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 $T=(g/2\pi)$. We study a generalization of this result for detectors moving with a time-dependent acceleration $g(\tau)$ along a given direction. After defining the rate of excitation of the detector appropriately, we evaluate this rate for time-dependent acceleration, $g(\tau)$, to linear order in the parameter $\eta = \dot g / g^2$. In this case, we have three length scales in the problem: $g^{-1}, (\dot g/g)^{-1}$ and $\omega^{-1}$ where $\omega$ is the energy difference between the two levels of the detector at which the spectrum is probed. We show that: (a) When $\omega^{-1} \ll g^{-1} \ll (\dot g/g)^{-1}$, the rate of transition of the detector corresponds to a slowly varying temperature $T(\tau) = g(\tau)/2 \pi$, as one would have expected. (b) However, when $g^{-1}\ll \omega^{-1} \ll (\dot g/g)^{-1}$, we find that the spectrum is modified \textit{even at the order $\mathcal{O}(\eta)$}. This is counter-intuitive because, in this case, the relevant frequency does not probe the rate of change of the acceleration since $(\dot g/g) \ll \omega$ and we certainly do not have deviation from the thermal spectrum when $\dot g =0$. This result shows that there is a subtle discontinuity in the behaviour of detectors with $\dot g = 0$ and $\dot g/g^2$ 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.