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Operator-Level Quantum Equivalence Principle

begüm yıldırım

Published Aug 10, 2026• 0 citations
Publisher

Abstract

The Einstein equivalence principle (EEP) links the universality of free fall, local Lorentz invariance, and local position invariance to the geometric description of gravity. In quantum systems, however, internal energy is represented by an operator and may occupy coherent superpositions; consequently, equality of classical mass parameters is not sufficient to establish equivalence at the operator level. This work develops a compact operator-level Quantum Equivalence Principle (QEP) residual and performs a literature-based experimental cross-validation using independent high-precision experiments. The theoretical basis follows the operator formulation of Zych and Brukner and its relativistic extension by Das, Fridman, and Lambiase. The quantum weak equivalence sector is tested using the 87Rb Bragg atom-interferometry experiment of Rosi et al., which compared two hyperfine internal-energy eigenstates and a coherent superposition, finding Eotvos ratios consistent with zero and constraining the off-diagonal quantum-WEP parameter to 5 x 10^-8. A classical diagonal WEP benchmark is supplied by the final MICROSCOPE result, eta(Ti,Pt) = [-1.5 +/- 2.3(stat) +/- 1.5(syst)] x 10^-15. The local Lorentz-invariance sector is cross-checked using fast 7Li+ optical clocks, which verified relativistic time dilation at 2 x 10^-10 accuracy, while the local-position-invariance sector is cross-checked using millimetre-scale 87Sr optical-clock gravitational-redshift measurements with fractional-frequency uncertainty 7.6 x 10^-21. Finally, published 2025 gravitational-wave QEP bounds are evaluated, with GW190814 providing the intervals [-0.59,0.54], [-0.40,0.37], and [-0.19,0.17] for WEP-, LLI-, and LPI-associated mass-energy differences, respectively. No statistically significant violation is identified in any analyzed sector. The combined evidence supports operator-level QEP consistency over laboratory quantum superpositions, relativistic clock kinematics, gravitational redshift, precision free fall, and strong-field gravitational-wave phenomenology, while also isolating the remaining experimental gap: direct off-diagonal tests of quantum LLI and quantum LPI.

