Just clean it with a dry cloth. You can also use a little silicon oil while cleaning it out with a cloth to make it a little easier to clean, but when you are done you need to go over and wipe it down with a clean and dry cloth. When done cleaning the barrel in your rifle it should be clean and dry.
Silicon oil in the barrel will only attract debris like dust and dirt, which will kill the rifle's accuracy and cause jams.
Depends on weather its polished or not, if it is then adding oils or silicon may mess up the polishing. If it is not polished then I suggest you look for the polishing guide that's on here some were and do that then messing with oils and silicon.
The reason I say that is if you leave a bit of oil or silicon in the barrel it can cause jams in a 6.01 because it would do the opposite of what you want. The dirt will collect at the spots you miss and make you have to clean it more frequently. So I dont know if it would be as bad in a .03 but this is just my two cents.