In linear open-quantum-system dynamics, the off-diagonal coherence of a spatially superposed source mass decays with a visibility envelope that is independent of the prepared relative phase between the two position amplitudes. This phase-blindness is a direct consequence of linearity: unitary and Lindblad generators commute with global phase rotations on the source branch, and the interference fringe takes the form P(φ,t) = ½[1 + v(t) cos φ] with v(t) = exp(−Γt). We propose a phase-scanning protocol to test this null hypothesis in a BMV-type nanodiamond interferometer. We parametrize a first-harmonic deviation by writing v(φ,t) = exp[−Γ₀t − Γ₁t(1 + η cos(φ − φ₀))], so that η = 0 recovers linear dynamics and η ≠ 0 signals phase-sensitive decoherence. No existing Markovian collapse model predicts η ≠ 0; the parameter is therefore a phenomenological null-test axis. For illustrative BMV target parameters (m = 10⁻¹⁴ kg, d = 250 μm, T = 10 mK) and a reference scale Γ₁ = 0.127 s⁻¹, idealized scaling gives η = 0.05 reachable at ~7 × 10⁴ shots and η = 0.01 at ~2 × 10⁶ shots for SNR = 3 at t = 1 s. The protocol bounds phase sensitivity, not the overall collapse rate, and is proposed, not demonstrated. I. Introduction The superposition principle is the structural core of quantum mechanics, yet its compatibility with gravitation remains untested in the regime where both the spatial extent of a superposition and its mass are macroscopically large. Proposals to prepare and interferometrically close spatial superpositions of massive nanoparticles—most prominently the BMV scheme [1,2]—have motivated experiments aimed at observing gravitationally mediated entanglement and at bounding exotic decoherence. In parallel, dynamical-collapse models [3,4] predict deviations from unitary evolution whose magnitude scales with mass and separation, and matter-wave interferometry has long been used to bound their parameters [5,6].
In linear open-quantum-system dynamics, the off-diagonal coherence of a spatially superposed source mass decays with a visibility envelope that is independent of the prepared relative phase between the two position amplitudes. This phase-blindness is a direct consequence of linearity: unitary and Lindblad generators commute with global phase rotations on the source branch, and the interference fringe takes the form P(φ,t) = ½[1 + v(t) cos φ] with v(t) = exp(−Γt). We propose a phase-scanning protocol to test this null hypothesis in a BMV-type nanodiamond interferometer. We parametrize a first-harmonic deviation by writing v(φ,t) = exp[−Γ₀t − Γ₁t(1 + η cos(φ − φ₀))], so that η = 0 recovers linear dynamics and η ≠ 0 signals phase-sensitive decoherence. No existing Markovian collapse model predicts η ≠ 0; the parameter is therefore a phenomenological null-test axis. For illustrative BMV target parameters (m = 10⁻¹⁴ kg, d = 250 μm, T = 10 mK) and a reference scale Γ₁ = 0.127 s⁻¹, idealized scaling gives η = 0.05 reachable at ~7 × 10⁴ shots and η = 0.01 at ~2 × 10⁶ shots for SNR = 3 at t = 1 s. The protocol bounds phase sensitivity, not the overall collapse rate, and is proposed, not demonstrated. I. Introduction The superposition principle is the structural core of quantum mechanics, yet its compatibility with gravitation remains untested in the regime where both the spatial extent of a superposition and its mass are macroscopically large. Proposals to prepare and interferometrically close spatial superpositions of massive nanoparticles—most prominently the BMV scheme [1,2]—have motivated experiments aimed at observing gravitationally mediated entanglement and at bounding exotic decoherence. In parallel, dynamical-collapse models [3,4] predict deviations from unitary evolution whose magnitude scales with mass and separation, and matter-wave interferometry has long been used to bound their parameters [5,6].
