Using gravitational decoherence to constrain the number of extended quantum systems

In the low energy regime, the effect of gravity on an evolving quantum system (clock) manifests as time dilation. A massive particle prepared in a spatial superposition near a composite particle with internal clocks decoheres due to entanglement between its spatial DOF and the states of the clocks. Here we expand this formalism to extended quantum systems (EQS) such as spin chains and construe the time dilation effect as a differential red-shift of the local Hamiltonian along the chain, which depends on the relative position of the massive particle. Hence, at zero temperature, by the mere dependence of the ground state of the spin chain on the position of the gravitating particle, we establish gravitational decoherence caused by such an EQS. Since this effect is additive for many such independent EQS in the vicinity of a massive particle, we identify the number of spin chains that can cause detectable decoherence in the spatial superposition of the massive particle. The coherence observed in an experiment can, therefore, be used to put a fundamental limit on the number of EQS surrounding the spatial superposition of a massive particle. In principle, the method introduced here can be extended to quantum fields, putting a bound on the number of extant fundamental particles.