Mode-resolved atomic gravitational cavity: realistic mirror potentials, loss budgets, non-paraxial modes, and mode-selective loading

In 1992 Wallis, Dalibard, and Cohen-Tannoudji analyzed a matter-wave resonator made of a single horizontal concave mirror closed by the Earth's gravitational field — the 'gravitational cavity' — deriving its stability criterion, its paraxial quantum modes, and estimates of how a dropped cold-atom cloud populates them, under deliberately idealizing assumptions: an infinite hard-wall parabolic mirror, near-vertical (paraxial) motion, no loss mechanisms, and only a qualitative suggestion of velocity-selective Raman mode selection. The open problem is to connect that idealized mode analysis to concrete experimental realizations: quantify the loss mechanisms and mode lifetimes (momentum diffusion at the reflecting evanescent potential, diffraction and leakage from a finite mirror aperture, density- and light-assisted collisional losses), compute the non-paraxial eigenmodes and their overlap integrals with realistic sources, analyze mode-selection schemes quantitatively, and include realistic finite-aperture, finite-penetration mirror models — enough to predict steady-state mode populations and achievable degeneracy factors (atoms per center-of-mass mode). Later literature has realized adjacent pieces of the program: bouncing-atom experiments with evanescent-wave and magnetic mirrors, gravito-optical surface traps reaching a regime of quantized vertical motion, and full gravitational-quantum-state spectroscopy for neutrons (GRANIT/qBounce). However, gravitational quantum states of atoms have never been observed — the GRASIAN collaboration is preparing a first attempt with a cryogenic hydrogen beam — and the mode-resolved, many-atoms-per-mode regime of the original atomic cavity remains unattained, leaving the compound question open.

Background

A 'gravitational cavity' is a matter-wave resonator in which a single horizontal mirror confines atoms from below while the Earth's gravitational potential $mgz$ confines them from above: an atom launched upward decelerates under gravity and returns to the mirror, repeating the cycle like a photon bouncing between the mirrors of an optical resonator, with gravity playing the role of the second mirror. The vertical part of this motion is the quantum 'bouncer': above a hard reflecting surface, the linear potential $V(z)=mgz$ supports bound stationary states that are Airy functions with characteristic length $z_0=(\hbar^2/2m^2g)^{1/3}$ and energies $E_n=mgz_n$, where $z_n$ is fixed by Airy-function zeros; for alkali atoms $z_0$ is of order a few micrometers. Atom mirrors can be built from evanescent waves — the exponentially decaying light field above a prism where a red-detuned laser undergoes total internal reflection, which acts as a steep repulsive potential of finite penetration depth (tens to hundreds of nanometers) — from patterned magnetic films, or, for very slow atoms, from quantum reflection, the nonclassical reflection of a matter wave off the attractive van der Waals–Casimir–Polder potential of a material surface before the atom reaches the surface.

Wallis, Dalibard, and Cohen-Tannoudji (1992) analyzed the cavity formed by a concave parabolic mirror of radius $R_\mathrm{M}$. Classically, paraxial (near-vertical) bouncing trajectories are stable when the trajectory apex height satisfies $h < R_\mathrm{M}/2$. Quantum mechanically, the three-dimensional problem is separable in parabolic coordinates anchored at the mirror focus, and in the paraxial regime the eigenmodes take a Gaussian-beam form analogous to the transverse-electromagnetic (TEM) modes of an optical resonator, with transverse waist $w_\mathrm{E}=[2z_0^3(R_\mathrm{M}-2z_\mathrm{E})]^{1/4}$ at the classical turning point $z_\mathrm{E}$, and with longitudinal spacings equal to the one-dimensional quantum-bouncer level spacings. Loading estimates for a dropped cold cloud showed that a typical magneto-optical trap (MOT) source populates on the order of $10^4$ longitudinal modes with a per-mode transverse ground-state overlap of order $10^{-7}$, so that accumulating many atoms per mode — the 'degeneracy factor' $N_v$ — requires mode-selective elements, for which velocity-selective Raman pumping between internal atomic states was suggested, but only qualitatively. The analysis was explicitly idealized: the mirror is treated as an infinite hard wall, detailed solutions are restricted to the paraxial regime, loss mechanisms are not treated, and the proposed mode-selection schemes are not analyzed quantitatively.

The adjacent status is asymmetric. For neutrons, the vertical quantum-bouncer spectrum above a material mirror — gravitational quantum states (GQS) — was observed in 2002 and is now spectroscopically manipulated in gravity resonance spectroscopy (the qBounce/GRANIT program, in which mechanically driven mirrors induce resonant transitions between gravitational states). For atoms, multiple-bounce confinement was demonstrated with evanescent-wave and magnetic mirrors in the 1990s, and gravito-optical surface traps confine atoms above an evanescent wave in a regime where the vertical motion is quantized; but gravitational quantum states of atoms have not been observed (the GRASIAN project is preparing a first attempt with hydrogen), and the coherent, mode-resolved regime of the original cavity — discrete transverse-plus-longitudinal modes populated by many atoms each — remains unattained. This asymmetry defines what remains open.

Problem Statement

The open problem left by the idealizations of the source paper is to connect the idealized mode analysis of the atomic gravitational cavity to concrete experimental realizations: develop the quantitative realistic theory of mode-resolved trapping of atoms in a single-mirror gravitational cavity. Concretely, for atoms bouncing on a concave mirror closed by the Earth's gravitational field:

(i) Loss mechanisms and finite lifetimes: quantify the momentum diffusion induced by the reflecting evanescent potential, the diffraction and leakage due to a finite-sized mirror, and density- or light-assisted collisional losses, so as to predict realistic mode lifetimes and achievable steady-state populations.

(ii) Non-paraxial mode structure: compute the explicit eigenvalues, eigenfunctions, and overlap integrals with realistic atom sources beyond the paraxial (near-vertical) regime, for the ideal infinite hard-wall parabolic cavity, where separability in parabolic coordinates survives although the Gaussian-beam form of the modes no longer applies; the realistic finite-aperture and finite-penetration mirror potentials that break this separability are the subject of item (iv).

(iii) Mode selection: analyze the proposed mode-selection schemes quantitatively, including their efficiency, state-preparation fidelity, perturbation of the motional states, and achievable repetition and filling rates.

(iv) Realistic mirror models: replace the infinite hard wall with the actual mirror physics — finite mirror aperture, realistic potential profiles (finite penetration depth of the evanescent wave, or van der Waals–Casimir–Polder potentials with quantum reflection for material mirrors), and mirror aberrations — and determine their impact on mode structure and trap stability, with the leakage loss caused by the finite aperture quantified under item (i).

These four items are all mandatory and jointly serve one objective: a quantitative prediction of the achievable degeneracy factors (atoms per center-of-mass eigenmode) that the solver validates against the adjacent experimental limits cited in the Background, for a single realistic operating configuration that the solver selects and specifies in full, with items (i)–(iv) as the required supporting components. The experimental realization of mode-resolved gravitational trapping of atoms — most immediately, the first observation of gravitational quantum states of atoms pursued by the GRASIAN project — is the downstream experimental motivation for this theory, not part of the objective itself.

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