  We consider an Unruh-DeWitt detector modeled as a harmonic oscillator that is
coupled to a massless quantum scalar field in the (2+1)-dimensional Minkowski
spacetime. We treat the detector as an open quantum system and employ a quantum
Langevin equation to describe its time evolution, with the field, which is
characterized by a frequency-independent spectral density, acting as a
stochastic force. We investigate a point-like detector moving with constant
acceleration through the Minkowski vacuum and an inertial one immersed in a
thermal reservoir at the Unruh temperature, exploring the implications of the
well-known non-equivalence between the two cases on their dynamics. We find
that both the accelerated detector's dissipation rate and the shift of its
frequency caused by the coupling to the field bath depend on the acceleration
temperature. Interestingly enough this is not only in contrast to the case of
inertial motion in a heat bath but also to any analogous quantum Brownian
motion model in open systems, where dissipation and frequency shifts are not
known to exhibit temperature dependencies. Nonetheless, we show that the
fluctuating-dissipation theorem still holds for the detector-field system and
in the weak-coupling limit an accelerated detector is driven at late times to a
thermal equilibrium state at the Unruh temperature.