In the Bose–Marletto–Vedral (BMV) class of tabletop gravity experiments, two mesoscopic masses are each placed in a spatial superposition and their Newtonian interaction is used as a witness of gravitationally induced entanglement. Aspelmeyer [5] has sharpened the standard argument: observing entanglement at the 10-15 kg scale already rules out current Diósi–Penrose (DP) collapse models. Here we ask a complementary question: when the source mass itself is the object in superposition, what is the sign of the gravity-induced decoherence *rate* as a function of M/r (mass over characteristic superposition scale), and where in the mass range is that rate actually measurable? We compare three branches: (A) linearized quantum field theory, which gives reversible phase accumulation and Gamma = 0; (B) DP / continuous spontaneous localization (CSL) collapse, for which Gamma > 0 and grows steeply with M/r (positive trend); and (C) a phenomenological strawman of variable vacuum-foam / stochastic-spacetime suppression, Gammafoam = Gammafbg exp(-3 kappa GM/(rc2)), for which the trend is negative. Branch C is not derived from any specific quantum-gravity model; it is introduced only to give the sign question a non-trivial alternative. We solve the rates for a uniform sphere (density rho = 2000 kg/m3, superposition separation d = 10 r) and find: (i) a rate-resolved window M ~ 2×10-16 to 10-14 kg in which the DP rate exceeds a BMV-class environmental floor of 0.05 s-1 but is still measurable — this window lies below the regime where entanglement-survival alone, à la Aspelmeyer, already excludes DP; (ii) the phenomenological strawman foam-suppression branch is observationally degenerate with QFT-zero on Earth, because d ln Gammafoam / d ln M = -2 kappa GM/(rc2) ~ -10-37 to -10-35 across the entire lab range. Discriminating this toy from QFT-zero would require the near-Schwarzschild regime x = GM/(rc2) ~ O(1). Shot-noise-limited run counts are ~6 at M = 10-15 kg, ~104 at M = 10-16 kg, and environment-limited below M = 10-16 kg. All numerical values are illustrative/synthetic order-of-magnitude estimates.
Laboratory-based experimental tests remain a central challenge in the study of the quantum-gravity interface. We present a quantitative, experimentally testable prediction for gravitationally induced quantum decoherence of spatial superpositions in levitated optomechanical systems. Starting from two independently well-tested physical elements—the quantum nature of material energy-momentum fluctuations (established in particle physics) and the linear response kernel of classical geometry (derived from the weak-field limit of general relativity and verified by gravitational-wave observations)—and supplementing them with a minimal extension postulate for superposition states, we obtain a rigid scaling law for the gravitational decoherence rate: \Gamma \sim G M^2 d^2 / (\hbar L^3) \cdot \Phi(l_*/L), where M is the mass of the object, d the spatial separation of the superposition branches, L the characteristic linear dimension, and \Phi a dimensionless function encoding the unknown ultraviolet cutoff scale l_* of the effective geometric noise. The structural form \Gamma \propto M^2 d^2 / L^3 contains no free adjustable parameters; the overall amplitude is set by the choice of l_*. Identifying l_* with the Planck length \ell_P = \sqrt{\hbar G / c^3}—the only known scale bridging quantum theory and gravity—yields \Phi \sim \mathcal{O}(1) and a specific numerical target. For current levitated nanoparticle experiments (M \sim 10^{-14}\,\text{kg}, d \sim 10^{-7}\,\text{m}, L \sim 10^{-6}\,\text{m}), this Planck-scale hypothesis predicts a decoherence rate of order 1\,\text{Hz}, placing it within the foreseeable sensitivity window of next-generation optomechanical setups. We discuss the exclusive physical signatures distinguishing this mechanism from environmental decoherence channels and outline explicit falsification conditions, providing experimental groups with a concrete target to probe the quantum-gravity interface in the laboratory.
Levitated optomechanical systems are approaching the regime where macroscopic quantum superpositions become experimentally accessible. A central question is whether gravity itself, through its unavoidable coupling to material quantum fluctuations, imposes a fundamental decoherence channel on such superpositions. Starting from two independently well-tested physical elements—the quantum vacuum fluctuations of the material energy-momentum tensor, and the weak-field linear response kernel of classical general relativity—we derive a structurally rigid scaling law for the gravitational decoherence rate: \Gamma \propto G M^2 d^2 / (\hbar L^3) \cdot \Phi(l_*/L). The exponents and dependence on mass M, superposition separation d, and object size L are fixed entirely by dimensional analysis and the semiclassical field equations, without tunable parameters. The overall amplitude is set by a single phenomenological input—the ultraviolet cutoff scale l_* of the effective geometric noise. Identifying l_* with the Planck length yields \Phi \sim \mathcal{O}(1) and a decoherence rate of order 1\,\text{Hz} for nanoparticles with M \sim 10^{-14}\,\text{kg}, d \sim 10^{-7}\,\text{m}, and L \sim 10^{-6}\,\text{m}, placing the prediction within the foreseeable sensitivity window of next-generation levitated optomechanical setups. We provide explicit falsification conditions, a three-path staged verification strategy that does not require simultaneous optimisation of all parameters, and operational signatures—including a distinctive pure-dephasing channel without energy relaxation—that distinguish this mechanism from environmental noise, zero-temperature electromagnetic vacuum fluctuations, and the Diósi-Penrose model.
