Apparatus for quantum-mixture research in microgravity

TL;DR

This paper demonstrates the high-flux creation and analysis of Bose-Einstein condensate mixtures of potassium-41 and rubidium-87 in a microgravity environment using a sounding rocket, advancing mobile quantum gas research.

Contribution

It introduces a fully integrated rocket-based setup for generating and studying ultracold quantum mixtures in microgravity, with a novel switch-off protocol and interaction characterization.

Findings

High-flux BEC mixture generation in space

Characterization of release dynamics under microgravity

Establishment of a new benchmark for mobile ultracold experiments

Abstract

Experiments with ultracold quantum gases are a rapidly advancing research field with many applications in fundamental physics and quantum technology. Here, we report on a high-flux generation of Bose-Einstein condensate mixtures of $^{41}$K and $^{87}$Rb, using a fully integrated sounding rocket setup. We investigate the release and the free expansion of the quantum mixtures for different orientations to gravity. The release dynamics are governed by the mixture interactions as well as the decaying magnetic field during the release. The latter can be minimized by a dedicated switch-off protocol of the trap generating currents where an exact model enabled us to characterize the interaction effects. Our results establish a new benchmark for generating ultracold mixtures on mobile platforms, with direct relevance for future experiments on interacting quantum gases and tests of the equivalence principle in space.

Equations

$$\Delta \mathbf{a}(t)={\mathbf{a}}_{\mathrm{K}}(t)-{\mathbf{a}}_{\mathrm{Rb}}(t)=\frac{m_{\mathrm{K}}-m_{\mathrm{Rb}}}{m_{\mathrm{K}}{m}_{\mathrm{Rb}}}\nabla \left(\mathbf{\mu }\cdot \mathbf{B}(\mathbf{r},t)\right).$$

$$\text{B}(\text{r},t)=\sum _j{\text{B}}_j(\text{r},t)=\frac{{\mu }_0}{4\pi }\sum _j{I}_j(t){\oint }_{C_j}\frac{d{\text{l}}_j\times {\text{r’}}_j}{r_j^{\prime 3}},$$

$$U(\text{r},t)={\mu }_B{g}_F{m}_{F}B(\text{r},t)$$

$$\ddot{q_j}=-\frac{1}{m}\frac{\partial U(\text{r},t)}{\partial q_j},$$

Full text

1 Introduction

Microgravity opens up new perspectives for experiments exploring mixtures of interacting quantum gases [1]. These experiments range from the enhanced generation of Feshbach molecules [2, 3], interaction-driven generation of mixed species quantum bubbles [4, 5] to high-precision tests of the Einstein equivalence principle and the search for new forces [6]. Microgravity experiments are either based on atom chips [7, 8, 9, 10, 11] or all-optical [12, 13, 14, 15] generation of quantum gases and mixtures. Due to their low power consumption, small size and high robustness, atom chip-based setups are the baseline of future space-borne setups such as BECCAL [16], CARIOQA [17], CAI [18], QGG [19] or STE-QUEST [6]. Magnetic lensing techniques led to picokelvin expansion energies in space-borne atom chip single-species experiments [20, 21]. While recent advancements have led to the creation of atomic shell potentials [22], atom interferometry [23, 24], and magnetometry [25], as well as the first observation of interacting ultracold mixtures in space [11], current microgravity-based BEC experiments fall short in the required atomic flux necessary for finding new Physics such as tests of the equivalence principle [6] or gravitational wave detection and dark matter search [26]. Another challenge in atom chip-based setups is that magnetic forces during and after release can cause side effects, which may obscure the actual measurement and complicate the data analysis [21]. In particular, differential release velocities have been shown to induce phase shifts that can mimic violations of the equivalence principle in interferometric tests, and are recognized as one of the leading systematic errors in high-precision experiments [27, 28]. Cancellation techniques have been suggested and successfully implemented experimentally but require precise knowledge of the differential velocities [29, 30, 31].

