A New Approach for Testing the Quantum Nature of Gravity

A New Approach for Testing the Quantum Nature of Gravity Shan Gao Research Center for Philosophy of Science and Technology, Shanxi University, Taiyuan 030006, P. R. China E-mail: [email protected] November 8, 2025 Abstract The quantum nature of gravity is one of physics’ greatest unsolved problems, with existing experimental proposals facing formidable challenges in maintaining spatial superpositions of massive objects against decoherence. We propose a fundamentally different approach that bypasses these difficulties. The central insight is that a quantum gravitational field can imprint a gauge-invariant phase on a particle’s wavefunction even along an open trajectory, arising from the quantum backreaction of the source. This phase, which manifests prior to the final recombination stage of an interferometer and cannot be generated by a classical gravitational field, provides an operationally direct witness to the quantumness of gravity. Our experimental design uses a single photon whose accumulated gravitational phase due to the Earth is transferred via absorption to auxiliary atoms near the end of the interferometer sequence, preserving the openpath character essential for the test. We scale the interferometer to yield a detectable phase of order 10−3rad and show how non-local correlation measurements on the atoms can access this gauge-invariant signature. Leveraging established quantum optical and atomic techniques, this proposal offers a feasible path to distinguish quantum from classical gravity. A positive result would provide the first experimental evidence for the quantum nature of gravitational fields, with profound implications for quantum gravity theories. 1 Introduction The unification of quantum theory and general relativity remains one of the most profound challenges in modern physics [1, 2]. While various quantum gravity theories have been proposed, experimental validation has remained elusive due to the extreme energy scales involved. A more fundamental question persists: does gravity itself possess quantum properties? This question transcends specific theoretical frameworks and addresses whether the gravitational field can exist in quantum superpositions and entangle physical systems—hallmarks of quantum behavior absent in classical fields [3, 4]. A direct experimental probe of gravity’s quantum character has long been considered beyond reach, owing to the extreme weakness of the gravitational interaction at microscopic scales. Recent proposals have sought to test gravity’s quantum nature through its capacity to generate entanglement. The Bose–Marletto–Vedral (BMV) protocol suggests that observing gravity-induced entanglement between two massive particles in spatial superposition would demonstrate quantum gravity, as classical fields cannot entangle quantum systems [5, 6, 7]. However, this approach faces formidable experimental challenges [8]. Achieving detectable gravitational phases requires superposing mesoscopic masses of order 10−14 kg for coherence times approaching one second, under 1 extreme isolation from thermal and collisional decoherence. Current optomechanical and interferometric platforms fall short of these requirements by several orders of magnitude, motivating the search for conceptually equivalent but experimentally more accessible tests [9, 10]. Here we propose a fundamentally different approach that circumvents these challenges. Rather than attempting to detect entanglement between two superposed masses, we employ a single photon in a Mach-Zehnder interferometer under Earth’s gravitational field with phase transfer to auxiliary quantum systems (e.g., atoms) via photon absorption near the end of the interferometer sequence [11, 12, 13, 14]. The central insight of our proposal is that quantum gravity can imprint a gauge-invariant phase on the photon’s wavefunction prior to the final recombination stage of the interferometer—arising from the quantum backreaction of the source—that cannot be generated by a classical gravitational field. This phase, which can be on the order of 10−3rad by scaling the interferometer, is accessible through non-local correlation