DOI
10.5281/zenodo.21876991

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The Einstein equivalence principle (EEP) links the universality of free fall, local Lorentz invariance, and local position invariance to the geometric description of gravity. In quantum systems, however, internal energy is represented by an operator and may occupy coherent superpositions; consequently, equality of classical mass parameters is not sufficient to establish equivalence at the operator level. This work develops a compact operator-level Quantum Equivalence Principle (QEP) residual and performs a literature-based experimental cross-validation using independent high-precision experiments. The theoretical basis follows the operator formulation of Zych and Brukner and its relativistic extension by Das, Fridman, and Lambiase. The quantum weak equivalence sector is tested using the 87Rb Bragg atom-interferometry experiment of Rosi et al., which compared two hyperfine internal-energy eigenstates and a coherent superposition, finding Eotvos ratios consistent with zero and constraining the off-diagonal quantum-WEP parameter to 5 x 10^-8. A classical diagonal WEP benchmark is supplied by the final MICROSCOPE result, eta(Ti,Pt) = [-1.5 +/- 2.3(stat) +/- 1.5(syst)] x 10^-15. The local Lorentz-invariance sector is cross-checked using fast 7Li+ optical clocks, which verified relativistic time dilation at 2 x 10^-10 accuracy, while the local-position-invariance sector is cross-checked using millimetre-scale 87Sr optical-clock gravitational-redshift measurements with fractional-frequency uncertainty 7.6 x 10^-21. Finally, published 2025 gravitational-wave QEP bounds are evaluated, with GW190814 providing the intervals [-0.59,0.54], [-0.40,0.37], and [-0.19,0.17] for WEP-, LLI-, and LPI-associated mass-energy differences, respectively. No statistically significant violation is identified in any analyzed sector. The combined evidence supports operator-level QEP consistency over laboratory quantum superpositions, relativistic clock kinematics, gravitational redshift, precision free fall, and strong-field gravitational-wave phenomenology, while also isolating the remaining experimental gap: direct off-diagonal tests of quantum LLI and quantum LPI.
The Einstein equivalence principle (EEP), in its quantum formulation due to Zych and Brukner, requires that the rest, inertial and gravitational mass-energy operators be identical as operators. Whereas diagonal (population) violations are bounded by dual-species atom interferometers at the 10⁻⁹ to 10⁻¹² level, the off-diagonal (internal-coherence) sector has so far been constrained only through excess phase noise in a single coherent superposition [Rosi et al., Nat. Commun. 8, 15529 (2017)]. We show that preparing the two spatial arms of a single-atom Mach–Zehnder interferometer with differential internal-state coherence c₊ ≠ c₋ opens a deterministic cross-term observable: a gravitationally coupled off-diagonal internal mass-energy operator σₓ produces a recombined fringe phase Δφ_cross = keff g T² ε (c₊ − c₋) that vanishes identically under quantum EEP (ε = 0) and is linear in the dimensionless off-diagonal violation parameter ε otherwise. Using real ⁸⁷Rb BEC parameters (hyperfine splitting 6.835 GHz, LMT order n = 100–1000, pulse separation T = 0.2–1.4 s on ground), we estimate projected shot-noise bounds ε_min ≈ 2.6×10⁻¹² (ground fountain), 1.3×10⁻¹⁴ (10 m tower, n = 1000) and 6.5×10⁻¹⁵ (drop tower, n = 1000, T = 1.4 s). The scheme converts the off-diagonal qVEP channel from a random-noise observable into a controllable, single-atom fringe shift linear in the LMT momentum transfer keff. All numerical values in figures are illustrative/synthetic projections. I. Introduction The weak equivalence principle (WEP) states that all test bodies fall with the same acceleration in a uniform gravitational field, independent of composition. Classical tests with torsion balances and lunar laser ranging bound differential accelerations at the 10⁻¹³ to 10⁻¹⁵ level, and the MICROSCOPE space mission reached 10⁻¹⁴. Atom interferometry has extended these tests to quantum objects: dual-species ⁸⁵Rb/⁸⁷Rb and ³⁹K/⁸⁷Rb comparisons reach the 10⁻⁷ to 10⁻⁹ range on the ground [Schlippert et al., PRL 112, 203002 (2014)], and the first in-orbit quantum WEP test aboard the China Space Station reports ≈ 2.8×10⁻⁸ [Zhang et al., Sci. Adv. (2026), doi:10.1126/sciadv.aeh4502]. Ground gradiometers in a ⁸⁷Rb |F = 1⟩ vs |F = 2⟩ configuration reach δg/g ≈ 10⁻¹².
Abstract The Einstein equivalence principle (EEP) has a central role in the understanding of gravity and space–time. In its weak form, or weak equivalence principle (WEP), it directly implies equivalence between inertial and gravitational mass. Verifying this principle in a regime where the relevant properties of the test body must be described by quantum theory has profound implications. Here we report on a novel WEP test for atoms: a Bragg atom interferometer in a gravity gradiometer configuration compares the free fall of rubidium atoms prepared in two hyperfine states and in their coherent superposition. The use of the superposition state allows testing genuine quantum aspects of EEP with no classical analogue, which have remained completely unexplored so far. In addition, we measure the Eötvös ratio of atoms in two hyperfine levels with relative uncertainty in the low 10−9, improving previous results by almost two orders of magnitude.
