Metrics play a central role in science and engineering. It is concerned with the final reachable accuracy of parameters or phase estimation and the construction of measurement schemes to achieve this accuracy. By combining quantum mechanics and basic theories of statistics, quantum metrics find that the final lower limit of estimation accuracy is related to input state preparation, phase accumulation modes, and measurement schemes, and the main goal is to break through the standard quantum limit and reach the Heisenberg limit of measurement accuracy. In recent years, due to the progress of experimental conditions, quantum metrics have been widely used in the frontier fields such as gravitational wave detection and atomic clocks. A major research direction of quantum metrics is phase estimation in optical interferometers, which was first proposed in research on the input coherent light and compressed light in Mach-Zende interferometers by Caves et al., and its theoretical phase sensitivity can reach the physical limit (Heisenberg limit). In recent years, other kinds of non-classical light sources have also been studied, such as the NOON state and twin-Fock state. The NOON state is a numerical light source that can theoretically reach the Heisenberg limit, while the twin-Fock state has theoretical phase sensitivity up to the Heisenberg scale and is more robust to photon loss than the NOON state. However, coincidence count detection for the twin-Fock state results in a multi-peak structure of the phase distribution (i.e., the likelihood function), which is the so-called phase ambiguity. Aiming at this problem, we propose a simple scheme to eliminate phase ambiguity and analyze its performance.
A binary-outcome photon counting and joint likelihood function measurement are employed in this work, where the detection event with an equal number of photons is a measurement outcome. All the other detection events are treated as another outcome. We generalize it to a multi-output scenario and use single-photon states for joint measurement. According to the relationship between the maximum likelihood estimator and the inverse function estimator in the case of multiple outputs, we have semi-analytically explained the reason why this method works. Using the Monte Carlo method, we simulate the measurement probabilities of the six-photon twin-Fock state and the single-photon state and get a numerical simulation of the measurement scheme, where the experimental imperfection is added artificially.
We propose a simple scheme to eliminate phase ambiguity of coincidence count detection for the twin-Fock state. Our scheme relies on a sequence of the N-photons Fock states and the single-photon state that are injected into the interferometer to realize a single-peak structure of the total phase distribution, which determines the maximum likelihood estimator. Phase uncertainty of the estimator can beat the standard quantum limit over the entire phase interval.