measurements after transferring it to auxiliary atoms. Crucially, the scheme relies on established quantum techniques, making it feasible with currently available experimental tools. By enabling a direct test of gravitationally induced quantum correlations, the approach provides a practical route to distinguish quantum from classical gravity and could offer the first experimental evidence of the quantum nature of gravitational interactions.1 2 Semiclassical Gravity: Classical Field, Quantum Matter In the semiclassical description, the gravitational field is treated classically, e.g sourced by the expectation value of the quantum matter stress tensor:2 Gµν[g]=8πGˆ Tµν.(1) Operationally, this amounts to evolving the particle wavefunction in an external (classical) potential Φext(x, t) produced by the source: i∂tψ(x, t) = −∇2 2m+mΦext(x, t)ψ(x, t).(2) The phase accumulated along an open particle history γis therefore S(sc) p[γ] = Zγ dt 1 2m˙ x2−mΦext(x, t).(3) Although the kinetic phase is gauge-invariant, the gravitational phase is gauge-dependent under linearized coordinate transformations in general because Φext 7→ Φext + ˙χadds the endpoint term −m[χ(tf)−χ(ti)]. In particular, if the particle moves entirely in a simply connected, curvature-free region where the external potential is a pure gauge (Φext = ˙χ), then by a gauge transformation one removes the potential and the particle evolution is identical to free evolution: the measurable gravitational phase for the open path is zero. Thus, no coherent relative phase is imprinted on the source; the particle accrues a gauge-dependent open-path phase which in a field-free simplyconnected region is operationally gauge-removable and hence yields zero measurable gravitational contribution. This is one key feature of the gravitational Aharonov-Bohm (AB) effect as predicted by semiclassical gravity [12, 16, 17, 18, 19, 20]. 1Here and in the following, by “classical gravity theories” we refer to the broad category of theories in which the gravitational field is not quantized, including semi-classical, stochastic, and hybrid models [15]. 2We set ℏ=c= 1 in this paper. 2 3 Quantum Gravity: Gauge-Invariant Backreaction Phases Now treat source, particle, and gravitational field as quantum degrees of freedom in weak-field, linearized gravity. In the path-integral formulation, integrating out the linearized field yields an effective action composed of two distinct terms representing the mutual interaction: Scross[Tp, Ts] = Sp←s+Ss←p,(4) where Sp←s=−4πG ZZ d4y d4x T αβ p(y)Gαβµν(y−x)Tµν s(x),(5) and Ss←p=−4πG ZZ d4y d4x T αβ s(y)Gαβµν(y−x)Tµν p(x),(6) with a symmetric Green’s function Gbeing the the graviton propagator (see Appendix). For closed paths, these terms are equal due to the symmetry of the Green’s function, but for non-closed paths they represent distinct physical processes and exhibit fundamentally different gauge behavior: Sp←s describes the phase acquired by the particle due to the source’s gravitational field and it is gaugedependent, while Ss←pdescribes the phase acquired by the source due to the particle’s gravitational field, and it is gauge-invariant under stress-energy conservation ∂µTµν s= 0 and standard boundary conditions.3This distinction is essential—the gauge-invariant Ss←penables detection of quantum gravitational effects for open interferometer paths. Starting from an initial product state |Ψ(0)⟩=1 √2|L⟩+|R⟩p⊗|S0⟩s,(7) unitary evolution under the effective interaction produces the branch-correlated state |Ψ(t)⟩=1 √2ei(Sp←s[L]+Ss←p[L])|L⟩⊗|SL⟩+ei(Sp←s[R]+Ss←p[R])|R⟩⊗|SR⟩.(8) The total relative phase between the two branches is: ∆ϕ= [(Sp←s[L]−Sp←s[R]) + (Ss←p[L]−Ss←p[R])] .(9) The crucial distinction lies in the physical observability of these two phase contributions. The Sp←scontribution, ∆Sp←s=Sp←s[L]−Sp←s[R], is gauge-dependent. For non-closed paths in a simply connected, curvature-free region, this phase can be completely eliminated by a gauge transformation and therefore has no observable consequence. In contrast, the Ss←pcontribution, ∆Ss←p=Ss←p[L]−Ss←p[R], is gauge-invariant. It represents the genuine, physical phase arising from the quantum backreaction of the particle on the source. Measuring the presence or absence of this relative phase for non-closed paths is the operational test distinguishing quantum mediation from a classical field. Note that the amplitude overlap between source states is |⟨SR|SL⟩| ≈ 1 for a macroscopic source like the Earth. For a full closed interference path for an atom under Earth’s gravitational field, the total relative phase is ∆ϕtot = ∆ϕatom + ∆ϕEarth.