{v}^{2})}{{\sigma }^{2}+{\sigma }_{r}^{2}+4{T}^{2}{\sigma }_{v}^{2}}\\ &&\quad +\displaystyle \frac{({k}_{{\rm{e}}{\rm{f}}{\rm{f}}}{T}^{2}\gamma )(\delta r({\sigma }^{2}+2{T}^{2}(1-\displaystyle \frac{\delta T}{\gamma {T}^{3}})\;{\sigma }_{v}^{2})+T\delta v((1+\displaystyle \frac{\delta T}{\gamma {T}^{3}})\;{\sigma }^{2}-(1-\displaystyle \frac{\delta T}{\gamma {T}^{3}})\;{\sigma }_{r}^{2}))}{{\sigma }^{2}+{\sigma }_{r}^{2}+4{T}^{2}{\sigma }_{v}^{2}})]\}.\end{eqnarray} \tag{ 11 }$$

The fringe contrast is thus identified as

$$\begin{eqnarray}&&\mathrm{exp}[-\displaystyle \frac{2{({k}_{{\rm{eff}}}{T}^{2}\gamma )}^{2}({\sigma }^{2}{\sigma }_{r}^{2}+{T}^{2}{(1-\displaystyle \frac{\delta T}{\gamma {T}^{3}})}^{2}{\sigma }_{v}^{2}{\sigma }_{r}^{2}+{T}^{2}{(1+\displaystyle \frac{\delta T}{\gamma {T}^{3}})}^{2}{\sigma }_{v}^{2}{\sigma }^{2})}{{\sigma }^{2}+{\sigma }_{r}^{2}+4{T}^{2}{\sigma }_{v}^{2}}].\end{eqnarray} \tag{ 12 }$$

Without imaging detection, where $\sigma \gg {\sigma }_{r}+{\sigma }_{v}T,$ the gravity-gradient dependent phase fringing reduces the contrast, practically limiting T ≤ 5 s [103]. The contrast can be partially recovered, as proposed in [110] and also demonstrated experimentally in [9], by compensating the initial velocity-dependent gravity gradient phase k_effT²γ (v_iT) with a choice of δT so that $1+\delta T/(\gamma {T}^{3})\ll 1.$ However, the remaining contribution from the gravity gradient across the initial cloud size still severely limits the useful range of T.

A possible mitigation of the remaining contrast loss comes from imaging detection. By choosing 'anti-compensation' of the open interferometer asymmetry so that β = $1-\delta T/(\gamma {T}^{3})\ll 1,$ together with the small pixel size approximation $\sigma \ll {\sigma }_{r}$ and σ_vT, contrast is significantly recovered as implied from the above expression of contrast. For instance, assuming σ_r ≅ σ_vT and moderate anti-compensation accuracy, β ≤ 10⁻², gravity gradient phase shifts across the initial cloud size of even k_effT²γσ_r = 30 rad correspond to a contrast of >96%.

Now consider directions orthogonal to the beam propagation direction (z), as shown in figure 1 in which the camera points along the y direction. The phase and fringes in the x direction can be treated similar to the z direction considered above. The analogy to the open interferometer asymmetry in the x direction is an additional tilt of the retroreflection mirror in the direction (rotation about y axis) for the last interferometer pulse, which can create spatial fringes in the x direction [7, 9]. A parameter similar to β can be applied to mitigate contrast loss caused by the kinematics in the x direction with comparable efficiency. The y direction, however, is unique in that there is no imaging resolution in this direction. Since the y direction is chosen to coincide with the rotation vector, leading phase shift terms are much smaller than those from other directions. The impact on contrast is thus negligible.

