Two pendulums hanging in the same room start swinging in step if an invisible thread connects them. In the world of quantum gravity, the gravitational field plays the role of that thread, and massive particles serve as the pendulums. In a new paper, physicists have traced how quantized gravitational waves, traveling between two particles in harmonic traps, give rise to quantum entanglement. It turns out the effect does not appear right away: the particles "feel" each other only after the signal has had time to travel from one to the other. The delay equals the time light takes to cover the distance between them.

The authors considered two identical particles, confined in spring-like potentials and separated by a distance d. They quantized the gravitational field in the linearized approximation on a flat spacetime background, keeping only the transverse traceless modes. The interaction with the particles proceeds through one component of the energy-momentum tensor, which describes momentum fluctuations along the axis of oscillation. To isolate the purely quantum contribution, the physicists averaged over all states of the gravitational field using the influence functional. This procedure splits the dynamics into two parts: one governs unitary evolution and is tied to the commutator of the field, while the other describes noise and is tied to the anticommutator. Only the first part matters for entanglement, because a classical channel cannot create quantum correlations.

In the ground state, the coherence, a measure of entanglement, grows linearly with time after the delay. For typical parameters (oscillator frequency 10⁸ Hz, distance 1 nanometer, observation time 1 second), it comes to about 10⁻⁵⁵. That is vanishingly small. However, if the particles are prepared in squeezed states, where quantum fluctuations of one quantity are reduced at the expense of another, the effect is amplified. With a squeezing parameter r = 1, the coherence rises to ~10⁻⁵². That is still far beyond any experimental sensitivity, but the very fact of the amplification matters: it points the way toward future tests.

The work shows explicitly for the first time that even propagating gravitational modes, and not just static energy shifts, are capable of entangling massive particles. Previously, the QGEM approach (quantum gravity through entanglement of masses) usually neglected retardation and treated the interaction as instantaneous. Now it has become clear that causality is built into the mechanism itself: correlations are established no faster than the speed of light allows. This is an important step toward understanding how quantum gravity manifests itself under laboratory conditions. The result has no practical applications yet, but it sharpens the theoretical picture and suggests where to look for an amplification of the effect.

Looking ahead, one could account for decoherence from the anticommutator contribution, go beyond the Markovian approximation, and consider a finite temperature of the gravitational field. Perhaps in more complex initial states of the particles or in other trap configurations the effect will prove more noticeable. For now, this work is a reminder that even an invisible conductor, as the quantized gravitational field is, cannot make pendulums swing in step faster than its "downbeat" reaches them.