Non-interferometric tests of gravitational decoherence models with mechanical oscillators

The paper in detail

All three master equations have the form \[\partial_t\hat\rho=-\tfrac{i}{\hbar}[\hat H_0,\hat\rho]-\frac{\tau}{\mathcal N}\bigl[\hat p^{\,j},[\hat p^{\,j},\hat\rho]\bigr],\] with \(j=1\) for the model of Arzano et al. (deformed Poincaré symmetry), \(j=2\) for Breuer et al. (a fluctuating metric, so a fluctuating kinetic energy) and \(j=4\) for Petruzziello and Illuminati (a fluctuating generalised-uncertainty parameter). The strength is a time \(\tau_X\), naturally compared with the Planck time \(t_P\approx5.4\times10^{-44}\) s.

For a free particle the dissipator commutes with the Hamiltonian, so momentum and energy are conserved. The effects are momentum-space decoherence, with rates scaling as \(\Delta p^2\), \(\bar p^2\Delta p^2\) and \(\bar p^6\Delta p^2\) for the three models, and a position variance that grows by an extra \((2\tau\hbar^2j^2/\mathcal N)\langle\hat p^{2j-2}\rangle t\). Both have the same origin: the equation is the average of a Schrödinger equation with a white-noise term \(\propto\hat p^{\,j}\xi(t)\), which adds a random velocity proportional to \(\hat p^{\,j-1}\).

A harmonic oscillator turns that velocity noise into heating, because random displacements feed potential energy. Zero-temperature damping leaves the ground state invariant, so any ground-state depletion is a clean signature. The heating laws differ: \(\dot n=-\gamma n+\Gamma_A/2\) for the first model, with stationary occupation \(\Gamma_A/2\gamma\); \(\dot n=-\gamma n+\Gamma_BK/\hbar\omega\) for the second, which is parametric heating and becomes unstable below a critical damping close to \(\Gamma_B/2\); and \((\Gamma_I/8)(20n^3+30n^2+40n+15)\) starting from Fock state \(n\) for the third. In steady state the first two produce the same excess position-spectrum lineshape, which differs from that of force noise.

Applied to experiments, the bounds depend on the correlation structure. For fluctuations correlated across a test mass but not with its reference, LISA Pathfinder gives \(\tau_A\lesssim2\times10^{-45}\) s and Advanced LIGO \(10^{-51}\) s; for fluctuations independent between atoms they weaken by the number of atoms. Ground-state gigahertz resonators would bound \(\tau_B\) up to fourteen orders of magnitude below levitated-nanoparticle heating. The residual heating of cold solids, \(\lesssim10^{-11}\) W/kg, gives \(\tau_A\lesssim2\times10^{-56}\) s for independent fluctuations and \(\tau_B\lesssim2\times10^{-44}\) s; the minimal-length model is bounded only at \(\tau_I\lesssim10^{49}\) s.

The paper corrects one coefficient in an earlier treatment of the third model, which changes only a phase and not any decoherence rate. Open issues are the composite-body extension, which is not unique, and dissipative versions of the models, which would replace the infinite-temperature Lindblad noise by a finite effective temperature and cap the heating.

Transferable methods

1. Closed moment equations for heating

   Decoherence rates require state preparation, but the energy of an oscillator is directly measurable. For dissipators linear or quadratic in \(\hat p\), the equations for the second moments close, which gives exact heating laws and exposes the instability of the metric-fluctuation model.

   With \(K_t=\langle\hat p^2\rangle/2m\), \(U_t=\tfrac12m\omega^2\langle\hat x^2\rangle\), \(S_t=\langle\hat x\hat p+\hat p\hat x\rangle\) and damping \(\gamma\), the Arzano dissipator adds the constant \(\hbar\omega\Gamma_A/2\) to \(\dot U\), so \(K\) and \(U\) add to \[\dot E=\frac{\hbar\omega}{2}\Gamma_A+\gamma\Bigl(\frac{\hbar\omega}{2}-E\Bigr),\qquad\Gamma_A=\frac{m\omega c^2\tau_A}{\hbar}.\] The Breuer dissipator instead adds \(\Gamma_BK_t\), with \(\Gamma_B=\tau_B\omega^2\). The undamped linear system for \((K,U,S)\) then has characteristic polynomial \(r^3+4\omega^2r-2\Gamma_B\omega^2\), whose single real root \[r=\frac{4\omega}{\sqrt3}\sinh\Bigl[\tfrac13\,\mathrm{arsinh}\Bigl(\frac{3\sqrt3}{8}\frac{\Gamma_B}{\omega}\Bigr)\Bigr]\approx\frac{\Gamma_B}{2}\] is the growth rate that damping must beat.