The analysis of the gravity-gradient-dependent contrast considers only the dimension along k_eff. In practice, if k_eff is not aligned to one of the principal axes of the gravity gradient tensor, the fringes will be tilted. On the one hand, for the fringes tilted in the imaging plane, detection and extraction of the phase difference would be similar to the one-dimensional case discussed above, assuming the choice of the propagation direction is known when calculating the relative phase. On the other hand, for the fringes tilted perpendicular to the imaging plane, the contrast will be degraded due to phase variations along the imaging line of sight. Such a sinusoidal phase variation of magnitude p would degrade an otherwise perfect contrast to sinc(p) ≈ 1 – p²/6, assuming a uniform atomic density distribution. With small misalignment angle between a principal axis of the gravity gradient tensor and the line of sight (θ) of the detector, the magnitude of the phase variation p would be p ≤ 2k_effγT²σ_rθ + 4k_effσ_rδΩTθ_rot, where similar effect of 0.1% rotation compensation imperfection (θ_rot) is also included. To maintain 90% (50%) contrast, θ  ≤ 13(30) mrad, θ_rot ≤ 172(385) mrad.

4.6.4. Initial condition sensitivity and mitigation

The phase terms in the cosine function in the above integral can be grouped into the gravity phase, the gravity gradient phase across the cloud, and the initial condition phase. The gravity phase (ϕ₀) is the target of the measurement, and the other terms are generally regarded as systematics. However, with imaging detection, the gravity gradient phase across cloud provides a natural and convenient fringe for phase extraction.

The initial condition phase is dependent on the centroids of the clouds (δr, δv). Its contribution to the observed excitation probability at position r_f, in the limits considered above, reduces to:

$$\begin{eqnarray}&&\displaystyle \frac{({k}_{{\rm{e}}{\rm{f}}{\rm{f}}}{T}^{2}\gamma )(2{T}^{2}{\sigma }_{v}^{2}\delta r-T{\sigma }_{r}^{2}\delta v)}{{\sigma }_{r}^{2}+4{T}^{2}{\sigma }_{v}^{2}}\beta \approx \displaystyle \frac{({k}_{{\rm{e}}{\rm{f}}{\rm{f}}}{T}^{2}\gamma )(2\delta r-T\delta v)}{5}\beta .\end{eqnarray} \tag{ 13 }$$

The sensitivity to the initial conditions is thus suppressed by β. For this fluctuation not to dominate the shot to shot SNR of 1000, the stability requirement on dual-species effective CM overlap (taking into account possible distortion of the cloud profiles) reduces to the limits: δr ≅ 34 μm and δv ≅ 0.4 μm s⁻¹. This level of systematic CM control should be relatively straightforward to achieve, as discussed in section 4.5. The gravity direction and k-reversal modulation will then average the WEP signal measurement precision down to the 10^-16 level in a few months of measurements.

Phase calculations for atom interferometers have been well studied for decades. Here we adopt the approach originally developed at Stanford [91], which is generally applicable for non-interacting atoms. In this approach, the classical trajectory of a point particle is calculated for each interferometer arm with a given initial position and velocity. The trajectory is symbolically computed iteratively as a power series of propagation time, which linearizes the influences of Taylor expanded nonlinear potentials while maintaining target accuracy. The photon recoil imparted at each beam splitter is included with magnitude and direction. The classical trajectories are then used to calculate the propagation phase, the laser phase, and the separation phase. The combination of these phases gives the total phase of the interferometer, and is broken down symbolically for physical understanding of the origin of the phase shifts. Table 6 lists the gravity signal and the largest phase shift terms with imperfect rotation compensation and with the assumptions in the table caption.

Table 6.  Leading phase terms. For T = 10 s, initial position (x₀, y₀, z₀) = (0.01, 0.01, 0.5) m, initial velocity (v_x, v_y, v_z) = (1, 1, 11.8) mm s⁻¹, ISS rotation Ω = (0, Ω_y, 0) = (0, Ω_ISS, 0), residual rotation δΩ = 10⁻³Ω_ISS = (1.13, 1.13, 1.13) μrad s⁻¹. The total potential is expanded as: $-({{\rm{\Phi }}}_{0}+g\;z+\frac{1}{2}{T}_{zz}{z}^{2}$ $+\frac{1}{3!}{Q}_{zzz}{z}^{3}+\frac{1}{4!}{S}_{zzzz}{z}^{4}$ $+\;\frac{1}{2}{T}_{xx}{x}^{2}+\frac{1}{2}{T}_{yy}{y}^{2}$ $+\;\hslash m\frac{\alpha }{2}({B}_{z}+z\;{\partial }_{z}B$ ${+x{\partial }_{x}B+y{\partial }_{y}B)}^{2})$ , where $g=-{R}_{ISS}{{\rm{\Omega }}}_{ISS}$ at the origin, ${B}_{z}=10mG,$ and $\partial B=10$ mG m⁻¹ in all directions. Parameters are evaluated for a ⁸⁷Rb F = 2, m_F = 0 interferometer at altitude 400 km above a spherical Earth of radius 6370 km. The pointing of the apparatus is assumed ${\theta }_{x},\;{\theta }_{y}=1mrad$ off the nadir. A common coefficient $\mathrm{cos}\;{\theta }_{x}\;\mathrm{cos}\;{\theta }_{v}$ for the terms listed on the table is not shown.

