Abstract
Extending the understanding of Bose–Einstein condensate (BEC) physics to new geometries and topologies has a long and varied history in ultracold atomic physics. One such new geometry is that of a bubble, where a condensate would be confined to the surface of an ellipsoidal shell. Study of this geometry would give insight into new collective modes, selfinterference effects, topologydependent vortex behavior, dimensionality crossovers from thick to thin shells, and the properties of condensates pushed into the ultradilute limit. Here we propose to implement a realistic experimental framework for generating shellgeometry BEC using radiofrequency dressing of magnetically trapped samples. Such a tantalizing state of matter is inaccessible terrestrially due to the distorting effect of gravity on experimentally feasible shell potentials. The debut of an orbital BEC machine (NASA Cold Atom Laboratory, aboard the International Space Station) has enabled the operation of quantumgas experiments in a regime of perpetual freefall, and thus has permitted the planning of microgravity shellgeometry BEC experiments. We discuss specific experimental configurations, applicable inhomogeneities and other experimental challenges, and outline potential experiments.
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Introduction
The study of quantumdegenerate ultracold atomic gases has historically been guided by explorations of geometry, dimensionality, topology, and interaction. Whenever the parameter space of dimensionality and geometry has been expanded, interesting physics has typically been unveiled. Studying Bose–Einstein condensates (BECs) in 2D has yielded insight into quasicondensation and the Berezinskii–Kosterlitz–Thouless (BKT) transition,^{1,2,3} and in 1D insight into fermionization and manybody systems out of equilibrium.^{4,5,6} Exploring toroidal condensates has driven progress in understanding persistent currents and uncovered links to cosmological inflation,^{7,8} and doublewell condensates have been used for many applications including matterwave interferometry and spin squeezing.^{9,10} A shell or bubblegeometry BEC, while physically interesting due to its distinct topology, has not been physically realized due to the distorting influence of gravity on typical atom traps. In this work, we present modeling related to proposed experiments with bubblegeometry BECs aboard the NASA Cold Atom Laboratory (CAL), currently in operation aboard the International Space Station (ISS).
An experimental path to creation of shell potentials for BECs was proposed not long after the first creation of BEC itself, focusing on socalled adiabatic potentials created with radiofrequency (rf)dressed magnetic traps.^{11} Alternate schemes for the study of shell BECs have focused on the specific study of the superfluid shells in opticallattice Mottinsulator systems,^{12} or in the exotic environment of a neutron star.^{13} More recent theoretical work has focused on the collective modes of shell condensates, and the signatures of a condensate transitioning to a hollow shell from a conventional topology.^{14,15} Interesting effects are predicted to occur when a shell condensate is released into timeofflight expansion; different regions of the shell BEC will interfere with each other, resulting in spatial matterwave interference patterns that are quite sensitive to the shape of the shell potential and (via meanfield interactions) the number of atoms in the condensate.^{16} Recent work has also been done exploring the basic physics of BEC on the surface of a sphere.^{17,18,19}
Further, the motivation for the study of shellshaped condensates stems from the drastic change in topology associated with expansion into a shell; vortex behavior (for example) shows promise as an avenue of investigation, including the potential study of vortex lattices in a curved background. Vortices in a shellshaped condensate will behave in a qualitatively different manner than those in a flat condensate (such as a disk) because of the curvature of the shell surface and because of the topology of the shell as an unbounded simply connected surface. Previous theoretical work has predicted that a single pair of vortices on the surface of a sphere will repel and therefore arrange themselves at polaropposite points.^{20} Vortices in the shellcondensate system can be induced through rotations or, if the shell is thin enough, they will be spontaneously produced near the thermal transition to a noncondensed gas.^{21} The effect of curvature on vortices in a thin condensate is a richer area for exploration; for example, defects (such as vortices) on a curved surface experience a force due to the local curvature.^{22,23}
