Species-selective lattice launch for precision atom interferometry

Long-baseline precision tests based on atom interferometry require drastic control over the initial external degrees of freedom of atomic ensembles to reduce systematic effects. The use of optical lattices (OLs) is a highly accurate method to manipulate atomic states in position and momentum allowing excellent control of the launch in atomic fountains. The simultaneous lattice launch of two atomic species, as required in a quantum test of the equivalence principle, is however problematic due to crosstalk effects. In this article, we propose to selectively address two species of alkalines by applying two OLs at or close to magic-zero wavelengths of the atoms. The proposed scheme applies in general for a pair of species with a vastly different ac Stark shift to a laser wavelength. We illustrate the principle by studying a fountain launch of condensed ensembles of 87Rb and 41K initially co-located. Numerical simulations confirm the fidelity of our scheme up to few nm and nm s−1 in inter-species differential position and velocity, respectively. This result is a pre-requisite for the next performance level in precision tests.


Introduction and motivations
Manipulating cold atomic ensembles with optical dipole traps is an exquisite tool to address their external degrees of freedom [1]. The Stark effect resulting from these beams realizes so-called optical tweezers and allows to precisely confine or move cold atoms or Bose-Einstein condensates (BEC) [2,3] and prepare them in desired position and momentum states. Interference of counter-propagating dipole beams creates a conservative periodic potential known as an optical lattice (OL) [4]. Atoms trapped at the potential minima of this periodic structure realize solid state physics-like systems with an unprecedented possibility to control the lattice properties. If the two interfering beams are relatively detuned, the lattice can displace the atoms. If the detuning is time-dependent, it accelerates them via Bloch oscillations [5,6] in analogy with electrons in a solid subject to an electric field. This method is very efficient in transferring large and quantized momenta to the atoms [7] putting them in well defined momentum states. Therefore, it is extensively used in precision atom interferometry [8][9][10][11][12].
The free evolution of an atomic ensemble in the gravitational field interrogated in a Mach-Zehnder lightpulse interferometer realizes a measurement of the gravitational acceleration g [13][14][15][16][17]. When two atomic ensembles of different masses are dropped in the gravity field, a comparison of their accelerations realizes a universality of free fall test or test of the weak Einstein's equivalence principle (WEP) [18]. Since violations of this principle are predicted at different levels in competing theories to unify fundamental interactions [19], such an experimental test can have a major impact in (in)validating these models. In recent years, atom interferometers performed WEP tests up to 10 −7 in the Eötvös ratio [20][21][22][23]  where a 1 and a 2 are the accelerations experienced by the test masses 1 and 2, respectively, δ a=(a 1 −a 2 ) and a a a 2.
Since the sensitivity of an atom interferometer typically scales quadratically with the pulse separation time T, there is an obvious incentive for using atomic fountains on ground to augment the available experimental time. This geometry is at the heart of recent proposals expected to open a new era of precision WEP tests (up to seven orders of magnitude expected improvement over state-of-the-art) by performing it in 10 m tall towers [24][25][26]. The idea is to launch two atomic ensembles of different atomic species using accelerated lattices before operating two simultaneous Mach-Zehnder atom interferometers, thus comparing the local gravity acceleration experienced by each.
In such a test performed in the vertical direction z, the differential velocity δ v z between the two species at the end of the acceleration stage couples e.g. to local gravity gradients (GG) resulting in a differential acceleration bias δ a. In line with recent and proposed experiments, we assume the typical Mach-Zehnder (π/2-π-π/2) pulse geometry. The full interferometer time 2T shall be the same for both species, but the effective wave numbers k 1 and k 2 may differ. The phase shift due to the GG in each interferometer i=1, 2 is [27] k T v T , 2 which divided by the scale factor k i T 2 leads to an acceleration equivalent bias of for a launch velocity v z, i . Thus, the differential acceleration is with δ v z =v z,1 −v z,2 . T zz denotes the first order GG tensor g .
z z ¶ In the considered case, the fountain is solely realized in the z-direction and therefore the other gradient tensor terms can safely be neglected. Phase shifts due to higher order tensors are orders of magnitude smaller than f i , thus not considered here. Dependent on the vibrational background, an additional mechanical accelerometer might be necessary to recover the interferometer signals of the weak equivalence principle as suggested in [28]. This GG coupling can be eliminated in a particular four-pulse atom interferometer scheme [29] at the cost, however, of a degraded (about a factor 5 lower) sensitivity to the gravity acceleration.
