Response of Unruh-DeWitt detector with time-dependent acceleration

Abstract

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^./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 order O(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 . This result shows that there is a subtle discontinuity in the behavior of detectors with and 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.