The interaction of rf or microwave radiation with a set of Zeemansplit hyperfine manifolds is a wellstudied system that is often characterized in terms of dressed states, which in the case of inhomogeneous magnetic fields, such as found in magnetic traps result in socalled adiabatic potentials.^{24,25} Figure 1 illustrates the general idea of rf dressing; lowerlying adiabatic potentials are associated with the “rf knife" techniques of evaporative cooling in magnetic traps, while the higherlying adiabatic potentials can be understood as doublewells in 1D, ring potentials in 2D, and shell potentials in 3D, as first proposed by Zobay and Garraway.^{11,26,27} Experimentally, ultracold gases in rfdressed shell potentials were first generated with the key observation that gravitational sag caused the shelltrapped samples to localize near the bottom of the shell potential;^{28,29} indeed, this localization could be considered a feature due to the possibilities of applying it to studies of effectively 2D quantum gases.^{30}
The detuning of the rf frequency from magnetic resonance acts to control the mean radius of the bubble potential, and the coupling strength \(\Omega\) (which, being proportional to the rf magnetic field amplitude, could have some weak spatial dependence) serves the twofold purpose of controlling the potential curvature of the local bubble minimum but also ensuring (through sufficiently large magnitude) stability against Landau–Zenertype nonadiabatic losses in this dressedstate picture. These losses have been explored in the context of magnetic traps^{31} and also are connected to the stability of condensates in rfdressed spindependent optical lattices.^{32,33}
The rfdressing process resulting in shelllike BEC could be performed in any ultracold atomic physics experimental framework featuring magnetic trapping and elimination of gravitational perturbation, and thus could be implemented in droptower,^{34,35} ballistic aircraft,^{36} or the most recently developed soundingrocket^{37} configurations (the latter representing the first BEC experiment in space). Our investigation has focused on planning experiments aboard the NASA CAL. CAL was developed by the Jet Propulsion Laboratory (JPL) beginning in 2013 and is currently in operation aboard the ISS after a 2018 delivery. CAL is an atomchipbased BEC machine equipped with a variety of experimental degrees of freedom permitting operation with multiple experimental PIs with a diversity of experimental frameworks, including Efimov physics,^{38} adiabatic expansion and deltakicked cooling to pK temperatures,^{39,40} novel atom lasers,^{41} and ongoing development of atominterferometer capabilities.
General capabilities of the instrument (see accompanying illustrations in Fig. 2) include providing ^{87}Rb BECs with \(N\;>\;1{0}^{4}\) in an initial highaspectratio trap configuration with approximate trap frequencies \(\{{\omega }_{x},{\omega }_{y},{\omega }_{z}\}=2\pi \times \{200,1000,1000\}\) Hz, where \(z\) is the direction perpendicular to the atom chip and \(x\) is the direction associated with the bias magnetic field at trap bottom. Condensates are obtained via rf evaporation of a sample held in the magnetic trap formed by a combination of currents flowing through the atomchip wires and three quasiuniform external bias fields. Details of system development and ground test status can be found in ref. ^{42} Specific design input was sought from prospective users; for example, significant guidance regarding the rf system design of dressedatom experiments can be found in the literature, specifically focusing on the need for direct digital synthesis (DDS) signal sources and very finegrained frequency ramps during the dressing process in order to avoid excess heating.^{43} A key capability to begin dressedatom experiments with CAL is the generation of traps of lower density and aspect ratio; hence, a trap expansion protocol that does not incur unwanted centerofmass motion is desired. Such paths have been developed in the context of shortcuts to adiabiaticity with droptower missions^{44} and in planning for CAL; in a semiclassical approach we have developed expansion ramps roughly in the form of a hyperbolic tangent, following the formalism of ref. ^{39}
Results
We present a general procedure for forming a shell condensate in a machine, such as CAL, illustrated and parametrized in Fig. 2c. First, the condensate would be prepared in a given initial starting condition (the “bare trap", in the internal state \(\leftF=2,{m}_{{F}}=2\right\rangle\)), at which point the rf dressing signal would be switched on with the detuning \(\Delta =\omega {\omega }_{0}\) negative and large compared with the Rabi frequency \(\Omega\), where \({\omega }_{0}\) is associated with magnetic resonance at trap bottom. Secondly, the rf frequency would be ramped upwards, forcing the condensate in the uppermost adiabatic potential into a shell geometry. The timescale of this ramp (~100 ms anticipated) would be enforced by mechanical adiabaticity of the BEC deformation and technical limits on the graining of the rf signal; timescales associated with motion perpendicular to the local shell surface are easily satisfied (dressedtrap frequencies perpendicular to the shell surface are \(\sim\)100–1000 Hz, depending on dressing parameters), but adiabaticity with respect to motion around the shell remains an open question. Coupling strengths \(\Omega /2\pi \sim\) 10 kHz are appropriate for these scenarios, chosen in the context of the suppression of Landau–Zener losses;^{30} this rf amplitude is well within the documented capability of the CAL instrument.