Phase Term	Phase (rad)	Relative Magnitude	Comment
${k}_{{\rm{eff}}}g{T}^{2}$	−1.40 × 1010	1	Reference
${k}_{{\rm{eff}}}{T}_{zz}{z}_{0}{T}^{2}$	2.07 × 103	1.48 × 10−7	Gravity gradient (initial position)
${k}_{{\rm{eff}}}{T}_{zz}{v}_{z}{T}^{3}$	4.87 × 102	3.48 × 10−8	Gravity gradient (initial velocity)
${k}_{{\rm{eff}}}^{2}\hslash {T}_{zz}{T}^{3}/2m$	2.44 × 102	1.74 × 10−8	Gravity gradient due to photon recoil
$-2{k}_{{\rm{eff}}}{v}_{x}\delta {{\rm{\Omega }}}_{y}{T}^{2}$	−3.65	2.61 × 10−10	Coriolis
$-2{k}_{{\rm{eff}}}{{\rm{\Omega }}}_{y}\delta {{\rm{\Omega }}}_{y}{z}_{0}{T}^{2}$	−2.07	1.48 × 10−10
$\frac{7}{6}{k}_{{\rm{eff}}}{T}_{zz}{{\rm{\Omega }}}_{y}{v}_{x}{T}^{4}$	5.47 × 10−1	3.91 × 10−11
${k}_{{\rm{eff}}}{B}_{z}{\partial }_{z}B\;\hslash m\alpha {T}^{2}$	4.25 × 10−1	3.04 × 10−11	Magnetic fields

With atom sources of phase space density of ∼10⁻⁴, we consider the mean field shift as additive and will discuss its influence later. Note that the phase terms are not directly measureable as the interferometer readout process averages the sum of all phase terms over initial conditions, as depicted in section 4.6. The systematic phase shifts of the WEP test are obtained after taking the difference of the corresponding phase terms between two species, as listed in table 7.

Table 7.  Leading terms for the phase difference between two species. The magnitude of the phase is calculated assuming initial condition mismatch of 1 μm and 1 μm s^-1 in all dimensions. ${c}_{x}=\;\mathrm{cos}\;{\theta }_{x},$ ${c}_{y}=\;\mathrm{cos}\;{\theta }_{y}.$ A common coefficient c_x c_y to all terms on the table is not shown.

