Gravitational Quantum States

• Gravitational quantum states are discrete energy levels arising from the interplay between quantum mechanics and gravitational forces above reflective surfaces.
• They are modeled by the Schrödinger equation with linear gravitational potential and Airy function solutions, facilitating high-precision tests like the Weak Equivalence Principle.
• Experiments with ultracold atoms, antimatter, and neutron setups leverage atom interferometry and spectroscopy to probe quantum gravity effects and decoherence.

Gravitational quantum states are discrete energy levels that arise when quantum systems, such as ultracold atoms, anti-atoms, or composite quantum probes, experience potential energies dominated by gravity—often in the presence of a material boundary that acts as a reflector or mirror. Theoretical and experimental investigation of such states not only provides a direct quantum probe of the gravitational interaction but also enables high-precision tests of fundamental principles, such as the Weak Equivalence Principle (WEP), and of quantum theory in extreme or novel contexts. Gravitational quantum states are described by solutions of the Schrödinger equation in a linear gravitational potential, typically resulting in Airy function wavefunctions with quantized energies. These states have been studied in the context of neutral antimatter (antihydrogen, positronium), helium in strong astrophysical gravitational fields, and as analogs for quantum systems in condensed matter and quantum gravity frameworks.

1. Theoretical Framework and Governing Equations

The canonical system for investigating gravitational quantum states is a quantum particle (for instance, an ultracold atom or antihydrogen) subject to the Earth's gravitational field while confined above a material surface. The vertical potential energy is given by the linear term $V_g(z) = Mgz$ (with $M$ the gravitational mass and $g$ the local gravitational acceleration), while the material surface induces an additional Casimir–Polder potential $V_{CP}(z) \sim -C_4/z^4$ for small $z$ (Voronin et al., 2011).

The governing equation for the vertical motion is the one-dimensional stationary Schrödinger equation: $$\left[ -\frac{\hbar^2}{2m} \frac{d^2}{dz^2} + V(z) + Mgz - E \right] \Psi(z) = 0,$$ where $m$ is the inertial mass and $V(z)$ combines gravity and the surface interaction.

For separations $z$ much larger than the characteristic Casimir–Polder length scale $l_{CP}$, the gravitational term dominates, and the solutions are Airy functions: $$\Psi(z) \propto \operatorname{Ai}\left(\frac{z}{l_0} - \frac{E}{\varepsilon_0}\right) + K(E)\operatorname{Bi}\left(\frac{z}{l_0} - \frac{E}{\varepsilon_0}\right),$$ with gravitational length and energy scales

$$l_0 = \left[\frac{\hbar^2}{2mMg}\right]^{1/3}, \qquad \varepsilon_0 = \left[\frac{\hbar^2 M^2 g^2}{2m}\right]^{1/3}.$$

The Casimir–Polder interaction is incorporated via a complex scattering length $a_{CP}$, yielding a modified boundary condition $\Psi(z\to 0) \propto (z - a_{CP})$ rather than the hard-wall condition. The energy eigenvalues for low-lying gravitational states are then

$E_n = \varepsilon_0 \lambda_n^0 + Mg a_{CP},$

where $\lambda_n^0$ are the zeros of the Airy function, i.e., $\operatorname{Ai}(-\lambda_n^0) = 0$.

2. Atom-Interferometric Measurement and Quantum Spectroscopy

Coherent superpositions of gravitational quantum states can be generated experimentally, facilitating interferometric and spectroscopic access to their energy differences. For antihydrogen, such superpositions produce time-dependent oscillations in observables linked to the annihilation rate at the surface, with a disappearance (annihilation) rate for a two-state superposition given by: $$\frac{dF_{12}}{dt} = -\frac{\Gamma}{\hbar} \exp\left(-\frac{\Gamma t}{\hbar}\right) [1 + \cos(\omega_{12} t)],$$ where $\Gamma$ is the decay width, and the beating frequency is $\omega_{12} = (E_2^0 - E_1^0)/\hbar$.

By measuring the transition frequencies or spatial density distribution, one can extract the gravitational mass $M$ via

$$\frac{M^2}{m} = \frac{2\hbar \omega_{12}^3}{g^2 (\lambda_2^0 - \lambda_1^0)^3},$$

enabling a direct test of the Weak Equivalence Principle for antimatter. The methodology is sensitive to deviations between gravitational and inertial mass, and these techniques can be further extended using time-resolved echo spectroscopy or by driving transitions with oscillatory or pulsed fields (Tutunnikov et al., 2020, Dufour et al., 2014).

3. Experimental Realizations and Feasibility

The observation of gravitational quantum states requires the preparation of ultracold particles with vertical velocities sufficiently low to be bound by gravity above the mirror. For antihydrogen, the predicted lifetimes in the lowest gravitational state are on the order of $0.1$ seconds, permitting approximately 30 quantum "bounces" before annihilation (Voronin et al., 2011).