(10) 3This result is also true for the electric and magnetic Aharonov–Bohm effects [21]. 3 By symmetry, each contributes exactly one-half of the total phase: ∆ϕatom = ∆ϕEarth =1 2∆ϕtot.(11) At the near-final stage of the interferometer evolution (t≈T), the Earth’s quantum contribution has accumulated to approximately one half of the total phase: ∆ϕQG ≃1 2∆ϕtot.(12) This quantum gravitational phase is gauge-invariant, and it represents a direct observable signal of the Earth’s quantum backreaction. 4 Classical Theories Cannot Reproduce the Quantum Phase We now present a minimal proof that establishes the impossibility of classical gravity theories reproducing the quantum gravity prediction of ∆ϕQG ≃1 2∆ϕtot for open interferometer paths at the near-final stage of the interferometer evolution (t≈T). Consider any classical gravity theory providing corrections to the AB phase predicted by standard semi-classical gravity (see Section 2). In order to be consistent with the full closed-loop AB phase, these corrections should be very small. Suppose at time t≈T, the AB phase difference between the two paths is ∆ϕclassical(t)=∆ϕ0(t) + ϵ(t), where ∆ϕ0(t) is the AB phase predicted by standard semi-classical gravity, and ϵ(t) is the correction.4There are only two possibilities for the additional phase ϵ(t): It is either gauge-invariant or gauge-dependent. In the first case, the total open-path phase remains gauge-dependent, and the measured AB phase at t≈Twill be only the gauge-invariant correction, namely ∆ϕclassical =ϵ(t)≈0. In the second case, since the correction ϵ(t) is very small, it cannot remove the gauge dependence of the leading order term ∆ϕ0(t). Thus ∆ϕclassical(t) remains gauge-dependent and cannot represent a physical observable, and any measurement at t≈Twould yield ∆ϕmeasured = 0. Therefore, in both cases, the classical prediction differs from the quantum prediction: ∆ϕmeasured ≃1 2∆ϕtot. This minimal proof, while simple, captures the essence of why our proposed approach constitutes a definitive test of quantum gravity. Measurement of ∆ϕmeasured ≈1 2∆ϕtot at t≈Twill be evidence for quantum gravity, while measurement of ∆ϕmeasured ≈0 at t≈Twill be evidence for classical gravity. Here we briefly comment on the recent article by Aziz and Howl in Nature [30]. These authors argue that classical theories of gravity can generate entanglement between the probe and the source through virtual matter exchange, challenging the conventional interpretation of gravitationally induced entanglement as unambiguous evidence for quantum gravity. While the Aziz-Howl mechanism represents an interesting theoretical possibility, it does not alter our conclusion. The source-term phase derived here arises already at the leading order of the quantum gravitational interaction, whereas the proposed virtual matter exchange correction represents a higher-order perturbation that contributes only a negligibly small modification to the full closed-loop AB phase. Since our predicted gauge-invariant quantum gravitational phase ∆ϕQG ≃1 2∆ϕtot, any such classical correction would be many orders of magnitude smaller and therefore irrelevant for our proposed experimental test of quantum gravity. 4For standard semi-classical gravity, ϵ= 0 [22]. For stochastic gravity [23], ϵcomes from ensemble averaging. For collapse models (e.g., Di´osi-Penrose models) [24, 25, 26, 27], ϵcomes from non-linear modifications. For hybrid theories [28], ϵcomes from effective interactions. And for emergent gravity [29], ϵcomes from thermodynamic corrections, and so on. 4 5 Experimental Proposal: Detecting Quantum Gravity We propose an experimental implementation using a single photon in a vertical Mach-Zehnder-type interferometer under Earth’s gravitational field, with relative phase transfer to auxiliary quantum systems to preserve the open-path character essential for isolating quantum gravitational effects. 