Here, we present the high-flux generation of a quantum degenerate K-Rb mixture in an atom chip apparatus suitable for space operation, surpassing current state-of-the-art realizations by an order of magnitude in atomic flux [10, 11]. Moreover, we control the differential trajectory of both atom species after release where the dynamics are governed by the interactions as well as the time-dependent release mechanism. We apply a realistic theoretical framework describing the 3-dimensional evolution of interacting quantum mixtures [32] and achieve a high level of agreement with the experimental results. Combining these efforts, we demonstrate a reliable generation of quantum mixtures for various inclination angles of the experimental setup offering a unique insight in the floating of interacting mixtures shaped by quantum repulsion in conjunction with gravity. These results demonstrate the potential of atom chip setups for field operation as well as for answering fundamental questions of quantum gas physics [33].

2 Results

2.1 High-flux generation of 41K and 87Rb BECs on an atom chip

The described experiments, unless explicitly stated otherwise, were performed on the fully integrated MAIUS-B sounding rocket payload. The payload has been thoroughly qualified for a sounding rocket flight carried out on the 2nd December 2023 at ESRANGE. A detailed description of the payload design and its qualification for a sounding rocket flight has been published previously [34].

The experimental setup is shown in Fig. 1. For extended use in the laboratory, the payload can be continuously operated with water cooling, external power supplies and control units. An experimental cycle starts by simultaneously loading 87Rb and 41K atoms from a 2D-magneto-optical trap (2D-MOT) into a 3D-MOT. The quadrupole field of the 3D-MOT is generated by external Helmholtz coils and a mesoscopic U-structure of the three-layer atom chip. The three layers consist of a mesoscopic chip (U-structure, H-structure), a base chip (Z-structure) and a science chip (Z-structure) [35]. Laser light addressing the D2-lines of 41K (767.7 nm) and 87Rb (780.2 nm) for cooling, optical pumping and absorption detection is generated by seven distinct external cavity diode lasers and guided to the collimators by optical fibers [36, 34]. The duration of the 41K- and 87Rb-MOTs can be tuned independently by switching the lasers with acousto-optical modulators. This allows to tune the ratio between the two species in the final condensates. A fixed loading time of 500 ms for the 41K-MOT with an adjustable duration of the 87Rb-MOT turned out to be a suitable choice for the generation of mixtures with tunable ratios. We emphasize that it is not decisive to use different MOT loading times and it is also possible to achieve tunable mixture ratios by other methods, for example by changing the loading rate using different cooling frequencies of the 2D-MOT lasers. As discussed in Section 4.2, higher numbers of trapped 87Rb atoms are possible but would prevent the generation of 41K condensates due to inelastic collisions between the co-trapped species.

Following the combined 3D-MOT stage, the atomic ensembles undergo spatial compression and additional cooling over $11\,\mathrm{ms}$ by adjusting the positions of the compressed MOTs (CMOTs) to maximize atoms loaded into the magnetic trap. By subsequent optical molasses cooling, we are able to reduce the temperatures to $43\,\mu$K for 41K and $10\,\mu$K for 87Rb, respectively. To reach sub-Doppler temperatures in the 41K molasses, we ramp the detuning of the cooling laser from $\delta=-3\,$MHz to $\delta=-8\,$MHz while switching the repumper to very low intensities $I_{rep}\approx I_{cool}/100$ with the cooling laser intensity $I_{cool}$ [37].

Before trapping the atoms magnetically, we apply a 1 ms pulse of circularly polarized laser light resonant to the transition $\lvert F=2\rangle\longleftrightarrow\lvert F^{\prime}=2\rangle$ for both species. This leads to an accumulation of atoms in the magnetically trappable state $\lvert F,m_{f}\rangle=\lvert 2,2\rangle$. By switching on the mesoscopic H-structure of the atom chip together with a Z-structure of the base chip, we load up to $7\cdot 10^{6}$ (41K) and $5\cdot 10^{8}$ (87Rb) atoms into a large volume magnetic trap with geometric mean trapping frequencies of $47\,$Hz (87Rb) and $69\,$Hz (41K). This magnetic trap is loaded into a harmonic cylindrical trap close to the atom chip, generated by the Z-structures of the base and science chip [35]. Due to the high trapping frequencies of $f_{Rb}=(23,908,911)\,$Hz and $f_{K}=(33,1324,1329)\,$Hz, this trap permits efficient evaporative cooling, benefiting from high thermalization rates.

