For a finite-mass quantum reference frame, centre-of-mass spreading and gravitational self-interaction define two Newtonian dephasing channels whose engineering crossover is operationally negligible for realistic optical probes; we recast this null result as a rate-based exclusion contour and experimental roadmap. We derive the operational coherence after tracing the frame, for an entangled probe split by internal energy Delta E_S, under (i) the Pikovski-type centre-of-mass wavepacket spread, which is mass-independent and grows as t^2, and (ii) a self-gravity channel in which the internal clock trap-position uncertainty sigma_c produces Gaussian dephasing linear in frame mass m and interrogation time. For a harmonic trap with correlation time tau_c = 1/omega_trap approx 1.6 microseconds (100 kHz trap), the laboratory interrogation T = 1 day satisfies T >> tau_c, so the white-noise Markovian regime applies and channel B grows linearly in t. The 1%-operational-dephasing contour is the analytic line Delta E_S,crit = 0.142 hbar R^2 c^2 /(G m sigma_c T). Optical probes (Delta E_S ~ 10^-19 J) require frame masses of order 10^7 kg, while nuclear/gamma probes (Delta E_S ~ 10^-13 J) require only tens of kg. We compute the engineering crossover m* = sqrt(hbar T R c^2 /(2 G sigma_0 L)) = 0.248 +/- 0.03 kg, the finite-temperature trap width, and Monte-Carlo uncertainty. No new physics is claimed; the result is a decidable null bound and a roadmap for next-generation experiments. PACS numbers: 03.65.Ta, 04.60.m, 03.65.Ud, 04.80.-y
For a finite-mass quantum reference frame, centre-of-mass spreading and gravitational self-interaction define two Newtonian dephasing channels whose engineering crossover is operationally negligible for realistic optical probes; we recast this null result as a rate-based exclusion contour and experimental roadmap. We derive the operational coherence after tracing the frame, for an entangled probe split by internal energy Delta E_S, under (i) the Pikovski-type centre-of-mass wavepacket spread, which is mass-independent and grows as t^2, and (ii) a self-gravity channel in which the internal clock trap-position uncertainty sigma_c produces Gaussian dephasing linear in frame mass m and interrogation time. For a harmonic trap with correlation time tau_c = 1/omega_trap approx 1.6 microseconds (100 kHz trap), the laboratory interrogation T = 1 day satisfies T >> tau_c, so the white-noise Markovian regime applies and channel B grows linearly in t. The 1%-operational-dephasing contour is the analytic line Delta E_S,crit = 0.142 hbar R^2 c^2 /(G m sigma_c T). Optical probes (Delta E_S ~ 10^-19 J) require frame masses of order 10^7 kg, while nuclear/gamma probes (Delta E_S ~ 10^-13 J) require only tens of kg. We compute the engineering crossover m* = sqrt(hbar T R c^2 /(2 G sigma_0 L)) = 0.248 +/- 0.03 kg, the finite-temperature trap width, and Monte-Carlo uncertainty. No new physics is claimed; the result is a decidable null bound and a roadmap for next-generation experiments. PACS numbers: 03.65.Ta, 04.60.m, 03.65.Ud, 04.80.-y
Reference frames are typically treated as idealized background structures. In reality, clocks, rulers, gyroscopes, and optical standards are extended physical systems with finite mass, size, and internal dynamics. This paper identifies the concrete operational mechanism by which such systems cease to function as reference frames for themselves and for other systems. Two unavoidable effects compete in any extended device used to transport time and phase information: • Quantum center-of-mass diffusion, which delocalizes light systems over long interrogation times.• Gravitational self-interaction, which produces redshift gradients across sufficiently massive systems and prevents internal synchronization. While related quantum–gravitational bounds on clocks and rods were identified by Salecker–Wigner and later developed by Ng, van Dam, and others, this work shows explicitly how a reference frame fails: it loses the ability to transport coherent time and phase information across its own spatial extent. Using optical cavities as an operational model, the paper demonstrates how gravitational phase gradients become irreducible dephasing channels once unavoidable endpoint motion is accounted for. This converts a deterministic redshift gradient into stochastic phase noise and provides the physical mechanism underlying the optimal quantum–gravitational timing bound. The resulting optimal mass scale for a meter-scale, day-long reference device is macroscopic (~0.3 kg), showing that the crossover from quantum-limited to gravity-limited reference frames occurs at everyday scales of mass and size, far from Planck physics. The analysis remains entirely within standard quantum mechanics and weak-field gravity and complements quantum reference frame formalisms by identifying the regime where the assumption of internal frame coherence itself breaks down.