5.1 Interferometer Setup and Phase Scaling Consider a single photon prepared in a coherent superposition of two vertically separated paths using an initial beam splitter: |Ψ(0)⟩=1 √2|L⟩+|R⟩,(13) where |L⟩and |R⟩represent the photon in the lower (height z) and upper (height z+h) paths respectively. The interferometer has horizontal arm length land vertical separation h. The total gravitational phase for a closed interferometer loop is given by: ∆ϕtot =2πNgA λc2, A =lh, (14) where N≈1.47 accounts for the refractive index in fiber implementations, g= 9.81 m/s2is Earth’s gravitational acceleration, λis the photon wavelength, and cis the speed of light. At the near-final stage of the interferometer evolution (t≈T), the gauge-invariant contribution from quantum backreaction—corresponding to the phase of the source due to the particle—has accumulated to approximately half of the total gravitational AB phase: ∆ϕQG ≃1 2∆ϕtot.(15) For practical detectability, we scale the interferometer to l= 100 km and h= 100 m, with λ= 1550 nm [14]. Then: ∆ϕtot =2π×1.47 ×9.81 ×100000 ×100 1.55 ×10−6×(3 ×108)2≈6.5×10−3rad,(16) and thus: ∆ϕQG ≈3.25 ×10−3rad.(17) This quantum gravitational phase for a non-closed path could be statistically measurable with ensemble measurements, feasible with high-repetition-rate sources [14]. 5.2 Phase Transfer to Auxiliary Atoms To preserve the open-path character while enabling phase measurement, we employ auxiliary quantum systems (e.g., Rydberg atoms) that absorb the photon locally in each path, transferring the relative phase to an entangled state of the auxiliaries [13]. At t≈T, the photon state is: |Ψphoton⟩=1 √2|L⟩+ei∆ϕQG |R⟩.(18) We place auxiliary two-level atoms in ground state |g⟩at fixed locations in both paths near the final point of the interferometer. The combined initial state is: |Ψtotal⟩=1 √2|L⟩+ei∆ϕQG |R⟩⊗|g⟩L|g⟩R.(19) 5 Expressing the photon in number basis: |L⟩ ≡ |1⟩L|0⟩R,|R⟩≡|0⟩L|1⟩R. Local resonant absorption via Jaynes-Cummings interaction (H=g(a†σ−+aσ+)) implements the transformation: |1⟩|g⟩→|0⟩|e⟩,|0⟩|g⟩→|0⟩|g⟩,(20) where |e⟩denotes the atomic excited state. Applying this locally yields: |Ψfinal⟩=1 √2|0⟩L|0⟩R|e⟩L|g⟩R+ei∆ϕQG |g⟩L|e⟩R.(21) The photon superposition is thus converted into an entangled state of the auxiliary atoms, with the quantum gravitational phase ∆ϕQG preserved in the atomic state. 5.3 Measuring the Phase via Atomic Correlations To extract the phase ∆ϕQG, we perform correlation measurements on the entangled atomic state: |Ψent⟩=1 √2|e⟩L|g⟩R+ei∆ϕQG |g⟩L|e⟩R.(22) The measurement procedure involves three key steps: Step 1: Unitary Rotation We apply local unitary rotations to both atoms:  Left atom: UL(θ) = exp(−iθσy/2)  Right atom: UR(ϕ) = exp(−iϕσy/2) Step 2: Projective Measurement After the rotations, we perform simultaneous projective measurements on both atoms in the |e⟩,|g⟩basis using state-selective detection techniques such as fluorescence detection or field ionization. Step 3: Probability Calculation We measure the joint probability Pee(θ, ϕ) of finding both atoms in excited states after the rotations. This probability is given by: Pee(θ, ϕ) = |⟨e|⟨e|(UL(θ)⊗UR(ϕ)) |Ψent⟩|2.(23) Substituting the expressions and computing the matrix elements yields: Pee(θ, ϕ) = 1 4[1 −cos θcos ϕ+ sin θsin ϕcos ∆ϕQG].(24) For maximum sensitivity, we set θ=ϕ=π/2 to measure: Pee(π/2, π/2) = 1 4(1 + cos ∆ϕQG).(25) Alternatively, fix θ=π/2 and vary ϕto measure: Pee(π/2, ϕ) = 1 4(1 + sin ϕcos ∆ϕQG).(26) By fitting the measured probabilities to these functional forms, ∆ϕQG can be extracted with standard statistical methods. This approach uses only routine atomic physics techniques—microwave rotations and projective measurements—making it experimentally straightforward while providing a clear signature of the quantum gravitational phase through the ∆ϕQG dependence in the joint excitation probability. 