Waiting around for Unruh

How long does a uniformly rotating observer need to interact with a quantum field in order to register an approximately thermal response due to the circular motion Unruh effect? We address this question for a massless scalar field in $2+1$ dimensions, defining the effective temperature via the ratio of excitation and de-excitation rates of an Unruh-DeWitt detector in the long interaction time limit. In this system, the effective temperature is known to be significantly smaller than the linear motion Unruh effect prediction when the detector's energy gap is small: the effective temperature tends to zero in the small gap limit, linearly in the gap. We show that a positive small gap temperature at long interaction times can be regained via a controlled long-time-small-gap double limit, provided the detector's coupling to the field is allowed to change sign. The resulting small gap temperature depends on the parameters of the circular motion but not on the details of the detector's switching. The results broaden the energy range for pursuing an experimental verification of the circular motion Unruh effect in analogue spacetime experiments. As a mathematical tool, we provide a new implementation of the long interaction time limit that controls in a precise way the asymptotics of both the switching function and its Fourier transform.