  We study the late-time asymptotic state of a stationary Unruh-DeWitt detector
interacting with a field in a thermal state. We work in an open system
framework, where the field plays the role of an environment for the detector.
The long-time interaction between the detector and the field is modelled with
the aid of a one-parameter family of switching functions that turn on and off
the interaction Hamiltonian between the two subsystems, such that the long-time
interaction limit is reached as the family parameter goes to infinity. In such
limit, we show that if the field is in a Kubo-Martin-Schwinger (KMS) state and
the detector is stationary with respect to the notion of positive frequency of
the field, in the Born-Markov approximation, the asymptotic state of the
detector is a Gibbs state at the KMS temperature. We then relax the KMS
condition for the field state, and require only that a frequency-dependent
version of the detailed balance condition for the Wightman function pulled back
to the detector worldline hold, in the sense that the inverse temperature
appearing in the detailed balance relation need not be constant. In this
setting, we show that the late-time asymptotic state of the detector has the
form of a thermal density matrix, but with a frequency-dependent temperature.
We present examples of these results, which include the classical Unruh effect
and idealised Hawking radiation (for fields in the HHI state), and also the
study of the late-time behaviour of detectors following stationary "cusped" and
circular motions in Minkowski space interacting with a massless Klein-Gordon
field in the Minkowski vacuum. In the cusped motion case, a
frequency-dependent, effective temperature for the asymptotic late-time
detector state is obtained analytically. In the circular motion case, such
effective temperature is obtained numerically.
