In 1993, Aminoff, Steane, Bouyer, Desbiolles, Dalibard and Cohen-Tannoudji dropped a cloud of laser-cooled cesium atoms onto a curved evanescent-wave mirror and observed roughly ten bounces in a stable gravitational cavity - atoms trapped between an atomic mirror and Earth's gravity - and stated as a far-off goal 'the realization of an atomic Fabry-Perot interferometer and the observation of its modes', which requires preserving atomic coherence, not just atom number, over many round trips. The three-decade lineage since then: classical multi-bounce storage was improved with magnetic mirrors; coherent reflection of condensates, pulsed-mirror 'quantum trampoline' interferometers, 200-round-trip waveguide interferometers, resonant many-pulse light-pulse interferometers, and (2026) a driven two-barrier matter-wave Fabry-Perot cavity were realized; and the discrete gravitational quantum states were observed for ultracold neutrons (2002) but, by explicit statements in 2023-2024 of the GRASIAN collaboration (a collaboration pursuing the first demonstration of gravitational quantum states of atoms with a cryogenic hydrogen beam), remain unobserved for atoms. The surviving open core is precise: no experiment demonstrates, for atoms, a single gravitational-cavity resonator in which atoms complete many round trips with matter-wave coherence preserved and the cavity's discrete quantum modes are observed and used interferometrically. The problem remains open as an experimental objective supported by mature theory.

Background

In the early 1990s, atom optics - the manipulation of the de Broglie matter waves of neutral atoms using light - had produced its first interferometers, in which an atomic wave is coherently split and recombined so that the relative phase of the matter waves is read out as interference fringes. A natural next step, articulated by Balykin and Letokhov in 1989, was to build the matter-wave analog of the optical Fabry-Perot cavity: a resonator in which a matter wave bounces many times between reflecting boundaries, so that only waves matching the cavity's discrete modes are reinforced. Aminoff, Steane, Bouyer, Desbiolles, Dalibard and Cohen-Tannoudji (Physical Review Letters 71, 3083 (1993)) took the first experimental step using a gravitational cavity: a cold cloud of cesium atoms dropped a few millimeters onto a curved mirror for atoms, so that the atoms bounce repeatedly under Earth's gravity like balls on a trampoline, with gravity itself acting as the second 'mirror' of the cavity.

The atomic mirror in that experiment was an evanescent-wave mirror: a laser beam tuned to the blue of an atomic resonance is totally internally reflected at a glass surface, and its evanescent field - light leaking from the surface with an intensity that decays exponentially over a fraction of the optical wavelength - produces, through the ac Stark (light) shift of the atom's ground state, a repulsive potential hill that reflects slow atoms. Reflection is only useful for coherent matter waves if the atom scatters essentially no photons during the bounce; the scattering probability per bounce falls only as the inverse of the laser detuning, which pushes operation far from resonance, where the weaker potential demands more power or slower atoms. The mirror was ground concave (radius of curvature about 2 cm), which makes near-vertical trajectories transversely stable, in the same way a curved mirror stabilizes an optical resonator. The 1993 experiment observed about eight to ten bounces of about 10^7 laser-cooled cesium atoms; a Monte Carlo simulation of classical ballistic trajectories reproduced all observations. The dynamics were thus entirely classical, with roughly 39% of atoms lost per bounce (photon scattering during reflection, stray light optically pumping atoms into non-reflective states, background-gas collisions, and residual misalignment).

The quantum regime of such a cavity is well defined. A particle above an ideal reflecting plane in a linear gravitational potential - the 'quantum bouncer' - has discrete bound states whose wavefunctions are Airy functions, with eigenvalues En=zn(ℏ2g2m/2)1/3E_n = z_n(\hbar^2 g^2 m/2)^{1/3} where znz_n are the zeros of the Airy function and mm the particle mass: picoelectronvolt-scale energies, with spacings between low-lying levels of order a kilohertz for cesium. Wallis, Dalibard and Cohen-Tannoudji (1992) analyzed these cavity modes and the cavity's stability, and Liston, Tan and Walls (1995) computed the quantum dynamics of bouncing atoms in this cavity: a localized wave packet disperses across the mode spectrum and partially rephases in quantum revivals - the matter-wave analog of the resonance spectrum through which an optical Fabry-Perot displays its modes. Reaching this regime demands something categorically harder than the 1993 demonstration: not only the atoms' survival but the survival of their coherence over many round trips, i.e. vacuum, mirror quality and mechanical stability such that the phase accumulated per bounce is controlled, and a detection method that can read the cavity without emptying it (the 1993 fluorescence probe was destructive).

The 1993 paper's closing perspectives therefore posed a single far-off goal - the realization of an atomic Fabry-Perot interferometer and the observation of its modes - with techniques 'envisaged by analogy with lasers': a pulsed mirror for mode locking of the cavity; transverse mode selection by reducing the mirror size; and longitudinal mode selection by Raman velocity selection, i.e. using velocity-selective Raman transitions between internal atomic states to load only atoms whose bounce frequency matches a cavity resonance.

