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arXiv:2505.24770 · Communications in Theoretical Physics

Phase estimation in lossy optical interferometry without a reference beam

Jun Tang, Dongqing Wang, Wei Zhong, Lan Zhou, Yu‐Bo Sheng

Published Jun 23, 2025• 1 citation
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Abstract

Abstract We investigate phase estimation in a lossy interferometer using entangled coherent states, with a particular focus on a scenario where no reference beam is employed. By calculating the quantum Fisher information, we reveal two key results: (1) the metrological equivalence between scenarios with and without a reference beam, established under ideal lossless conditions for the two-phase-shifting configuration, breaks down in the presence of photon loss, and (2) the pronounced inferior performance of ECSs relative to NOON states, observed in the presence of a reference beam, disappears in its absence.

DOI
10.1088/1572-9494/ade6da

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Abstract We investigate phase estimation in a lossy interferometer using entangled coherent states, with a particular focus on a scenario where no reference beam is employed. By calculating the quantum Fisher information, we reveal two key results: (1) the metrological equivalence between scenarios with and without a reference beam, established under ideal lossless conditions for the two-phase-shifting configuration, breaks down in the presence of photon loss, and (2) the pronounced inferior performance of ECSs relative to NOON states, observed in the presence of a reference beam, disappears in its absence.
Quantum entanglement offers the possibility of making measurements beyond the classical limit, however some issues still need to be overcome before it can be applied in realistic lossy systems. Recent work has used the quantum Fisher information (QFI) to show that entangled coherent states (ECSs) may be useful for this purpose as they combine sub-classical phase precision capabilities with robustness (Joo et al., 2011). However, to date no effective scheme for measuring a phase in lossy systems using an ECS has been devised. Here we present a scheme that does just this. We show how one could measure a phase to a precision significantly better than that attainable by both unentangled "classical" states and highly-entangled NOON states over a wide range of different losses. This brings quantum metrology closer to being a realistic and practical technology.
We investigate an optimal distance of two components in an entangled coherent state for quantum phase estimation in lossy interferometry. The optimal distance is obtained by an economical point, representing the quantum Fisher information that we can extract per input energy. Maximizing the formula of the quantum Fisher information over an input mean photon number, we show that, as the losses of the interferometer increase, it can be more beneficial to prepare an initially entangled coherent state that is less entangled. This represents that the optimal distance of the two-mode components decreases with more loss in the interferometry. Under the constraint of the input mean photon number, we obtain that the optimal entangled coherent state is more robust than a separable coherent state, even in a high photon loss rate. The optimal entangled coherent state preserves quantum advantage over the standard interferometric limit of the separable coherent state. We also show that the corresponding optimal measurement requires correlation measurement bases.
We study the precise phase estimation using squeezed states in the presence of photon loss. Our exact quantum Fisher information calculation shows significant enhancement even though the loss is very large and sets a benchmark for experimental realization. We show that if we blindly use the existing parity measurement scheme [Anisimov et al., Phys. Rev. Lett. 104, 103602 (2010)] for the ideal case, the result will be even worse than the classical case given very small loss. Using our formulas, we can optimize the measurement result by making loss-dependent phase shift and choosing appropriate squeezed states (average photon number of squeezed states).
Optical interferometry is a cornerstone technique for precise phase measurements across various fields. In many applications, for example, biological imaging, it often necessitates stringent limits on light intensity to prevent adverse effects on light-sensitive samples, a condition known as dose-limited regimes. Maximizing the precision per dose is therefore crucial. In quantum metrology, quantum correlations enable high precision in phase estimation while adhering to dose constraints. Nevertheless, photon loss, including absorption by a sample, substantially diminishes the benefits of quantum enhancement in interferometry. In this work, we experimentally investigate a dose-efficient approach to quantum phase estimation using sequential strategies in the presence of loss. Performance of sequential strategies with and without control is evaluated through quantum Fisher information (QFI) per dose. Experimental results show that both sequential strategies exceed the classical limit and outperform the parallel strategy using unbalanced N00N states. Notably, the control-enhanced sequential strategy attains superior QFI per dose, approaching the quantum limit. These results highlight the promise of sequential strategy for imaging and sensing in resource-constrained scenarios, marking a significant step toward practical and efficient quantum metrology in lossy environments.
We investigate the phase enhancement of quantum states subject to nonlinear phase shifts. The optimal phase estimation of even entangled coherent states (ECSs) is shown to be better than that of NOON states with the same average particle number $\ensuremath{\langle}n\ensuremath{\rangle}$ and nonlinearity exponent kkk. We investigate the creation of an approximate entangled coherent state (AECS) from a photon-subtracted squeezed vacuum with current optical technology methods and show that a pure AECS is even better than an even ECS for large $\ensuremath{\langle}n\ensuremath{\rangle}$. Finally, we examine the simple, but physically relevant, cases of loss in the nonlinear interferometer for a fixed average photon number $\ensuremath{\langle}n\ensuremath{\rangle}$.
We investigate the performance of entangled coherent states for quantum-enhanced phase estimation. An exact analytical expression of quantum Fisher information is derived to show the role of photon losses on the ultimate phase sensitivity. We find a transition of the sensitivity from the Heisenberg scaling to the classical scaling due to quantum decoherence of the photon state. This quantum-classical transition is uniquely determined by the number of photons being lost, instead of the number of incident photons or the photon loss rate alone. Our results also reveal that a crossover of the sensitivity between the entangled coherent state and the NOON state can occur even for very small photon loss rate.