Following the formalism discussed in the “Methods” section we calculate adiabatic potentials for several different cases of interest. In particular, it is useful to investigate the effects of rf detuning and atom number on the planned experiments, and explore the consequences of various inhomogeneities associated with the experiment. The most dominant inhomogeneity associated with such experiments on Earth is gravitational potential energy \(mgz\) (absent in the “Methods” section Eq. (1)), which for ^{87}Rb corresponds to a tilt of \(h\times\)2.14 kHz/μm (or \({k}_{{\mathrm {B}}}\times\)103 nK/μm). Taken into account across a typical condensate profile, this dwarfs the ability of BEC interaction energy to “fill up" a gravitationally tilted shell, although some initial progress has been made in using ac Stark shift gradients as compensation.^{45} In freefall this effect is negligible (~1 μg) and we are left with several confounding factors orders of magnitude smaller, framed as follows: inhomogeneity A, associated with the ellipsoidal aspect ratio of the shell potential, inhomogeneity B, associated with the difference in direction of the local magnetic field vector across the trapped atomic cloud (impacting dressing via departure from orthogonality with the dressing field), and inhomogeneity C, associated with the difference in \(\Omega\) across the sample.
While slices of condensate density along principal directions are useful modeling checks, experimental data will come in the form of column density along a particular direction, obtained via absorption imaging. Fig. 3 shows calculated condensate density slices (Fig. 3d–f) and column densities (Fig. 3g, h) for planned magnetic field configurations. The example trap chosen has identical atomchip currents to the “tight trap" where the CAL BEC first forms, but has had external bias field reduced to 0.2\(\times\) their initial value, resulting in trap frequencies of approximately (30, 100, 100) Hz as suggested by our model and by initial calibration experiments aboard CAL. Also shown are examples of column densities taken without accounting for the inhomogeneities A, B, C as defined above, to illustrate their impact. The couplingrelated inhomogeneity C pulls the condensate toward \(+z\) (away from the chip); inhomogeneity B pulls toward \(+y\), and inhomogeneity A results in pooling of atoms at the tips of the trap ellipsoid.
These calculations confirm typical intuition, that the Gross–Pitaevskii nonlinearity driven by repulsive atom–atom interaction (i.e. the chemical potential \(\mu\)) serves to some degree to conceal nonuniformities that are of order \(\mu\). However, this benefit is limited; the groundstate energies associated with the the scenarios in Fig. 3g and h range from \(h\times\) 100 to 200 Hz (or k_{B} × 5–10 nK), an order of magnitude smaller than the groundstate energy of the original condensates. In general, the atom number is not large enough in the CAL scenario to drive a shelltrapped condensate into the interactiondominated Thomas–Fermi regime. Nevertheless, the Gross–Pitaevskii ground states show that a shelltrapped BEC (of size ~50 μm) is within the capabilities of the CAL system to observe, with the caveat that complete density coverage around the surface of the shell will be strongly sensitive to the atom number made available. The effects of terrestrial gravitational tilt is absent in these plots, given the planned microgravity environment; were it present, the modeled clouds would be very strongly pinned to one end of the trap as in the terrestrial experiments.^{28,29,46} The significant inhomogeneity C (that of the rf coupling) can be reduced to some degree by moving to lower absolute coupling strength, given that the tilt is proportional to \(\Omega\); this would be at the eventual cost of reduced dressedstate lifetime due to Landau–Zener nonadiabaticity. For future experiments, it also could be mitigated through experimental design (e.g., rf loop radius and placement).
Discussion
Following 5 years of development, NASA CAL was recently commissioned aboard the ISS after a 2018 launch. It has undergone testing and is in active userfacility mode with several PI groups, including the authors. Initial work will focus on calibration of the various models used to predict trap fields, trap frequencies, and other properties of the atomchip system, followed by exploration of residual motion in given trap configurations, where the magnetic trap has been expanded and translated away from the chip surface. Assuming sufficiently stable BEC production, stable trap position, and repeatable magnetic resonance observations, rf dressing of the CAL atomchip trap is within reach.