Three decades of related work followed. Magnetic mirrors for atoms (magnetized or current-carrying surfaces that reflect atoms via the magnetic dipole potential) enabled classical multi-bounce storage with far lower loss; coherent reflection of Bose-Einstein condensates (ultracold, coherent atomic ensembles) from evanescent-wave mirrors was demonstrated, including diffraction in the time domain; a pulsed 'quantum trampoline' interferometer suspends a falling condensate with repeated diffraction pulses and shows high-contrast multi-wave interference; guided atom interferometers have preserved coherence over 200 round trips in a magnetic waveguide; resonant many-pulse light-pulse interferometers have achieved large phase amplification; a driven two-barrier matter-wave Fabry-Perot cavity was realized in 2026; and the gravitational quantum states - the quantum-bouncer mode structure - have been observed for ultracold neutrons above a mirror (2002) and spectroscopically resolved (2014). None of these, however, delivers for atoms the combination the 1993 goal requires: a mirror-and-gravity cavity in which atoms complete many round trips with coherence preserved and the cavity's discrete quantum modes are observed and used interferometrically.

Problem Statement

Realize, for neutral atoms, an atomic Fabry-Perot interferometer: a matter-wave resonator in which atoms complete many round trips of a gravitational cavity - an atomic mirror closed by Earth's gravity, in the original conception a curved evanescent-wave mirror, with magnetic mirrors a later-developed alternative and a pulsed mirror the identified mode-locked variant - and observe the cavity's quantum modes. This restates the far-off goal of the closing perspectives of the 1993 experiment: 'the realization of an atomic Fabry-Perot interferometer and the observation of its modes', which requires preserving atomic coherence, not just atom number, over many round trips. The objective therefore combines, in a single gravitational-cavity resonator for atoms, coherence-preserving multi-round-trip operation with the experimental observation of the cavity's discrete quantum modes - the quantum-bouncer eigenstates defined in the Background - rather than merely classical bouncing-ball dynamics. The mirror technology (evanescent-wave, magnetic, or pulsed mode-locked), atom species, temperature regime (thermal cloud or quantum-degenerate gas), and continuous-wave versus pulsed-mirror mode of operation are free choices of the solver; they are means to the objective, not parts of its statement.