With the atom chip, we deploy two microwave fields resonant to the $\lvert F,m_{f}\rangle=\lvert 2,2\rangle\longleftrightarrow\lvert 1,1\rangle$ and $\lvert 2,1\rangle\longleftrightarrow\lvert 1,1\rangle$ transitions of 87Rb. The former is used for evaporation while the latter is used to continuously depump remaining 87Rb-atoms trapped in the $\lvert 2,1\rangle$ state. These impurity states are immune to the evaporation microwave at later evaporation stages and would prevent the formation of a 87Rb BEC due to the additional thermal load. Additionally, as detailed in the methods section, 87Rb-atoms in $\lvert 2,1\rangle$ are highly detrimental to the co-trapped 41K-atoms due to inelastic collisions. Microwave depumping is a widely used technique to limit interspecies collisional losses [10, 38, 39, 40, 41, 42, 43]. The co-trapped 41K-atoms are cooled sympathetically by fast thermalization with the 87Rb-atoms. The trajectory of the evaporation microwave consists of 9 concatenated ramps and an intermediate decompression step. The decompression is achieved by reducing the bias field from $28.8\,$G to $20.3\,$G and reduces losses due to three-body collisions between the 87Rb atoms. The decompressed trap has trap frequencies of $f_{Rb}=(25,396,398)\,$Hz and $f_{K}=(36,577,580)\,$Hz.

For the mixture generation, only the final frequency of the last ramp was increased by (70$\pm$10) kHz compared to the 87Rb-only case. We account this for the few µm-shift due to the repulsion with cold 41K atoms leading to an energy shift of the 87Rb atoms.

After $1.7\,$s of evaporation, we generate BEC mixtures with a tuning range between $1.5\cdot 10^{5}$ 87Rb-atoms and $5.0\cdot 10^{4}$ 41K-atoms. By changing the duration of the 87Rb MOT, different mixture ratios can be achieved (Fig. 2). The trend of the curve demonstrates that an increased number of 87Rb atoms leads to a decrease of condensed 41K atoms. In a single species optimized sequence, we generate up to $2.5\cdot 10^{5}$ atoms in a pure 87Rb BEC. The total duration of one experimental cycle amounts to 2.3 s. Fig. 3 shows the comparison of the performance of the apparatus to other BEC experiments working with the same atomic species. Our approach of combining a 2D-MOT and a three-layer atom chip enables us to prepare mixtures with both high flux and atom numbers. It surpasses the performance of other mobile and compact setups by an order of magnitude, and sets the stage for the upcoming generation of atom chip-based space-borne mixture experiments.

2.2 Release dynamics of atom chip-based traps

Trapped atoms are commonly released into free fall by turning off the confining potentials created by a Z-shaped wire and an external bias field. Our magnetic Ioffe-Pritchard trap is formed by $I_{\textrm{SC-z}}=2\,$A on the science chip and $I_{\textrm{BC-z}}=5.12\,$A on the base chip combined with a bias field of $B_{y}=20.3\,$G and an offset field of $B_{x}=-1.3\,$G to lift the trap bottom and prevent Majorana losses.

We observe that a naive switch-off, reducing the chip currents to zero while keeping the bias field on, results in a kick of the atoms towards the atom chip. Despite their low inductance, atom chip wires have typical switching times of a few $10\,$µs. These short transients, along with additional eddy currents induced in the copper mount of the atom chip holder and the titanium vacuum chamber, generate time-dependent magnetic gradients which lead to a finite release velocity of the atoms although they were at rest in the trap.

This has low effect in most ground-based single species experiments as the resulting kick is much smaller than the gravity induced motion for typical free fall times.

However, in mixture experiments, time-dependent magnetic gradients lead to a differential acceleration $\Delta a$ which separates both atomic species with masses $m_{\mathrm{Rb}}$ and $m_{\mathrm{K}}$ and equal magnetic moment $\boldsymbol{\mu}$ according to

[TABLE]

In case of a vanishing magnetic field gradient, there is no differential acceleration while, for a non-vanishing gradient at any time $t$ during the switch-off, we obtain different trajectories for each species. Velocity and position differences between species lead to systematic errors in atom interferometry [44, 28, 45] and limit applications requiring precise co-location, such as shell-shaped condensates [4, 46], fundamental tests [27] or studies on interspecies interactions [3, 47, 2].