To date, no framework combining quantum field theory and general relativity and hence unifying all four fundamental interactions, exists. Violations of the Einstein's equivalence principle (EEP), being the foundation of general relativity, may hold the key to a theory of quantum gravity. The universality of free fall (UFF), which is one of the three pillars of the EEP, has been extensively tested with classical bodies. Quantum tests of the UFF, e.g. by exploiting matter wave interferometry, allow for complementary sets of test masses, orders of magnitude larger test mass coherence lengths and investigation of spin-gravity coupling. We review our recent work towards highly sensitive matter wave tests of the UFF on ground. In this scope, the first quantum test of the UFF utilizing two different chemical elements, Rb-87 and K-39, yielding an Eötvös ratio η Rb,K=(0.3±5.4)×10−7η_{\,\text{Rb,K}}=(0.3\pm 5.4)\times 10^{-7}ηRb,K​=(0.3±5.4)×10−7 has been performed. We assess systematic effects currently limiting the measurement at a level of parts in 10810^8108 and finally present our strategies to improve the current state-of-the-art with a test comparing the free fall of rubidium and ytterbium in a very long baseline atom interferometry setup. Here, a 10 m baseline combined with a precise control of systematic effects will enable a determination of the Eötvös ratio at a level of parts in 101310^{13}1013 and beyond, thus reaching and overcoming the performance limit of the best classical tests.
The weak equivalence principle (WEP), that is, the universality of free fall, states that all point-like neutral particles in a gravitational field fall in the same way. This is the basis of the geometrization of the gravitational interaction. Together with further requirements on the behavior of point particles, light propagation and clocks one can show that gravity is modeled by a Riemannian geometry. Since in the quantum domain all objects are extended, it is not clear whether the notion of a WEP in the quantum domain makes sense at all. We show that for matter wave interferometry the notion of WEP still can be given a meaning. We give a short overview over schemes which allows a violation of the WEP and emphasize that there are also schemes which show that there might be violations of the WEP in the quantum regime which are not present classically. This makes a test of the WEP with quantum matter necessary. We also give a brief outline of the efforts made for testing the WEP with interferometry with cold atoms in the Bremen drop tower carried out by the QUANTUS and PRIMUS collaboration.
The search for a fundamental, self-consistent theoretical framework to cover phenomena over all energy scales is possibly the most challenging quest of contemporary physics. Approaches to reconcile quantum mechanics and general relativity entail the modification of their foundations such as the equivalence principle. This corner stone of general relativity is suspected to be violated in various scenarios and is therefore under close scrutiny. Experiments based on the manipulation of cold atoms are excellently suited to challenge its different facets. Freely falling atoms constitute ideal test masses for tests of the universality of free fall in interferometric setups. Moreover, the superposition of internal energy eigenstates provides the notion of a clock, which allows to perform tests of the gravitational redshift. Furthermore, as atom interferometers constitute outstanding phasemeters, they hold the promise to detect gravitational waves, another integral aspect of general relativity. In recent years, atom interferometers have developed into versatile sensors with excellent accuracy and stability, and allow to probe physics at the interface of quantum mechanics and general relativity without classical analog. In this thesis, various aspects regarding tests of general relativity with atom interferometry have been theoretically investigated. This includes the analysis of fundamental effects as well as feasibility studies of experimental configurations. The work is partially focussed on a space-borne mission scenario for a dedicated quantum test of the universality of free fall beyond state-of-the-art by dropping matter waves of different elements. To enable the target accuracy at the level of 10^(−17), a compensation scheme has been developed and discussed, mitigating the detrimental effects of imperfect test mass co-location upon release and relaxing the requirements on the source preparation by several orders of magnitude. In addition, it was demonstrated that the careful design of quantum degenerate sources is indispensable for these experiments, requiring tailored schemes to prepare miscible binary sources. The possibility to test the gravitational redshift with atom interferometers has also been examined in this thesis and connected to the ideas of clock interferometry. With the proof that closed light pulse atom interferometers without transitions between internal internal states are not sensitive to gravitational time dilation, an ongoing scientific debate has been resolved. Instead, certain configurations were shown to implement a quantum version of the special-relativistic twin paradox, for which an experiment has been proposed. Finally, requirements on atomic sources and atom optics for scenarios of gravitational wave detection on ground and in space have been investigated.
The Einstein equivalence principle is based on the equality of gravitational and inertial mass, which has led to the universality of a free-fall concept. The principle has been extremely well tested so far and has been tested with a great precision. However, all these tests and the corresponding arguments are based on a classical setup where the notion of position and velocity of the mass is associated with a classical value as opposed to the quantum entities.Here, we provide a simple quantum protocol based on creating large spatial superposition states in a laboratory to test the quantum regime of the equivalence principle where both matter and gravity are treated at par as a quantum entity. The two gravitational masses of the two spatial superpositions source the gravitational potential for each other. We argue that such a quantum protocol is unique with regard to testing especially the generalisation of the weak equivalence principle by constraining the equality of gravitational and inertial mass via witnessing quantum entanglement.