  We study Bell inequality tests with entanglement between a coherent-state
field in a cavity and a two-level atom. In order to detect the cavity field for
such a test, photon on/off measurements and photon number parity measurements,
respectively, are investigated. When photon on/off measurements are used, at
least 50% of detec- tion efficiency is required to demonstrate violation of the
Bell inequality. Photon number parity measurements for the cavity field can be
effectively performed using ancillary atoms and an atomic detector, which leads
to large degrees of Bell violations up to Cirel'son's bound. We also analyze
decoherence effects in both field and atomic modes and discuss conditions
required to perform a Bell inequality test free from the locality loophole.
