According to the current paradigm of circumstellar disk evolution, gas-rich primordial disks evolve into gas-poor debris disks compose of second-generation dust. To explore the transition between these phases, we searched for $^{12}$CO, $^{13}$CO, and C$^{18}$O emission in seven dust-rich debris disks around young A-type stars, using ALMA in Band 6. We discovered molecular gas in three debris disks. In all these disks, the $^{12}$CO line was optically thick, highlighting the importance of less abundant molecules in reliable mass estimates. Supplementing our target list by literature data, we compiled a volume-limited sample of dust-rich debris disks around young A-type stars within 150 pc. We obtained a CO detection rate of 11/16 above a $^{12}$CO J=2$-$1 line luminosity threshold of $sim 1.4 times 10 ^4$ Jykms$^{-1}$pc$^2$ in the sample. This high incidence implies that the presence of CO gas in bright debris disks around young A-type stars is likely more the rule than the exception. Interestingly, dust-rich debris disks around young FG-type stars exhibit, with the same detectability threshold as for A-type stars, significantly lower gas incidence. While the transition from protoplanetary to debris phase is associated with a drop of dust content, our results exhibit a large spread in the CO mass in our debris sample, with peak values comparable to those in protoplanetary Herbig Ae disks. In the particularly CO-rich debris systems the gas may have primordial origin, characteristic of a hybrid disk.
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