  The origin and evolution of gas in debris disks is still not well understood.
Secondary gas production from cometary material or a primordial origin have
been proposed. So far, observations have mostly concentrated on CO, with only
few C observations available. We create an overview of the C and CO content of
debris disk gas and use it 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.
We find that current state-of-the-art models of secondary gas production
overpredict the neutral carbon content of debris disk gas. This does not rule
out a secondary origin, but might indicate that the models require an
additional C removal process. Alternatively, the gas might be produced in
transient events rather than a steady-state collisional cascade. We also test a
primordial gas origin by comparing our results to a simplified thermo-chemical
model. This yields promising results, but more detailed work is required before
a conclusion can be reached. Our work demonstrates that the combination of C
and CO data is a powerful tool to advance our understanding of debris disk gas.