Consequently, it is critical for those experiments to realize an atom release where both species wouldn’t acquire a differential velocity and remain co-located over the time of flight. To guarantee co-location during the free fall time, the transient differential accelerations $\Delta\boldsymbol{a}(t)$ must integrate to zero at the end of the release. Assuming that the magnetic field remains constant over the size of the two overlapping BECs, $B(r_{K},t)=B(r_{Rb},t)\,\,\forall\,t$, the masses and accelerations of K and Rb are related through $a_{\mathrm{Rb}}(t)/a_{\mathrm{K}}(t)=m_{\mathrm{K}}/m_{\mathrm{Rb}}$. It follows that, if the measured relative position between the BECs $\Delta z$ is zero after a fixed time of flight, the associated relative velocity has to be zero as well. The latter has been confirmed by the simulation described in Sec. 4.3.

To minimize residual forces due to the release, we introduce a tunable delay $\Delta t$ between switching off the slow, high-inductance coils and the fast chip structures, while keeping the $B_{x}=-1.3\,$G offset field to preserve internal states (see Fig. 6). By changing the value of $\Delta t$, it is possible to tune the magnetic forces sensed by the atoms during the switch off transient. In Fig. 4 we show the differential position of the clouds after a fixed time of flight for different $\Delta t$ and reach a zero crossing, realizing an effectively force-free release. We model the release by solving the classical equations of motion of 41K and 87Rb in a time-dependent magnetic field (see methods Section 4.3). The temporal evolution of this field is determined using a calibrated model of the atom chip and the external coils but does not account for the material surroundings. With a magnetic field prober (Aim TTi Iprober 420) we observe that the coil current follows an approximately linear transient while the chip structures can be modeled by an exponential decay. Since the magnetic field near the atoms depends on additional eddy currents which cannot be measured externally, we treat the decay times of the currents in the simulation as free parameters without altering the general linear or exponential shape.

The model successfully reproduces the observed data using an exponential decay with $\tau=(38\pm 2)\,$µs for the atom chip and a linear ramp with $\tau=(470\pm 24)\,$µs for the bias coils. The final drift velocity $\Delta v=v_{K}-v_{Rb}$ between the two species depends on the switch off time with a linear dependency of $\delta\Delta v/\delta\Delta t=(205\pm 11)\,$µm/ms2. Limited by our time-step quantization of $20\,$µs, the lowest achievable shift of $26\pm 10\,$µm after a time of flight of $20.38\,$ms is below their Thomas-Fermi radii ($R_{z,Rb}=161\,$µm, $R_{z,K}=127\,$µm) and reached for $\Delta t=60\,$µs.

To estimate the differential kick under microgravity conditions, we simulate the release from a weak trap, ideally suited for dual-species equivalence principle tests in microgravity (red curve in Fig. 4). Due to the lower magnetic forces of weak traps, the effect is naturally suppressed by an order of magnitude and amounts to $\delta\Delta v/\delta\Delta t=(10.2\pm 0.7)\,$µm/ms2. With an assumed time-resolution of $100\,$ns of a modern control system this would result in $\Delta v=1.0\,$µm/s which is far below previously reported values [48, 21] and meets the requirements for tests of the equivalence principle with Eötvös parameters in the range of $10^{-15}$ [44].