A GG T zz with an uncertainty Δ T zz parallel to the effective wave vectors couples to a differential velocity of the two ensembles δ v z oriented in the same direction with an uncertainty Δδv z . Since both the GG and the differential velocity are known within their uncertainties, a post-correction reduces the GG contribution to an uncertainty of Using compensation masses within the region where the atom interferometer is formed as proposed in [30] would only reduce the contribution of the first term in equation (5) proportional to T zz without relaxing the knowledge level required for Δ T zz .
If one considers the case of Earth's GG on ground T zz =−3×10 −6 s −2 , and a differential velocity of δv=0 μm s −1 with an uncertainty Δδv=±0.2 μm s −1 , one would reach an uncertainty in Δη of 10 −13 even with a modest knowledge of the GG of Δ T zz =3×10 −6 s −2 . If δ v z =100 μm s −1 and Δδv=±0.2 μm s −1 , then a characterization of T zz to 0.2% corresponding to Δ T zz =6×10 −9 s −2 would be necessary to reach the same uncertainty of the WEP test.
Due to the different recoil velocities, the two atom interferometers will cover slightly different trajectories. The related recoil phase terms can be suppressed by inverting the direction of momentum transfer for subsequent cycles and calculating the half difference [27]. This requires a sufficient homogeneity over the baseline of the atom interferometer. To reach Δ η=10 −13 , the homogeneity requirements of 3×10 −11 s −2 in GG and 5×10 −12 m −1 s −2 are compatible with Earth's contributions. The mass distribution of the experimental apparatus itself has to be designed appropriately to avoid contributions exceeding these thresholds. Figure 1 illustrates this scaling by indicating the required GG level of characterization as a function of the starting differential velocity for various target accuracies of the WEP test parametrized by Δ η. In principle, the GG could be characterized with the atom interferometer in a gradiometer mode [31]. It becomes clear that an inherently small differential velocity would relax the requirements for a GG knowledge or even make it obsolete. Consequently, the systematics assessment would be considerably relieved, especially in scenarios, where the GG might drift unpredictably. The figure suggests that with a characterization of the GG within its magnitude on ground (horizontal line), reaching state-of-the-art WEP test performances of Δη=10 −13 with quantum objects requires a bias in differential velocities of a fraction of μm s −1 . This proposal is motivated by the absence, to our knowledge, of appropriate methods to achieve this accuracy in a dual atomic launch.
To understand the origin of the differential velocity between two launched masses, we briefly recall the principle of a fountain launch. A lattice accelerated from v=0 to v=v f induces a quantized momentum transfer via Bloch oscillations to an atom initially at rest. To avoid the population of multiple orders, the final lattice velocity v f should match a targeted atomic velocity N k m  where N  Î with the modified Planck constant ,  the lattice light wave vector k and the atomic mass m. When the lattice transports two species of masses m 1 and m 2 , the differential velocity at the end of the launch reads: where N 1 and N 2 are the numbers of photon kicks transferred to species 1 and 2, respectively. From equation (6), it becomes clear that minimizing δ v translates to finding a minimum of (N 1 /m 1 −N 2 /m 2 ). With a single OL, choosing a final v f matching a certain kN m 1 1  completely fixes the choice of N 2 . Thus, the bias δv cannot be minimized to better than a few tens of μ m s −1 in agreement with the predictions of [27] for the pair of isotopes 87 Rb and 85 Rb.
The idea of individually addressing each species with separate lattices is problematic since this would require vastly different ac-Stark shifts from each lattice. For sufficient lifetimes in the lattice however, high power and large detunings from the targeted electronic excitation are required. This generally means that other transitions, which are not much stronger detuned, e.g. of the other species, will have a considerable ac-Stark shift. Each lattice would have an effect on both species and the clean lattice dynamics are replaced by strongly time-varying potentials and undesired excitations. In this proposal, we solve this issue by choosing two lattices with zero-magic or tune-out wavelengths to selectively accelerate the two atomic clouds. At the start and the end of the acceleration phase, the wavepackets of different atomic masses m 1 and m 2 shall ideally have matching positions and velocities.

