The weak equivalence principle (WEP) has been tested to extraordinary precision for classical bodies, but only recently for matter prepared in nonclassical internal states. Onofrio, Smith, and Viola proposed testing the WEP with a torsion balance whose test masses are ensembles of two-level systems in coherent superpositions of internal energy eigenstates, with possible violations encoded in an operator M^gM^i−1\hat{M}_g\hat{M}_i^{-1} relating gravitational and inertial mass operators; the mean and variance of a quantized torque operator would bound the off-diagonal element ∣r∣|r| and the diagonal deviations r1,r2r_1,r_2. The proposal is explicit that its own foundations and feasibility are unresolved: the operator treatment of mass remains a working hypothesis, large-ensemble coherence over torsion-balance timescales is far beyond current platforms, no complete dynamical modeling with backaction and realistic noise exists, and the analysis is restricted to two-level product states. This problem asks for the theoretical groundwork — foundational, many-body, and dynamical — and the coherence/noise conditions needed to make such a test, or a definitive assessment of its impossibility, a reality.

Background

The weak equivalence principle (WEP) states that the free-fall trajectory of an uncharged test body is independent of its internal structure and composition; in Newtonian language it equates inertial and gravitational mass. Violations are quantified by the Eötvös ratio, η=2∣aA−aB∣/∣aA+aB∣\eta=2|\mathbf{a}_A-\mathbf{a}_B|/|\mathbf{a}_A+\mathbf{a}_B|, comparing the accelerations of two test bodies. Classical tests — rotating torsion balances (a horizontal beam suspended from a fiber, in which a torque proportional to η\eta appears if the two bodies fall differently) and the MICROSCOPE satellite — constrain η≲10−13\eta\lesssim 10^{-13}–10−1510^{-15}. Quantum mechanics raises a new question: the WEP for a body whose internal state is a coherent superposition of energy eigenstates, since mass-energy equivalence makes both the inertial and the gravitational mass internal-energy-dependent quantities. Zych and Brukner formulated a quantum version of the equivalence principle in which the inertial and gravitational masses of such a body are operators, M^i\hat{M}_i and M^g\hat{M}_g; the object M^gM^i−1\hat{M}_g\hat{M}_i^{-1} is then a matrix on the space of internal energy eigenstates. Its diagonal elements equal unity if the classical WEP holds, but an off-diagonal element rr (with modulus ∣r∣|r| and phase φr\varphi_r) would represent a WEP violation with no classical counterpart, tied to quantum coherence itself. Atom-interferometric experiments have begun to bound these parameters: Rosi et al. obtained ∣r∣≲10−8|r|\lesssim 10^{-8}, and Asenbaum et al. reached the 10−1210^{-12} level on the diagonal sector. Building on this operator framework, Onofrio, Smith, and Viola (Phys. Rev. D 112, 124014 (2025)) proposed WEP tests using a torsion balance (of Eötvös/Dicke–Braginsky type) in which each arm carries NN coherently controlled two-level systems — a candidate platform being ensembles of NV centers (nitrogen-vacancy color defects in diamond, whose electron spin states form an addressable two-level system) — and showed that the mean and the variance of a quantized torque operator τ^∥\hat{\tau}_{\parallel} encode the mass-matrix parameters r1,r2,rr_1,r_2,r, with a proposed quantum signal-to-noise indicator qSNR quantifying detectability and a N\sqrt{N} enhancement from large ensembles. Because torsion balances self-balance on timescales of minutes to hours while superposition coherence in dense solid-state ensembles survives only milliseconds (the ensemble dephasing time T2∗T_2^{*} and coherence time T2T_2 being limited by strain and spin-bath noise, only partially extendable by multipulse dynamical decoupling — sequences of control pulses that average out system–environment couplings — or by engineered dissipative stabilization into pointer states, i.e. states robust against decoherence), the authors also proposed a dynamical, Cavendish-like variant with a time-dependent gravitational source and lock-in detection at the modulation frequency. Their feasibility estimates (e.g. a thermal-noise-limited reach of ∣r∣≲1.6×10−4|r|\lesssim 1.6\times 10^{-4} under optimistic cryogenic assumptions) remain analytic and leading-order; the balance is modeled through its classical angular dynamics with a classical thermal (Brownian) torque noise budget, so a complete dynamical treatment must additionally address measurement backaction on the balance, which the paper's analysis does not include. The paper itself flags what remains open: the operator nature of inertial and gravitational mass is assumed as a working hypothesis (with the related Bargmann mass-superselection issue — the question of whether coherent superpositions of different masses are physical at all — unsettled); preserving high-purity coherence in large ensembles over the required timescales is undemonstrated; no complete dynamical numerical model of the balance — including its full Newtonian arm dynamics and technical noise — exists; the noise floors needed for the quantum variance to dominate are quantitatively uncertain; and the analysis is limited to two-level product states, leaving entangled or correlated many-body preparations unexplored.

