Postselection and quantum energetics

We investigate the anomalous energy change of the measurement apparatus when a qubit is measured in bases that do not commute with energy. We model two possible measurement implementations: one is a quantum clock model with a completely time-independent Hamiltonian, while the other is a Jaynes-Cummings model that is time-dependent but conserves the total excitation number. We look at the mean energy change of the measurement apparatus in both models, conditioned on the qubit postselection, and we find that this change can be much greater than the level spacing of the qubit, like an anomalous weak value. In the clock model, the expression for the apparatus energy shift explicitly contains the weak value of the qubit Hamiltonian. However, in our case, no explicit weak measurements are carried out. Our two models give different results, which we explain to be a consequence of the nondegenerate spectrum of the Jaynes-Cummings model. We compare our calculations in the Jaynes-Cummings model with the experimental data of J. Stevens et al., [Phys. Rev. Lett. 129, 110601 (2022)], and we find good agreement when the conditions of our derivation are valid.

Automated summary

We investigate the anomalous energy change of the measurement apparatus when a qubit is measured in bases that do not commute with energy. We model two possible measurement implementations and find that the change in energy can be much greater than the level spacing of the qubit, resembling an anomalous weak value. Our models provide different results due to the nondegenerate spectrum of the Jaynes-Cummings model. Comparing our calculations with experimental data, we find good agreement when the conditions of our derivation are valid.