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arXiv:0908.3008 · Physical Review A

Optimization of quantum interferometric metrological sensors in the presence of photon loss

Tae‐Woo Lee, Sean D. Huver, Hwang Lee, Lev Kaplan, S. Blane McCracken, Changjun Min, Dmitry B. Uskov, Christoph F. Wildfeuer, Georgios Veronis, Jonathan P. Dowling

Published Dec 2, 2009• 92 citations
arXivHTMLPDF

Abstract

We optimize two-mode entangled number states of light in the presence of loss in order to maximize the extraction of the available phase information in an interferometer. Our approach optimizes over the entire available input Hilbert space with no constraints, other than fixed total initial photon number. We optimize to maximize the Fisher information, which is equivalent to minimizing the phase uncertainty. We find that in the limit of zero loss, the optimal state is the maximally path-entangled (so-called N00N) state, for small loss, the optimal state gradually deviates from the N00N state, and in the limit of large loss, the optimal state converges to a generalized two-mode coherent state, with a finite total number of photons. The results provide a general protocol for optimizing the performance of a quantum optical interferometer in the presence of photon loss, with applications to quantum imaging, metrology, sensing, and information processing.

DOI
10.1103/physreva.80.063803

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We optimize two-mode entangled number states of light in the presence of loss in order to maximize the extraction of the available phase information in an interferometer. Our approach optimizes over the entire available input Hilbert space with no constraints, other than fixed total initial photon number. We optimize to maximize the Fisher information, which is equivalent to minimizing the phase uncertainty. We find that in the limit of zero loss, the optimal state is the maximally path-entangled (so-called N00N) state, for small loss, the optimal state gradually deviates from the N00N state, and in the limit of large loss, the optimal state converges to a generalized two-mode coherent state, with a finite total number of photons. The results provide a general protocol for optimizing the performance of a quantum optical interferometer in the presence of photon loss, with applications to quantum imaging, metrology, sensing, and information processing.
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 investigate interferometric phase-estimation using separable photon inputs that evolve into number-path entangled states through linear optical networks, followed by photon-number-resolving detection. A simple analytical expression for the classical Fisher information at zero phase is derived for arbitrary NNN-photon states distributed across 2NNN optical modes, partitioned into phase-encoding and reference blocks. Among all possible photon distributions between these blocks, the balanced configuration maximizes the phase sensitivity for every photon number NNN and uniquely exhibits a phase-independent response. The achievable sensitivity degrades monotonically with increasing asymmetry in the photon distribution. We further investigate the robustness of the protocol in the presence of realistic photon loss and extend the analysis to distributed architectures with multiple receivers. In the low photon-flux regime, vacuum fluctuations fundamentally limit local quadrature measurements, whereas nonlocal photon-number-resolving measurements exploit multiphoton interference to mitigate loss-induced sensitivity degradation. Together, these results establish a scalable framework for quantum-enhanced distributed multimode metrology.
To acquire the best path-entangled photon Fock states for robust quantum-optical metrology with parity detection, we calculate phase information from a lossy interferometer by using twin entangled Fock states. We show that (a)(\text{a})(a) when loss is less than 50% twin entangled Fock states with large photon number difference give higher visibility, while when loss is higher than 50%, the ones with less photon number difference give higher visibility; and (b)(\text{b})(b) twin entangled Fock states with large photon number difference give sub--shot-noise limit sensitivity for phase detection in a lossy environment. This result provides a reference on what particular path-entangled Fock states are useful for real world metrology applications.
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 give a detailed discussion of optimal quantum states for optical two-mode interferometry in the presence of photon losses. We derive analytical formulae for the precision of phase estimation obtainable using quantum states of light with a definite photon number and prove that maximization of the precision is a convex optimization problem. The corresponding optimal precision, i.e., the lowest possible uncertainty, is shown to beat the standard quantum limit thus outperforming classical interferometry. Furthermore, we discuss more general inputs: states with indefinite photon number and states with photons distributed between distinguishable time bins. We prove that neither of these is helpful in improving phase estimation precision.
Quantum-enhanced interferometry is often discussed in terms of ideal resources and asymptotic scalings, whereas in practice its performance is set by a delicate interplay between losses, state imbalance, and photon number. We address this interplay in a folded Franson interferometer fed with partially entangled N00N states, treating asymmetric losses and tunable input imbalance on equal footing. From exact detection probabilities we obtain closed-form expressions for the fringe visibility and the Fisher information, and show that these two figures of merit respond very differently to imperfections. In particular, we demonstrate that perfect interference contrast can always be recovered by compensating loss asymmetry with an appropriate input imbalance, while the Fisher information generally peaks at a distinct operating point, reflecting the irreducible trade-off between coherence restoration and signal attenuation. By determining the exact optima and benchmarking against single-photon strategies, we identify the critical loss and minimum entanglement required to maintain a genuine quantum advantage over optimized single-photon strategies under identical loss conditions, and establish their scaling with the photon number N . Beyond delineating the fundamental trade-offs between loss, entanglement, and sensitivity, this work establishes a comprehensive theoretical framework that both underpins and extends the experimental demonstration of quantum advantage reported in [1], providing a unified description of the relevant operating regimes.