Energy Conservation with Post-selection

We investigate the energy change of the measurement apparatus when a quantum system is measured. For the qubit case, we model two possible measurement implementations: a clock model with a time-independent Hamiltonian, and a Jaynes-Cummings model with a time-dependent Hamiltonian that conserves the total excitation number. We find that, in both models, the mean energy change of the measurement apparatus, conditioned on the qubit post-selection, can be much greater than the level spacing of the qubit, like an anomalous weak value. Owing to similarities to weak measurement, the expression for the clock energy shift explicitly contains the weak value of the qubit Hamiltonian. Our two models give different results, which we explain to be a consequence of the non-degenerate 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 find good agreement in the appropriate regime.