To stop sound from going through surfaces you need mass. With a concrete floor, that is probably the best way to stop the sound. In the US, we don't typically have concrete floors except for the foundation level. I've seen houses built in Greece, and they use very thick poured concrete for all interior walls and floors.
Is the person below you from your family or another family? If your own family, could you ask them if they hear all noises, or maybe just the breaks, etc?
I live in an apartment, and I have another family below me. I try to limit my pool to the daytime, and if I do play at night, I stop by 2230. I also try to limit hard break shots, as they are the loudest.
I have a drop pocket table, and I put socks inside the pockets so when the balls fall in they don't continue to roll around. But ball return tables are louder still, and cutting out some of that noise will help.
Carpet I think will help more for footstep noise (on an American style wood frame floor). I don't know if it will do anything for noise at pool table height. Those soft materials reduce the echos from the sounds, but they don't do much to prevent the original sound from going through a surface. (that is what the mass does).
Oh, I also play some music when I play to try to disguise the sound of the balls.
A carpet on the tile certainly won't hurt the noise transmission, and is worth a try. It can also keep your balls from chipping when they will eventually fly off the table (and protect the tiles as well).
Good luck, and welcome to the forum.