Phase Term	Phase (rad)	Relative Magnitude	Comment
$\frac{1}{2}{k}_{{\rm{eff}}}^{2}\hslash {T}_{zz}{T}^{3}(\frac{1}{{m}_{85}}-\frac{1}{{m}_{87}}){c}_{x}{c}_{y}$	5.7	4.07 × 10−10	Photon recoil
${k}_{{\rm{eff}}}{B}_{z}{\partial }_{z}B\;\hslash {T}^{2}(\frac{{\alpha }_{85}}{{m}_{85}}-\frac{{\alpha }_{87}}{{m}_{87}})$	5.54 × 10−1	3.96 × 10−11	Magnetic field
${k}_{{\rm{eff}}}{z}_{0}{({\partial }_{z}B)}^{2}\hslash {T}^{2}(\frac{{\alpha }_{85}}{{m}_{85}}-\frac{{\alpha }_{87}}{{m}_{87}})$	2.77 × 10−1	1.98 × 10−11
${k}_{{\rm{eff}}}{v}_{z}{({\partial }_{z}B)}^{2}\hslash {T}^{3}(\frac{{\alpha }_{85}}{{m}_{85}}-\frac{{\alpha }_{87}}{{m}_{87}})$	6.52 × 10−2	4.66 × 10−12
${k}_{{\rm{eff}}}{T}_{zz}\delta {v}_{z}{T}^{3}$	4.14 × 10−2	2.95 × 10−12	Gravity gradient (initial velocity)
$\frac{1}{2{m}_{87}}{k}_{{\rm{eff}}}^{2}{({\partial }_{z}B)}^{2}\hslash {T}^{3}(\frac{{\alpha }_{85}}{{m}_{85}}-\frac{{\alpha }_{87}}{{m}_{87}})$	3.26 × 10−2	2.33 × 10−12	Cross term between photon recoil and Zeeman shift
$-{k}_{{\rm{eff}}}{B}_{z}{\partial }_{x}B\;{{\rm{\Omega }}}_{y}\hslash {T}^{3}(\frac{{\alpha }_{85}}{{m}_{85}}-\frac{{\alpha }_{87}}{{m}_{87}})$	−6.28 × 10−3	4.48 × 10−13
${k}_{{\rm{eff}}}{y}_{0}{\partial }_{y}B\;{\partial }_{z}B\;\hslash {T}^{2}(\frac{{\alpha }_{85}}{{m}_{85}}-\frac{{\alpha }_{87}}{{m}_{87}})$	5.54 × 10−3	3.96 × 10−13

5.1.1. Suppression of the dominant phase-shift terms

Suppression and mitigation of each term in table 7 can be understood intuitively: For terms proportional to k_eff², which originate from the trajectory difference due to differential photon recoil velocity ħk_eff²/m, simultaneous k-reversal measurements will remove all of them because the WEP signal k_effgT² changes sign while these terms don't.

The magnetic field gradient dependent shifts in the magnetic insensitive (m_F = 0) states are given by the quadratic Zeeman effect, described by the Breit-Rabi formula as Δω_mF=0 = ±½αB², with α=(g_J-g_I)² μ_B²/(hΔE_HFS), where μ_B the Bohr magneton, g_J and g_I are the Landé g-factors, ΔE_HFS is the hyperfine splitting, and the sign corresponds to the atomic hyperfine state (F = I ± ½) with nuclear spin I. The difference between the isotopes is Δα = 2π × 718.6 Hz G^-2. Earth's field is ∼0.5 G and we assume it's shielded to achieve gradients less than 10 mG m⁻¹ with a 10 mG bias field (longitudinally, to set the quantization axis) inside the science chamber. Because the internal state of the atoms in both interferometer arms are the same, only spatial variations of fields affect the interferometer.

To suppress the B-field dependent systematics, the atom interferometer will be operated subsequently in the two-hyperfine ground state pairs (clock states: F = 1, m_F = 0 and F = 2, m_F = 0 for ⁸⁷Rb and F = 2, m_F = 0 and F = 3, m_F = 0 for ⁸⁵Rb). The sign change of the quadratic Zeeman effect will lead to averaging out of the systematic. However, temporal fluctuations not correlated with the state change cycle lead to stochastic noise. To suppress such noise to below the atom shot noise, the field fluctuations Δ(BδB) need to be less than 0.45 mG² m⁻¹. Systematics and drifts can then be removed in data analysis. However, for any B-field dependent systematics correlated with the modulation of the atomic states, the associated component of the field gradient in the interferometer drift tube would have to be suppressed to less than 100 nG m⁻¹ with ∼10 mG bias.

After demodulating the data from k-reversal and hyperfine state switching, the remaining terms, given in table 8, depend on linear combinations of differential initial conditions and gravity gradient or imperfect rotation compensation. Note that the initial position z₀ does not appear in the list, which validates the effectiveness of simultaneous k-reversal interferometers using independent dual sources. With tight constraints on differential initial conditions, these phase terms will be below 10⁻⁶ rad. As discussed in section 4.6.4, imaging of the phase structure and controlled open atom interferometry suppresses the dominant gravity-gradient-dependent systematics. Further, with rotating platform modulation, differential initial conditions will be modulated out while leaving the rest of the terms unchanged. The requirements on initial conditions are thus relaxed, as discussed in section 3.3.