6 5.4 Summary The above experimental proposal for detecting quantum gravity leverages several established technologies. The principal challenge involves achieving sufficient phase sensitivity given ∆ϕQG ∼10−3 rad. This requires integration over a large number of trials. Crucially, this implementation preserves the essential feature of open interferometer paths, ensuring that any measured phase ∆ϕQG represents a genuine gauge-invariant signature of quantum gravitational backreaction, unobtainable through classical gravity mediation. If the Earth (source) is classical, there will be no gaugeinvariant gravitational phase for an open trajectory (the apparent particle phase is gauge-removable and operationally zero). If gravity is quantum and mediates coherent interactions, then the source term can imprint a gauge-invariant phase on the source (and correspondingly on joint atom–source correlations) even before recombination; this phase is accessible by the correlation measurements described above. 6 Implications and Advantages of the Proposed Approach The proposed experiment represents a significant departure from previous approaches to testing the quantum nature of gravity. Unlike the BMV protocol that requires maintaining spatial superpositions of two massive objects, our scheme leverages the Earth’s gravitational field as a fixed quantum source, dramatically reducing experimental complexity while preserving the essential quantum gravitational signature. The key innovation lies in detecting the gauge-invariant quantum gravitational phase ∆ϕQG = 1 2∆ϕtot, which arises from fundamental quantum backreaction effects. This phase has several distinctive characteristics that make it an unambiguous signature of quantum gravity: Gauge Invariance and Physical Significance: The phase ∆ϕQG remains invariant under coordinate transformations, unlike gauge-dependent phases that can be removed by appropriate choice of reference frame. This gauge independence ensures the phase represents a genuine physical effect rather than an artifact of mathematical description. In classical gravity, only closed interferometer paths yield measurable phases; open paths produce only gauge-dependent terms that are operationally unobservable in curvature-free regions. Quantum Backreaction Mechanism: The quantum gravitational phase emerges from the symmetric structure of the quantum gravitational interaction Scross[Tp, Ts] = Sp←s+Ss←p. While Sp←srepresents the conventional phase acquired by the particle in the source’s gravitational field, Ss←pcaptures the quantum backreaction—the phase acquired by the source due to the particle’s gravitational field. It is this backreaction term that enables gauge-invariant phase accumulation even for open interferometer paths. Theoretical Implications: A positive result would demonstrate that gravitational interactions can generate quantum correlations and entanglement, establishing gravity as a quantum interaction rather than a purely classical field. This would have profound implications for quantum gravity theories, particularly those proposing emergent or classical-like behavior of gravity at laboratory scales. A null result, while not definitively ruling out quantum gravity, would place strong constraints on models where gravitational decoherence or classical behavior emerges at accessible energy scales. Experimental Feasibility: Current quantum optical technology already achieves the necessary coherence times and phase stability for this experiment. The use of fiber-based interferometers with km-scale arm lengths and high-finesse quantum memories provides a robust platform for phase accumulation. The non-local measurement scheme, adapted from Aharonov and Vaidman’s protocol [13], eliminates the need for path recombination while enabling precise phase detection 7 through correlation measurements. The principal challenges involve maintaining phase stability over extended integration times and controlling systematic effects, which could be soon addressable with the rapid developments of the experimental techniques in quantum optics and atomic physics [12, 14, 31, 32, 33, 34, 35, 36]. The proposed test thus offers a new window into quantum gravitational phenomena, bridging the gap between theoretical speculation and experimental verification. 