Maximum-likelihood statistics of multiple quantum phase measurements

Abstract

Recent theoretical studies have considered various possibilities for experimental measurements of phase using quantum states of a finite mean number of photons. Currently, the most experimentally accessible of these possibilities is an interferometric phase measurement using a squeezed state that has minimum variance for the phase quadrature; this possibility can yield, in principle, a phase measurement with uncertainty of order l/(mean number of photons). Shapiro, Shepard, and Wong1 (SSW) have proposed another possibility. They maximize the peak likelihood in an abstract measurement of the Susskind-Glogower phase operator, and they claim a phase sensitivity that goes as l/(mean number of photons).2 The SSW phase probability distribution consists of an extremely narrow peak sitting on a broad base that covers the entire 2 radians of phase; as a result, a single SSW phase measurement has poor sensitivity. Shapiro, Shepard, and Wong argue that their claimed sensitivity can be achieved in a sequence of measurements. We study the statistics of the maximum-likelihood estimator in a sequence of SSW phase measurements, and we report how the uncertainty in the estimator scales with the mean number of photons used in the entire sequence.