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.
This work presents numerical and theoretical evidence for a testable regime in which semiclassical Schrödinger–Newton dynamics predict a state-dependent gravitational self-interaction in mesoscopic quantum systems. Using large-scale Split-Step Fourier Method (SSFM) simulations, we identify a mass window (≈10⁻¹¹–10⁻¹⁰ kg) where quantum coherence and gravitational self-interaction coexist without immediate decoherence. In this regime, the gravitational self-energy depends smoothly on the spatial structure of the quantum state, leading to a differential phase evolution between superposition states with identical external potentials but different mass-density distributions. The predicted phase signal scales as m² and reaches experimentally relevant magnitudes (Δφ ≈ 10⁻³ rad) for millisecond evolution times, making the effect accessible to next-generation matter-wave and optomechanical interferometry. The work systematically validates the numerical approach, demonstrates energy conservation and convergence of the nonlinear dynamics, identifies a stable “breathing-mode” regime near 10⁻¹⁰ kg, and isolates the gravitational origin of the effect via null tests with the Schrödinger–Newton term removed. This preprint makes no claim regarding spacetime engineering, geometry manipulation, or violations of the equivalence principle. It is explicitly framed as a falsifiable test of semiclassical gravity in a previously inaccessible experimental regime. Confirmation or null detection would provide strong constraints on Schrödinger–Newton–type models and inform the ongoing search for the correct quantum–gravity interface.
Quantum mechanics exhibits a broad collection of theoretical results in complete agreement with experimental evidence. Besides showing unquestionable success in the description of isolated systems, it can be also successfully used to characterize non-isolated quantum system. In such a case, phenomena like dissipation, diffusion or decoherence. The theory of open quantum systems provides the framework where such features can be conveniently explained. \nThis thesis is about decoherence and collapse models. Albeit conceptually they are very far from each other, they both belong the framework of open quantum systems. Indeed, they consider systems interacting with an external entity: an environment for decoherence models, a noise for collapse models. Although the external influence has different origin, they can be described by similar dynamical equations and, to confirm or falsify them, the same experimental tests can be performed. \nThe quantum Brownian motion model can be considered the paradigm of decoherence models. We provide an exact and analytic equation for the time evolution of the operators and we show that the corresponding equation for the states is equivalent to well-known results in the literature. Our derivation allows to compute the time evolution of physically relevant quantities in a much easier way than previous formulations. Moreover, we are not bound to compute the time evolution of the state of the system, which in general is a complicated task. The explicit dependence on the initial state appears only in the initial expectation values and not in the dynamics. This makes possible the derivation of expectation values also for nontrivial states. \nAnother decoherence model we considered is a recently proposed model, based on the mass-energy equivalence. The model describes a decoherence source acting universally on every system whose superposition is extended on positions experiencing different gravitational potentials. We studied the conditions under which this mechanism becomes the dominant decoherence effect in typical interferometric experiments. We show that current experiments are off by several orders of magnitude. New ideas are needed to achieve the necessary requirements. \nThe second part of this thesis concerns collapse models, in particular the Continuous Spontaneous Localization (CSL) model. We focus on experimental tests that can probe it. In this respect, experiments can be grouped in two classes: interferometric tests and non-interferometric ones. The first class includes those experiments, which directly create and detect quantum superpositions of the center of mass of massive systems. The strongest bounds on the CSL parameters come from the second class of non-interferometric experiments, which are sensitive to small position displacements and detect CSL-induced diffusion in position. \nWe investigate how we can benefit from the non-interferometric perspective given by optomechanical setups, which have reached high sensitivities as force and position sensors. Three examples are considered.\nFirst, we compute the upper bounds on the CSL parameters, which can be inferred by the gravitational wave detectors LIGO, LISA Pathfinder and AURIGA.\nSecond, we report new results from an experiment based on a high-quality cantilever cooled to millikelvin temperature. High accuracy measurements of the cantilever thermal fluctuations reveal a nonthermal force noise of unknown origin. This excess noise is compatible with the CSL heating predicted by Adler. Several physical mechanisms able to explain the observed noise have been ruled out.\nThird, we propose an unattempted non-interferometric test aimed to investigate the still unexplored region of the CSL parameter space. Our proposal relies on torsional degrees of freedom rather than the usual vibrational ones. We believe that the test that has been put forward here, will eventually probe the unexplored CSL parameter space.