The origin and evolution of gas in debris disks are still controversial; secondary gas production from cometary material or gas leftover from protoplanetary disks (i.e. primordial origin) have been proposed. So far, observations have mostly concentrated on CO, with only few C (atomic carbon) observations available. We aim to get an overview of the C and CO content of debris disk gas and test state-of-the-art models. We use new and archival ALMA observations of CO and CI emission, complemented by CII data from Herschel, for a sample of 14 debris disks. This expands the number of disks with ALMA measurements of both CO and CI by ten disks. We present new detections of CI emission towards three disks: HD 21997, HD 121191, and HD 121617. We use a simple disk model to derive gas masses and column densities in a uniform way.
We find a bimodal distribution of CO and C column densities, which suggests that self- and mutual shielding of UV radiation play an important role in gas-rich disks. We compare our results with state-of-the-art models of secondary gas production and find that the models overpredict the atomic C content of debris disk gas. This might indicate that the models are requiring an additional C removal process, or that the gas is produced in transient events rather than a steady-state collisional cascade. A comparison to a simplified thermo-chemical model representing a primordial origin yields promising results that motivate further investigation. Our work demonstrates that the combination of C and CO data is a powerful tool to advance our understanding of debris disk gas.