  We report microwave-driven photon-assisted tunneling in a suspended carbon
nanotube double quantum dot. From the resonant linewidth at a temperature of 13
mK, the charge dephasing time is determined to be 280 +- 30 ps. The linewidth
is independent of driving frequency, but increases with increasing temperature.
The moderate temperature dependence is inconsistent with expectations from
electron-phonon coupling alone, but consistent with charge noise arising in the
device. The extracted level of charge noise is comparable with that expected
from previous measurements of a valley-spin qubit, where it was hypothesized to
be the main cause of qubit decoherence. Our results suggest a possible route
towards improved valley-spin qubits.