Problem Statement

Establish the complete theoretical and experimental groundwork required to test the weak equivalence principle for matter prepared in coherent superpositions of internal energy eigenstates using a torsion balance, as proposed by Onofrio, Smith, and Viola — that is, within the mass-operator framework in which M^gM^i−1\hat{M}_g\hat{M}_i^{-1} acts on the internal-energy space of the test masses, determine whether and under what conditions the mean and variance of the quantized torque operator can bound the mass-matrix parameters r1r_1, r2r_2, and the off-diagonal rr at sensitivities competitive with atom-interferometric tests, that is, at or below the level of the atom-interferometric bounds quoted in the Background (the off-diagonal ∣r∣≲10−8|r|\lesssim 10^{-8} of Rosi et al. and the diagonal sector at 10−1210^{-12} of Asenbaum et al.). This jointly requires: (i) settling the foundational status of treating inertial and gravitational masses as operators — either deriving the treatment from established physics or exhibiting its inconsistency — together with the question of whether bona fide mass-superposition states can be physically prepared (the mass-superselection issue); (ii) identifying how highly coherent internal states can be prepared and preserved in large (NN-particle) ensembles over the torsion-balance-relevant timescales, or over the modulation timescale of a dynamical-source variant, including the quantum-control requirements this imposes; (iii) constructing a complete dynamical model of the torsion balance — including quantum backaction, the full Newtonian dynamics of the balance arms, realistic environment coupling, and technical noise sources (acoustic, electrostatic, vibrational, thermal) — that converts the leading-order analytic estimates into an experimentally actionable design with quantified reach in ∣r∣|r|, r1r_1, r2r_2; (iv) establishing the noise-floor conditions (thermal torque noise, mechanical quality factor, cryogenic and laser-cooling requirements, detector performance) under which the quantum variance dominates and the qSNR becomes sufficient to bound ∣r∣|r|, r1r_1, and r2r_2 at the competitive level fixed above (taken at the 3σ3\sigma level); and (v) extending the two-level, product-state analysis of the mean and variance of the acceleration and torque operators to general many-body preparations, including entangled or correlated states, quantifying how such correlations modify the mean/variance scaling (e.g. beyond the N\sqrt{N} enhancement) and the decoherence susceptibility of the signal. The components are linked by the single objective of making the nonclassical-matter WEP test with a torsion balance theoretically complete and experimentally actionable, and they are jointly required: components (i)–(v) complete the task exactly when, taken together, they establish a quantified reach at the competitive level fixed above (off-diagonal ∣r∣≲10−8|r|\lesssim 10^{-8}, diagonal 10−1210^{-12}), so a package that quantifies a weaker reach documents feasibility but does not by itself resolve the task, although it may support the impossibility argument below. Alternatively, an answer may demonstrate, on physical grounds, that no such experiment can reach that competitive level; such an impossibility argument is a self-contained alternative that must treat the static and the dynamical, time-dependent-source variants of the proposal separately and cover the full parameter space of each. One exception to joint requirement: a demonstration that the operator treatment of mass is inconsistent, or that the required mass superpositions cannot be physically prepared, refutes the premise on which components (ii)–(v) rest and is by itself a complete negative resolution of the task.

References

  1. Roberto Onofrio, Alexander R. H. Smith, Lorenza Viola, 'Testing the weak equivalence principle for nonclassical matter with torsion balances', Phys. Rev. D 112, 124014 (2025); DOI 10.1103/r2ts-fsjj; arXiv:2512.06333; https://arxiv.org/abs/2512.06333
  2. Magdalena Zych, Časlav Brukner, 'Quantum formulation of the Einstein equivalence principle', Nature Physics 14, 1027–1031 (2018); DOI 10.1038/s41567-018-0197-6; https://doi.org/10.1038/s41567-018-0197-6
  3. G. Rosi, G. D'Amico, L. Cacciapuoti, F. Sorrentino, M. Prevedelli, M. Zych, Č. Brukner, G. M. Tino, 'Quantum test of the equivalence principle for atoms in coherent superposition of internal energy states', Nature Communications 8, 15529 (2017); DOI 10.1038/ncomms15529; https://doi.org/10.1038/ncomms15529
  4. Peter Asenbaum, Chris Overstreet, Minjeong Kim, Joseph Curti, Mark A. Kasevich, 'Atom-Interferometric Test of the Equivalence Principle at the 10−1210^{-12} Level', Phys. Rev. Lett. 125, 191101 (2020); DOI 10.1103/PhysRevLett.125.191101; https://doi.org/10.1103/PhysRevLett.125.191101
  5. Lorenza Viola, Roberto Onofrio, 'Testing the equivalence principle through freely falling quantum objects', Phys. Rev. D 55, 455 (1997); DOI 10.1103/PhysRevD.55.455; https://doi.org/10.1103/PhysRevD.55.455
  6. Or Dobkowski, Barak Trok, Peter Skakunenko, Yonathan Japha, David Groswasser, Maxim Efremov, Chiara Marletto, Ivette Fuentes, Roger Penrose, Vlatko Vedral, Wolfgang P. Schleich, Ron Folman, 'Observation of quantum free fall and the consistency with the equivalence principle', arXiv:2502.14535 (2025); https://arxiv.org/abs/2502.14535
  7. Peter Asenbaum, Chris Overstreet, 'Magnetic recoil interferometer in a uniform gravitational field. Comment on Observation of the quantum equivalence principle for matter-waves', arXiv:2504.15409 (2025); https://arxiv.org/abs/2504.15409
  8. Or Dobkowski, Barak Trok, Peter Skakunenko, Yonathan Japha, David Groswasser, Maxim Efremov, Chiara Marletto, Ivette Fuentes, Roger Penrose, Vlatko Vedral, Wolfgang P. Schleich, Ron Folman, 'Reply to comment on: Observation of the quantum equivalence principle for matter-waves', arXiv:2504.21626 (2025); https://arxiv.org/abs/2504.21626
  9. Cyrus Zeledon, Benjamin Pingault, Jonathan C. Marcks, Mykyta Onizhuk, Yeghishe Tsaturyan, Yu-xin Wang, Benjamin S. Soloway, Hiroshi Abe, Misagh Ghezellou, Jawad Ul-Hassan, Takeshi Ohshima, Nguyen T. Son, F. Joseph Heremans, Giulia Galli, Christopher P. Anderson, David D. Awschalom, 'Minute-long quantum coherence enabled by electrical depletion of magnetic noise', arXiv:2504.13164 (2025); https://arxiv.org/abs/2504.13164