The papers are by and large critical of recent reforms, including school choice. New Orleans Recovery School District has mostly charter schools.
And here's a paper just published in the AER that looks at the effects of charter schools, using the fact that some non-charters have been taken over, which changes the selection criteria (students already in a takeover school are grandfathered in...)
American Economic Review 2016, 106(7): 1878–1920
Atila Abdulkadiroğlu, Joshua D. Angrist, Peter D. Hull and Parag A. Pathak
Charter takeovers are traditional public schools restarted as charter schools. We develop a grandfathering instrument for takeover attendance that compares students at schools designated for takeover with a matched sample of students attending similar schools not yet taken over. Grandfathering estimates from New Orleans show substantial gains from takeover enrollment. In Boston, grandfathered students see achievement gains at least as large as the gains for students assigned charter seats in lotteries. A non-charter Boston turnaround intervention that had much in common with the takeover strategy generated gains as large as those seen for takeovers, while other more modest turnaround interventions yielded smaller effects.