Using the IRAM Plateau-de-Bure Interferometer (PdBI) we have searched for the upper fine structure line of neutral carbon (CI(2->1), nu_rest = 809 GHz) and 12CO(J=7->6) (nu_rest=806GHz) towards the submillimetre galaxies (SMGs) GN20 (SMM J123711.9+622212, z = 4.055) and GN20.2 (SMM J123708.8+622202, z = 4.051). The far-infrared (FIR) continuum is detected at 8sigma significance in GN20, with a flux density of S_1.8mm = 1.9+-0.2mJy, while no continuum is detected in GN20.2. Both sources are statistically undetected in both CI and CO(7-6) lines; we derive line luminosity limits for both CI and CO of L<~2x10^10K km s^-1 pc^2. Assuming carbon excitation temperatures of T_ex = 30K (the galaxies measured dust temperatures), we infer CI mass limits of M_CI < 5.4x10^6 Msun (GN20) and M_CI < 6.8x10^6 Msun (GN20.2). The derived CI abundance limits are <1.8x10^-5 for GN20 and <3.8x10^-5 for GN20.2 implying that the systems have Milky Way level neutral carbon enrichment (X[CI]/X[H2]) or lower, similar to high-redshift carbon-detected systems (at 5x10^-5) but about 50 times less than the neutral carbon enrichment of local starburst galaxies. Observations of GN20 and GN20.2 in high-resolution MERLIN+VLA radio maps of GOODS-N are used to further constrain the sizes and locations of active regions. We conclude that the physical gas properties of young rapidly evolving systems like GN20 and GN20.2 are likely significantly different than starburst/ULIRG environments in the local Universe yet similar to z~2 SMGs. Unless gravitationally amplified examples can be found, observations of galaxies like GN20 will require the order of magnitude increase in sensitivity of the Atacama Large Millimetre Array (ALMA) to constrain their CI and high-J CO content, despite the fact that they are the brightest systems at z~4.
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