The effects of time delays in adaptive phase measurements

Abstract

It is not possible to make measurements of the phase of an optical mode using linear optics without introducing an extra phase uncertainty. This extra phase variance is quite large for heterodyne measurements, however it is possible to reduce it to the theoretical limit of log [nbar]/(4[nbar]2) using adaptive measurements. These measurements are quite sensitive to experimental inaccuracies, especially time delays and inefficient detectors. Here it is shown that the minimum introduced phase variance when there is a time delay of log τ is τ/(8[nbar]). This result is verified numerically, showing that the phase variance introduced approaches this limit for most of the adaptive schemes using the best final phase estimate. The main exception is the adaptive mark II scheme with simplified feedback, which is extremely sensitive to time delays. The extra phase variance due to time delays is considered for the mark I case with simplified feedback, verifying the τ/2 result obtained by Wiseman and Killip both by a more rigorous analytic technique and numerically.