Scheme and method
For alkaline atoms, the contribution of the D 1 and D 2 lines to the dipole potential for one species of atoms reads are the detuning of the laser ω L from the D 1 and D 2 atomic transition lines, respectively, and g F is the hyperfine Landé factor. P is the polarization of the laser, m F the Zeeman state of the atom and c the speed of light. When the atomic dynamic polarizability switchs sign between two resonances, a magic-zero or tune-out wavelength could be found [32][33][34][35][36][37][38][39]. Several magic-zero wavelengths have  Figure 1. Required knowledge of the gravity gradient (GG) as a function of the inter-species differential velocity for several levels of performance Δ η of a WEP test. The case of a 10 m fountain is considered and implies an interferometry time of 2T=2.74 s. The interferometer is assumed to operate with two-photon transitions at about 780 nm each. Earth's GG affects the measurement and consequently needs to be determined in the shaded area. The corresponding differential acceleration is δ a=−T zz (±Δ T zz )δv z (±Δ δv z )T, where Δ δv z denotes the uncertainty in differential velocity δv z , and Δ T zz denotes the uncertainty in GG T zz . This leads to the plotted contribution to Δ η ∝ Δ T zz δv z . For each curve, a maximum Δ δv z is assumed (see legend) to keep, on ground, the other contributions (equation (5)) below the represented Δ η.
been implemented in different experimental contexts [40][41][42]. In table 1, we provide some key values for commonly used alkalines and alkaline-Earth-metal atoms in the metrology context with identified tune-out wevelengths.
Recently very precise-uncertainty below 1 pm-measurements of these wavelengths were done for rubidium [49] and potassium [50]. We shall consider these two most used species in cold atoms laboratories as a study case to illustrate our method. Figure 1(a) shows that at the wavelength λ 1 (λ 2 ), the contributions of the two lines cancel out the dipole potential for 41 K ( 87 Rb). These special wavelengths were already implemented in mixture experiments to selectively manipulate 87 Rb and 41 K degenerate ensembles [51].
Based on these investigations, a magic-zero wavelength λ 1 could be used to create a lattice OL 1 accelerating the Rb cloud (species 1) without any effect on the 41 K atoms (species 2), and vice-versa with λ 2 (lattice OL 2 ), therefore realizing a dual-species fountain, which principle is illustrated in figure 2(b). The atomic ensembles are guided during the time t f from the same initial position z=0. OL 1 transports species 1 (in red) and OL 2 conveys species 2 (in blue) to z v , we achieve a fountain launch that greatly relaxes the required knowledge of GG uncertainties as discussed in the previous section.
The simultaneous and independent use of lattices with two wavenumbers k L 1 and k L 2 allows for more flexibility towards cancelling δ v, which reads now: After choosing a final velocity, thus the corresponding couple of integers N 1 and N 2 , we could slightly tune one or both wavelengths to strictly reach a zero differential velocity. This is illustrated for the study isotopes case in Table 1. Magic-zero or tune-out wavelengths for alkaline atoms with neutralizing D 1 and D 2 contibutions according to equation (7). The variables P parametrizing the laser polarizations are assumed to be zero for simplicity. Theoretical calculations for alkaline-Earth-metal atoms are made in [35] and predict tune-out wavelengths for Be, Mg   respectively. This allows one lattice to bring one of the atomic clouds to any desired final position or velocity without altering those of the other species. In order to reach the precision target for this manuscript, a slight shift Δ λ from the rubidium magic wavelength is introduced (see inset) for the potassium lattice, leading to a small perturbation of rubidium dynamics.