Beyond confirmation of shell structure with microgravity BECs, discoveryoriented user time should focus on elucidation of the adiabaticity requirements of shell creation, possible exploration of collectivemode dynamics, and studies of the lifetime of BEC shell structures. We also anticipate observation and characterization of the inhomogeneities predicted and discussed above, and exploration of the behavior of noncondensed thermal atoms in the dressed potential, depending on what condensate fractions are available on orbit. To frame future design considerations, we note that shell thickness might potentially be tunable through use of the nonlinear Zeeman shift,^{47} and that the inhomogeneity associated with the rf loop could potentially be compensated through application of a similarly inhomogeneous microwave dressing field, i.e. a compensatory ac Zeeman shift.^{48,49,50}
CAL is currently scheduled to remain in operation until late 2019, whereupon a major hardware replacement is scheduled to occur, after which the facility should return to userfacility operations for additional time. A secondgeneration orbital microgravity atomchip ultracold atomic physics facility, BECCAL, is currently under development of in Germany as a joint DLR/NASA venture.^{51} This successor machine should share CAL’s capabilities for generation of rfdressed systems, potentially enabling a second generation of shellBEC physics and permitting extended exploration of this novel quantumgas topology.
Methods
To calculate the dressed potentials associated with a single rf driving frequency \(\omega\), we operate in the usual rotatingframe rotatingwaveapproximation formalism summarized in refs, ^{24,25} which results in the Hamiltonian
where \({{\mathcal{H}}}_{{\rm{Zeeman}}}({\bf{r}})\) is diagonal and represents the Zeeman shifts of the states in use, which for the purposes of this work are the ^{87}Rb upper hyperfine ground state denoted by \(\leftF=2,\;{m}_{{{F}}}\right\rangle\), with \({m}_{{{F}}}\) taking values from −2 to 2. Here, the coupling strength \(\Omega \simeq {g}_{{{F}}}{\mu }_{{{B}}}{B}_{{\rm{rf}}}/\hslash\), where \({g}_{{{F}}}\) is approximately \(1/2\) and \({B}_{{\rm{rf}}}\) is the amplitude of the driving rf field. Modeling of terrestrial experiments would require the addition of an \(mgz\) term to \({\mathcal{H}}\). The spatially varying eigenvalues of the Hamiltonian in Eq. (1) represent the adiabatic potentials and the eigenvectors represent the spatially varying decomposition of the dressed state in the labspin basis. The magnetic field \({\bf{B}}({\bf{r}})\) that yields the bare potentials depicted at left in Fig. 1 is calculated using Biot–Savart integration using finitewidth and finitelength wires informed by experimental design specifications (see Fig. 2).
Condensate densities are calculated using an imaginarytime propagationbased Gross–Pitaevskii solver^{52,53} using the uppermost dressedstated potential \(U({\bf{r}})\) as input, along with illustrative condensate numbers \(N\) chosen to be different by a factor of 5 and consistent with CAL specifications.
Data availability
Relevant simulation results are available upon reasonable request from the corresponding author.
Code availability
Calculation codes (written in MATLAB 2017 and Mathematica 11) are available upon reasonable request from the corresponding author.
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Acknowledgements
We thank Ethan Elliott, Barry Garraway, Jeffrey Oishi, German SinucoLeón, Smitha Vishveshwara, and Karmela Padavić for useful discussion. This work was supported by Jet Propulsion Laboratory (JPL) Research Support Agreement no. 1502172 under a contract with the National Aeronautics and Space Administration (NASA), Division of Space Life and Physical Sciences Research and Applications (SLPSRA).
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N.L. wrote the main text and conceived of the proposal. R.A.C. performed simulations and provided commentary. C.L. provided simulation code and provided commentary. D.C.A. provided experimental guidance and helped direct the research. D.P., X.J., N.S., and M.J.G. all contributed to simulations and discussion.
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Lundblad, N., Carollo, R.A., Lannert, C. et al. Shell potentials for microgravity Bose–Einstein condensates. npj Microgravity 5, 30 (2019). https://doi.org/10.1038/s415260190087y
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DOI: https://doi.org/10.1038/s415260190087y
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