2.3 Immiscible quantum mixtures in gravity

Using the above described methods to find a suitable switch-off protocol, we study interacting quantum mixtures in presence of the Earth’s gravitational field. While being partially suppressed, the system remains sensitive to the shutdown of the magnetic trapping potentials, as illustrated in this section. This gives rise to a complex interplay between gravity, interatomic repulsion and magnetic release forces. Having a solid understanding of this interplay is an important step towards future space-borne experiments with high demands on absolute or differential release velocities [6, 17, 5]. The inherent portability of our experiment offers the unique possibility to adjust the orientation of the experimental chamber (and so the atom chip) with respect to gravity. For that purpose, we mount the physics package into a rotatable frame which defines the rotation axis as shown in Fig. 1 b,c). In this way we disentangle the forces due to the decaying magnetic field, which mainly acts perpendicular to the atom chip surface, from the forces due to gravity and due to the mutual repulsion of the gases. The latter is aligned with gravity as the two gases being located one on top of the other. In Fig. 5a-c, we show absorption images of 41K (cyan) and 87Rb (red) BEC mixtures released from a magnetic trap for three different angles to gravity. In all cases, the 41K BECs are well separated from the 87Rb BECs and aligned to the gravity vector. In our release trap, the gravitational sag-induced displacement of the trap minima is only on the order of 1 µm. Thus, the center-of-mass separation of both species is primarily due to the repulsive interaction, as both condensates align in the trap along gravity, thereby increasing the shift of the clouds in this direction. We perform a comprehensive simulation of the ground state and evolution of the BEC mixture taking into account the previously described dynamics of the magnetic trap release. The simulation is based on a numerical toolkit solving the 3D time-dependent Gross-Pitaevskii equations [32] and has been extended for time-varying potentials to capture the magnetic field transients.

The calculated density profiles are shown in Fig. 5 on top of the experimental data. Although the mechanical mount allows to tune the angle in steps of $\Delta\varphi=2.5^{\circ}$, we find the exact angle by evaluating the absolute $x$ and $y$ position of the 87Rb cloud after 25 ms time of flight to account for possible deviations. Apart from the refinement of the angle, the simulation does not contain any free parameters and shows the expected absolute positions of the atomic densities projected to the detection system axis (see sec. 4.4 for more details).

The simulated density profiles for rotation angles of $30.9^{\circ}$ and $60.6^{\circ}$ show excellent agreement with the experimental data in positions and shapes. For the $0.0^{\circ}$ configuration (Fig. 5a), a deviation of 57$\,\textrm{\textmu m}\pm 10\,\textrm{\textmu m}$(exp) between the relative distances of the two density maxima remains between simulation and experiment. To quantify the impact of the decaying magnetic fields on the atoms, we compare our simulated density distributions with a second simulation assuming an immediate release without decaying magnetic fields, shown in Fig. 5d)-f). By comparing the calculated positions of the atoms in a-c) with the respective positions in d-f) we can directly quantify the impact of the transient magnetic fields. For all angles, the decaying magnetic fields are acting in the direction normal to the atom chip surface while the interaction driven acceleration is directed along the separation due to the differential gravitational sag. The positions horizontally to the atom chip are not affected by the release. For $\varphi=0.0^{\circ}$, the transient magnetic fields account for $18\,\textrm{\textmu m}$ of the observed separation of the two species.

We also consider the impact of background magnetic field gradients along the direction of free fall and systematic errors in the estimation of absolute atom numbers which would impact the simulated repulsion. From time of flight measurements of differential mixture positions we obtain an upper limit for a residual magnetic field gradient of $B_{z}^{\prime}<0.025\,$G/cm which leads to an additional relative separation of $<5.6\,\textrm{\textmu m}$. By increasing the simulated Rb (K) atom number by $10\%$, the differential position changes by $5.8\,\textrm{\textmu m}$ ($-2.3\,\textrm{\textmu m}$). Given the magnitude of the different systematic effects, we attribute the observed deviations in the 0.0∘ case primarily to an oversimplified release model and, to a lesser extent, to underestimated atom numbers. We emphasize that the observed inconsistency in absolute position is below the size of the expanded BECs and only present in the $0.0^{\circ}$ configuration for Rb.

3 Discussion

We have developed a fully integrated sounding rocket payload for a two-species quantum gas experiment which is able to produce BEC mixtures of 87Rb and 41K with up to $2.5\cdot 10^{5}$ (87Rb) and $5\cdot 10^{4}$ (41K) atoms in a single species optimized sequence. By tuning the duration of the 87Rb-MOT we are able to tune the ratio of the BEC mixtures, allowing to work with $2\cdot 10^{4}$ atoms in each condensate with a short preparation time of 2.3 s. This represents, to the best of our knowledge, the highest flux of BEC mixture generation.