the inset of figure 2(a) by the shift Δ λ=2.9 pm corresponding to Δ k=29.22 m −1 from the tune-out value k L 2 . This comes at the price of introducing a perturbation by OL 2 to the dynamics of the 87 Rb cloud which now is subject to the accelerating effect of this lattice ramped in time to transport 41 K. In the following sections, numerical simulations are implemented to check whether this parasitic contribution affects the fidelity and performance of the selective acceleration process. Both chosen species 87 Rb and 41 K have positive scattering lengths: 99 a 0 and 60 a 0 , respectively, where a 0 is the Bohr radius. They can be cooled down to degeneracy independently [52] or taking advantage of sympathetic cooling [53]. Another advantage of this pair in the context of precision measurements is the existence of a lowmagnetic field (79 G) Feshbach resonance allowing to tune the inter-species interaction to zero [40] and maximize the overlap between the two BECs while in the Zeeman state F=1, m F =1. This manipulation is very useful since inter-component interactions lead otherwise to a shell structure (in the case of the chosen isotopes) or could in immiscible phases form mixture ground states with broken spatial symmetry. Both of the last geometries lead to complex and coupled expansion dynamics of the BECs, thus to significant wave fronts-related systematic effects. These effects being leading systematics in most atom interferometry experiments, it is vital for precision differential atom interferometry to minimize or neutralize the inter-species interactions. This motivates our choice to consider in this study the intra-species interactions only. As a consequence, during the proposed acceleration ramps, the atoms are assumed to remain in magnetic sensitive sub-states. Couplings to the Feshbach field lead to magnetic field gradients and induce a differential velocity due to the different atomic properties. In principle, a characterization of the gradients could be performed by dedicated measurements with the atom interferometer [54] and the resulting differential velocity cancelled by accounting for it in the proposed lattice launch sequence. Assuming a time 10 ms before transfer to the magnetically insensitive states and switching-off the Feshbach field, the magnetic field gradient should be below 1 mG m −1 in average to reach Δη=10 −13 or 10 μG m −1 to reach Δη=10 −15 .

Theoretical model
Justified by the cancelled inter-species interactions, the treatment is described for one species and could be applied to the other simply by accounting for its different temporal control sequence of external potentials. The momentum shift Δk=29.22 m −1 chosen to perfectly match the two final velocities leads to a parasitic contribution of OL 2 in the dynamics of species 1 (lattice depth of few nK). Moreover, the assumed finite uncertainty in the definition of the tune-out wavelengths (1 pm) leads to a small ac Stark effect even in the case of the atoms for which it should be magic-zero. None of these effects is neglected by solving the dynamics equations of both BECs in presence of the two lattice potentials weighted by their exact numerical magnitude.

Gross-Pitaevskii equations (GPEs)
In the mean-field regime, a BEC is well described by the GPE. This often-called nonlinear Schrödinger equation features an additional term describing the interactions between atoms. At low temperature, the system of N bosons is described by a single wavefunction Ψ(r) [55] solution of the stationary GPE: where m is the mass of the bosonic species considered, V(r) the external potential seen by the atoms and μ is the chemical potential. The magnitude of the nonlinear term is proportional to the total number of condensed bosons N and the atom-atom interaction magnitude. When assuming s-wave scattering only, the interaction term reads where a s is the s-wave scattering length of the atomic species. In this proposal, we consider atomic species with repulsive interatomic interactions. In a fountain configuration, the relevant physical effects (acceleration, center-of-mass motion, etc) triggered by the lattice accelerating potentials V OL 1 and V OL 2 occur mainly in the gravity direction z justifying a one-dimentional treatment. The ground state of the problem is found by solving the effective one-dimensional GPE m z m z Ng z z z 2 where g 1D and 1D m are the effective 1D interaction strength and chemical potential, respectively.
The time-dependent behavior of the BEC before and during vertical acceleration, is followed while solving the time-dependent GPE and V z t , OL 2 ( )are time-dependent potentials accounting for a complete sequence of loading a BEC from a harmonic trap into two OLs that accelerate it.

Loading, release and acceleration ramps
During any fountain launch proposed in this manuscript, The total potential reads: The expressions taken by the potential at each time step are presented in the following.
Loading and release In order to adiabatically load a BEC in the OLs considered, the magnitude of their potentials is increased smoothly. We model this step by multiplying the potential with the function f ON (t) defined as follows: where t ON denotes the starting time of ramping up the lattice and the characteristic loading duration is set by τ. A complementary behavior regulates an adiabatic switch-off through the function: During the loading phase, the initial harmonic trap is switched off while the lattices are ramped up. We label this time interval [t 0 , t 1 ]. After acceleration, the BEC is released adiabatically in order to recover its single-peaked distribution in momentum space. This step is performed over the time interval [t 4 , t 5 ]. The total external potential for a species of atoms within these initial and final steps reads:   Acceleration ramps. In order to accelerate the condensates without leaving the first band, the OLs must be tuned on and off adiabatically. A common method of doing this is to use the following lattice acceleration profile a(t) during a time sequence [t 0 , t 5 ]: This sequences determines the lattice phases. Once the common target final velocity and the acceleration sequence duration are chosen, one needs to determine the constant acceleration a max and the different time intervals of the acceleration sequence. To find out these characteristic times, we choose a couple of total numbers of recoils cancelling δv in equation (8) and a value of a max for each species that is not too large for an optimal acceleration [56].
