Apparent heating due to imperfect calorimetric measurements

Performing imperfect or noisy measurements on a quantum mechanical system both impacts the measurement outcome and the state of the system after the measurement. In this paper we are concerned with imperfect calorimetric measurements. In calorimetric measurements one typically measures the energy of a thermal environment to extract information about the system. In our setting the measurement is imperfect due to noise directly acting on the detector. Concretely, we consider an additional noise bath such that its energy is measured simultaneously with the calorimeter energy. Under weak coupling assumptions, we derive a hybrid master equation for the state of the system and the detected energy and find that the presence of the noise bath manifests itself by modifying the jump rates of the reduced system dynamics. We study an example of a driven qubit interacting with a resonant boson calorimeter and demonstrate that increasing the additional noise leads to an apparent reduction in the power flowing from qubit to calorimeter and thus to a seemingly detected heating up of the calorimeter.