We revisit the idea of quantum nondemolition measurement (QND) of optical quanta via a resonantly enhanced Kerr nonlinearity taking into account quantum backaction. We show that the monolithic microcavities enable QND measurement of the number of quanta in a weak signal field using a classical probe field spatially overlapping with the signal. The phase of the probe field acquires information about the signal number of quanta without altering it due to the cross-phase modulation effect. We find the exact solution to the Heisenberg equations of motion of this system and calculate the measurement error, accounting for the optical losses in the measurement path. We identify a realistic approximation to obtain the explicit form of the final conditional quantum state of the signal field, accounting for the undesirable self-phase modulation effect and designing the optimal homodyne measurement of the probe beam to evade this effect. We show that the best modern monolithic microcavities allow achieving the measurement imprecision several times better than the standard quantum limit.
By analyzing a single-photon interference experiment together with a quantum nondemolition measurement scheme for the which-path information, we prove that given the total mean number
〈n〉 of available photons, the fundamental limit in precision measurement of a phase shift is the Heisenberg limit, i.e.,
1/〈n〉. The analysis is based on the complementarity principle and is independent of the scheme for the measurement of the phase shift. We also show that to achieve the Heisenberg limit states with photon number fluctuations of the order of or larger than the mean photon number have to be exploited.