References

  1. V. I. Balykin and V. S. Letokhov, 'Atomic cavity with light-induced mirrors', Applied Physics B 48, 517-523 (1989). DOI: 10.1007/BF00694689, https://doi.org/10.1007/BF00694689
  2. H. Wallis, J. Dalibard and C. Cohen-Tannoudji, 'Trapping atoms in a gravitational cavity', Applied Physics B 54, 407-419 (1992). DOI: 10.1007/BF00325387, https://doi.org/10.1007/BF00325387
  3. C. G. Aminoff, A. M. Steane, P. Bouyer, P. Desbiolles, J. Dalibard and C. Cohen-Tannoudji, 'Cesium atoms bouncing in a stable gravitational cavity', Physical Review Letters 71, 3083-3086 (1993). DOI: 10.1103/PhysRevLett.71.3083, https://doi.org/10.1103/PhysRevLett.71.3083
  4. G. J. Liston, S. M. Tan and D. F. Walls, 'Quantum dynamics of bouncing atoms in a stable gravitational cavity', Applied Physics B 60, 211-227 (1995). DOI: 10.1007/BF01135865, https://doi.org/10.1007/BF01135865
  5. T. M. Roach, H. Abele, M. G. Boshier, H. L. Grossman, K. P. Zetie and E. A. Hinds, 'Realization of a magnetic mirror for cold atoms', Physical Review Letters 75, 629-632 (1995). DOI: 10.1103/PhysRevLett.75.629, https://doi.org/10.1103/PhysRevLett.75.629
  6. P. Ryytty, M. Kaivola and C. G. Aminoff, 'Reflection of atoms from a pulsed evanescent wave', Europhysics Letters 36, 343-348 (1996). DOI: 10.1209/EPL/I1996-00233-9, https://doi.org/10.1209/EPL/I1996-00233-9
  7. I. G. Hughes, P. A. Barton, T. M. Roach, M. G. Boshier and E. A. Hinds, 'Atom optics with magnetic surfaces: I. Storage of cold atoms in a curved floppy disk', Journal of Physics B: Atomic, Molecular and Optical Physics 30, 647-658 (1997). DOI: 10.1088/0953-4075/30/3/018, https://doi.org/10.1088/0953-4075/30/3/018
  8. Valery V. Nesvizhevsky, Hans G. Boerner, Alexander K. Petukhov, Hartmut Abele, Stephan Baessler, Frank J. Ruess, Thilo Stoeferle, Alexander Westphal, Alexei M. Gagarski, Guennady A. Petrov and Alexander V. Strelkov, 'Quantum states of neutrons in the Earth's gravitational field', Nature 415, 297-299 (2002). DOI: 10.1038/415297a, https://doi.org/10.1038/415297a
  9. Yves Colombe, Brigitte Mercier, Helene Perrin and Vincent Lorent, 'Diffraction of a Bose-Einstein condensate in the time domain', Physical Review A 72, 061601 (2005). DOI: 10.1103/PhysRevA.72.061601, https://doi.org/10.1103/PhysRevA.72.061601
  10. F. Impens, P. Bouyer and C. J. Borde, 'Matter-wave cavity gravimeter', Applied Physics B 84, 603-615 (2006). DOI: 10.1007/s00340-006-2399-3, https://doi.org/10.1007/s00340-006-2399-3
  11. M. Robert-de-Saint-Vincent, J.-P. Brantut, Ch. J. Borde, A. Aspect, T. Bourdel and P. Bouyer, 'A quantum trampoline for ultra-cold atoms', EPL (Europhysics Letters) 89, 10002 (2010). DOI: 10.1209/0295-5075/89/10002, https://doi.org/10.1209/0295-5075/89/10002
  12. T. Jenke, G. Cronenberg, J. Burgdorfer, L. A. Chizhova, P. Geltenbort, A. N. Ivanov, T. Lauer, T. Lins, S. Rotter, H. Saul, U. Schmidt and H. Abele, 'Gravity Resonance Spectroscopy Constrains Dark Energy and Dark Matter Scenarios', Physical Review Letters 112, 151105 (2014). DOI: 10.1103/PhysRevLett.112.151105, https://doi.org/10.1103/PhysRevLett.112.151105
  13. P. Manju, K. S. Hardman, P. B. Wigley, J. D. Close, N. P. Robins and S. S. Szigeti, 'An atomic Fabry-Perot interferometer using a pulsed interacting Bose-Einstein condensate', Scientific Reports 10, 15052 (2020). DOI: 10.1038/s41598-020-71973-0, https://doi.org/10.1038/s41598-020-71973-0
  14. Weronika Golletz, Andrzej Czarnecki, Krzysztof Sacha and Arkadiusz Kuros, 'Basis for time crystal phenomena in ultra-cold atoms bouncing on an oscillating mirror', New Journal of Physics 24, 093002 (2022). DOI: 10.1088/1367-2630/ac8a0d, https://doi.org/10.1088/1367-2630/ac8a0d
  15. Hyosub Kim, Katarzyna Krzyzanowska, K. C. Henderson, C. Ryu, Eddy Timmermans and Malcolm Boshier, 'One second interrogation time in a 200 round-trip waveguide atom interferometer' (2022). arXiv:2201.11888, https://arxiv.org/abs/2201.11888
  16. Carina Killian, Zakary Burkley, Philipp Blumer, Paolo Crivelli, Fredrik Gustafsson, Otto Hanski, Amit Nanda, Francois Nez, Valery Nesvizhevsky, Serge Reynaud, Katharina Schreiner, Martin Simon, Sergey Vasiliev, Eberhard Widmann and Pauline Yzombard, 'GRASIAN: towards the first demonstration of gravitational quantum states of atoms with a cryogenic hydrogen beam', The European Physical Journal D 77, 50 (2023). DOI: 10.1140/epjd/s10053-023-00634-4, https://doi.org/10.1140/epjd/s10053-023-00634-4; preprint arXiv:2301.12731, https://arxiv.org/abs/2301.12731
  17. Yiping Wang, Jonah Glick, T. Deshpande, Kenneth DeRose, Sharika Saraf, N. Sachdeva, Kefeng Jiang, Zilin Chen and Tim Kovachy, 'Robust quantum control via multipath interference for thousandfold phase amplification in a resonant atom interferometer', Physical Review Letters 133, 243403 (2024). DOI: 10.1103/PhysRevLett.133.243403, https://doi.org/10.1103/PhysRevLett.133.243403
  18. Carina Killian, Philipp Blumer, Paolo Crivelli, Otto Hanski, Daniel Kloppenburg, Francois Nez, Valery Nesvizhevsky, Serge Reynaud, Katharina Schreiner, Martin Simon, Sergey Vasiliev, Eberhard Widmann and Pauline Yzombard, 'GRASIAN: shaping and characterization of the cold hydrogen and deuterium beams for the forthcoming first demonstration of gravitational quantum states of atoms', The European Physical Journal D 78, 132 (2024). DOI: 10.1140/epjd/s10053-024-00916-5, https://doi.org/10.1140/epjd/s10053-024-00916-5
  19. Muhammad Naeem, A. Sohail, Suleman Mumtaz, Muhammad Ayub, S. A. Hassan and Khalid Naseer, 'Mean-field interaction effects on dynamical localization of cold atoms in a modulated gravitational cavity', Chaos, Solitons and Fractals 211, 118724 (2026). DOI: 10.1016/j.chaos.2026.118724, https://doi.org/10.1016/j.chaos.2026.118724
  20. Jeremy L. Tanlimco, Eber Nolasco-Martinez, Xiao Chai, S. Nicole Halawani, Eric Zhu, Ivar Martin and David M. Weld, 'Matter-wave Fabry-Pérot cavity in the ultrastrong driving regime', Physical Review Letters (2026). DOI: 10.1103/44mq-jfp1, https://doi.org/10.1103/44mq-jfp1; preprint arXiv:2601.21122, https://arxiv.org/abs/2601.21122 (published 2026-09-02)