Gravitational diffusion in optomechanical probes is conventionally modelled as Markovian white noise, but a finite environmental memory can invalidate the interrogation-time optimisation that underlies precision estimates of the gravitational decoherence rate Λg. We treat the diffusion source as an Ornstein–Uhlenbeck colored-noise channel with memory time τc, generalising the single-mode Gaussian covariance of Souza et al. [arXiv:2602.14841] and recovering the Markovian baseline exactly as τc→0, and evaluate the quantum Fisher information (QFI) for Λg under a fixed total experimental budget including preparation overhead. In the underdamped regime of a feedback-cooled femtogram levitated oscillator, we find that the steady-state QFI is independent of τc, whereas the optimal interrogation time at fixed budget shifts by a factor 15–34 and the QFI rate degrades by up to an order of magnitude as τc grows from the Markovian limit to 0.3 s. A two-parameter Fisher analysis further shows that Λg and τc are nearly degenerate during the transient, so that memory acts as a convergence-rate penalty rather than a precision-ceiling penalty. We stress that this work is a protocol-design and quantum-estimation benchmark, not a near-term detection proposal: even at an effective occupation n̄=50 resolving Λg requires of order 10²⁹–10³⁰ repetitions. All numerical results are illustrative/synthetic benchmarks for a 5 K, 100 kHz femtogram oscillator. I. Introduction Gravitationally induced decoherence [1,2] leaves measurable signatures in the Wigner function of a mechanical probe, and quantum metrology asks how precisely the associated diffusion rate Λg can be inferred. Recent work by Souza, de Sá Neto, Russo, Lo Franco and Adesso [3] established the ultimate quantum-Cramér–Rao sensitivity of a single-mode Gaussian probe to Λg within the Diósi/Kafri–Taylor–Milburn master equation, showing that squeezed-vacuum preparation maximises the transient QFI and that the long-time state is independent of the probe. Their analysis, like essentially all optomechanical decoherence metrology, assumes a delta-correlated (Markovian) environment.
We analytically solve the entangling quantum dynamics of two interacting Stern-Gerlach Interferometers~(SGI). Each SGI exploits an operator-valued force applied by a qubit to create and recombine a non-Gaussian state of matter. The entangling phase between the two qubits generated by the leading-order gravitational interaction of the massive degrees of freedom is found to be mass-independent, both for unitary and open dynamics, irrespective of the temperature and squeezing of the initial states. Further, we show that the solution of the four interferometric paths reveals that the mere presence of the interaction does not allow for a perfect recombination of the centre of mass. This second-order effect, alongside higher-order interaction terms, can be used to bound the mass from above and below, thus restricting the experiment's regime to mesoscopic masses. By solving the open dynamics which includes diffusion and dephasing with initial squeezed thermal states, the bounds are tightened by the inclusion of realistic experimental noise. We discuss diamagnetic levitated masses with embedded NV-centres as a specific physical implementation.
