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It just occurred to me that the graviton is still a hypothetical quantum of gravity yet gravitational waves are proven and measurable. Should we not expect that gravity follows the same wave/particle duality of other occurring quanta (radiation) given its ability to behave like a wave?
What limitations exist for establishing a "double-slit" graviton experiment?
The effective theories of quantized gravitation do give the same function to the graviton as the gauge boson of gravity, a zero mass spin 2 particle. As an elementary particle individual gravitons should also have a probability-wave nature , and hypothetically a double slit experiment should give the similar results as the single photon at a time experiment , if it were not for the very very small coupling constant in gravitational interactions. in the effective quantum field theory of gravitation.
"Should we not expect that gravity follows the same wave/particle duality of other occurring quanta (radiation) given its ability to behave like a wave?"
We expect it and we do it. we can quantize the gravitational wave using regular QFT and call it graviton. The problem is when we calculate the cross section of graviton, we get infinities. One way to get rid of these infinities is string theory.
"What limitations exist for establishing a "double-slit" graviton experiment?"
I do not think that it is a possibility. In order to do the double split experiment, you put a sheet which BLOCKS the light and put two holes in it. There is nothing to block gravitational wave. The gravitational wave passes through everything.