Table 8.  Phase terms after k-reversal and internal state modulation, up to 10⁻¹⁶ relative magnitude. Imaging and control of the phase-fringes of the ⁸⁷Rb and ⁸⁵Rb clouds further reduces the dominant CM overlap requirements by β. Μodulation of gravity with the rotating platform makes all remaining systematics negligible for measuring η to a sensitivity on the order of 10⁻¹⁶. ${s}_{x}=\;\mathrm{sin}\;{\theta }_{x},\;\;{s}_{y}=\;\mathrm{sin}\;{\theta }_{y}.\;$

Phase term	Phase (rad)	Relative magnitude	w/imaging (β = 10−2)	w/rotation modulation
${k}_{{\rm{eff}}}{T}_{zz}\delta {v}_{z}{T}^{3}{c}_{x}{c}_{y}$	4.14 × 10−2	2.95 × 10−12	2.95 × 10−14	<3 × 10−18
${k}_{{\rm{eff}}}{T}_{zz}\delta {z}_{0}{T}^{2}{c}_{x}{c}_{y}$	4.14 × 10−3	2.95 × 10−13	2.95 × 10−15	<3 × 10−19
$-2{k}_{{\rm{eff}}}\delta {{\rm{\Omega }}}_{y}\delta {v}_{x}{T}^{2}{c}_{x}{c}_{y}$	−3.65 × 10−3	2.61 × 10−13	2.61 × 10−15	<3 × 10−19
$\frac{7}{6}{k}_{{\rm{eff}}}{T}_{zz}{{\rm{\Omega }}}_{y}\delta {v}_{x}{T}^{4}{c}_{x}{c}_{y}$	5.47 × 10−4	3.91 × 10−14	3.91 × 10−16	<4 × 10−20
$-\frac{7}{6}{k}_{{\rm{eff}}}{T}_{xx}{{\rm{\Omega }}}_{y}\delta {v}_{x}{T}^{4}{c}_{x}{c}_{y}$	2.73 × 10−4	1.95 × 10−14	1.95 × 10−16	<2 × 10−20
$-6{k}_{{\rm{eff}}}{{\rm{\delta }}{\rm{\Omega }}}_{z}{{\rm{\Omega }}}_{y}\delta {v}_{y}{T}^{3}{c}_{x}{c}_{y}$	−1.24 × 10−4	8.86 × 10−15		<9 × 10−19
$-{k}_{{\rm{eff}}}{T}_{xx}{{\rm{\Omega }}}_{y}\delta {x}_{0}{T}^{3}{c}_{x}{c}_{y}$	2.34 × 10−5	1.67 × 10−15	1.67 × 10−17	<2 × 10−21
$-{k}_{{\rm{eff}}}{T}_{xx}\delta {v}_{x}{T}^{3}{s}_{x}$	2.07 × 10−5	1.48 × 10−15	1.48 × 10−17	<2 × 10−21
${k}_{{\rm{eff}}}{T}_{yy}\delta {v}_{y}{T}^{3}{c}_{x}{s}_{y}$	−2.07 × 10−5	1.48 × 10−15		<2 × 10−19
$-\frac{3}{2}{k}_{{\rm{eff}}}{T}_{zz}{{\rm{\Omega }}}_{y}^{2}\delta {v}_{z}{T}^{5}{c}_{x}{c}_{y}$	−7.97 × 10−6	5.69 × 10−16		<6 × 10−20
$-2{k}_{{\rm{eff}}}{{\rm{\delta }}{\rm{\Omega }}}_{z}{{\rm{\Omega }}}_{y}\delta {y}_{0}{T}^{2}{c}_{x}{c}_{y}$	−4.14 × 10−6	2.95 × 10−16		<3 × 10−20
$-2{k}_{{\rm{eff}}}{{\rm{\delta }}{\rm{\Omega }}}_{y}{{\rm{\Omega }}}_{y}\delta {z}_{0}{T}^{2}{c}_{x}{c}_{y}$	−4.14 × 10−6	2.95 × 10−16		<3 × 10−20
$\frac{3}{2}{k}_{{\rm{eff}}}{T}_{xx}{{\rm{\Omega }}}_{y}^{2}\delta {v}_{z}{T}^{5}{c}_{x}{c}_{y}$	−3.98 × 10−6	2.85 × 10−16		<3 × 10−20
$-2{k}_{{\rm{eff}}}\delta {{\rm{\Omega }}}_{y}\delta {v}_{z}{T}^{2}{s}_{x}$	−3.65 × 10−6	2.61 × 10−16		<3 × 10−20
$\frac{1}{4}{k}_{{\rm{eff}}}{{\rm{T}}}_{zz}^{2}\delta {v}_{z}{T}^{5}{c}_{x}{c}_{y}$	2.66 × 10−6	1.9 × 10−16		<2 × 10−20
$-{k}_{{\rm{eff}}}{T}_{xx}\delta {x}_{0}{T}^{2}{s}_{x}$	2.07 × 10−6	1.48 × 10−16		<2 × 10−20
${k}_{{\rm{eff}}}{T}_{yy}\delta {y}_{0}{T}^{2}{c}_{x}{s}_{y}$	−2.07 × 10−6	1.48 × 10−16		<2 × 10−20
$-{k}_{{\rm{eff}}}{{\rm{\delta }}{\rm{\Omega }}}_{z}^{2}\delta {v}_{x}{T}^{2}/{{\rm{\Omega }}}_{y}{c}_{x}{c}_{y}$	−1.83 × 10−6	1.3 × 10−16		<2 × 10−20
$-{k}_{{\rm{eff}}}{{\rm{\delta }}{\rm{\Omega }}}_{x}^{2}\delta {v}_{x}{T}^{2}/{{\rm{\Omega }}}_{y}{c}_{x}{c}_{y}$	−1.83 × 10−6	1.3 × 10−16		<2 × 10−20
$7{k}_{{\rm{eff}}}{{\rm{\delta }}{\rm{\Omega }}}_{x}{{\rm{\Omega }}}_{y}^{2}\delta {v}_{y}{T}^{4}{c}_{x}{c}_{y}$	1.64 × 10−6	1.17 × 10−16		<2 × 10−20

