Theoretically, nothing prevents a really massive object to be in a superposition of two spatial locations, even far away one from the other. Then I guess spacetime would also show the superposition of two corresponding gravitational wells. Could this be observed somehow ? Now measuring the gravitational field would provide a way to measure the object position and thus give it a specific one. Could this impose a limit to the mass of an object in superposition (in the line of: when heavy enough its position gets automatically measured by whatever is subject to its gravity) ? If yes, what would be the order of magnitude of such a limit ?
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You say:
Theoretically, nothing prevents a really massive object to be in a superposition of two spatial locations, even far away one from the other.
but a massive object cannot be maintained in a superposition for very long due to quantum decoherence. Strictly speaking it's not the mass that matters but rather the number of degrees of freedom, but in practice the more massive an object is the more degrees of freedom it has.
Anyhow, any macroscopic object has such a short decoherence time that it is never likely to be observed in a superposition of states. As far as I know, the largest object ever to show quantum behaviour is an oscillator built by Andrew Cleland's group at Santa Barbara, which was around 50 - 100 microns in size.
Having said this, it has been suggested that a gravitational field would act as one of the mechanisms for decoherence, and this is known as gravitational decoherence. So far there is little in the way of experimental studies, but a Chinese team will shortly be launching a satellite to test whether entangled states can be maintained between the ground and a satellite.
John Rennie
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I'm answering my own question by refering to a paper linked by glS in a comment (Is Gravity Quantum ?, Bahrami & al., 2015).
The authors feel that although a satisfying theory of quantum gravity is still missing at the moment, they "can safely claim that, should gravity be quantum, [the spatial superposition in a massive quantum system] would be manifested by the superposition of gravitational fields".
They go on proposing an experimental setup that could, amazingly, actually test that possibility in the not too far future by directly probing the gravitational field of a mesoscopic system (of mass about 100 ng) prepared in a superposition of two different positions. They seem confident that it should be possible to keep all non-gravitational interactions (notably van der Waals) negligible in that "technically demanding" experiment.
Edit:
A new paper appeared: Probing a Gravitational Cat State: Experimental Possibilities.
Stéphane Rollandin
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