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arXiv:2308.08113

Enhanced super-Heisenberg scaling precision by nonlinear coupling and postselection

Lupei Qin, Jialin Li, Yazhi Niu, Xin-Qi Li

Published Aug 16, 2023• 0 citations
arXivHTMLPDF

Abstract

In quantum precision metrology, the famous result of Heisenberg limit scaling as $1/N$ (with $N$ the number of probes) can be surpassed by considering nonlinear coupling measurement. In this work, we consider the most practice-relevant quadratic nonlinear coupling and show that the metrological precision can be enhanced from the $1/N^{\frac{3}{2}}$ super-Heisenberg scaling to $1/N^2$, by simply employing a pre- and post-selection (PPS) technique, but not using any expensive quantum resources such as quantum entangled state of probes.

DOI
10.48550/arxiv.2308.08113

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Abstract The Heisenberg limit is acknowledged as the ultimate precision limit for parameter estimation in quantum metrology. Recent studies have demonstrated super-Heisenberg-scaling limits, where the root mean square errors of estimation decrease faster-than-linearly with the number of probes or evolution processes. This casts doubt on what the ultimate precision limit of quantum metrology really is and, in particular, whether there is a deeper connection with Heisenberg's uncertainty principle. By linking the dynamics of the metrological process with properties in parameter space, we construct a Heisenberg's uncertainty relation between the unknown parameter and its canonical momentum. We propose a novel quantum metrological scheme that utilizes a generating process with indefinite time direction to increase the canonical momentum nonlinearly. This approach achieves a nonlinear-scaling precision limit that improves quadratically with the increase in evolution time length or the number of processes, in accordance with Heisenberg's uncertainty relation. Then we experimentally demonstrate in quantum optical systems that this nonlinear-scaling enhancement can be achieved with a fixed amount of energy in the probes. Consequently, our results provide a deeper insight into the Heisenberg limit in quantum metrology, and shed new light on improving the precision of practical metrological and sensing tasks in realistic quantum systems.
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Nonlinear interactions are recognized as potential resources for quantum metrology, facilitating parameter estimation precisions that scale as the exponential Heisenberg limit of ${2}^{\ensuremath{-}N}$. We explore such nonlinearity and propose an associated quantum measurement scenario based on the nonlinear interaction of NNN-probe entanglement-generating form. This scenario provides an enhanced precision scaling of ${D}^{\ensuremath{-}N}/(N\ensuremath{-}1)!$ with D>2D>2D>2 a tunable parameter. In addition, it can be readily implemented in a variety of experimental platforms and applied to measurements of a wide range of quantities, including local gravitational acceleration ggg, magnetic field, and its higher-order gradients.
It has been suggested that both quantum superpositions and nonlinear interactions are important resources for quantum metrology. However, to date the different roles that these two resources play in the precision enhancement are not well understood. Here, we experimentally demonstrate a Heisenberg-scaling metrology to measure the parameter governing the nonlinear coupling between two different optical modes. The intense mode with n (more than 10^{6} in our work) photons manifests its effect through the nonlinear interaction strength which is proportional to its average photon number. The superposition state of the weak mode, which contains only a single photon, is responsible for both the linear Hamiltonian and the scaling of the measurement precision. By properly preparing the initial state of single photon and making projective photon-counting measurements, the extracted classical Fisher information (FI) can saturate the quantum FI embedded in the combined state after coupling, which is ∼n^{2} and leads to a practical precision ≃1.2/n. Free from the utilization of entanglement, our work paves a way to realize Heisenberg-scaling precision when only a linear Hamiltonian is involved.
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We scrutinize the role of quantum entanglement in quantum metrology and discuss recent advances in nonlinear quantum metrology that allow improved scalings of the measurement precision with respect to the available resources. Such schemes can surpass the conventional Heisenberg limited scaling of 1/N of quantum enhanced metrology. Without investing in the preparation of entangled states, we review how systems with intrinsic nonlinearities such as Bose–Einstein condensates and light-matter interfaces can provide improved scaling in single parameter estimation.