Table 9.  Interspecies and intraspecies mean-field parameters for the low-temperature, nondegenerate ⁸⁵Rb-⁸⁷Rb mixture. The magnitudes of the phase shifts are further suppressed to be negligible with k-reversal.

	abg	Phase (rad)	Relative magnitude
87Rb	100 a0	2 × 10−6	1.2 × 10−16
85Rb	−450 a0	−9.1 × 10−6	−5.8 × 10−16
87Rb-85Rb	213 a0	4.2 × 10−6	2.7 × 10−16

Finally, we consider forces from magnetic field gradients external to the rotating AIPP that will limit the usefulness of the rotating platform by imposing CM offsets in position or velocity that are not reversed with respect to gravity by modulation. After transfer to the clock states, to release the atoms from the QUIC trap, stray magnetic field gradients (δB) will still exert differential accelerations on the ⁸⁷Rb and ⁸⁵Rb gases, given by the second-order Zeeman shift. One second of free fall time before interferometry, in the bias field and field gradient considered above, leads to an initial center of mass offset in position and velocity of only δr < 90 pm and δv < 170 pm s⁻¹. These effects can be ignored as they are far below any other relevant length scale in the system.

In two-photon Raman interferometers, vibrational noises affect ⁸⁵Rb and ⁸⁷Rb differently, as detailed in [102]. The differential sensitivity of vibration comes from different k_eff for ⁸⁵Rb and ⁸⁷Rb, due to their different hyperfine spacings of ground states in Raman transitions. k_eff can be matched to 2 parts per billion (2 × 10⁻⁹) by choosing laser wavelengths properly. With the QTEST science objective of <10⁻¹⁵ g precision performed on ISS, the unsuppressed vibrations are still significant. With the Bragg interferometers proposed for QTEST, ⁸⁵Rb and ⁸⁷Rb can be driven by identical frequency components with matching diffraction efficiency, due to the accessibility of the 'magic' wavelength. Thus, k_eff and the momentum transferred to the atoms are identical, directly translating to identical sensitivities to accelerations. The vibrational noise induced phase shifts are then strictly common to the two species. Detunings deviating from the magic wavelength will affect the relative diffraction efficiency, while k_eff is always perfectly matched. The impact on contrast will be second order to diffraction efficiency variation when the Mach-Zehnder interferometer operates at the optimal condition. The requirement on environmental vibration is thus relaxed to the Doppler shift when the atom interferometer light is on and bounces off of moving optics. Let the allowance of Doppler shift noise be 400 Hz (assumed to be 10% of the two-photon Rabi frequency), corresponding to an RMS velocity of 157 μm s⁻¹. For comparison, a vibrational noise of 10⁻⁴ m s⁻²/√Hz would have RMS velocity noise of 71 μm s⁻¹, integrating from 0.05 Hz to infinity.

