Response of Unruh-DeWitt detector with time-dependent acceleration

It is well known that a detector, coupled linearly to a quantum field and accelerating through the inertial vacuum with a constant acceleration g, will behave as though it is immersed in a radiation field with temperature T=(g/2π). We study a generalization of this result for detectors moving with a time-dependent acceleration g(τ) along a given direction. After defining the rate of excitation of the detector appropriately, we evaluate this rate for time-dependent acceleration, g(τ), to linear order in the parameter η=g˙/g2. In this case, we have three length scales in the problem: g−1, (g˙/g)−1 and ω−1 where ω is the energy difference between the two levels of the detector at which the spectrum is probed. We show that: (a) When ω−1≪g−1≪(g˙/g)−1, the rate of transition of the detector corresponds to a slowly varying temperature T(τ)=g(τ)/2π, as one would have expected. (b) However, when g−1≪ω−1≪(g˙/g)−1, we find that the spectrum is modified even at the orderO(η). This is counter-intuitive because, in this case, the relevant frequency does not probe the rate of change of the acceleration since (g˙/g)≪ω and we certainly do not have deviation from the thermal spectrum when g˙=0. This result shows that there is a subtle discontinuity in the behavior of detectors with g˙=0 and g˙/g2 being arbitrarily small. We corroborate this result by evaluating the detector response for a particular trajectory which admits an analytic expression for the poles of the Wightman function.