To further increase the atom numbers in the context of future projects, it will be necessary to increase the number of initially trapped 41K atoms in the magnetic trap but also to further limit losses due to inelastic collisions between the species during evaporation. Improvements in laser cooling of 41K, for example by D1-cooling techniques, could significantly increase the number of magnetically trapped atoms [49]. Inelastic collisions can be further reduced by increasing the power of the depumper microwave. Techniques such as optical shielding have also been suggested as an effective way to reduce inelastic collisions in ultracold atomic mixtures[50].

We have shown the impact of realistic magnetic field decays of the external coils and chip structures and observed a differential release velocity between both species. By introducing a time delay between switching off the magnetic coil and the atom chip we were able to reduce the differential release velocity. The technique is compatible with requirements on the differential velocity of future high-precision measurements such as the tests of the Einstein equivalence principle.

We note that the spatial evolution of expanding BEC mixtures in different interaction regimes has been studied in a number of experiments [51, 52, 53, 54]. In these realizations, atoms were trapped in optical dipole traps and are naturally not affected by magnetic forces during release. Although recent progress has demonstrated the feasibility of dipole traps in microgravity [13, 55], atom chips are widely used in compact experiments and represent the tool of choice in environments with high demands on robustness and low power consumption.

We could demonstrate the robustness of the system and the sequences by rotating the payload with respect to gravity between 0 and 75∘ and generated BEC mixtures in each configuration without changing sequence parameters. The rotation allowed us to study the different forces acting on the atoms, in particular gravitational forces, interspecies repulsion and magnetic forces generated by eddy currents during release.

By applying our switch-off protocol in microgravity with this payload, the dynamics of the atoms would only be governed by their intra- and interspecies interactions. In absence of the gravitational sag, the mixture ground state can show various geometric configurations, such as symmetric splitting of the 41K-atoms in cigar-shaped traps [32] or closed shells of 41K forming bubble traps [4]. The interaction driven expansion in microgravity thus depends on the details of the underlying ground state geometry. Our findings offer therefore a detailed understanding of the release processes and minimization of additional transient forces that are necessary for future quantum mixture experiments relevant for many-body physics as well as for quantum sensing.

4 Methods

4.1 Apparatus design and sequence details

All experiments reported here were performed on ground with the sounding rocket payload MAIUS-B, designed for the generation of 41K and 87Rb BECs in space. The payload was launched within the sounding rocket flight campaign MAIUS-2 on 2nd of December 2023. Details on the overall payload design can be found in [34].

The BEC generation is performed in a two chamber design, consisting of a source and a science chamber. The source chamber generates a cold, dual-species atomic beam by overlapping a 41K and a 87Rb 2D*+*-MOT. The pusher laser beam guides the cold atoms through a differential pumping stage towards the science chamber, where they are trapped by a three-dimensional mirror MOT, provided by four laser beams, an atom chip and three pairs of Helmholtz coils along all spatial directions. In a subsequent step, the atom chip Z-structures form the Ioffe-Pritchard trap together with a bias field generated by the y-coils for magnetic confinement of 41K and a 87Rb (see Fig. 6). At this stage, evaporative cooling of 87Rb is performed with two microwave frequencies that are coupled via U-shaped antennas on the base chip layer. The atom clouds are imaged in consecutive experimental runs using absorption imaging [56] with two lenses in $F-2F-F$ configuration ($F=50\,$mm) and circularly polarized light. After a variable time of flight, the atoms are imaged with a 20 µs imaging pulse resonant to the $\lvert S_{1/2},F=2\rangle\rightarrow\lvert P_{3/2},F=3\rangle$ transition of 41K or 87Rb, respectively.

Before integration of the full payload, the cold atom apparatus was mounted in a rotatable aluminium frame to characterize the influence of different gravity directions. This allowed a rotation around the weak (longitudinal) trapping axis of the cylindrical trap in steps of 2.5°, resulting in relative change of the direction of gravity with respect to the magnetic trap. This way, gravity never points along the weak axis which makes it possible to trap the atoms for all angles.