Mean field systematic shifts arise from inter- and intra-particle collisions among the atoms in ultracold, dual-species atomic gases. Here, only s-wave scattering is generally allowed, and the interactions are fully described by the s-wave scattering length. For non-degenerate, dual-species ⁸⁵Rb-⁸⁷Rb gases, the mean-field energy shift per particle is given by:

$$\begin{eqnarray}&&\hslash {\omega }_{87}=\displaystyle \frac{4\pi {\hslash }^{2}}{{m}_{87}}({a}_{\mathrm{87-87}}{n}_{87}+{a}_{\mathrm{85-87}}{n}_{85}),\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&\hslash {\omega }_{85}=\displaystyle \frac{4\pi {\hslash }^{2}}{{m}_{85}}({a}_{\mathrm{85-85}}{n}_{85}+{a}_{\mathrm{85-87}}{n}_{87}),\end{eqnarray} \tag{ 15 }$$

where n_x is the average density of the rubidium atomic gas (species x), m_x is the atomic mass, and a_x-y are the interspecies and intraspecies scattering lengths. In the atom interferometer, imperfect beam-splitting efficiency of the first π/2 pulse leads to a phase-shift: ${\rm{\Delta }}{\phi }_{{\rm{x}}}={\int }_{0}^{2T}{\omega }_{{\rm{x}}}{\chi }_{{\rm{x}}}\;{\rm{d}}t,\; where$ T = 10 s is the interferometer time and χ_x is the fractional imbalance of atomic populations in the two interferometer arms. Table 9, characterizing the magnitude of the interaction shift for the QTEST dual-species gases assuming χ_x = 10⁻², integrates the time-dependent density to account for ballistic expansion but doesn't include corrections due to the separation of the two species' CM during interferometry (inter-species scattering). Note that the mean-field effect is very small without additional suppression. However, since the phase term is independent of k_eff, the mean-field effect will be modulated further with k-reversal (limited only by our knowledge of the average atomic gas density at each trap).

The atomic phase during interferometry is greatly affected by fluctuations of the Bragg-beam collimation and motion of the atoms across the effective Bragg-laser wavefront. Such motion can induce sizable systematic shifts and statistical fluctuations attributed to insufficient collimation and purity of the incident and reflected lasers [108, 131, 132], and can also write phase gratings onto the atomic clouds to reduce the contrast (notably in a direction perpendicular to the imaging plane). To first order, the wavefront-curvature-dependent phase shift is given by:

$$\begin{eqnarray}&&{\rm{\Delta }}{\phi }_{WF}(z,t)=-\displaystyle \frac{1}{2}\displaystyle \frac{{k}_{{\rm{eff}}}}{{R}_{{\rm{eff}}}(z)}{({\sigma }_{r}+{\sigma }_{v}t)}^{2},\end{eqnarray} \tag{ 16 }$$

where R_eff(z) = (1/R_I(z) + 1/R_R(z))⁻¹ is the sum of the incident (R_I) and reflected (R_R) radii of curvature of the Gaussian beam at a distance (z) from the waist and t is the free expansion time. In the ideal case, the 24 mm waist beam is collimated at the retro-mirror, such that R_eff(z) ≅ z[1 + (z_R/z)]² > 5 × 10⁶ m over a z = 1 m beam path-length. Here, the thermal velocity for ⁸⁷Rb and ⁸⁵Rb atoms at 1 nK (σ_v ≅ 310 mm s⁻¹), and assuming initial cloud widths σ_r = 6 mm, leads to a phase-shift across the cloud of approximately 6 × 10⁻⁵ radians, negligibly effecting the contrast.

