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Exact conditions for antiUnruh effect in (1+1)-dimensional spacetime areobtained. For detectors with Gaussian switching functions, the analytic resultsare similar to previous ones, indicating that antiUnruh effect occurs when theenergy gap matches the characteristic time scale. However, this conclusion doesnot hold for detectors with square wave switching functions, in which case thecondition turns out to depend on both the energy gap and the characteristictime scale in some nontrivial way. We also show analytically that there is noantiUnruh effect for detectors with Gaussian switching functions in(3+1)-dimensional spacetime.