4.2 Atom losses and collisions

There are different mechanisms that lead to a decrease of 41K atoms in the BEC with increasing number of 87Rb atoms trapped initially in the MOT. If loaded simultaneously in the MOT, collisions between 41K and 87Rb lead to losses within the 41K-MOT. As shown in Fig. 7 a) we see an impeding effect of the 87Rb-MOT to the 41K-MOT. In the magnetic trap, the most important loss process is inelastic collisions between 41K atoms in the state $\lvert F=2,m_{F}=2\rangle$ and 87Rb atoms in the impurity state $\lvert F=2,m_{F}=1\rangle$ [57]. Detrimental effects of remaining 87Rb atoms in $\lvert F=2,m_{F}=1\rangle$ were observed previously in other setups [58, 59]. Fig. 7 shows the comparison of the different lifetimes of magnetically trapped 87Rb atoms (3.5 s), 41K atoms (2.3 s) and 41K in presence of $4.7\cdot 10^{7}$ 87Rb atoms (0.91 s). By switching on the depumper microwave which is used to depump co-trapped 87Rb atoms in the impurity state $\lvert F=2,m_{f}=1\rangle$, the lifetime of 41K increases from 0.91 s to 1.4 s. We attribute the remaining difference in lifetime to 87Rb atoms in the state $\lvert F=2,m_{F}=1\rangle$ getting continuously repumped from the anti-trapped state $\lvert F=1,m_{F}=1\rangle$ by the evaporation microwave after they moved by some distance.

4.3 Simulation of center-of-mass dynamics and trap configurations

The center-of-mass motion of the released atoms, which models the trap release discussed in Section 2.2, is determined by solving the classical equations of motion. These include the time-varying potential generated by the magnetic field $\textbf{B}(\textbf{r},t)$. For a given time $t$, the magnetic field configuration is calculated by adding up the magnetic fields of the chip structures and coils using the Biot-Savart law

[TABLE]

with the time-dependent currents $I_{j}(t)$ of the atom chip structures and external coils, the magnetic constant $\mu_{0}$, the line element $d\textbf{l}_{j}$ along the structure $C_{j}$ and the distance $\textbf{r'}_{j}$ between the line element $d\textbf{l}_{j}$ and r. We calibrated our magnetic field model by evaluating positions of trap minima and their respective magnetic field magnitudes and trap frequencies over a wide range of relevant traps. As gauging parameters, we used scaling factors for the X- and Y-coil bias fields, constant magnetic field offsets $B_{x},B_{y}$ and z-direction shifts of the science and base chip layers. The residual deviation between model predictions and experimental measurements in the trap frequencies and positions is below $2\%$ across the range of traps studied in the Results section.

The potential is given by the Zeeman energy shift

[TABLE]

with $B(\textbf{r},t)$ being the modulus of the magnetic field $\textbf{B}(\textbf{r},t)$, $\mu_{B}$ the Bohr magneton, $g_{F}=1/2$ the Landé g-factor and $m_{F}=2$ the magnetic quantum number. The equations of motion for the 87Rb and 41K atoms are given by:

[TABLE]

where $q_{j}\in\left\{x,y,z\right\}$ and $m$ the respective atomic mass. These equations are respecting the initial conditions $q_{j}(0)=q_{j,0}$ and $\dot{q}_{j}(0)=\ddot{q}_{j}(0)=0$ for both species where $q_{j,0}$ is the trap minimum at $t=0$. To solve the equations of motion, we use the leap-frog integration method [60] with time steps of $\delta t=10\,$µs.

4.4 3D-modelling of BEC mixture

The simulation of the absorption images is based on a three-step process: calculation of the BEC ground state, propagation during free-fall and finally, projection onto the imaging plane. The ground state is determined using the imaginary time propagation method to solve the coupled Gross-Pitaevskii equation (GPE) describing an interacting BEC mixture [61, 62]. The s-wave scattering lengths used to calculate the interspecies and intraspecies interactions are $a_{Rb}=98.96\,a_{0}$ [63], $a_{K}=60.54\,a_{0}$ [64] and $a_{KRb}=165.3\,a_{0}$ [65] in units of the Bohr radii $a_{0}$. The trapping potential is approximated harmonically around the potential minimum using the magnetic field from the chip model. This model provides the characteristics of the trapping potential such as the minimum position, frequencies and orientation of the eigenvectors. In general, minimum positions and eigenvectors are different for both species 41K and 87Rb due to the gravitational sag. The gravity vector is set according to the tunable orientation of the apparatus. The atom numbers of the simulated ensembles are determined experimentally using absorption imaging of the two clouds [56].

