Post-selection and Quantum Energetics

When an unread measurement of an energy-noncommuting property causes the mean energy of a system to change, the measurement apparatus's mean energy change should be equal and opposite. How this balance works out when we post-select on a particular measurement outcome is less obvious. Then the shift of the energy of the measurement apparatus is similar to a weak value, since energy conservation weakly correlates the energies of the apparatus and measured system. We calculate this shift for a realistic qubit measurement protocol involving coupling to an oscillator. We show that this shift depends on the particular form of the coupling interaction: for the same target qubit dynamics, a Jaynes-Cummings interaction gives different results from a similar interaction with advantageous degeneracies. For the latter interaction type, we give a formula for the conditional mean energy shift of the measurement apparatus, in terms of qubit properties, including the energy weak value. This formula has intuitive consequences, particularly when the qubit is prepared in an energy eigenstate. The Jaynes-Cummings interaction, by contrast, has surprising experimental effects.