At night when they roost, do chickens experience a state of sleep (meaning do they move from NREM1 to NREM2, NREM3 and REM) or are they in a state of torpor (reduced metabolic rate and decreased physiological activity)?
Birds provide a unique opportunity to evaluate current theories for the function of sleep. Like mammalian sleep, avian sleep is composed of two states, slow-wave sleep (SWS) and rapid eye-movement (REM) sleep that apparently evolved independently in mammals and birds. Despite this resemblance, however, it has been unclear whether avian SWS shows a compensatory response to sleep loss (i.e., homeostatic regulation), a fundamental aspect of mammalian sleep potentially linked to the function of SWS. Here, we prevented pigeons (Columba livia) from taking their normal naps during the last 8 h of the day. Although time spent in SWS did not change significantly following short-term sleep deprivation, electroencephalogram (EEG) slow-wave activity (SWA; i.e., 0.78-2.34 Hz power density) during SWS increased significantly during the first 3 h of the recovery night when compared with the undisturbed night, and progressively declined thereafter in a manner comparable to that observed in similarly sleep-deprived mammals. SWA was also elevated during REM sleep on the recovery night, a response that might reflect increased SWS pressure and the concomitant 'spill-over' of SWS-related EEG activity into short episodes of REM sleep. As in rodents, power density during SWS also increased in higher frequencies (9-25 Hz) in response to short-term sleep deprivation. Finally, time spent in REM sleep increased following sleep deprivation. The mammalian-like increase in EEG spectral power density across both low and high frequencies, and the increase in time spent in REM sleep following sleep deprivation suggest that some aspects of avian and mammalian sleep are regulated in a similar manner .
From the Abstract and reference  above, we see that EEGs have been taking on birds and show that birds enter REM and SWS similarly to mammals.
In response to Chris Stronks, researchers have also studied chickens see Evolution of Sleep: Phylogenetic and Functional Perspective by Patrick McNamara, Robert A. Barton, Charles L. Nunn. In the section for REM sleep in birds, the studies were conducted on pigeons and chickens see pages 204-205.