7 Conclusions We propose a novel test for quantum gravity using a single photon in a Mach-Zehnder interferometer with phase transfer to auxiliary quantum systems, detecting a gauge-invariant quantum gravitational phase via non-local correlation measurements. Unlike the BMV proposal, our approach leverages the Earth’s gravitational field and existing quantum technology, offering a practical path to probe quantum gravity. A measurable phase would provide strong evidence for the quantum nature of gravity, opening new avenues for theoretical and experimental exploration. References [1] Rovelli, C. Notes for a brief history of quantum gravity. In Proc. The Ninth Marcel Grossmann Meeting 742–768. World Scientific, 2002. [2] Weinstein, S. & D. Rickles. Quantum Gravity. The Stanford Encyclopedia of Philosophy (Spring 2024 Edition), Edward N. Zalta & Uri Nodelman (eds.), https://plato.stanford.edu/archives/spr2024/entries/quantum-gravity/. [3] Eppley, K. & E. Hannah, The necessity of quantizing the gravitational field. Found. Phys. 7, 51–68 (1977). [4] Kent, A. Simple refutation of the Eppley-Hannah argument. Class. Quant. Grav. 35, 245008 (2018). [5] Bose, S. et al. Spin entanglement witness for quantum gravity. Phys. Rev. Lett. 119, 240401 (2017). [6] Marletto, C. & Vedral, V. Gravitationally induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity. Phys. Rev. Lett. 119, 240402 (2017). [7] Marletto, C. & Vedral, V. Witnessing nonclassicality beyond quantum theory. Phys. Rev. D 102, 086012 (2020). [8] Bose, S. et al. Massive quantum systems as interfaces of quantum mechanics and gravity. Rev. Mod. Phys. 97, 015003 (2025). [9] Carney, D., Stamp, P. C. E. & Taylor, J. M. Tabletop experiments for quantum gravity: a user’s manual. Class. Quant. Grav. 36, 034001 (2019). [10] Huggett, N., Linnemann, N. & Schneider, M. D. Quantum Gravity in a Laboratory? Cambridge, UK: Cambridge University Press, 2023. [11] Colella, R., Overhauser, A. W. & Werner, S. A. Observation of gravitationally induced quantum interference. Phys. Rev. Lett. 34, 1472 (1975). 8 [12] Overstreet, C., et al. Observation of a gravitational Aharonov–Bohm effect. Science, 375, 226 (2022). [13] Aharonov, Y. & Vaidman, L. Nonlocal aspects of a quantum wave. Phys. Rev. A 61, 052108 (2000). [14] Hilweg, C. et al. Gravitationally induced phase shift on a single photon. New J. Phys. 19, 033028 (2017). [15] Carlip, S. Is quantum gravity necessary? Class. Quant. Grav. 25, 154010 (2008). [16] Ehrenberg, W. & R. E. Siday, The Refractive Index in Electron Optics and the Principles of Dynamics. Proc. Phys. Soc. B 62, 8-21 (1949). [17] Aharonov, Y. & D. Bohm, Significance of Electromagnetic Potentials in Quantum Theory, Phys. Rev. 115, 485–491 (1959). [18] Tonomura, A. et al., Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave, Phys. Rev. Lett. 56, 792–795 (1986). [19] Peshkin, M. & A. Tonomura, The Aharonov-Bohm Effect, Lecture Notes in Physics, Springer (1989). [20] Audretsch, J. & C. Lammerzahl. J. Phys. A: Math. Gen. 16 2457 (1983). [21] Gao, S. Half Gauge-Dependent, Half Gauge-Invariant: The Aharonov–Bohm Effect Reconsidered. Preprint (2025). [22] Rosenfeld, L. On quantization of fields. Nucl. Phys. 40, 353–356 (1963). [23] Hu, B.L., Verdaguer, E. Stochastic Gravity: Theory and Applications. Living Rev. Relativ. 11, 3 (2008). [24] Di´osi, L. A universal master equation for the gravitational violation of quantum mechanics. Phys. Lett. A 120, 377–381 (1987). [25] Penrose, R. On Gravity’s role in Quantum State Reduction. Gen. Relat. Gravit 28, 581–600 (1996). [26] Tilloy, A. & Di´osi, L. Sourcing semiclassical gravity from spontaneously localized quantum matter. Phys. Rev. D 93, 024026 (2016). [27] Howl, R., Penrose, R. & Fuentes, I. Exploring the unification of quantum theory and general relativity with a Bose-Einstein condensate. New J. Phys. 21, 043047 (2019). [28] Oppenheim, J. A Postquantum Theory of Classical Gravity? Phys. Rev. X 13, 041040 (2023). [29] Verlinde, E. On the Origin of Gravity and the Laws of Newton. J. High Energy Phys. 2011, 29 (2011). [30] Aziz, J. & Howl, R. Classical theories of gravity produce entanglement. Nature 646, 813–818 (2025). [31] Mohageg, M. et al. The deep space quantum link: prospective fundamental physics experiments using long-baseline quantum optics. EPJ Quantum Technology 9, 25 (2022). 9