We present a survey of atomic carbon (CI) emission in high-redshift (z>2) submillimeter galaxies (SMGs) and quasar host galaxies (QSOs). Sensitive observations of the CI(3P_1->3P_0) and CI(3P_2->3P_1) lines have been obtained at the IRAM Plateau de Bure interferometer and the IRAM 30m telescope. A total of 16 CI lines have been targeted in 10 sources, leading to a total of 10 detected lines --- this doubles the number of CI observations at high redshift to date. We include previously published CI observations (an additional 5 detected sources) in our analysis. Our main finding is that the CI properties of the studied high-redshift systems do not differ significantly from what is found in low-redshift systems, including the Milky Way. The CI(3P_2->3P_1)/CI(3P_1->3P_0) and the CI(3P_1->3P_0)/12CO(3-2) line luminosity (L) ratios change little in our sample, with respective ratios of 0.55+/-0.15 and 0.32+/-0.13. The CI lines are not an important contributor to cooling of the molecular gas (average L_CI/L_FIR ~ (7.7+/-4.6) x 10^-6). We derive a mean carbon excitation temperature of 29.1+/-6.3 K, broadly consistent with dust temperatures derived for high-redshift starforming systems, but lower than gas temperatures typically derived for starbursts in the local universe. The carbon abundance of X_CI/X_H2~8.4+/-3.5 x 10^-5 is of the same order as found in the Milky Way and nearby galaxies. This implies that the high-z galaxies studied here are significantly enriched in carbon on galactic scales, even though the look-back times are considerable (the average redshift of the sample sources corresponds to an age of the universe of ~2 Gyr).
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