Quantum energetics of non-commuting measurements in a circuit quantum electrodynamics system

When the measurement observable does not commute with the system Hamiltonian, the energy of the measured system is typically not conserved during the measurement. Instead, energy is transferred between the measured system and the measuring meter. In this work, we have studied this effect in a circuit quantum electrodynamics system containing a transmon qubit embedded in a 3D microwave cavity. The qubit is dispersively coupled to the cavity, which enables quantum non-demolition (energy-conserving) measurement of the qubit states. By applying an additional microwave drive on resonance with the qubit frequency, the Hamiltonian of the qubit is modified, leading to a non-commuting measurement that is not energy-conserving. We investigate the energy exchange between the qubit and the cavity by performing a spectral analysis of the cavity field during the measurement. In the spectra, due to the microwave drive, the Mollow triplets can be clearly observed. The two side peaks show different intensities, indicating an effective frequency shift that compensates for the energy change of the qubit during the measurement. While the energy is not conserved for the qubit, the total energy of the qubit and the cavity field is still a conserved quantity. This energy transfer occurs without the need to know the measurement outcome, which could potentially be utilized for autonomous quantum engines.