Measuring the vibrational energy of a trapped ion

We show that an approximate quantum-nondemolition measurement of the vibrational energy of a trapped ion in a standing wave may be made by monitoring the fluorescent intensity from a probe transition. The ac component of the mean photocurrent signal is directly proportional to the average vibrational quantum number. The power spectrum of the photocurrent also contains information on the vibrational energy. We show that the phase of the vibrational motion undergoes diffusion as expected for a quantum-limited measurement of the energy of oscillation