Experimental strategies involve:

• Ultracold neutron experiments as tests beds and prototypes, with similar mass scales and state preparation challenges.
• The use of filtering schemes employing horizontal slits formed by mirrors and absorbers, as in ultracold neutron and positronium setups, to spatially select particular gravitational states (Crivelli et al., 2014, Dufour et al., 2014).
• Atom interferometric approaches for dynamic measurement of quantum beats, enabling high-precision extraction of transition frequencies.

In antimatter systems such as positronium, Rydberg excitation, Stark deceleration, and transition into high angular momentum states are crucial for suppressing motional effects and expanding the state lifetime to enable detection (Crivelli et al., 2014). For astrophysical gravitational fields, such as those near neutron stars, the enormous gravitational acceleration ($g \sim 10^{12}\,\mathrm{m/s}^2$) permits gravitational quantum states of helium, resulting in energy splittings in the GHz range, with observable transitions induced by the neutron star's rotating magnetic field (Dalkarov et al., 2015).

4. Gravitational Quantum States and Fundamental Principles

The paper of gravitational quantum states provides a unique platform for tests of fundamental physics:

• Equivalence Principle: Directly probing the ratio of gravitational to inertial mass by spectroscopically measuring transition frequencies, independent of detailed surface interactions.
• Quantum-classical interface: The occurrence of universal phase shifts in atom interferometers reflects the quantum generalization of the equivalence principle, with the ratio $m_g / m_i$ entering the phase (Unnikrishnan et al., 2011).
• Mass-energy relation: For hydrogen-like systems, the equivalence between passive gravitational mass and energy holds for stationary states, but not for superpositions, where time-dependent oscillations of gravitational mass expectation values can arise (Lebed, 2013).

In addition, deviations from additivity of observable distances in superposed quantum gravitational states signal that non-classical geometry manifests only at large scales, while locally the metric remains regular (Piazza, 2022).

5. Extensions to Quantum Gravity, Quantum Information, and Topological Matter

Canonical and path-integral quantization approaches for gravity introduce gravitational quantum states as states in the Hilbert space of the gravitational field, constructed via tetrad and conjugate variables or as spin-network states in manifestly Lorentz-covariant formulations (Matwi, 2019, Cianfrani, 2021). States built from finite-dimensional representations of the Lorentz group enforce invariance under boosts, rotations, and discrete symmetries.

In the context of quantum information and quantum optics, gravitational cat states (massive quantum superpositions) exhibit distinctive robustness to classical and power-law dephasing channels (Rahman et al., 2023, Haddadi et al., 20 Apr 2024). The dynamical preservation of coherence, purity, and one-way steerability demonstrates gravity's role as a resource in maintaining and recovering quantum correlations under noise. Weak measurement reversal protocols further enhance this preservation.

Topological quantum states of matter display quantized gravitational responses, as exemplified by the quantized gravitational coupling constant (GCC) in the charge-curvature response of disclinated quantum Hall states. The GCC is a topological invariant, robust against disorder and geometric imperfections, and analogous to the Chern number for quantum Hall conductance (Jiang et al., 2022).

6. Quantum-Classical Transition and Probing Gravity-Induced Decoherence

Self-gravitational effects become significant in large, massive superposed quantum systems, leading to decoherence and eventual localization on timescales inversely proportional to the system's gravitational self-energy (Bruschi et al., 2020). For individual particles and photons, gravitational coherence remains robust, while for macroscopic bodies, gravitational backreaction is responsible for the classical limit. Gravitational cat states serve as testbeds for gravity-induced decoherence models and for distinguishing among competing theories (standard quantum mechanics plus classical gravity, collapse models, Newton–Schrödinger–type approaches).

7. Broader Implications and Future Directions

Precision studies of gravitational quantum states across platforms—antihydrogen, positronium, helium in neutron star fields, ultracold neutrons, quantum many-body states under gravitational waves—provide essential data for understanding the universality of the equivalence principle, the nature of dark matter and dark energy, and the operational consequences of quantum superpositions in spacetime geometry (Voronin et al., 2011, Crivelli et al., 2014, Dalkarov et al., 2015, Yan et al., 2023). Techniques combining magnetic, optical, and gravitational confinement (e.g., magneto-gravitational traps) enable longer observation times and higher measurement accuracy (Nesvizhevsky et al., 2020).

The interplay of gravitational interaction with quantum information protocols, quantum metrology, and quantum gravity phenomenology is the subject of ongoing theoretical and experimental investigation, with future prospects depending on advances in cooling, state preparation, and detection technologies for both neutral and antimatter systems.