is the recoil velocity of one of the species of mass m driven by the OL of wave vector k. In general, a different choice of t  (or t 2 and t 3 ) and a max for each species could be done leading to a different phase in the time-dependent potentials parametrized with the position offset between the two OL previously introduced in equation (17) by c .
where z t OL 1 2 ( ) are completely set after the choice of t 1 , t 2 , t 3 , t 4 and a max for each of the lattices OL 1/2 . We would like to stress out that due to the deliberately introduced shift Δ k, it is not guaranteed that the effect of the magic-zero lattice OL 2 on species 1, which magnitude scales with V 2 , is negligible. Any effect on the center of mass and momentum distribution of BEC 1 would set a limit on our proposed method. This is essentially the motivation for the in-depth numerical simulations made, the results being detailed in the Results section.

Frame transformation
The typical realizations we would like to model involve tall fountains of 10 m or more. Corresponding acceleration ramps would accelerate the BECs over distances of few cm. To efficiently treat this problem numerically, we employ a moving frame description, reducing the required grid extension. All of the dynamics takes place around the translated center of the BECs, which in the ideal case follows closely the trajectory z OL of the accelerated OL. We take advantage of this situation to perform a transformation to an accelerated co-moving frame with the OL. This classical so-called extended Galilean transformation [57][58][59] The advantage of this system of coordinates becomes clear when writing the transformed accelerating potential terms (20) to 2 1 cos 2 2 1 cos 2 , 24 where the choice is made here to center the grid on z . OL 1 Since the two accelerated lattices of the problem are not allowed to acquire a large position offset all along the acceleration sequence, the term z t z t OL OL 2 1 ( ) ( ) appearing in (24) does not lead to a need for a larger grid than the one centered on z . OL 1

Numerical techniques
The method used to find the ground state and dynamics of the condensates is based on a split-operator treatment initially reported in [60] and previously applied in a similar context in [61,62]. It consists in breaking the evolution operator within a time step δ t in a product of two kinetic propagators separated by a potential one. The Hamiltonian is assumed to be time-independent during this time step and the error made during it scales with (δ t) 3 . The potential propagator is applied in a straightforward way to the wave function whereas the kinetic one is only diagonal in the momentum space. For each application of the latter, the BEC wave function is transformed to the momentum space beforehand. Transforming back and forth from these spaces is done using an optimized fast Fourier transform (FFT) developed by Intel for FORTRAN compilers [63]. Finding out the ground states is done following the same recipe but in complex time [64]. Typical grids used feature 2 17 points and extend over 700 μm. A full loading and acceleration sequence of about 10 ms requires a numerical propagation of few hours on a standard desktop computer (Intel i5 processor with 8 GB RAM).

Loading to and release from the OL
As previously stated, the choice of a long enough time τ is crucial to allow an adiabatic transfer of the BECs from their initial harmonic traps to the OLs and at the release step. In order to stress out the dramatic effect of this choice, we contrast in figure 3 the extreme case of a sudden switch-on and -off of the lattice potential (τ=0) with an adiabatic ramping.
The gallery shows the BEC momentum distribution at three different times (i) in the harmonic trap just before loading in the OL, (ii) after loading and (iii) after release from the OL. A too short or zero value of τ leads to a released BEC with several momentum classes populated (blue dashed peaks at k 2  in figure 3(c)). This effect is limiting the population of the chosen momentum at the end of the ramp thereby reducing the number of atoms involved in the AI. An adiabatic loading and release at τ=80 μs (red plain curve) guarantees a singlepeaked density in momentum space k 2 | ( )| Y after release from the lattice. This is observed in both cases of a static or accelerated lattice. The sequences considered in this manuscript are all characterized by a choice of τ guarantying a final single momentum peak, thus a maximum efficiency of the coherent transport.