   For the quartic model the source is \(64a_P^2\tau_I\omega^2\langle\hat K^3\rangle\), a third moment, so the hierarchy does not close; from a Fock state, \(\langle n\vert\hat K^3\vert n\rangle=(\hbar\omega)^3(20n^3+30n^2+40n+15)/64\). From the ground state, \(\hat K^2\) couples \(\ket0\) to \(\ket2\) and \(\ket4\) at rates \(9\Gamma_I/16\) and \(3\Gamma_I/16\), and weighting by the phonons created gives \(15\Gamma_I/8\).

   Editorial checks: in a truncated Fock space the diagonal of \(\hat K^3\) reproduces the polynomial for \(n=0,\dots,3\); the transition rates come out as \(9/16\), \(3/16\) (quartic), \(1/8\) (Breuer, \(0\to2\)) and \(1/2\) (Arzano, \(0\to1\)) in units of the respective \(\Gamma\); the sinh formula matches the numerical root at \(\Gamma_B/\omega=0.01,0.5,3\); and solving the stationary linear system at \(\gamma=0.2\omega\), \(\Gamma_B=0.1\omega\) reproduces Eqs. (52)–(53).

   How to reuse it

   For any Lindblad term built from a function of momentum, the energy injection equals \(\tfrac12m\omega^2\hbar^2\langle f'(\hat p)^2\rangle\). Computing that one average tells whether heating is constant, parametric or runaway before solving any dynamics.

2. Calorimetric bounds from summing over all phonon modes

   A readout measures one mode, but the same dissipator heats every normal mode of a solid. Summing the injection over the whole phonon spectrum turns the residual heat leak of a cold detector into a bound that no single-mode experiment reaches.

   For independent fluctuations on each atom, \(\dot E_A=\tfrac12\tau_Ac^2\sum_a\nabla_a^2U\), and \(\sum_a\nabla_a^2U=m\sum_\kappa\omega_\kappa^2\). A Debye spectrum has \(\langle\omega^2\rangle=3\omega_D^2/5\), so \[\frac{\dot E_A}{M}=\frac32c^2\tau_A\langle\omega^2\rangle.\] For the Breuer model every mode carries at least its zero-point kinetic energy \(\hbar\omega_\kappa/4\), which gives a temperature-independent floor \[\frac{\dot E_B}{M}\ge\frac{3\tau_B\hbar\omega_D^3}{8m}.\]

   Editorial check with silicon (\(\omega_D=8.4\times10^{13}\,\mathrm{s^{-1}}\), atomic mass \(28\,u\)) and \(\dot E/M\le10^{-11}\) W/kg: \(\tau_A\le1.75\times10^{-56}\) s, \(\tau_B\le1.99\times10^{-44}\) s \(=0.37\,t_P\), and for Gaussian correlations of range \(1\,\mu\)m, \(\tau_A\le5.4\times10^{-49}\) s, matching Eq. (95).

   How to reuse it

   Collapse and decoherence models that inject energy at a state-independent or zero-point-limited rate can be bounded by the calorimetry of cold solids, summing the injection over the full mode spectrum instead of reading out one mode.

Source: Eqs. (1)–(3), (13)–(17); Sec. IV.2 (Eqs. (33)–(65)); Eqs. (79)–(84); Sec. V.6 (Eqs. (91)–(95)). Based on the linked paper version. Derivations marked as editorial, and suggested uses in other problems, are editorial guidance rather than results of the paper.

Context & limitations

Why it matters. The free parameter of each model becomes a number fixed by data that already exist, without superposition states. Because the heating rates depend differently on excitation, frequency, mass and correlation length, a positive signal could in principle be attributed to one model rather than another, or to technical noise.

Limitations. The bounds depend on how the single-particle equations are extended to composite bodies: which constituents are fundamental and over what length the fluctuations are correlated. Between independent and collective fluctuations the bounds move by many orders of magnitude, and the paper tabulates both. Bulk-heating bounds assume white noise up to the Debye frequency and the Lindblad form, which corresponds to infinite-temperature noise; dissipative extensions would cap the heating. For the minimal-length model the moment equations do not close, its finite-time divergence is an artefact of a secular approximation used beyond its validity, and the bounds remain far from the Planck scale. Several platform values assume a calibration precision of 10%. The paper has no AI-use statement.

Description based on the full arXiv HTML of v1, read 2026-11-14: Sections I–VI, including the stochastic unravelling and all bounds in Section V and Table 1; the appendices with the exact solutions and the composite-body derivation were not read. The Fock-state matrix elements, transition rates, the instability root, the stationary moments and the silicon bulk-heating bounds were recomputed.