At least according to the TFA, it seems that the breakthrough is that they got 2.5 mJ out vs. the 2.1 mJ that was used to power the laser.
Fun fact that Wolfram alpha just informed me of: a phone uses between 10 and 20 MJ a year: multiple kilos of TNT. 4000mAh * 3.7V * 365: yep, it's about right.
Also, interesting fun fact indeed.