Dual-species launch
In this section, we illustrate the implementation of the dual-fountain launch by propagating the two BECs using the ramps shown in figure 4(a). We choose to imprint 2280 and 1104 kicks for 87 Rb and 41 K. This choice would drive the atoms to acquire a final velocity of 13.6 m s −1 realizing a fountain of about 10 m as planned in three facilities so far [24][25][26]. After the acceleration phase, the two BECs are transferred to the exact target momentum class as shown in figures 4(c) and (e) (red and blue dashed lines) and spatially lifted off to the same height of 5.5 cm (figures 4 (b) and (d)). The fidelity of this process is subject to a final adiabatic release from the OLs. A sudden or imperfect release leads to the a loss of atoms in other momentum classes ( k k 2 , 4     and k 6   (black plain lines in (b) and (d)).
To check, in the adiabatic loading and release case, if the acceleration process is free from parasitic effects caused by the simultaneous application of two OLs, we zoom-in in figure 5 around the central and unique momentum peaks obtained. Whereas the effect of the presence of two lattices is negligible for 41 K as expected (lower graph), it is clearly visible that it amplifies the fluctuations around the maximum of the 87 Rb momentum distribution. This stems from the use of two OL in the latter case where both are not at the magic-zero wavelengths. The parasitic effect of OL 2 seems, however, to be simply modulating the momentum distribution around the targeted central value in a symmetric fashion. The numerical analysis conducted in the next sections will confirm this statement. Such a perturbed but symmetric momentum distribution (around the target value) would not lead to an additional dephasing in a WEP test since the differential velocity between the two BECs relies solely on the centers of the momentum wave packets at the input of the dual interferometer.

Dual-lattice dynamics and role of the interactions
To interprete the momentum density distribution of 87 Rb, we contrast in figure 6 the cases of an acceleration following the ramps previously applied (right column) and a simple expansion lasting for the same duration (left column) for different regimes of interactions. In all numerical experiments, the 87 Rb BEC wave packets are released adiabatically from the OLs. The first row (a) and (b) shows that the BEC loaded in OL 1 has the same momentum density whether accelerated by this lattice or simply expanded in it for the same time. The momentum width in both cases is, however, larger than the initial one (by a factor of 10 to 20) driven by interaction dephasing over the lattice sites. Indeed, depending on the number of atoms per site, the chemical potential leads to different phase winding in every lattice well. This causes momentum broadening even in the case where atoms are loaded and released adiabatically [65,66]. When Bloch oscillations are involved, atomic interactions lead to a dephasing and a broadening of the quasi-momentum width as observed in [67] and analyzed in [68]. Recently, the same effect was observed in the context of a metrology-oriented latticeaccelerated BEC experiment [69]. In [68], the increase in momentum width is proportional to g t , 1D · ) where t is the evolution time. By varying the number of atoms in the BEC or the time spent in the lattice, we could check that our results are consistent with this scaling in the case of a simple expansion in the lattice. Since the momentum width of a wave packet is a critical quantity for the contrast of an atom interferometer [70], it is of interest to keep it as low as possible. This can be realized by utilizing delta-kick cooling techniques [69,[71][72][73] or by taking advantage of the existence of Feshbach resonances to tune down the interactions magnitude. The next row (c) and (d) illustrates the case when two OLs are in presence and shows that the main perturbation and broadening of the momentum peak stems from the evolution in the bichromatic lattice configuration imposed to 87 Rb. Although, the second lattice OL 2 is several orders of magnitude weaker than the main accelerating one, it introduces a dephasing of the atomic cloud spatially extending over bichromatic lattice sites starting co-located at the origin but with potential minima that spatially separate the further the atoms are off-centered from z=0. By contrasting (c) and (d), it becomes clear that the accelerated case with different ramps applied to the two OLs leads to an averaging of this dephasing effect reducing perturbations and broadening of the momentum density distribution. Since the case of graph (d) is the one of interest in this article, the suitability of our proposal is confirmed. It is obvious that the size of the wave function during the expansion or acceleration process is a key quantity to account for since it determines the lattice sites occupancy. The size being shaped, in the BEC case, by the atomic interactions, it is necessary to clarify and distinguish the roles of size and interactions. To this end, we plot in the lower row (e) and (f), the momentum densities of the 87 Rb BEC with tuned-off interactions. In plot (e) and for the case of two lattices, we start with the same BEC wave function (same initial extension than the one in plot (c) before switching off the interactions for the complete evolution time. This suggests that, initially, the same number of bichromatic lattice sites are occupied. The similarity in the behavior of the two momentum structures (same peaks each k k 2 and relative magnitude than (c) confirms our interpretation of the dephasing due to more spatial separation between the sites of OL 1 and OL 2 . The interactions in the case (c) simply broaden every momentum peak already visible in (e). When a single lattice is present (black plain curves of graphs (e) and (f), the broadening of the initial ground state observed in (a) and (b) disappears with the vanishing interactions. In this case, the local lattice sites density does not play any role in altering the overall BEC phase. As for (a) and (b), there is no difference whether the BEC is accelerated or kept expanding in the lattice. The broadening observed in the expanding case of the bichromatic static lattice (red dashed curve in (e) is averaged out (red dashed in (f)) thanks to the different accelerating ramps and matches the single lattice case (solid black curve in (f)). As a conclusion, two effects are altering the momentum distributions of the BECs: (i) momentum broadening driven by interactions dephasing with complex shapes of the envelopes and (ii) bichromatic lattice dephasing, however, averaged out by the different acceleration ramps proposed in our scheme.

Effect of the number of kicks
In the case of the proposed scheme, the parasitic effects discussed above do not seem to be related to the number of momentum kicks transferred. It is, however, important to check if the fidelity of the process is harmed for higher velocity ramps. Indeed, if it is the case, the dual-lattice dephasing will add up for longer sequences or taller fountains setting a limit on the practically realizable interferometry times. Figure 7 does not support the occurrence of such effects. Comparing the momentum probability density of the two species (upper and lower rows) for largely different accelerations (left versus right column), we observe no difference in the shape or density magnitude of the wave functions. Numerical estimations of the differential velocity between the two species confirm this statement. This demonstrates the scalability of the method since no detrimental effects are observed for longer baselines. Only realistic experimental constraints are expected to set a limit to the proposed dual-species launch as homogeneity of the optical or magnetic traps involved.

Quantitative evaluation of the differential velocity
The minimum differential velocity between the two species is solely limited by the effect of the potassium lattice perturbing the rubidium atoms distribution, being non-magic-zero. In order to evaluate this effect, we estimate the velocity difference of rubidium when the second lattice is present compared to the ideal case of OL 1 alone. The analogous effect of OL 1 on potassium is strictly absent since its wavelength is exactly the magic-zero one. By changing the power of OL 2 , we estimate in figure 8 the bias velocity offset of the 87 Rb BEC from the reference ideal one of k m 2280 .
For an OL 2 beam waist of 1 mm, The perturbation starts to be important for several Watts. Keeping the lattice power between typical experimental values of 0.5 and 1 W, we bound the velocity perturbation below the low limit (few tens of nm s −1 ) identified for the high-precision measurements motivating the actual proposal. Within this range of parameters, the inter-species differential velocity lies similarly around few tens of nm s −1 . To match the launching velocities on a level of μm s −1 (nm s −1 ) for Δη=10 −13 [10 −15 ], the lattice frequencies have to be controlled to 0.1 GHz [0.1 MHz], which can be done e.g. by a frequency comb. Assuming retro reflected lattices, the relative angle between the lattices has to be below 40 μrad to allow for maximum differential velocities of nm s −1 . A mitigation strategy is using common optics for both lattices. The method adopted to evaluate these velocities was to find the expectation values of the momentum operator for each of the BECs and compare them. The finite velocity width implies a statistical uncertainty in the center of mass velocity. Coupled to GGs this leads to a noise contribution in the interferometer which has to be kept below the shot noise limit. For 10 6 atoms of each species, two photon beam splitters, contrasts near unity, a free evolution time of T=1.37 s, Earth's GG, and velocity widths as depicted in figure 5 the related noise would be smaller than the shot noise by one order of magnitude. The differential center of mass motion can be assessed by spatially resolved imaging directly after launch and subsequently after 2T. Repeating these measurements for about 20 times is sufficient to reach the precision required for a target of Δ η=10 −13 . Higher precisions require either a higher number of measurements or a reduction in velocity width. Even in a space-borne experiment and assuming the parameters from [74], the noise associated with the statistical uncertainty in the center of mass velocity would be below the anticipated shot noise limit.