The ground state is then used as the initial state to simulate the dynamics of the mixture during its free expansion by solving the coupled GPE with the scaled grid method presented in [32]. This enables an efficient calculation of the dynamics as the simulation volume increases by $10^{4}$ during the 25.38 ms expansion. The CPU time required for the free expansion dynamics is roughly 50 minutes of CPU time, and 4 minutes of real time on a cluster using 16 cores in parallel, for a spatial grid of $(N_{x}=64,N_{y}=64,N_{z}=128)$ points.

The evolution of the BEC mixture is described in the coordinates of the atom chip (see Fig. 4). During the switch-off ramp, the trap characteristics (frequencies and position) are extracted from the chip model by evaluating the Hessian matrix of the minimum position. However, within this sequence, this estimate shows divergences over very short time intervals (less than 10 µs), due to a brief and transient dominance of potential anharmonicities. Given their extremely short duration, these quasi-instantaneous irregularities are smoothed out, which enables agreement between simulation and experimental results. The free expansion of the atoms is considered as a free fall in the calculations, the final position of the atoms takes into account the distance travelled by the atoms during the time of flight. The chip edge is measured on the camera and calibrates the origin of the coordinate frame. To visualize the atoms in the camera frame, we integrate the 3D density along the optical axis of the detection system $\textbf{n}=(\cos(\theta),\sin(\theta),0)$ with $\theta=46.1^{\circ}$. To take into account the finite resolution of the detection system, the acquired density images are convoluted with a Gaussian distribution of width $\sigma=15$ µm.

For better comparability, the thermal fractions of both ensembles are extracted from the Gaussian part of a bimodal fit from the absorption images and added to the simulated data, using the same method as in [32, 11].

5 Data availability

Simulation results and absorption image data are available on request.

6 Code availability

Codes for the performed simulations are available under reasonable request.

7 Acknowledgements

The QUANTUS IV - MAIUS project is a collaboration of Zentrum für angewandte Raumfahrttechnologie und Mikrogravitation Bremen, Leibniz Universität Hannover, Humboldt-Universität zu Berlin, Johannes Gutenberg-Universität Mainz and Ferdinand-Braun-Institut, Leibniz-Institut für Höchstfrequenztechnik. We acknowledge support from Deutsches Zentrum für Luft- und Raumfahrt - Raumfahrtbetrieb, Oberpfaffenhofen, Deutsches Zentrum für Luft- und Raumfahrt - Simulations- und Softwaretechnik, Braunschweig.

8 Competing interests

The authors declare no competing interests.

9 Author contributions

B.P, J.B., P.G. and D.L. conducted the experiments under supervision of E.M.R. Data analysis and interpretation were performed by B.P., J.B., T.E., A.Pi., P.G., N.G. and E.M.R. Numerical simulations were provided by T.E., A.Pi. and G.M. under supervision of N.G. and E.C. The development and commissioning of the apparatus was led by J.G. with contributions from B.P, J.B., P.A., W.B., S.B., K.D., M.E., O.H., C.K., A.Pa., C.R., A.W. and T.W.

B.P, J.B., T.E. and A.Pi. wrote the manuscript with feedback from all authors. E.C., N.G., J.G., A.Pe. K.S., A.W, P.W. and E.M.R. are principal investigators of the contributing groups. J.G. is the principal investigator of the project.

10 Funding

This work is supported by the German Space Agency DLR with funds provided by the Federal Ministry for economic affairs and climate action (BMWK) under grant number DLR 50WP 1431-1435. T.E. and G.M. acknowledge support from DLR grants 50WM2245-A (CAL-II), 50WM2545A (CAL-III). We acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC-2123 QuantumFrontiers - 390837967 and SFB 1227 (DQ-mat).