Ray LaRaja and Brian Schaffner have posted this draft on SSRN.  Here is the abstract:

This paper explores whether a voting system that rewards voters with a lottery ticket would boost turnout. Using an experimental design with data from the 2011 Cooperative Congressional Elections Study, it demonstrates that lotteries have the potential to boost turnout significantly for under-represented groups. The study also suggests that lottery systems with numerous winners and moderate payoffs would be more successful than systems with very few winners and large payouts.

Of course, payments for turnout are absolutely illegal under current federal law in any election in which a federal candidate is on the ballot.