Conclusion and discussion
In this article, the idea of using two OLs at the zero-magic wavelengths of 87 Rb and 41 K allowed to manipulate each of them selectively. The motivation behind this scheme is to achieve a perfectly zero inter-species differential velocity required in precision tests of the WEP in fountain geometries. To the best of our knowledge, no acceleration method of two different atoms or isotopes to a common precise velocity was reported so far. In order to strictly cancel the differential velocity between the species, one of the lattices had to be slightly shifted leading to a perturbing effect on one of the atoms ( 87 Rb). Numerical simulations of the dynamics of two BECs of Figure 6. Expansion of the 87 Rb BEC for various situations after an adiabatic release from the involved lattice(s). The final wavefunction is shown for a single lattice OL 1 (solid black) and with a second one (OL 1 +OL 2 ) (dashed red). The effects of lattice(s) acceleration (right column) are contrasted with static lattice(s) (left column). (a) and (b) The proposed acceleration ramp in a single lattice does not lead to a broadening of the momentum distribution. (c) and (d) The evolution of the BEC in the bichromatic lattice potential (OL 1 +OL 2 ) leads to a broadening of the momentum distribution due to dephasing occurring between multiple bichromatic lattice sites. This effect is dramatically reduced in the accelerated case (d) as we chose slightly different acceleration ramps for the two lattices, causing the perturbation in each site to oscillate and average down. (e) Even with vanishing interactions, the same BEC structure than case (c) is observed for the bichromatic lattice case. The dephasing effect leads to the same momentum peaks (with less broadening) and relative magnitudes. (f) The different acceleration ramps make the bichromatic dephasing average down and we observe the same momentum distribution for one or two lattices. Note that the BECs of (b) and (d) are the same plotted in figure 5(a). Figure 7. Effect of the number of momentum kicks. The right column differ from the left one by more than one order of magnitude in the total number of momentum kicks transferred. The upper panel illustrates the 87 Rb case whereas the lower ones corresponds to the acceleration of the potassium 41 isotope. This figure demonstrates the scalability of the method since different fountains of variable sizes could be operated using the same principle. No alteration of the momentum of the launched species is observed whether the fountain realized is few tens of cm or has a height of 10 m. Note that the duration of the two ramps (less than 8 ms) is chosen to be the same in the two cases (left versus right) in order to decouple the effect of momentum transfer from interaction dephasing effects, which scale linearly with time [68]. and plotted against the power of the perturbing lattice OL 2 . Since the lowest differential inter-species velocity is limited by the effect of OL 2 on 87 Rb, the velocity offset introduced for this species is the relevant quantity to compare with the target differential velocity between the species. The waist of the dipole lasers forming the lattice is chosen to be 1 mm for all the simulated cases of this figure. It is clear that for typical experimental values of the power of OL 2 (about 0.5 W), the systematic velocity offset introduced is about 10 nm s −1 required in high-precision tests. the above-mentioned atoms with realistic parameters lead to the conclusion that this perturbation does not alter the efficiency of the method to more than few tens of nm s −1 . Moreover, the effects of atomic interactions were extensively assessed and contrasted to the ideal collision-less case making this study valuable for the noncondensed regime as well. The method proposed is not bound to a particular experimental arrangement and covers a wide range of fountain baselines from few mm to several meter-tall chambers without suffering from any performance deterioration. In general, the acceleration ramps can be engineered to account for an initial spatial offset between the two atomic clouds, which was for simplicity omitted in the treated example. This feature is an intrinsic advantage of the scheme that provides a solution to the gravitational sag issue complicating Earth-bound inertial precision measurements. The choice of the atomic test pair is not restricted to the study case considered but can be made among the multitude of alkaline and alkaline-Earth-metal species possessing tune-out wavelengths [32][33][34][35][36][37][38][39][40][41][42]50]. The fountain concept presented is the baseline of a launch stage in an atomic interferometry test of the equivalence principle requiring the two species to start with velocities as close as few nm s −1 . This result puts a WEP test with an uncertainty of 10 −15 within reach in already existing fountain facilities.