We re-examine whether the internal-clock Page–Wootters (PaW) coupling of an optical lattice clock contributes an additional gravitational-redshift bias when the clock's center of mass (COM) is delocalized in a gravitational potential. We first identify and correct a dimensional error in a frequently used branch propagator: the mass-defect interaction must read H_int=(H_c/mc^2)V_k, so the gravitational and motional branch unitaries are U_g/e=exp(± i (omega_0/2mc^2) V_k t), with a dimensionless coefficient beta=omega_0/(2mc^2) ~ 10^23 J^-1. A propagation factor exp(± i V_k t/(hbar m c^2)) used in earlier estimates carries units of 1/J and is numerically ~2/(hbar omega_0)~10^19 too large, producing spurious order-unity phase wrapping. Using the corrected propagator, exact diagonalization on a Fock basis for ^24Mg and ^87Sr lattice clocks shows that (i) the unconditional reduced-COM mean of V_k equals the bare expectation to numerical accuracy (~10^-42 J), a trivial V_k-with-its-own-exponential identity; (ii) the internal–COM entanglement entropy remains below 10^-3 bit at interrogation times up to 1 s; (iii) the only nonzero PaW quantity is a conditional mean shift whose well-conditioned (high-norm) part scales as beta^2 t^2 Cov(V_k,V_k^2)/(mc^2) ~ 10^-22 (Mg, t=50 ms) and oscillates at the optical frequency omega_0 (period ~2 fs); (iv) this conditional shift averages to ~Delta omega/omega_0 ~10^-6 under any phase-insensitive optical readout and is therefore not a clock observable; (v) finite-temperature thermal moments scale the shift as ~T^3 but remain below 10^-25 at 1 mK. We conclude that internal PaW entanglement introduces no resolvable redshift bias at current or near-future optical clocks; Hu-type COM delocalization decoherence remains the leading quantum-relativistic correction. We provide a full parameter-space exclusion map and a sensitivity roadmap. All numerical results in this paper are illustrative/synthetic, computed with the corrected propagator and realistic Mg/Sr parameters.
ABSTRACT Background The gravity–quantum interface lacks a minimal, operational theory for the weak-field, non-relativistic, mesoscopic regime. Semiclassical gravity (SCG), which sources the classical Einstein equations from the quantum expectation value of stress–energy, is the dominant approximation but was designed for cosmological and astrophysical contexts, not for two-body entanglement experiments in the laboratory. A structural silence exists: no framework specifies how gravity couples to quantum matter when entanglement between spatially delocalised mesoscopic masses is at stake. Gap The Named Binary distinguishing Semiclassical Gravity Interface (SCGI) from Quantum-Mediator Gravity Interface (QMGI) does not appear in the literature as a formal operational distinction with computable consequences. No framework simultaneously satisfies: standard quantum mechanics for matter, no-signalling, reduction to Newtonian gravity for classical configurations, and entanglement-generation capability. This four-constraint conjunction defines the structural gap. Approach We formalise the two-body BMV (Bose–Marletto–Vedral) configuration using Hilbert-space language and quantum channels, derive the effective entangling Hamiltonian, prove that C(t) = |sin(Δφ(t)/2)| where Δφ(t) is the entangling phase determined by geometry and interaction time, and derive the design inequality τ_ent ≪ τ_dec as the admissibility condition for entanglement detection. We establish the Strongest Formulation in the five-part programme template, specify a pre-registerable CCS with quantitative decision rule, confirm structural invariance across three independent physics domains, ground the framework in IGT theory, and provide a complete Weil Protocol practitioner review pack. Results Under the four-constraint conjunction (standard QM + no-signalling + Newtonian reduction + entanglement capability), gravity in the mesoscopic weak-field regime must be modelled as a quantum channel: SCGI fails because classical channels cannot increase entanglement between initially separable systems. The entangling phase scales as Δφ(t) ~ Gm²dt/(ℏD²); detectable entanglement requires τ_ent/τ_dec < 1. Three independent distinguishing predictions follow that SCGI cannot make. Structural invariance is confirmed in quantum electrodynamics (photon-mediated entanglement), nuclear physics (exchange-mediated forces), and quantum optics (cavity-mediated entanglement). Implications Any tabletop detection of gravitationally induced entanglement between two mesoscopic masses in a BMV-type experiment would constitute a direct, model-independent, low-energy signature of the quantum nature of gravity. Any confirmed null result provides the strongest direct evidence for SCGI-type models and bounds the parameter space of quantum mediator theories. Weil Protocol practitioner review is required before policy adoption of experimental specifications; status INCOMPLETE. Human-stakes level: L2 · Weil Protocol: INCOMPLETE
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.