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arXiv:2405.17115 · Physical Review A

Heisenberg-scaling precision in the estimation of two parameters in a Mach-Zehnder interferometer

Atmadev Rai, Danilo Triggiani, Paolo Facchi A, Vincenzo Tamma

Published Jun 5, 2025• 4 citations
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Abstract

Achieving the ultimate quantum precision in the estimation of multiple physical parameters simultaneously is a challenge in quantum metrology due to fundamental limitations and experimental challenges in harnessing the necessary quantum resources. We propose an experimentally feasible scheme to reach Heisenberg-scaling precision in the simultaneous estimation of two unknown phase parameters in a Mach-Zehnder interferometer by using squeezed and coherent states of light as input and homodyne detections at the outputs. Our results open a new research paradigm in harnessing the full quantum advantage of multiparameter estimation in distributed quantum sensing and quantum information processing, as enabled by multiphoton quantum interference with scalable experimental resources.

DOI
10.1103/physreva.111.062408

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Achieving the ultimate quantum precision in the estimation of multiple physical parameters simultaneously is a challenge in quantum metrology due to fundamental limitations and experimental challenges in harnessing the necessary quantum resources. We propose an experimentally feasible scheme to reach Heisenberg-scaling precision in the simultaneous estimation of two unknown phase parameters in a Mach-Zehnder interferometer by using squeezed and coherent states of light as input and homodyne detections at the outputs. Our results open a new research paradigm in harnessing the full quantum advantage of multiparameter estimation in distributed quantum sensing and quantum information processing, as enabled by multiphoton quantum interference with scalable experimental resources.
We present a framework for simultaneously estimating all four real parameters of a general two-channel unitary U(2)\mathrm{U}(2)U(2) with Heisenberg-scaling precision. We derive analytical expressions for the quantum Fisher information matrix and show that all parameters attain the 1/N1/N1/N scaling in the precision by using experimentally feasible Gaussian probes, such as two-mode squeezed states or two single-mode squeezed states. Our results extend multiparameter metrology to its most general two-mode setting and establish concrete design principles for experimental implementations of Heisenberg-scaling, multiparameter optical interferometry with experimentally feasible resources. It not only sheds light on the fundamental interface between quantum interference of squeezed light and quantum metrological advantage in multiparameter estimation, but it also provides an important stepping stone towards the development of a wide range of quantum technologies based on distributed quantum metrology in arbitrary optical networks.
Abstract This work presents a quantum sensing interferometric scheme for the simultaneous estimation of multiple parameters at the ultimate quantum scaling precision in a two-mode optical network. This scheme is experimentally feasible because it employs input Gaussian states such as squeezed and coherent states and a relatively low number of experimental runs. We focus on the precise and simultaneous measurement of two unknown phase shifts and an unknown beam splitter reflectivity, achieving Heisenberg scaling sensitivity in all three parameters without any constraints on the parameter values.
Quantum sensing and quantum metrology propose schemes for the estimation of physical properties, such as lengths, time intervals, and temperatures, achieving enhanced levels of precision beyond the possibilities of classical strategies. However, such an enhanced sensitivity usually comes at a price: the use of probes in highly fragile states, the need to adaptively optimise the estimation schemes to the value of the unknown property we want to estimate, and the limited working range, are some examples of challenges which prevent quantum sensing protocols to be practical for applications. This work reviews two feasible estimation schemes which address these challenges, employing easily realisable resources, i.e., squeezed light, and achieve the desired quantum enhancement of the precision, namely the Heisenberg-scaling sensitivity. In more detail, it is here shown how to overcome, in the estimation of any parameter affecting in a distributed manner multiple components of an arbitrary M-channel linear optical network, the need to iteratively optimise the network. In particular, we show that this is possible with a single-step adaptation of the network based only on a prior knowledge of the parameter achievable through a “classical” shot-noise limited estimation strategy. Furthermore, homodyne measurements with only one detector allow us to achieve Heisenberg-limited estimation of the parameter. We further demonstrate that one can avoid the use of any auxiliary network at the price of simultaneously employing multiple detectors.
Continuous-wave (CW) interferometry has stood at the frontier of precision measurement science since its inception, where it was used to search for the luminiferous ether, to the present day, where it forms the basis of interferometric gravitational-wave detection. Quantum theory predicts that this frontier can be expanded more rapidly by employing certain quantum resources, compared with the case of using only classical resources. In the quantum case, we can achieve ``Heisenberg scaling'', which manifests as a quadratic improvement over the best possible classical precision scaling. Although Heisenberg scaling has been demonstrated in pulsed operation, it has not been demonstrated for continuous operation. The challenge in doing so is two-fold: continuous measurements capable of Heisenberg scaling were previously unknown, and the requisite CW quantum states are fragile. Here we overcome these challenges and demonstrate the first CW interferometer exhibiting resource efficiency approaching Heisenberg scaling. Our scheme comprises a Mach-Zehnder interferometer illuminated with a pair of squeezed light sources, followed by a nonlinear estimator of the output homodyne record to estimate a differential phase modulation signal that drives the interferometer. We observe that this signal can be extracted with a precision that scales faster than what is allowed classically, and approaches the Heisenberg scaling limit.
Quantum metrology is the state-of-the-art measurement technology. It uses quantum resources to enhance the sensitivity of phase estimation beyond what reachable within classical physics. While single parameter estimation theory has been widely investigated, much less is known about the simultaneous estimation of multiple phases, which finds key applications in imaging and sensing. In this manuscript we provide conditions of useful entanglement (among multimode particles, qudits) for multiphase estimation and adapt them to multiarm Mach-Zehnder interferometry. We discuss benchmark multimode Fock states containing useful qudit entanglement and overcoming the sensitivity of separable qudit states in three and four arm Mach-Zehnder-like interferometers - currently within the reach of integrated photonics technology.