Yes other papers have studied phononic qubits after that one. I could just list them here, but I think it would be better for you to learn how to find such papers yourself, so here is how I found the papers:
Find the paper on Google, then click the link I have highlighted in yellow:
Out of the 39 papers that cited this one, I have listed some (but not all) of the relevant ones below:
As for advantages and disadvantages, perhaps the only advantage would be robustness to decoherence, which is claimed in the paper you mentioned, but not yet backed up very much.
There are lots of disadvantages though:
- Phonons are quasi-particles, not fundamental particles. This gives phonons a similar disadvantage as anyons. We all know how hard it is to build a quantum computer with anyons.
- You almost never find one isolated phonon, you get a huge spectrum of phonons at the same time.
- The technology required for phononic qubits seems to have had its beginnings in 2012, whereas for NMR qubits it goes back to at least the 1940s and for superconducting qubits it goes back to at least the 1960s. Phononic qubits are far behind any other candidate for quantum computers, so there is lot of catching up to do.