A more reasonable approximation for the effective Bragg wavefront curvature considers the effective wavefront dependence on the initial collimation [133]. For a fiber-based collimator producing a 24 mm waist beam (fiber numerical aperture = 0.12, focal length = 0.2 m, λ = 780 nm), a longitudinal displacement of the fiber-lens spacing of even 100 μm still allows R_eff(z) > 10⁵ m over the ∼0.5 m interaction region. The resultant phase shift on each rubidium AI is calculated as:

$$\begin{eqnarray}&&{\rm{\Delta }}{\phi }_{{\rm{WF}}}({z}_{0},{t}_{0})-2{\rm{\Delta }}{\phi }_{{\rm{WF}}}({z}_{1},{t}_{0}+T)+{\rm{\Delta }}{\phi }_{{\rm{WF}}}({z}_{2},{t}_{0}+2T),\end{eqnarray} \tag{ 17 }$$

where z₀, z₁, and z₂, are the cloud CM positions at the first, second, and third interferometer pulses, respectively. The level of collimation assumed leads to a systematic error Δη_WF ≅ 8 × 10⁻¹⁴. Wave-front curvature at this level will moderately affect the contrast (see table 5), random fluctuations of the fiber-lens spacing of even ±10 μm RMS will be below shot-noise by over an order of magnitude, and systematic offsets attributed to differential CM of the clouds and differential cloud widths/temperatures will be removed by k-reversal and demodulation with the rotation platform. After demodulation, wavefront related shifts will be suppressed to Δη_WF < 10⁻¹⁷.

Laser frequency noise introduces a relative phase shift between the incident and time-delayed return components constituting the Bragg lattice. The two rubidium species are identically affected by the laser phase noise, suppressing its influence for differential acceleration measurements. However, CM separations of the gases during interferometry give, in the case of white frequency noise, a residual sensitivity of [134]:

$$\begin{eqnarray}&&{\sigma }_{\phi }^{2}\approx \displaystyle \frac{4{\pi }^{3}}{\tau }\displaystyle \frac{{L}^{2}}{{c}^{2}}{\rm{\Gamma }}{\rm{.}}\end{eqnarray} \tag{ 18 }$$

where L is the mean displacement of the atomic clouds of each species along the laser k-vector, c is the speed of light in vacuum, and 2τ = π/Ω with Ω the two-photon Rabi frequency. Assuming that the Bragg laser has a line width Γ < 1 MHz and τ > 10 μs, gives ${\sigma }_{\phi }\lt {10}^{-4}\;$rad, which is much smaller than the atomic shot noise.

Interferometers using Bragg diffraction have been known to exhibit diffraction phases [24, 96, 128]. In light-pulse AIs, these systematic shifts arise from operating in the quasi-Bragg regime, where atom-light interactions cause atom losses from the effective two-level system and thereby break down the AI time-reversal symmetry: in absence of such losses, the second half of the AI is a time-reversed mirror image of the first, and any diffraction phases cancel, unless the symmetry is broken by technical imperfections. The presence of coherent loss channels breaks this symmetry. A model of these effects can be obtained by numerically integrating the Schrödinger equation for the atom-light interaction, and has been shown to agree with experimental data. For $\hslash {k}_{{\rm{eff}}}$ Bragg diffraction in the short-pulse regime used in QTEST, the uncompensated diffraction phase per beam splitter can amount to hundreds of milliradians [97].

In the QTEST implementation, several factors will reduce the diffraction phase to a level below the science requirement. First, Mach-Zehnder interferometers are inherently free of diffraction phases if the atoms are detected at the output that moves with the same momentum as the input (detecting the other output leads to the same results only if there are no losses). QTEST will detect the fringes from both ports for each interferometer. In addition, QTEST will use much longer pulse separation times than used in [97], thereby reducing the relative influence of the shift, and longer pulse durations, therefore strongly suppressing the diffraction phases. According to the diffraction phase model, simultaneously using two species having the same Ω will mostly have a common diffraction phase. Residual diffraction phase shifts will be suppressed with k-reversal. All remaining systematics are independent of the influence of gravity (e.g. Doppler dependent terms) and therefore will be averaged to zero with the rotating platform modulation measurements.